text:
\section{Introduction}
Coagulation processes are ubiquitous in nature, from the movement of cells to the atmosphere, and in general complicated to understand experimentally and mathematically. In this work, we are motivated by the question whether a turbulent velocity field in the atmosphere enhances the coagulation of small rain droplets, and therefore favors rain fall. Physics literature on this topic has been vast, at least as early as Saffman-Turner \cite{SaTu} in the fifties, see also \cite{FFS, PW, Shaw}, arguing in favor of such coagulation enhancement. Here and in subsequent works, we give a novel approach to this problem that is fully mathematical. In particular, we take a kinetic viewpoint and study rigorously a variant of Smoluchowski's coagulation equation with velocity dependence that is akin to Boltzmann equation. It arises as the scaling limit of a system of second-order (microscopic) coagulating particles, modelling the interactions of rain droplets in the clouds, which are subjected to a common noise of transport type. Such a noise, constructed in recent mathematical works \cite{Galeati, FGL}, possesses several characteristics of real turbulence, such as it enhances diffusion of passive scalars. In the present work, we focus on the existence, uniqueness and regularity of this new \abbr{PDE}, after briefly introducing its origin.

Smoluchowski's classical equation \cite{smoluchowski1916} provides a first model for the time evolution of the probability distribution $\{f_m(t,x)\}_{m=1}^\infty$ of diffusing particles
of different sizes (or masses) $m\in\mathbb N$, say in $\mathbb{T}^d:=(\mathbb{R}/\mathbb{Z})^d$, when they undergo pairwise coagulation with certain coagulation rate $\alpha(m,n)$:
\begin{align*}
    \partial_t f_m(t, x)=\Delta f_m(t,x)+&\sum_{n=1}^{m-1}\alpha(n,m-n)
    f_n(t,x)f_{m-n}(t,x)\\
    -&2    \sum_{n=1}^\infty \alpha(m,n)
    f_m(t,x)f_n(t,x), \quad t>0,\, x\in\mathbb{T}^d, \, m\in\mathbb N.
\end{align*}
The nonlinearity has two parts, a gain term and a loss term. Such a system of equations has been derived from scaling limits of Brownian particle systems by Hammond-Rezakhanlou \cite{HR, HR2}. To model the influence of a large-scale turbulent flow, it is natural to introduce a common noise. If we adopt a transport noise of the type
in \cite{FGL}
\begin{align*}
        \dot{\mathcal W}(t,x)    =\sum_{k\in K}\sigma_k(x)\dot{W}_t^k
\end{align*}
where $\{\sigma_k(x)\}_{k\in K}$ is a countable collection of divergence-free smooth vector fields and $\{W_t^k\}_{k\in K}$ independent one-dimensional Brownian motions, then we get a stochastic version of Smoluchowski's equation (an \abbr{SPDE})

\begin{equation}
\begin{aligned}\label{old-smol}
    d f_m(t,x)=&
    \Delta f_m(t, x)dt  +   \sum_{n=1}^{m-1}\alpha(n,m-n)
    f_n(t,x)f_{m-n}(t,x)dt    \\
    -& 2  \sum_{n=1}^\infty \alpha(m,n)    f_m(t,x)f_n(t,x)dt
    -\sum_{k\in K}\nabla f_m(t,x)\cdot\sigma_k(x)\, dW_t^k\\
    &+ \text{div}\left(\mathcal{Q}(x,x)\nabla f_m(t,x)\right),\quad m\in\mathbb{N}
\end{aligned}
\end{equation}
where $\mathcal{Q}(x,x):=\sum_{k\in K}\sigma_k(x)\otimes\sigma_k(x)$ coming from the transport-type noise. Under specific choice of $\{\sigma_k\}_{k\in K}$, we can have that $\mathcal{Q}(x,x)\equiv \kappa I_d$, for an enhanced diffusion coefficient $\kappa>0$, the so-called ``eddy diffusion" \cite{Boussinesq}. This picture, in its special case of finitely many mass levels $m=1,2,...,M$
and unit coagulation rate $\alpha(m,n)\equiv 1$, has been derived from particle systems in \cite{FH}. Another version of the \abbr{SPDE} with continuous mass variables $m\in\mathbb{R}$ has also been derived from particle systems with mean-field interactions in \cite{AP}.

However, conceptually and mathematically, the most difficult step
in this program
is to verify
that diffusion enhancement leads to
coagulation enhancement, namely, the fast increase of probability densities $f_m$ for $m\gg 1$ (large masses) for large diffusion coefficient case. In fact, the model \eqref{old-smol} turns out to be too crude, and even numerically we cannot verify a coagulation enhancement.

The problem lies in the fact that quick diffusion of masses may not lead to enhanced collision unless the coagulation rate depends on the velocity variable. Otherwise, the masses merely move around.
We introduce a new system with both position and velocity variables. In the atmospheric physics literature, e.g. \cite{FFS, PW, Shaw, GW}, it is also common to consider cloud particles coagulate with a rate that is proportional to (when $d=3$)
\begin{align*}
    \alpha(m_1,m_2):=|v_1-v_2|(r_1+r_2)^2
\end{align*}
where $v_i$, $i=1,2$ are the velocities of two colliding rain droplets and $r_i:=m_i^{1/3}$, $i=1,2$ their respective radius. Under certain simplifications, it leads to the following kinetic version of Smoluchowski's equation (cf. Appendix \ref{appen-particle})
\begin{align}\label{main-eq-intro}
\begin{cases}
\partial_t f_m (t,x,v)
= -v\cdot \nabla_xf_m+c(m)\text{div}_v\left(vf_m\right)+\kappa c(m)^2\Delta_vf_m + Q_m(f,f)\\[5pt]
f_m|_{t=0}=f^0_m(x,v),  \quad m=1,...,M,
\end{cases}
\end{align}
where $(t,x,v)\in[0,T]\times\mathbb{T}^d\times\mathbb{R}^d$, and
\begin{equation}\label{collison-term}
\begin{aligned}
Q_m(f,f)(t,x,v) &:= \sum_{n=1}^{m-1}\iint_{\{nw'+(m-n)w=mv\}} s(n,m-n)f_n(t,x,w')f_{m-n}(t,x,w)|w-w'|dwdw'\\
&\quad -2\sum_{n=1}^M \int s(n,m)f_m(t,x,v)f_n(t,x,w)|v-w|dw,
\end{aligned}
\end{equation}
where
\begin{align}\label{mass-const}
c(m):=\alpha m^{(1-d)/d}, \quad s(n,m):=(n^{1/d}+m^{1/d})^{d-1}.
\end{align}
In Appendix \ref{appen-particle}, we sketch the proof of the scaling limit from a coagulating microscopic particle system subjected to a common noise, to an \abbr{SPDE} that eventually gives rise to this \abbr{PDE}. Although it is not fully rigorous, it should justify the interest of this equation. Here again, turbulence contributes to large $\kappa$ versus small $\kappa$ when no turbulence. The eddy diffusion occurs now in the velocity variable.

The aim of our research is two-fold. Theoretically, we are interested in proving the well-posedness of the \abbr{PDE} and associated \abbr{SPDE} cf. \eqref{smol-spde}, as well as the passage from the one to the other. Then, both theoretically and numerically we aim to demonstrate that the larger the $\kappa$, i.e. the more intense is the turbulence, the faster masses coagulate. See \cite{FHP} for a first numerical study in this direction. In the present article, we focus on the \abbr{PDE} system \eqref{main-eq-intro} in the spatially-homogeneous case, i.e. by considering the initial conditions $f_0^m$ constant in $x$ for every $m$, we can reduce \eqref{main-eq-intro} to
\begin{align}\label{reduced-eq-intro}
\begin{cases}
\partial_t f_m (t,v)
= c(m)\text{div}_v\left(vf_m(t,v)\right)+\kappa c(m)^2\Delta_vf_m(t,v) + Q_m(f,f)(t,v)\\[5pt]
f_m|_{t=0}=f^0_m(v), \quad m=1,...,M,
\end{cases}
\end{align}
where $(t,v)\in[0,T]\times\mathbb{R}^d$ and $Q_m(f,f)$ is as in \eqref{collison-term} but without the $x$-dependence, and we
prove existence, uniqueness and regularity of the solutions of \eqref{reduced-eq-intro}, for every fixed $\kappa>0$.

Denote a weighted $L^p$ space
\begin{align}\label{weighted-Lp}
    L^p_k(\mathbb{R}^d):=\left\{f:\mathbb{R}^d\rightarrow\mathbb{R}\ \text{ s.t. }\ f\brak{v}^k\in L^p(\mathbb{R}^d)\right\}, \quad p\in[1,\infty],\, k\in\mathbb{N},
\end{align}
where
\[
\brak{v}:=\sqrt{1+|v|^2}
\]
and a weighted Sobolev space
\begin{align}\label{weighted-sobolev}
    H^n_k(\mathbb{R}^d):=\left\{f\in L^2_k\ \text{ s.t. } \nabla^\ell f\in L^2_k(\mathbb{R}^d), \; \forall 1\leq \ell\leq n\right\}.
\end{align}
The main result of this article is as follows.
\begin{theorem}
Fix any finite $T$ and $\kappa>0$.
Suppose that initial conditions $f_m^0(v)\in (L^2\cap L^1_2)(\mathbb{R}^d)$ and nonnegative, for every $m=1,...,M$, then there exists at least one nonnegative solution in the class $L^\infty\left([0,T]; L^1_2(\mathbb{R}^d)\right)^{\otimes M}$.

If the initial conditions $f_m^0(v)\in (H^1_1\cap L^1_2)(\mathbb{R}^d)$ and nonnegative, then there exists a unique nonnegative solution, and in this case $f_m(t)\in C_b^\infty(\mathbb{R}^d)$ for any $t>0$.
\end{theorem}
The most difficult part in our opinion is uniqueness, due mainly to the presence of $|w-w'|$ in the nonlinear term, and the fact that the velocity variable $v\in\mathbb{R}^d$ is unbounded, hence in the presence of Laplacian, $f_m(t,v)$ is never compactly supported even if starting with so. These together with the fact that we have a system rather than just one equation, cause a severe difficulty in closing a Gronwall inequality for uniqueness. As far as we are able, the weighted $L^1$ space is the only one in which a Gronwall argument can work, even if one is willing to assume that solutions are Schwartz functions. (The problem is related to integrability rather than smoothness.) Indeed, with weighted $L^1$ we can find certain cancellations that remove those terms with higher weights brought by the kernel, and this seems not achievable with other spaces such as weighted $L^2$. Equally essential to this cancellation is considering the sum over the norms of all the densities $f_m$, $m=1,..,M$, rather than treating them individually. Indeed, this is already essential to derive various apriori estimates.

On the other hand, existence is proved by constructing a family of approximating equations each corresponding to a truncation of the kernel $|w-w'|$ in the nonlinearity. These approximating problems are more amenable to study since the difficulty related to the kernel is no longer severe, for each fixed truncation parameter.

There is an unexpected connection to the vast area of Boltzmann equations \cite{Villani}. Our \eqref{main-eq-intro} may be viewed as a Boltzmann-type equation with perfectly inelastic collision, rather than the classical elastic collision. Indeed, it is derived from particles undergoing pairwise coagulation, hence two particles merge into one based on the principle of conservation of momentum (and not energy). It is also local in nature in that the nonlinearity acts on the velocity variable, per $(t,x)$. The closest works in the Boltzmann literature seem to be the ones on excited granular media, see \cite{GPV} and references therein, and on multi-species Boltzmann equations, see \cite{BD} and references therein. In a sense our equation combines the features of both of them. From a technical point of view, the aforementioned difficulty with uniqueness are also present in \cite{MW, GPV} and some references therein, and we have learned from these sources. On the other hand, there are various differences that set our model apart from these references. Since $M<\infty$, we do not have the conservation of mass and momentum, and we do not expect nontrivial stationary solutions -- indeed all $f_m(t)$ should decay to zero as $t\to\infty$ (i.e. eventually all masses are transferred out of the system). In general, our nonliearity $Q_m(f,f)$ does not enjoy any particular kind of symmetry. Our derivation of the apriori estimates is also quite different, in particular we need not invoke entropy estimates and Povzner-type inequalities, as are standard in the Boltzmann literature.

Without loss of generality and simplifying notations, in the main part of the paper, we set $\kappa=1$, $c(m)=1$, $s(n,m)=1$.

\section{Deterministic Smoluchowski system with Velocity}\label{sec:apriori-est}
\subsection{Non-local Smoluchowski Equations}
Recall our system of nonlinear integral-differential equations for $\{f_m(t,x,v)\}_{m=1}^M$ defined as:
\begin{align}\label{main-eq}
\begin{cases}
\partial_t f_m = -v\cdot \nabla_xf_m+\text{div}_v\left(vf_m\right)+\Delta_vf_m + Q_m(f,f)\\[5pt]
f_m|_{t=0}=f^0_m
\end{cases}
,\ \quad m=1,...,M
\end{align}
where $(t,x,v)\in [0,T]\times \mathbb{T}^d\times \mathbb{R}^d$. The nonlinear term $Q_m(f,f)$, representing the masses interactions is given by \eqref{collison-term}.
Since we expect the system to represent density of particles with different masses we take $f^0_m\geq 0,\forall\  m=1,...,M$.

We focus on the case when the initial condition is constant in the space variable $x$, and so this property pass to the solution of $(\ref{main-eq})$. We get a new set of equations solved by $f_m(t,v), m=1,...,M$, where we omit the subscript $v$ in the derivatives in the sequel (since there is no ambiguity)
\begin{align}\label{main-eq1}
\begin{cases}
\partial_t f_m = \text{div}\left(vf_m\right)+\Delta f_m + Q_m(f,f)\\[5pt]
f_m|_{t=0}=f^0_m
\end{cases}
, \quad m=1,...,M
\end{align}
where $(t,v)\in [0,T]\times \mathbb{R}^d$ and $f^0_m\geq 0,\ \forall\ m$.
By a change of variables
\[
w'=\varphi(v,w):=\frac{mv-(m-n)w}{n},
\]
the nonlinear term can be written as:
\begin{align*}
Q_m(f,f)(t,v) &:= \sum_{n=1}^{m-1}\iint_{\{nw'+(m-n)w=mv\}} f_n(t,w')f_{m-n}(t,w)|w-w'|dwdw'\\
&\quad -2\sum_{n=1}^M \int  f_m(t,v)f_n(t,w)|v-w|dw\\
&\ =\sum_{n=1}^{m-1}\int f_n\left(t,\varphi(v,w)\right)f_{m-n}(t,w)\left(\frac{m}{n}\right)^2|v-w|dw\\
&\quad -2\sum_{n=1}^M \int f_m(t,v)f_n(t,w)|v-w|dw.
\end{align*}
\iffalse\begin{align*}
&\iint_{\{nw+(m-n)w'=mv\}}f_n(t,w)f_{m-n}(t,w')|w-w'|dwdw'\\
&=\int f_n\left(t,\frac{mv-(m-n)w'}{n}\right)f_{m-n}(t,w')\left(\frac{m}{n}\right)^2|v-w'|dw'\\
&=\int f_n\left(t,\frac{mv-(m-n)w}{n}\right)f_{m-n}(t,w)\left(\frac{m}{n}\right)^2|v-w|dw
\end{align*}\fi
We will consider the following notion of weak solutions to \eqref{main-eq1}.
\begin{definition}[$L^\infty_t L_v^1$-weak solution]\label{def:weak-sol}
Fix $T>0$, and $f^0_m\in L^1_2\cap L^2$. A solution for equation $(\ref{main-eq1})$ is a set of functions $f_m\in L^\infty(0,T;L_2^1(\mathbb{R}^d))$ indexed by the masses $m=1,...,M$ such that $\forall m,\ Q_m(f,f)\in L^\infty(0,T;L^1(\mathbb{R}^d))$ and the following holds:
\begin{align*}
   \sum_{m=1}^M\brak{f_m(t),\phi}&-\sum_{m=1}^M\brak{f_m^0,\phi}=\int_0^t \sum_{m=1}^M\brak{f_m(s),\Delta\phi}ds\\
   &-\int_0^t\sum_{m=1}^M\brak{f_m(s),v\cdot\nabla\phi}ds
   + \int_0^t \sum_{m=1}^M\brak{Q_m(f(s),f(s)),\phi}ds,
\end{align*}
for a.e. $t\in [0,T]$, and $\forall\phi\in C^\infty_c(\mathbb{R}^d)$.
\end{definition}
\subsection{A priori estimate and conserved quantities for system $(\ref{main-eq1})$}
We prove for the system with finite number of masses a decay in time of the total mass and the conservation of such quantity in the infinite system.

Our first result is valid even in the system with space variable, hence we state it more generally.
\begin{lemma}\label{lem:conserved}
Suppose that $f_m^0(x,v)\in L^1(\mathbb{T}^d\times\mathbb{R}^d)$ and $\{f_m(t)\}_{m=1}^M$ is a nonnegative solution of the equation $(\ref{main-eq})$, such that $f_m(t)$ and its gradient $\nabla f_m(t)$ have a fast decay at infinity. Then if $M<\infty$, then the quantity
\begin{align}\label{total_rain}
\mathcal T(t):=\iint \sum_{m=1}^M m\, f_m(t,x,v)dxdv
\end{align}
is non-increasing in $t$; if $M=\infty$, then it is constant in $t$.
\end{lemma}
\begin{rmk}\label{mass-leave}
It is not clear how to relate the finite- and infinite-level systems. We only work with $M<\infty$. Since $t\mapsto \mathcal T(t)$ is decreasing in this case, we interpret $\mathcal T(0)-\mathcal T(t)$ as the amount of mass falling out of the system. It can be viewed as an indicator of how efficient the coagulation is (by transferring densities from small masses to large masses, see \cite{FHP}).
\end{rmk}
\begin{proof}
Consider first the nonlinear part. For every fixed $t,x$ we have that
\begin{align*}
&\int\sum_{m=1}^M m\, Q_m(f,f)dv\\
&= \sum_{m=1}^M \sum_{n=1}^{m-1}m\int dv \iint_{\{nw+(m-n)w'=mv\}} f_n(t,x,w)f_{m-n}(t,x,w')|w-w'|dwdw'\\
&\quad -2\sum_{m=1}^M\sum_{n=1}^M m\iint f_m(t,x,v)f_n(t,x,w)|v-w|dwdv\\
&=\sum_{m=1}^M \sum_{n=1}^{m-1}m\iint f_n(t,x,w)f_{m-n}(t,x,w')|w-w'|dwdw'\\
&\quad -2\sum_{m=1}^M\sum_{n=1}^M m\iint f_m(t,x,v)f_n(t,x,w)|v-w|dwdv
\end{align*}
Exchanging the order of summations in the first term, and setting $k=m-n$, the above equals
\begin{align*}
&\sum_{n=1}^M \sum_{k=1}^{M-n} (k+n)\iint f_n(t,x,w)f_k(t,x,w')|w-w'| dwdw'\\
&\quad -2\sum_{m=1}^M\sum_{n=1}^M m\iint f_m(t,x,v)f_n(t,x,w)|v-w|dwdv\\
&\le \sum_{n=1}^M \sum_{k=1}^{M} (k+n)\iint f_n(t,x,w)f_k(t,x,w')|w-w'| dwdw'\\
&\quad -2\sum_{m=1}^M\sum_{n=1}^M m\iint f_m(t,x,v)f_n(t,x,w)|v-w|dwdv=0
\end{align*}
where equality is achieved if and only if $M=\infty$. For the linear part, we note that for every $m, t,x$ we have that
\begin{align*}
\int_{\mathbb{R}^d} \text{div}_v\left(vf_m+\nabla_v f_m\right)dv = 0
\end{align*}
due to integration by parts and $|vf_m|+|\nabla_vf_m|\to0$ as $|v|\to\infty$ for every fixed $t,x$. Also, for every $m,t,v$ we have that
\begin{align*}
\int_{\mathbb{R}^d} v\cdot\nabla_xf_m dx = 0
\end{align*}
due to integration by parts and $|f_m|\to0$ as $|x|\to\infty$ for every fixed $t,v$. Thus we can conclude since $\mathcal T(t)$ \eqref{total_rain} can be written as
\begin{align*}
\partial_t\mathcal T &= \int_{\mathbb{R}^d}\int_{\mathbb{R}^d} \sum_{m=1}^M m \; \partial_tf_m(t,x,v)dxdv= -\sum_{m=1}^M m\iint v\cdot \nabla_x f_m dxdv \\
&+\sum_{m=1}^M m\iint  \text{div}_v\left(vf_m+\nabla_v f_m\right)dvdx + \iint \sum_{m=1}^M m\, Q_m(f,f)dvdx\le 0
\end{align*}
with equality if and only if $M=\infty$.
\end{proof}
We specialize now to consider the spatially-homogeneous case, i.e. $f_m(t,x,v)=f_m(t,v)$ independent of $x\in\mathbb{T}^d$.

Recall the weighted $L^p$ spaces $L^p_k(\mathbb{R}^d)$ \eqref{weighted-Lp}
and we will indicate by $L^p_{k,M}$ the space in which the sequence $\{f_m(t)\}_{m=1}^M$ lives, with norm given by
\begin{align*}
\|f\|_{p,M,k}:=\sum_{m=1}^M\|f_m\|_{p,k}=\sum_{m=1}^M\left(\int |f_m(v)|^p\brak{v}^{pk} dv\right)^{1/p}.
\end{align*}
The reason we are going to introduce such spaces is motivated not only to understand the regularity of the solution of the system in study, but also to deal with the terms $|v-w|$ in the coagulation kernel since, as we have already noted in the explicit formulation of the nonlinearity $Q_m(f,f)$.

There are a few ways to control the norm of such a quantity, and we prove here the main estimates that we apply in all the a priori estimates for the equation.
\begin{proposition}\label{sobolev}
For every $p\in[1,\infty]$ and $k\geq0$, there exists a constant $C$ depending only on $p,k, M$ and the dimension $d$, such that we have the following bound on the nonlinearity
$$
\|Q_m(g,f)\|_{p,k}\leq C\left(\|g\|_{p,M,k+1}\|f\|_{1,M,k+1}\right)
$$
and more general
$$
\|Q(g,f)\|_{p,M,k}\leq MC\left(\|g\|_{p,M,k+1}\|f\|_{1,M,k+1}\right).
$$
\end{proposition}
\begin{proof}
We prove this with a simple computation, using the duality of the norm for $L^p$ spaces. For simplicity of notation during the proof we omit whenever is possible the appendix $M$ on the norm. Fix $m\in \{1,...,M\}$ and consider $Q_m(g,f)$ for $(g_n)_{n=1,...,M}$ and $(f_n)_{n=1,...,M}$:
\begin{align*}
    \|Q_m(g,f)\|_{p,k}&=\sup_{\phi\in L^{p'},\ \|\phi\|_{p'}=1}\int Q_m(g,f)\brak{v}^{k}\phi(v)dv,
\end{align*}
where $1/p+1/p'=1$. We now work directly on each term of the summation in the nonlinearity both in the positive and negative part. For the negative part we have:
\begin{align*}
    \iint g_m(v)f_n(w)|v-w|\brak{v}^k\phi(v)dwdv&\leq \iint g_m(v)f_n(w)\brak{v}^{k+1}\brak{w}\phi(v)dwdv\\
    &=\|f_n||_{1,1}\int g_m(v)\brak{v}^{k+1}\phi(v)dv\\
    &\underbrace{\leq}_{\|\phi\|_{p'}=1} \|g_m\|_{p,k+1}\|f_n||_{1,1}\leq \|g\|_{p,k+1}\|f\|_{1,k+1}.
\end{align*}
And now for the positive:
\begin{align*}
    &\iint c_m^n g_n(h(v,w))f_{m-n}(w)|v-w|\brak{v}^k\phi(v)dwdv\\
    &\lesssim \iint g_n(h)f_{m-n}(w)\brak{v}^{k+1}\brak{w}\phi(v)dwdv\\
    &\underbrace{=}_{h(v,w)=z}\iint g_n(z)f_{m-n}(w)\brak{\tilde{h}(z,w)}^{k+1}\brak{w}\phi(\tilde{h})dwdv\\
    &\leq \iint g_n(z)f_{m-n}(w)(\brak{z}^{k+1}\brak{w}+\brak{w}^{k+1})\phi(\tilde{h})dwdv\\
    &=\iint g_n(z)f_{m-n}(w)\brak{z}^{k+1}\brak{w}\phi(\tilde{h})dwdv+\iint g_n(z)f_{m-n}(w)\brak{w}^{k+1}\phi(\tilde{h})dwdv\\
    &=\int f_{m-n}(w)\int g_n(z)\brak{z}^{k+1}\phi(\tilde{h})dv\brak{w}dv+\int f_{m-n}(w)\brak{w}^{k+1}\int g_n(z)\phi(\tilde{h})dvdw\\
    &\underbrace{\leq}_{\text{H\"older\ inequality}}\int f_{m-n}(w)\|\brak{w}\|g_n\|_{p,k+1}dv+\int f_{m-n}(w)\brak{w}^{k+1}\|g_n\|_{p}dw\\
    &\underbrace{\leq}_{\text{Translation\ invariance}} 2\|g_n\|_{p,k+1}\|f_{m-n}||_{1,k+1}\leq 2\|g\|_{p,k+1}\|f\|_{1,k+1}.
\end{align*}
Putting everything together we conclude the proof.
\end{proof}
We begin now proving the a priori estimates for the standard moments of regular enough solution to the system of coagulation equations.
\begin{lemma}\label{lem:moments}
For any $\ell\in\mathbb N$, suppose that $f_m^0\in L^1_\ell$ and $\{f_m(t)\}_{m=1}^M$ is a nonnegative solution of the equation $(\ref{main-eq1})$, such that $f_m(t)$ and its gradient $\nabla f_m(t)$ have a fast decay at infinity. Then, there exists some finite constant $C_\ell$ depending only on $\ell, d,$ and the initial data $\{f_m(0)\}_{m=1}^M$ such that for any $t\ge 0$,
\begin{align}\label{eq:moment}
\sum_{m=1}^M\int  |v|^{\ell}f_m(t,v)dv\le C_\ell.
\end{align}
\end{lemma}
\begin{proof}
We first consider $\ell=2k$ even, and prove by induction on $k\in\mathbb{N}$. The case $k=0$ follows from  Lemma \ref{lem:conserved}. Now assume $k\ge 1$ and assume \eqref{eq:moment} is proved for the case $\ell=2(k-1)$. Note that for each $m=1,...,M$, by \eqref{main-eq1}
\begin{align*}
&\partial_t\int  |v|^{2k}f_m(t,v)dv=\int  |v|^{2k}\partial_tf_m(t,v)dv\\
=&\int |v|^{2k} \text{div}_v\left(vf_m\right)dv+\int |v|^{2k} \Delta_v f_mdv\\
& +\int dv \sum_{n=1}^{m-1}\iint_{\{nw+(m-n)w'=mv\}}|v|^{2k} f_n(t,w)f_{m-n}(t,w')|w-w'|dwdw'\\
&-2\int dv \sum_{n=1}^M \int |v|^{2k} f_m(t,v)f_n(t,w)|v-w|dw\\
= & -2k\int|v|^{2k} f_mdv+2k(2k+d-2)\int |v|^{2k-2}f_m dv \\
& + \sum_{n=1}^{m-1}\iint \left|\frac{nw+(m-n)w'}{m}\right|^{2k} f_n(t,w)f_{m-n}(t,w')|w-w'|dwdw'\\
&- 2\sum_{n=1}^M\iint |v|^{2k}  f_m(t,v)f_n(t,w)|v-w|dwdv.
\end{align*}
Above, we assumed that for every $t>0$,
\begin{align*}
|v|^{2k+1}|f_m|\to0, \quad |v|^{2k}|\nabla_vf_m|\to0,  \quad \text{ as }|v|\to\infty
\end{align*}
such that the boundary terms vanish in the integration by parts. Summing the above in $m=1,...,M$ and noticing by Jensen's inequality
\begin{align*}
 &\left|\frac{nw+(m-n)w'}{m}\right|^{2k}\le \left(\frac{n|w|+(m-n)|w'|}{m}\right)^{2k}\\
& \le \frac{n}{m}|w|^{2k}+\frac{m-n}{m}|w'|^{2k}\le |w|^{2k}+|w'|^{2k},
\end{align*}
whenever $1\le n\le m-1$, we obtain that
\begin{align*}
&\partial_t\sum_{m=1}^M\int  |v|^{2k} f_m(t,v)dv\\
&\le  -2k\sum_{m=1}^M\int  |v|^{2k} f_m(t,v)dv+2k(2k+d-2)\sum_{m=1}^M\int |v|^{2k-2}f_m(t,v) dv\\
&+\sum_{m=1}^M \sum_{n=1}^{m-1}\iint \left(|w|^{2k}+|w'|^{2k}\right) f_n(t,w)f_{m-n}(t,w')|w-w'|dwdw'\\
&- 2\sum_{m=1}^M\sum_{n=1}^M\iint |v|^{2k} f_m(t,v)f_n(t,w)|v-w|dwdv\\
&\le -2k\sum_{m=1}^M\int  |v|^{2k} f_m(t,v)dv+2k(2k+d-2) C_{2(k-1)}\\
&+\sum_{n=1}^M \sum_{m=n+1}^M\iint \left(|w|^{2k}+|w'|^{2k}\right) f_n(t,w)f_{m-n}(t,w')|w-w'|dwdw'\\
&- 2\sum_{m=1}^M\sum_{n=1}^M\iint |v|^{2k} f_m(t,v)f_n(t,w)|v-w|dwdv,
\end{align*}
where we used the induction hypothesis that for some finite constant $C_{2(k-1)}$ independent of $t$,
\begin{align*}
\sum_{m=1}^M\int |v|^{2k-2}f_m(t,v) dv\le C_{2(k-1)}.
\end{align*}
Further, setting $\ell=m-n$ we see that
\begin{align*}
&\sum_{n=1}^M \sum_{m=n+1}^M\iint \left(|w|^{2k}+|w'|^{2k}\right) f_n(t,w)f_{m-n}(t,w')|w-w'|dwdw'\\
&- 2\sum_{m=1}^M\sum_{n=1}^M\iint |v|^{2k} f_m(t,v)f_n(t,w)|v-w|dwdv\\
&\le\sum_{n=1}^M \sum_{\ell=1}^{M-n}\iint \left(|w|^{2k}+|w'|^{2k}\right) f_n(t,w)f_{\ell}(t,w')|w-w'|dwdw'\\
&- 2\sum_{m=1}^M\sum_{n=1}^M\iint |v|^{2k} f_m(t,v)f_n(t,w)|v-w|dwdv\; \le 0.
\end{align*}
Thus we conclude that
\begin{align*}
\partial_t\sum_{m=1}^M\int  |v|^{2k}f_m(t,v)dv\le &-2k \sum_{m=1}^M\int  |v|^{2k}f_m(t,v)dv+2k(2k+d-2)C_{2(k-1)}.
\end{align*}
Assuming the initial data is such that
\begin{align*}
A_0(k):=\sum_{m=1}^M\int |v|^{2k}f_m(0,v)dv
\end{align*}
is finite, it follows that for any $t\ge0$ we have that
\begin{align*}
\sum_{m=1}^M\int  |v|^{2k}f_m(t,v)dv&\le A_0(k)e^{-2kt}+(2k+d-2) C_{k-1}(1-e^{-2kt})\\
&\le A_0(k)+(2k+d-2) C_{2(k-1)}=:C_{2k}.
\end{align*}
This completes the induction for $n=2k$ even. Finally, we turn to the case $n=2k-1$ for $k\ge 1$. Note that
\begin{align*}
\sum_{m=1}^M\int |v|^{2k-1}f_m(v)dv&\le \sum_{m=1}^M\int_{|v|\le 1}|v|^{2k-1}f_m(v)dv+\sum_{m=1}^M\int_{|v|>1}|v|^{2k-1}f_m(v)dv\\
&\le \sum_{m=1}^M\int f_m(v)dv+\sum_{m=1}^M\int|v|^{2k}f_m(v)dv\le C_0+C_{2k}.
\end{align*}
\end{proof}

\begin{rmk}
Call $A_k:=\sum_{m=1}^M A_k^m:=\sum_{m=1}^M\int f_m\brak{v}^kdv$ and consider $\int Q(f,f)\brak{v}^k dv$. Although we have used other properties, this quantity can be also expanded in the following way:
\begin{align*}
    \int Q(f,f)\brak{v}^k dv\lesssim -2\sum_{m=1}^M\sum_{n=M-n+1}^M\iint f_m(v) f_n(w)|v-w|\brak{v}^k dwdv
\end{align*}
Using Proposition \ref{sobolev} and the modified triangular inequality: $-2|v-w|\brak{v}^k\leq -2\brak{v}^{k+1}+2|w|\brak{v}^k$, we get
\begin{align*}
    \lesssim -2\sum_{m=1}^M\sum_{n=M-n+1}^M\iint f_m(v) f_n(w)|v-w|\brak{v}^k dwdv\leq 2 A_1 A_k-2A_{k+1}C_k^M
\end{align*}
where $C_k^M:=\inf_t \int f_M(w)dw\geq0$. This give us an equation for $A_k$ of the form:
$$
\frac{d}{dt}A_k\leq -2C_k^MA_{k+1}+ C_kA_k(A_0+A_1)+C_{k,k-1}A_{k-1})
$$
Using the already proved bound on $A_k$ and the aforementioned equation, integrating in time, is enough to show that moments of order $k
\geq2$ are controlled for $t_0>0$, however small, even if $f_0$ has not bounded moments of higher order.
\end{rmk}
We can prove also an a priori bound in $L^2_tH^1_v\cap L^\infty_tL^2_v$ of solutions of \eqref{main-eq1}, under regularity assumption.
\begin{lemma}\label{lem:H_1}
Suppose that $f_m^0\in L^1_2\cap L^2$ and $\{f_m(t)\}_{m=1}^M$ is a nonnegative solution of the equation $(\ref{main-eq1})$, such that $f_m(t)$ and its gradient $\nabla f_m(t)$ have a fast decay at infinity. There exists some constant $C>0$ depending on the initial data $\{f_m(0),\nabla f_m(0)\}_{m=1}^M$, $T<\infty$, $M<\infty$  such that
\begin{align*}
\sup_{t\in[0,T]}\left( \sum_{m=1}^M\int  f^2_m(t,v)dv +\int_0^t\int |\nabla f_m(s,v)|^2dv\ ds\right)\le C.
\end{align*}
\end{lemma}
\begin{proof}
Note that for each $m=1,...,M$, by \eqref{main-eq}
\begin{align*}
&\frac{1}{2}\partial_t\int  f^2_m(t,v)dv=\int f_m(t,v)\partial_t f_m(t,v) dv\\
&= \int f_m \text{div}_v\left(vf_m\right)dv+\int f_m\Delta_v f_mdv\\
& +\int dv \sum_{n=1}^{m-1}\iint_{\{nw+(m-n)w'=mv\}} f_m(t,v)f_n(t,w)f_{m-n}(t,w')|w-w'|dwdw'\\
&-2\int dv \sum_{n=1}^M \int  f_m(t,v)f_m(t,v)f_n(t,w)|v-w|dw\\
&= \frac d2 \int f_m^2dv-\int |\nabla f_m|^2 dv \\
& + \sum_{n=1}^{m-1}\iint f_m\left(t,\frac{nw+(m-n)w'}{m}\right)f_n(t,w)f_{m-n}(t,w')|w-w'|dwdw'\\
&- 2\sum_{n=1}^M\iint f^2_m(t,v)f_n(t,w)|v-w|dwdv.
\end{align*}
Here, we assumed that
$|v\nabla_vf_m|\to0$,
as $|v|\to\infty$ for every $t$ such that the boundary terms vanish in the integration by parts.

\iffalse
roba
\begin{align*}
\partial_t\sum_{m=1}^M\int  f^2_m(t,v)dv+&\kappa\sum_{m=1}^M\int |\nabla f_m(t,v)|^2 dv\le  \frac d2\sum_{m=1}^M\int  f^2_m(t,v)dv+\\
&+\sum_{m=1}^M \sum_{n=1}^{m-1}\iint f_m\left(t,\frac{nw+(m-n)w'}{m}\right)f_n(t,w)f_{m-n}(t,w')|w-w'|dwdw'\\
&- 2\sum_{m=1}^M\sum_{n=1}^M\iint f^2_m(t,v)f_n(t,w)|v-w|dw\\
=&-2\sum_{m=1}^M\int  v^2f_m(t,v)dv+C_0\\
&+\sum_{n=1}^M \sum_{m=n+1}^M\iint \left(w^2+(w')^2\right)f_n(t,w)f_{m-n}(t,w')|w-w'|dwdw'\\
&- 2\sum_{m=1}^M\sum_{n=1}^M\iint v^2f_m(t,v)f_n(t,w)|v-w|dw\\
=&-2\sum_{m=1}^M\int  v^2f_m(t,v)dv+C_0\\
&+\sum_{n=1}^M \sum_{k=1}^{M-n}\iint \left(w^2+(w')^2\right)f_n(t,w)f_{k}(t,w')|w-w'|dwdw'\quad (\text{we set }k=m-n)\\
&- 2\sum_{m=1}^M\sum_{n=1}^M\iint v^2f_m(t,v)f_n(t,w)|v-w|dw,
\end{align*}
\fi

Further, we see that each term on the positive nonlinearity can be written using the Young's inequality has (same for $v$ or $w$ variable):
\begin{align*}
&\iint f_m\left(t,\frac{nv+(m-n)w}{m}\right)f_n(t,v)f_{m-n}(t,w)|v|dvdw=\\
&\le \int \left[\frac12 \left(\int f_m\Big(t,\frac{nv+(m-n)w}{m}\Big)f_n(t,v)|v|dv\right)^2+\frac12 f_{m-n}(t,w)^2\right]dw\\
&= \int \frac12 \left(\int f_m\left(t,\frac{nv+(m-n)w}{m}\right)f_n(t,v)|v|dv\right)^2dw+\frac12\|f_{m-n}(t,w)\|_{L^2}^2
\end{align*}
Thus we need to take care of the squared factor and we can do this using Jensen and normalizing respect the measure $(\int f_n(t,v)dv)^{-1}f_n(t,v)dv$, so that we obtain:
\begin{align*}
&\int \frac12 \left(\int f_m\left(t,\frac{nv+(m-n)w}{m}\right)f_n(t,v)|v|dv\right)^2dw\\
&\leq  \frac12 C_0\int\int f_m^2\left(t,\frac{nv+(m-n)w}{m}\right)|v|^2f_n(t,v)dvdw\\
&= (A)
\end{align*}
We have then, using translation invariance of Lebesgue measure:
\begin{align*}
&(A)=\frac12 C_0\int\left(\int f_m^2\left(t,\frac{nv+(m-n)w}{m}\right)dw\right)f_n(t,v)|v|^2dv\\
&\lesssim \|f_m\|^2_{L^2}\int |v|^2f_n(t,v)dv \leq C_1\|f_m\|^2_{L^2}
\end{align*}
where the constant $C_1:=C_1(M)$ depend also on $M$, using the moment bound Lemma \ref{lem:moments} with $\ell=2$, since we assumed that $f_m^0\in L^1_2$.

We can repeat the same argument for the $|w|$ part.
Thus we conclude that, erasing the negative term of the nonlinearity,
\begin{align*}
&\partial_t\sum_{m=1}^M\int  f^2_m(t,v)dv+\sum_{m=1}^M\int |\nabla f_m(t,v)|^2 dv\le  C(M,0,1,A)\sum_{m=1}^M\int  f^2_m(t,v)dv
\end{align*}
where $C(M,0,1,A)>0$, and assuming the initial data is such that
\begin{align*}
C_1:=\sum_{m=1}^M\int |v|^2f_m(0,v)dv,\;  C_0:=\sum_{m=1}^M\int f_m(0,v)dv,\;  C_l:=\sum_{m=1}^M\int f^2_m(0,v)dv< \infty.
\end{align*}
It follows that for any $t\in [0,T]$ we pass to integral form
\begin{align*}
&\sum_{m=1}^M\int  f^2_m(t,v)dv+\kappa c(m)^2\int_0^t\ \sum_{m=1}^M\int |\nabla f_m(s,v)|^2 dv\ ds\\
&\le C_{l} + C(M,0,1,A)\int_0^t\sum_{m=1}^M\int  f^2_m(s,v)dvds\\
&\leq C_{l} + C(M,0,1,A)\int_0^t\left(\sum_{m=1}^M\int  f^2_m(s,v)dv + \int_0^s\ \sum_{m=1}^M\int |\nabla f_m(r,v)|^2 dv\ dr \right)ds
\end{align*}
From Gronwall's lemma, it follows:
\begin{align*}
&\sup_{t\in [0,T]}\left(\sum_{m=1}^M\int  f^2_m(t,v)dv+\int_0^t\ \sum_{m=1}^M\int |\nabla f_m(s,v)|^2 dv\ ds\right)\leq C_{l}\ e^{TC_{M,0,A,1}}\leq C(T)<\infty.
\end{align*}
\end{proof}
\begin{rmk}
Consider the constant in the previous lemma on the right hand side of the Gronwall inequality and consider $C_T:=\sup_{t\leq T}C_t<\infty$. Then from the lemma we can recover the following:
$$
\partial_t \left(e^{-C_Tt}\|f_t\|_{2,M}^2\right)+\int_0^te^{-C_Tt}\|f_s\|_{H^1,M}^2ds\leq 0
$$
using Gronwall' lemma and using the property of sup and exponential, we have that exists $C:=C_Te^{C_TT}<\infty$ depending only on the initial condition $f_0$ and not his gradient, such that:
$$
\sup_{t\leq T}\|f_t\|_{2,M}^2+\int_0^t\|f_s\|_{H^1,M}^2ds\leq C.
$$
\qed
\end{rmk}
Using the already proven a priori estimate and the bound on the nonlinearity we can also show an a priori bound on the solution in the weighted space $L^\infty_t(L^2_k)_v\cap L^2_t(H^1_{k})_v$,
noting also that $|v|^k\le \langle v\rangle^k\le 2^{k-1}(1+|v|^k)$.

\begin{lemma}\label{lem:H_1-weighted}
For every $k\in\mathbb{N}$, suppose that $f^0_m\in L^1_{4k+2}\cap L^2_{k}$ and $\{f_m(t)\}_{m=1}^M$ is a nonnegative solution of the equation $(\ref{main-eq1})$, such that $f_m(t)$ and its gradient $\nabla f_m(t)$ have a fast decay at infinity. Then there exists some constant $C>0$ depending on the initial data $\{f_m(0),\nabla f_m(0)\}_{m=1}^M$, $T<\infty$, $M<\infty$ such that
\begin{align*}
\sup_{t\in[0,T]}\left( \sum_{m=1}^M\int  f^2_m(t,v)|v|^{2k}dv + \int_0^t\int |\nabla f_m(s,v)|^2|v|^{2k}dv\ ds\right)\le C.
\end{align*}
\end{lemma}
\begin{proof}
We perform induction on $k\in\mathbb{N}$. The case $k=0$ is proved in Lemma \ref{lem:H_1}.
Suppose that the thesis has been proved for the case $k-1$. Then, for each $m=1,...,M$, by \eqref{main-eq}
\begin{align*}
&\frac{1}{2}\partial_t\int  |v|^{2k}f^2_m(t,v)dv=\int |v|^{2k}f_m(t,v)\partial_t f_m(t,v) dv\\
&= \int |v|^{2k}f_m \text{div}_v\left(vf_m\right)dv+\int|v|^{2k} f_m\Delta_v f_mdv\\
& +\int dv \sum_{n=1}^{m-1}\iint_{\{nw+(m-n)w'=mv\}}|v|^{2k}f_m(t,v)f_n(t,w)f_{m-n}(t,w')|w-w'|dwdw'\\
&-2\int dv \sum_{n=1}^M \int  |v|^{2k}f_m(t,v)f_m(t,v)f_n(t,w)|v-w|dw\\
&\le (\frac{d}{2}-k)\int |v|^{2k}f_m^2dv-\frac{1}{2}\int |v|^{2k}|\nabla f_m|^2 dv +2 k^2\int |v|^{2k-2}f_m^2 dv\\
& + \sum_{n=1}^{m-1}\iint \left|\frac{nw+(m-n)w'}{m}\right|^{2k}f_m\left(t,\frac{nw+(m-n)w'}{m}\right)f_n(t,w)f_{m-n}(t,w')|w-w'|dwdw'\\
&- 2\sum_{n=1}^M\iint |v|^{2k}f^2_m(t,v)f_n(t,w)|v-w|dwdv,
\end{align*}
where we used Young's inequality
\begin{align*}
    -2k\int |v|^{2k-2} v\cdot\nabla f_m f_m dv &\le 2k\int |v|^{2k-1}|\nabla f_m|f_mdv\\
    &\le \frac{1}{2}\int |v|^{2k}|\nabla f_m|^2dv+2k^2\int |v|^{2k-2}f_m^2dv
\end{align*}
where we assumed that $|v|^{2k}|f_m\nabla f_m|\to0$, $|v|^{2k+1}|f_m|^2\to0$,
as $|v|\to\infty$ for every $t$ such that the boundary terms vanish in the integration by parts.

\iffalse
roba
\begin{align*}
\partial_t\sum_{m=1}^M\int  f^2_m(t,v)dv+&\kappa\sum_{m=1}^M\int |\nabla f_m(t,v)|^2 dv\le  \frac d2\sum_{m=1}^M\int  f^2_m(t,v)dv+\\
&+\sum_{m=1}^M \sum_{n=1}^{m-1}\iint f_m\left(t,\frac{nw+(m-n)w'}{m}\right)f_n(t,w)f_{m-n}(t,w')|w-w'|dwdw'\\
&- 2\sum_{m=1}^M\sum_{n=1}^M\iint f^2_m(t,v)f_n(t,w)|v-w|dw\\
=&-2\sum_{m=1}^M\int  v^2f_m(t,v)dv+C_0\\
&+\sum_{n=1}^M \sum_{m=n+1}^M\iint \left(w^2+(w')^2\right)f_n(t,w)f_{m-n}(t,w')|w-w'|dwdw'\\
&- 2\sum_{m=1}^M\sum_{n=1}^M\iint v^2f_m(t,v)f_n(t,w)|v-w|dw\\
=&-2\sum_{m=1}^M\int  v^2f_m(t,v)dv+C_0\\
&+\sum_{n=1}^M \sum_{k=1}^{M-n}\iint \left(w^2+(w')^2\right)f_n(t,w)f_{k}(t,w')|w-w'|dwdw'\quad (\text{we set }k=m-n)\\
&- 2\sum_{m=1}^M\sum_{n=1}^M\iint v^2f_m(t,v)f_n(t,w)|v-w|dw,
\end{align*}
\fi

Further, since
\begin{align*}
    \left|\frac{nw+(m-n)w'}{m}\right|^{2k}(|w|+|w'|)\le |w|^{2k+1}+|w'|^{2k+1}+|w|^{2k}|w'|+|w'|^{2k}|w|,
\end{align*}
we see that each term on the positive nonlinearity can be written using the Young's inequality has (same for $v$ or $w$ variable):
\begin{align*}
&\iint f_m\left(t,\frac{nv+(m-n)w}{m}\right)f_n(t,v)f_{m-n}(t,w)|v|^{2k+1}dvdw\\
&\le \int \left[\frac12 \Big(\int f_m\left(t,\frac{nv+(m-n)w}{m}\right)f_n(t,v)|v|^{2k+1}dv\right)^2+\frac12\left(f_{m-n}(t,w)\Big)^2\right]dw\\
&= \int \frac12 \left(\int f_m\left(t,\frac{nv+(m-n)w}{m}\right)f_n(t,v)|v|^{2k+1}dv\right)^2dw+\frac12\|f_{m-n}(t,w)\|_{L^2}^2
\end{align*}
Thus we need to take care of the squared factor and we can do this using Jensen and normalizing respect the measure $(\int f_n(t,v)dv)^{-1}f_n(t,v)dv$, so that we obtain:
\begin{align*}
&\frac12 \left(\int f_m\left(t,\frac{nv+(m-n)w}{m}\right)f_n(t,v)|v|^{2k+1}dv\right)^2dw\\
&\leq  \frac12 C_0\int\int f_m^2\left(t,\frac{nv+(m-n)w}{m}\right)|v|^{4k+2}f_n(t,v)dvdw\\
&= (A)
\end{align*}
We have then, using translation invariance of Lebesgue measure:
\begin{align*}
&(A)=\frac{1}{2}C_0\int\left(\int f_m^2\left(t,\frac{nv+(m-n)w}{m}\right)dw\right)f_n(t,v)|v|^{4k+2}dv\\
&\le C_0\|f_m\|^2_{L^2}\int |v|^{4k+2}f_n(t,v)dv \leq C_1\|f_m\|^2_{L^2}
\end{align*}
where the constant $C_1:=C_1(M)$ depend also on $M$ and comes from the momentum bound of $f_m$, Lemma \ref{lem:moments} since we assumed that $f_m^0\in L^1_{4k+2}$.

Similarly, we have another term
\begin{align*}
    &\iint f_m\left(t,\frac{nv+(m-n)w}{m}\right)f_n(t,v)f_{m-n}(t,w)|v|^{2k}|w|dw\\
    &\le\int \frac12 \left(\int f_m\left(t,\frac{nv+(m-n)w}{m}\right)f_n(t,v)|v|^{2k}dv\right)^2dw +\int \frac12 f^2_{m-n}(t,w)|w|^2dw\\
    &\le C_0\int \frac12 \int f^2_m\left(t,\frac{nv+(m-n)w}{m}\right)f_n(t,v)|v|^{4k}dvdw +\int \frac12 f^2_{m-n}(t,w)|w|^2dw\\
    &\le C_0||f_m||_{L^2}^2\int f_n(t,v)|v|^{4k}dv+ \int \frac12 f^2_{m-n}(t,w)|w|^2dw\\
    &\le C_1||f_m||_{L^2}^2+\int \frac12 f^2_{m-n}(t,w)|w|^2dw.
\end{align*}
Thus we conclude that, erasing the negative term of the nonlinearity and summing over $n=1,...,m-1$, using the induction hypothesis,
\begin{align*}
&\partial_t\sum_{m=1}^M\int  |v|^{2k}f^2_m(t,v)dv+\frac{1}{2}\sum_{m=1}^M\int |v|^{2k}|\nabla f_m(t,v)|^2 dv\\
&\le  C(M,0,1,A)\sum_{m=1}^M\int  f^2_m(t,v)|v|^{2k}dv+ C_B,
\end{align*}
where $C(M,0,1,A),C_B>0$, and assuming the initial data is such that
\begin{align*}
C_1:=\sum_{m=1}^M\int |v|^{4k+2}f_m(0,v)dv, \; C_0:=\sum_{m=1}^M\int f_m(0,v)dv,\;  C_l:=\sum_{m=1}^M\int |v|^{2k} f^2_m(0,v) dv< \infty
\end{align*}
It follows that for any $t\in [0,T]$ we pass to integral form
\begin{align*}
&\sum_{m=1}^M\int  |v|^{2k}f^2_m(t,v)dv+\frac{1}{2}\int_0^t\ \sum_{m=1}^M\int |v|^{2k}|\nabla f_m(s,v)|^2 dv\ ds\\
&\le C_{l,B} + C(M,0,1,A)\int_0^t\sum_{m=1}^M\int  |v|^{2k}f^2_m(s,v)dvds\\
&\leq C_{l,B} + C(M,0,1,A)\int_0^t\left(\sum_{m=1}^M\int  |v|^{2k}f^2_m(s,v)dv + \int_0^s\ \sum_{m=1}^M\int |v|^{2k}|\nabla f_m(r,v)|^2 dv\ dr \right)ds
\end{align*}
From Gronwall's lemma, it follows:
\begin{align*}
\sup_{t\in [0,T]}&\left(\sum_{m=1}^M\int  |v|^{2k}f^2_m(t,v)dv+ \int_0^t\ \sum_{m=1}^M\int |v|^{2k}|\nabla f_m(s,v)|^2 dv\ ds\right)\\
&\leq C_{l,B}\ e^{TC_{M,0,A,1}}\leq C(T)<\infty.
\end{align*}
\end{proof}
We want to extend the inequality to higher derivatives and higher moments, to this end we consider the following proposition regarding the nonlinearity $Q$:

\begin{proposition}\label{ppn:lebeniz}
Let $f,g$ smooth and rapidly decaying function in $v$ at infinity, then
\begin{align*}
    \nabla Q_m(f,g)=Q_m(c_m^n\nabla f,g)+Q_m(f,c_m^n\nabla g),
\end{align*}
for each $m=1,..,M$.
\end{proposition}
\begin{proof}
First of all we split the nonlinearity into the positive and negative part and we change variables to get
\begin{align*}
    Q^+_m(f,g)&:=\sum_{n=1}^{m-1}\int \left(\frac{m}{n}\right)^2f_{n}(\varphi(v,w))g_{m-n}(w)|v-w| dw\\
    &=\sum_{n=1}^{m-1}\int  \left(\frac{m}{n}\right)^2f_{n}(\tilde{\varphi}(v,w))g_{m-n}(\theta(v,w))|w| dw\\
    Q^-_m(f,g)&:=\sum_{n=1}^M  f_m(v)\int g_n(w)|v-w|dw=\sum_{n=1}^M f_m(v)\int g_n(\theta(v,w))|w|dw,
\end{align*}
where we have set
$$
\varphi(v,w):=\frac{mv-(m-n)w}{n},\quad \theta(v,w):=v+w,\quad \tilde{\varphi}(v,w):=v-\frac{m-n}{n}w.
$$
Whit this we can use the differentiation under integral sign and since the dependence on $v$ is only on $f$ and $g$ we have
\begin{align*}
    &\nabla_v \left(\int f_{n}(\tilde{\varphi}(v,w))g_{m-n}(\theta(v,w))|w| dw\right)\\
    &=\int \left(\nabla_v f_{n}(\tilde{\varphi}(v,w))\right)g_{m-n}(\theta(v,w))|w| dw+\int f_{n}(\tilde{\varphi}(v,w))\nabla_v g_{m-n}(\theta(v,w))|w| dw.\\
    &\nabla_v  \left(f_m(v)\int g_n(w)|v-w|dw\right) = \nabla_v \left(f_m(v)\int g_n(\theta(v,w))|w|dw\right)=\\
    &=\nabla_v f_m(v)\int g_n(\theta(v,w))|w|dw+  f_m(v)\int \nabla_v g_n(\theta(v,w))|w|dw.
\end{align*}
Taking out the constant in the differentiation, depending only on $m$ and $n$, and summing all together we conclude the proof.
\end{proof}
\begin{rmk}
As a corollary, iterating Proposition \ref{ppn:lebeniz}, higher-order derivatives of $Q$ can be
calculated using the following formula:
\begin{align*}
    \partial^jQ(f,g)=\sum_{0\leq l\leq j} \binom{j}{l}Q_m\left(c_{m,n}\partial^{j-l} f,c_{m,n}\partial^l g\right)
\end{align*}
where $j$ is a multi-index $|j|=n$ such that $j=j_1...j_d$ (with indices possibly being zeroes) and
$$
\partial^j:=\partial_{v_1}^{j_1}...\partial^{j_d}_{v_d}.
$$
The factor $\binom{j}{l}$ is a multinomial coefficients and the sum is intended in the sense of ordering of multi-indices.
\end{rmk}
With this in mind we are able to state the following a priori estimate on the space $H^n_k$ for all $k,n\in \mathbb{N}$, defined in \eqref{weighted-sobolev}
equipped with the norm $\|f\|_{H_k^n}:=\left(\sum_{0\le|j|\le n}\|\partial^j f\|_{2,k}^2\right)^{1/2}$.
\begin{lemma}\label{lem:H^n_k}
Suppose that $\{f_m\}_{m=1}^M$ is a nonnegative solution of the equation $(\ref{main-eq-trunc1})$, such that $f_m$ is regular enough. For all $n\in\mathbb{N},\ k\in\mathbb{N}$, there exists some constant $C>0$ depending on the initial data $\{f_m(0)\}_{m=1}^M$, $T<\infty$, $M<\infty$, $n,k$ such tha
\begin{align*}
\sup_{t\in[0,T]}\left( \sum_{m=1}^M \|f_m\|^2_{H^{n}_{k}}\right)\le C.
\end{align*}
\end{lemma}
\begin{proof}
We work by induction. We know form previous lemma that $n=0,1$ we have already proven the a priori bound. Now we suppose that for every $p<n$ it's true that $f\in H^p_k$ for all $k$, and we prove that $f\in H^n_k$ for all $k$. To this end, we consider a multi index $j, |j|=n$ and the quantity $\partial^j f_m$. Using the equation for $f_m$ we ge
\begin{align*}
    \partial_t \left(\partial^j f_m\right)=\Delta\left(\partial^j f_m\right)+ \div\left(v\partial^j f_m\right)+C_j\left(\partial^j f_m\right)+\sum_{0\leq l\leq j} \binom{j}{l}Q_m\left(\partial^{j-l} f,\partial^l f\right)
\end{align*}
where $C_j$ is either $0$ or $1$ depending if the multi index has the $i$-th component of the derivative of $\sum_{i=1}^d \partial^j\left(\partial_i f_m v_i\right)$. We now consider the quantity
$$
\sum_{m=1}^M\int |\partial^j f_m|^2\brak{v}^{2k} dv
$$
and derivate in time we have for each $m$:
\begin{align*}
\frac{1}{2}\partial_t\int  &\brak{v}^{2k}|\partial^j f_m|^2(t,v)dv=\int \brak{v}^{2k}\partial^jf_m(t,v)\partial_t \partial^jf_m(t,v) dv\\
&= \int \brak{v}^{2k}\partial^jf_m \text{div}\left(v\partial^jf_m\right)dv+\int\brak{v}^{2k} \partial^jf_m\Delta (\partial^jf_m) dv+\\
&+ \int C_j\brak{v}^{2k}|\partial^jf_m|^2dv
 +\sum_{0\leq l\leq j}\int Q_m(\partial^{j-l}f,\partial^lf)\partial^j f_m \brak{v}^{2k} dv\\
&\leq \frac d2 \int \brak{v}^{2k}|\partial^jf_m|^2 dv- k\int \brak{v}^{2k-2}|\partial^jf_m|^2 dv +\\
&+k \int \brak{v}^{2k}|\partial^jf_m|^2 dv- k\int \brak{v}^{k-2}|v||\partial^jf_m|^2 dv+\\
&-\int |\nabla (\partial^j f_m)|^2 \brak{v}^{2k}dv+\sum_{0\leq l\leq j}\int Q_m(\partial^{j-l}f,\partial^lf)\partial^j f_m \brak{v}^{2k} dv\\
&\leq C'_{d,k}\int \brak{v}^{2k}|\partial^jf_m|^2 dv -\int |\nabla (\partial^j f_m)|^2 \brak{v}^{2k}dv+\\
&+\underbrace{\sum_{0\leq l\leq j}\int Q_m(\partial^{j-l}f,\partial^lf)\partial^j f_m \brak{v}^{2k} dv}_{A}.
\end{align*}
We miss to understand now the nonlinearity. To this end we consider:
\begin{align*}
    \| Q_m(f,g)\|_2\leq  C\left(\sum_{n=1}^{m-1}\|f_{m-n}\|_{2,k+1}\|g_n\|_{1,k+1}+\|f_m\|_{1,k+1}\sum_{n=1}^M\|g_n\|_{2,k+1}\right).
\end{align*}
In particular we have:
\begin{align*}
    &\sup_{\|\phi\|_2=1}\iint C_{m,n}f_n(h(v,w))f_{m-n}(w)|v-w|\phi(v)dvdw\\
    &\leq \sup_{\|\phi\|_2=1}\iint C'_{m,n}f_n(z))f_{m-n}(w)|z-w|\phi(h'(z,w))dzdw\\
    &\lesssim\begin{cases}
        &\sup_{\|\phi\|_2=1}\int \int f_{m-n}(w)\phi(h'(z,w))\brak{w}dwf_n(z))\brak{z}dz\leq C_{n,m}\|f_n\|_{2,1}\|f_{m-n}\|_{1,1}\\[5pt]
        &\sup_{\|\phi\|_2=1}\int \int f_n(z))\phi(h'(z,w))\brak{z}dzf_{m-n}(w)\brak{w}dw\leq C_{n,m}\|f_{m-n}\|_{2,1}\|f_{n}\|_{1,1}.
    \end{cases}
\end{align*}
With this, in particular, we can select where to put the norm and the weight depending on the index of the derivative and we get summation of quantity that depends on lower order in $H^p_k$ with $p<n$ and a quantity depending on $\partial^j$.
\begin{align*}
  \sum_{0\leq l\leq j}\int &Q_m(\partial^{j-l}f,\partial^lf)\partial^j f_m \brak{v}^{2k} dv\leq \int Q_m(\partial^{j}f,f)\partial^j f_m \brak{v}^{2k} dv \\
  &+\int Q_m(f,\partial^j f)\partial^j f_m \brak{v}^{2k} dv  + \sum_{1\leq l\leq j-1}\int Q_m(\partial^{j-l}f,\partial^lf)\partial^j f_m \brak{v}^{2k} dv\\
  &\leq C_{m,n}(k,M)\sum_{m=1}^M\|\partial^j f_m\|^2_{2,k+\mu}.
\end{align*}
Now we note that:
\begin{align*}
    \|\partial^j f_m\|^2_{2,k+\mu}&\leq \delta \|\nabla\partial^j f_m\|^2_{2}+ c_\delta\|\partial^{j-1} f_m\|^2_{2,2(k+\mu)}\\
    &\leq  \delta \|\nabla\partial^j f_m\|^2_{2,2k}+ c_\delta\|\partial^{j-1} f_m\|^2_{2,2(k+\mu)}\\
    &\leq \delta \|\nabla\partial^j f_m\|^2_{2,2k} + C_{\delta, M, n-1}.
\end{align*}
We put together all the estimate and we get:
\begin{align*}
\frac{1}{2}&\partial_t\sum_{m=1}^M\int  \brak{v}^{2k}|\partial^j f_m|^2(t,v)dv-C'_{M,d,k}\sum_{m=1}^M\int \brak{v}^{2k}|\partial^jf_m|^2 dv \leq\\
&-\sum_{m=1}^M\int |\nabla (\partial^j f_m)|^2 \brak{v}^{2k}dv+\delta C(k,M)\sum_{m=1}^M\int |\nabla (\partial^j f_m)|^2 \brak{v}^{2k}dv+ C_M(n-1,k)
\end{align*}
Taking $\delta$ small enough, up to constant $C_\delta(M,k,n-1)$ we obtain
\begin{align*}
\frac{1}{2}\partial_t\sum_{m=1}^M\int  \brak{v}^{2k}|\partial^j f_m|^2(t,v)dv&-C'_{M,d,k}\sum_{m=1}^M\int \brak{v}^{2k}|\partial^jf_m|^2 dv \leq C_\delta(M,k,n-1).
\end{align*}
Thanks to Gronwall's lemma we conclude the proof.
\end{proof}
With all this a priori estimates follow also estimates on the $L^2_k$ norm of the solution. At the level of the a priori estimate, Lemmas \ref{lem:H^n_k} and \ref{lem:H_1} guarantee us that we have a bound for every $T>0$ on the derivatives in $L^2([0,T]\times\mathbb{R}^d)$ implying that after a arbitrary short time the derivatives $\partial^n f_m(t)\in L^2(\mathbb{R}^d)$ for any $n$ and thus they propagate in time. On the level of a priori estimates the solution results to be immediately infinitely smooth in $v$ and decay faster than any negative power of $|v|$ at infinity.\\
Using the a priori regularity in $H^n_k$ and the aforementioned propagation, we note that for every $t>0$ and $m=1,...,M$, $f_m(t)$ is in fact a Schwartz function  for $\forall t$ and this decay is enough to get that $\partial_t f\in L^2([0,T]\times\mathbb{R}^d)$.
\section{Approximating Problems and preliminary}\label{sec:approx-plm}
We define a sequence of approximating problems, indexed by $R>0$, in the following way:
\begin{align}\label{main-eq-trunc1}
\begin{cases}\partial_t f^R_m = \text{div}\left(vf^R_m\right)+\Delta f^R_m + Q^R_m(f^R,f^R)\\[5pt]
f^R_m|_{t=0}=f^0_m
\end{cases}
, \quad m=1,...,M
\end{align}
where $(t,v)\in [0,T]\times\mathbb{R}^d$ and $f^0_m$, $\forall m=1,...,M$ are the same as $(\ref{main-eq1})$. The approximated nonlinear term is expressed as follows:
\begin{align*}
Q^R_m(f,f)(t,v) &:= \sum_{n=1}^{m-1}\chi_R(v)\iint_{\{nw+(m-n)w'=mv\}} f_n(t,w)f_{m-n}(t,w')|w-w'|\chi_R(w) dwdw'\\
&\quad -2\sum_{n=1}^M \chi_R(v)\int  f_m(t,v)f_n(t,w)|v-w|\chi_R(w)dw\\
&= \sum_{n=1}^{m-1}\chi_R(v)\int f_n\Big(t,\varphi(v,w)\Big)f_{m-n}(t,w)|v-w|\big(\frac mn \big)^2\chi_R(w) dw\\
&\quad -2\sum_{n=1}^M \chi_R(v)\int f_m(t,v)f_n(t,w)|v-w|\chi_R(w)dw
\end{align*}
where $\chi_R(z):=\chi_{\mathbb{B}(0,R)}(z)$ is the indicator function on the ball of radius $R>0$ centered at the origin in $\mathbb{R}^d$.
\subsection{A priori estimate for the approximating equations}
We start with the same a priori estimate for the moments and energy, under the same regularity assumption of $(\ref{main-eq1})$, that we'll verify once existence and uniqueness is shown.\\
We suppose that $f^0_m\geq0,\ \forall\ m=1,..,M$, the solution of $(\ref{main-eq-trunc1})$ is nonnegative and  regular enough and has a fast decay at infinity, then we have for the nonlinear term:
\iffalse\begin{align*}
&\chi_{\mathbb{B}(0,R)}(v)\int \xi_R(f_n(t,\frac{mv-(m-n)w}{n}))\xi_R((f_{m-n}(t,w))|v-w|\frac mn \chi_{\mathbb{B}(0,R)}(w) dw\leq\\
&\leq\int f_n(t,\frac{mv-(m-n)w}{n})(f_{m-n}(t,w)|v-w|\frac mn dw
\end{align*}\fi
\begin{align*}
&\chi_R(v)\int f_n\Big(t,\varphi(v,w)\Big)f_{m-n}(t,w)|v-w|\chi_R(w) dw\\
&\leq\int f_n\Big(t,\varphi(v,w)\Big)f_{m-n}(t,w)|v-w| dw
\end{align*}
And thus we can recover the same a priori estimates, as in Lemmas \ref{lem:moments}, \ref{lem:H_1}, \ref{lem:H_1-weighted}, \ref{lem:H^n_k}, on the total mass, moments and energy in both $H^1$ and $H^n_k$ for the approximated problems with the same constant independent of $R$.
\begin{lemma}
For any $\ell\in\mathbb N\cup\{0\}$, there exists some finite constant $C_\ell$ depending only on $n,d$ and the initial data $\{f_m(0)\}_{m=1}^M$ such tha
\begin{align*
\sum_{m=1}^M\int  |v|^{\ell}f^R_m(t,v)dv\le C_\ell, \quad \forall R,
\end{align*}
as soon as $\{f_m^R\}_m\in L^2.$
\end{lemma}

\begin{lemma}\label{lem:H_1-trunc}
As soon as $\{f_m^R\}_m$ is regular enough, there exists some constant $C>0$ depending on the initial data $\{f_m(0),\nabla f_m(0)\}_{m=1}^M$, $T<\infty$, $M<\infty$ such that
\begin{align*}
\sup_{t\in[0,T]}\left( \sum_{m=1}^M\int  |f^{R}_m(t,v)|^2dv +\int_0^t\int |\nabla f^R_m(s,v)|^2dv\ ds\right)\le C.
\end{align*}
The constant $C$ does not depend on R and  is the same as the one in the full problem.
\end{lemma}

\begin{lemma}
As soon as $\{f_m^R\}_m$ is regular enough, $\forall k\geq 0$, there exists some constant $C>0$ depending on the initial data $\{f_m(0),\nabla f_m(0)\}_{m=1}^M$, $T<\infty$, $M<\infty$ such tha
\begin{align*}
\sup_{t\in[0,T]}\left( \sum_{m=1}^M\int
|f^{R}_m(t,v)|^2|v|^{2k}dv +\int_0^t\int |\nabla f^R_m(s,v)|^2|v|^{2k}dv\ ds\right)\le C.
\end{align*}
The constant $C$ does not depend on R and  is the same as the one in the full problem.
\end{lemma}
\begin{lemma}
Suppose that $\{f^R_m\}_{m=1}^M$ is a nonnegative solution of the equation $(\ref{main-eq-trunc1})$, such that $f^R_m$ is regular enough. For all $n\in\mathbb{N},\ k\in\mathbb{N}$, there exists some constant $C>0$ depending on the initial data $\{f_m(0)\}_{m=1}^M$, $T<\infty$, $M<\infty$, $n,k$  such that
\begin{align*}
\sup_{t\in[0,T]}\left( \sum_{m=1}^M \|f^R_m\|^2_{H^{n}_{k}}\right)\le C.
\end{align*}
The constant C does not depend on R and is the same as the one in the full problem.
\end{lemma}
\subsection{Preliminary on semigroups
We want to construct mild solution for our approximated problems. With this in mind,  we define the following operator on function defined on $\mathbb{R}^d$:
\begin{align*}
\mathcal{L}f:= \Delta f+  v\cdot\nabla f
\end{align*}
and rewrite our system of equations as:
\begin{align*}
\begin{cases}\partial_t f^R_m -\mathcal{L} f_m^R= d f^R_m + Q^R_m(f^R,f^R)\\
f^R_m|_{t=0}=f^0_m
\end{cases}
, \quad m=1,...,M
\end{align*}
with the usual definition of space, time and initial condition.

We know, \cite{MPV
, that endowed with its maximal domain
$$
D_{p,max}(\mathcal{L}):=\left\{u\in L^p(\mathbb{R}^d)\cap W^{2,p}_{loc}(\mathbb{R}^d):\ \mathcal{L}u\in L^p(\mathbb{R}^d)\right\}
$$
the operator $\mathcal{L}$ is the generator of a strongly continuous semigroup $(P_t)_{t\geq0}$ in $L^p(\mathbb{R}^d)$. In particular we can characterize the domain of the operator as follow:
\begin{lemma}[{\cite[Theorem 1]{MPV}}]
The domain $D_{p,max}(\mathcal{L})$ of the generator of the semigroup $(P_t)_{t\geq0}$ coincides with:
$$
D_p(\mathcal{L}):=\left\{u\in W^{2,p}(\mathbb{R}^d):\ v\cdot \nabla u\in L^p(\mathbb{R}^d)\right\}.
$$
\end{lemma}
Fur such a semigroup $(P_t)_{t\ge 0}$, one can derive the following properties \cite[Theorem 3.3]{MPV}.
\begin{proposition}
The operator $(\mathcal{L}, D_{p,max}(\mathcal{L}))$ generates a semigroup $(P_t)_{t\geq0}$ in $L^p(\mathbb{R}^d)$ which satisfies the estimate
$$
\|P_tf\|_p\leq \|f\|_p
$$
for every $f\in L^p(\mathbb{R}^d)$.
\end{proposition}
\begin{proposition}\label{ppn:grad-estim}
For every $f\in L^p(\mathbb{R}^d)$ and $T>0$, the function $P(\cdot)f$ belongs to $C((0,T],W^{2,p}(\mathbb{R}^d))\cap C^1((0,T],L^{p}_{\text{loc}}(\mathbb{R}^d))$, and satisfies the estimates for $t\in(0,T]$,
\begin{align*}
    \|D^2P_tf\|_p\leq \frac{C_T}{t}\|f\|_p,\ \|\nabla P_tf\|_p\leq\frac{C_T}{\sqrt{t}}\|f\|_p.
\end{align*}
\end{proposition}
\begin{proposition}\label{ppn:reg-semigp}
Let $T>0$ and $g\in C([0,T],L^p(\mathbb{R}^d))$ be given, and consider the mild solution $u$ of the
Cauchy problem
\begin{align*}
    \begin{cases}
    \partial_t u-\mathcal{L} u = g\ &in\ [0,T]\times\mathbb{R}^d\\
    u(0) = f\ &in\ \mathbb{R}^d,
    \end{cases}
\end{align*}
with $f=0$. Then, $u$ belongs to $C([0,T],W^{2,p}(\mathbb{R}^d))\cap W^{1,p}_{\text{loc}}([0,T]\times\mathbb{R}^d)$.
\end{proposition}
Thus, we can define the notion of mild solution for our system. For simplicity of notation we omit the dependence on $m$.
\begin{definition}[$L_v^2$-mild solution for approximated problems]
We rewrite first our approximated problems as an evolution equation in $L^p$ space
\begin{align*}
    \Dot{f}+\mathcal{L}f=df+Q(f,f),\ t>0\,\ f(0)=f^0.
\end{align*}
Fix $T>0$, a function $f\in C([0,T],L^2)$ is said to be a $L^2$-mild solution to $(\ref{main-eq-trunc1})$ on $[0,T]$ if $f$ solves the integral equation
\begin{align*}
    f(t)=P_tf^0+\int_0^tP_{t-s}\left(df(s)+Q(f(s),f(s))\right)ds,\ t\in[0,T].
\end{align*}
\end{definition}
\subsection{Aubin-Lions' Theorem in full Space}
\begin{lemma}[Aubin-Lions weighted] \label{lem:aubin}
Consider $H^1(\mathbb{R}^d)$, the usual Sobolev space, and $L^2(\mathbb{R}^d,|v|^2dx)$, the weighted Lebesgue space containing functions $f$ for which $\int|f|^2|v|^2dv<\infty$. Then
$$
H^1(\mathbb{R}^d)\cap L^2(\mathbb{R}^d,|v|^2dv)\subset \joinrel\subset L^2(\mathbb{R}^d)
$$
and the embedding is compact.
As a consequence for every $q,k\geq1$ we have that
$$
L^p\left(0,T;H^1(\mathbb{R}^d)\cap L^2(\mathbb{R}^d,|v|^2dv)\right)\cap W^{1,q}(0,T;H^{-k}(\mathbb{R}^d))\subset\joinrel\subset L^p(0,T;L^2(\mathbb{R}^d))
$$
 for $p\in [1,\infty]$ and the embedding is compact.
\end{lemma}
\begin{proof}
Call $X:=H^1(\mathbb{R}^d)\cap L^2(\mathbb{R}^d,|v|^2dv)$ and consider $Y\subset X$ a bounded set.\\
We claim that $\forall \varepsilon>0,\exists N>0$ such that for all $f\in Y$:
$$
\int_{B^c_N} f^2<\varepsilon/2,
$$
Where $B^c_N$ is the complement in $\mathbb{R}^d$ of the ball of radius $N$ and centered at the origin.
Otherwise, assume this is not the case, then for some $\varepsilon>0$, for every $N>0$ there is some $f\in Y$ such that $\|f\|_{L^2(B^c_N)}\geq\varepsilon$.\\
This implies that $\|f\|_{X}\geq \|f^2|v|^2\|_{L^1(B^c_N)}\geq N^2\varepsilon$ so that the set $Y$ is not bounded, and this contradicts the hypotheses. So we consider the ball $B_N$ and we can apply Rellich-Kondrachov theorem to ensure the compact embedding to $L^2(B_N)$.\\
Now we recall that compactness of a set $A\subset X$ means that $\forall\varepsilon>0$ we find $f_1,...,f_k\in X$, where $k=k(\varepsilon)$, such that $A\subset\cup_{i=1}^k B(f_i,\varepsilon)$.\\
This means that $\forall \varepsilon>0$ we find functions $f_1,...,f_k\in L^2(B_N),\ k=k(\varepsilon)$, such that $\forall f\in Y$ we have some $f_j$ such that
$$\|f-f_j\|_{L^2(B_N)}<\varepsilon/2$$
We recall then that $\|f\|_{L^2(B^c_N)}<\varepsilon/2$, so we even have
$\|f-f_j\|_{L^2(\mathbb{R}^d)}<\varepsilon$,
which implies the compact embedding $Y\subset\joinrel\subset L^2(\mathbb{R}^d)$.\\
The second part of the theorem now follows from the usual Aubin-Lions Theorem cf. \cite[Corollary 5]{Simon}, thus concluding the proof.
\end{proof}

\section{Existence and Uniqueness for the approximating problems}\label{sec:exist-uniq}
\begin{theorem}[Existence and Uniqueness]\label{thm:approx-plm}
Consider $d\geq 1$. Given any initial value $f^0_m\in L^2,\forall
 m=1,...,M$, problem $(\ref{main-eq-trunc1})$ possesses a unique maximal $L^2$-mild solution $f:= f(\cdot; f^0)$ on $[0,T(f^0))$. The maximal interval is such that $[0,T(f^0))$ is an open interval in $\mathbb{R}_+$. In addition,
 $$
 f\in L^\infty([0,T(f^0)), L^2(\mathbb{R}^d)^{\otimes M})\cap W_{loc}^{1,2}((0,T(f^0)), H^2(\mathbb{R}^d)^{\otimes M}).
 $$
In addition: if $T(f^0)<\infty$, then
$$
\sup_{T(f^0)/2<t<T(f^0)}\|f_t\|_2=\infty.
$$
\end{theorem}
\begin{proof}
Let $T_0>0$ be arbitrary and define $X_T:=L^\infty([0,T],L_M^2)$ for $T\in(0,T_0]$, where we endowed the space with the usual sup norm in time and $L_M^2$ norm in velocity as $\|f\|_{L^2_M}:=\sum_{m=1}^M\|f_m\|_{L^2}.$\\
Consider $f^0\in L_M^2$ then from Proposition \ref{ppn:grad-estim} we know that $P_tf^0\in X_T$. From the same theorem it is implied that exist a constant $\kappa:=\kappa(T_0)>0$ such that $\|P_t f^=\|\leq e^\kappa(T_0)t$; we show that this is enough to define a map $\Gamma$ from $X_T\rightarrow X_T$, given by
$$
\Gamma(f):=P_\cdot f^0+\int_0^\cdot P_{\cdot-s}(df-Q^R(f,f))ds,\ f\in X_T.
$$
First, thanks to the estimate on the semigroup and the property of the truncated nonlinearity, we have that, as soon as $f\in X_T$
\begin{align*}
\begin{cases}
    &\|Q^R_m(f,f)\|_2^2\leq C_R (\sum_{m=1}^M \|f_m\|_2)^2<\infty.\\[5pt]
    & \int_0^\cdot P_{\cdot-s}(df-Q^R(f,f))ds\in X_T.
\end{cases}
\end{align*}
In particular we obtain the first estimate from the following computation on each of the terms of the nonlinearity
$$
\int \left|\int |v-w|_Rf_{n}(h(v,w))f_{m-n}(w)dw\right|^2dv\leq 2R\int_{B_R} \|f_n\|_2^2\|f_{m-n}\|_2^2dv= 2R|B_R|\|f_n\|_2^2\|f_{m-n}\|_2^2.
$$
Now we notice that the function $\Gamma$ also satisfies
\begin{align*}
    \|\Gamma(f)-P_\cdot f^0\|_{X_T}&=\left\|\int_0^\cdot P_{\cdot-s}(df-Q^R(f,f))ds\right\|_{X_T}\\
    &\leq \kappa T\left(d+C_R(\sum_{m=1}^M \|f_m\|_2)\right)(\sum_{m=1}^M \|f_m\|_2),
\end{align*}
for all $f\in X_T$. And we can prove the continuity of the map
\begin{align*}
    \|\Gamma(f)-\Gamma(g)\|_{X_T}&=\left\|\int_0^\cdot P_{\cdot-s}(d(f-g)+\left(Q^R(g,g)-Q^R(f,f)\right)ds\right\|_{X_T}\\
    &\leq T\left(d+C_R\|f\|_{X_T}+C_R\|g\|_{X_T}\right)\|f-g\|_{X_T},
\end{align*}
for all $f,g\in X_T$.\\
If we find a way to restrict the mapping $\Gamma$ to some closed subset, and prove that it is a contraction then, with a standard fix point argument, we conclude local existence and uniqueness for each approximating problems.\\
Consider to this extent the norm of the initial condition $\|P_\cdot f^0\|_{X_T}=\gamma$, we choose the  ball in $X_T$, call it $B_T$, centered in $P_\cdot f^0$ with radius $\gamma$. So that every function $f\in B_T$ as norm $\|f\|_{X_T}\leq 2\gamma$. Then selecting
$$
T<\min\left\{ \frac{1}{d+4C_R\gamma},\frac{1}{\kappa(d+C_R 2\gamma)}\right\}
$$
The two inequality implies that the map $\Gamma$ send function in $B_T$ to itself and it is a contraction. Therefore there exist a unique $\tilde{f}\in B_T$ such that $\Gamma(\tilde{f})=\tilde{f}$, that is,
$$
\tilde{f}\in L^\infty([0,T],L_M^2)\ \text{ and }\ \tilde{f}=P_{\cdot}f^0+\int_0^{\cdot}P_{\cdot-s}\left(d\tilde{f}+Q^R(\tilde{f},\tilde{f})\right)ds.
$$
and $\tilde{f}$ is a mild-$L^2$ solution to the approximated problems.\\

Clearly we can extend the solution $\tilde{f}$ to a unique maximal solution $\tilde{f}:=\tilde{f}(f^0)$ with maximal interval $[0,T(f^0))$ that must be open in $\mathbb{R}^+$.\\
Consider $T^0:=T(f^0)<\infty$ and suppose there exist an increasing sequence $t_i\rightarrow T^0$ such that $\|f_{t_i}\|_p\leq r<\infty,\ \forall i\in\mathbb{N}$. Fix now $\overline{T}>T^0$ and fix $\gamma(\overline{T})$ the constant defined in the existence part of the theorem for the ball in which we perform the contraction. We have from the semigroup $P_t$ in the interval $[0,\overline{T}]$ that
$$
\|P_t (f_{t_i})\|_{X_{\overline{T}}}\lesssim re^{-\overline{T}}.
$$
We choose then $\overline{T}>0$ such that it holds $\|P_t (f_{t_i})\|_{X_{\overline{T}}}\leq \gamma(T^0)$, for $i\in \mathbb{N}$.\\
We can now repeat the same existence scheme and we obtain that $f$ exists at least on $[t_i,t_i+\overline{T}],\ \forall i\in\mathbb{N}$ contradicting the maximal extension of the interval. Thus, for $f^0\in L^2$ we must have
$$
\sup_{T(f^0)-\epsilon<t<T(f^0)}\|f_t\|_{L^2_M}=\infty.
$$
This concludes the proof.
\end{proof}
\begin{theorem}
Given the solution $f\in X_T$ of the truncated equation from Theorem \ref{thm:approx-plm}, we have that $f(t; f^0) \in L_+^2$ for $t \in [0,T(f^0))$ provided that $f^0 \in L_+^2$.
\end{theorem}
\begin{proof}
Consider $f^0\in L^2_+,$ and let $T\in [0,T^0)$ be arbitrary. Then there is a constant $\omega:=\omega(T)>0$ such that, for $t\in [0, T]$,
\begin{align*}
    \left|\sum_{n=1}^M\int |v-w|\chi_R(w)\chi_R(v)f_n(w)dw\right|\leq \sum_{n=1}^M C_R\|f_n\|_{X_T}:=\omega(T),\ a.e.\ v\in\mathbb{R}^d.
\end{align*}
Now, for $g\in L^2$ and $0\leq t\leq T$ set for $m=1,..,M$
$$
\overline{Q}^m_t(g):=dg_m+Q_m^R(g,g)+\omega(T)_Rg_m,
$$
so, for $t\in[0,T],\ a.e.v\in\mathbb{R}^d$ and $g\in L^p_+$, we have $\overline{Q}(g,g)\geq0$.\\
We consider now the evolution equation
\begin{align}\label{poseq}
    \Dot{g}+\left(\omega(T)+\mathcal{L}\right)g=\overline{Q}(g,g),\ t\in[0,T],\ g(0)=f^0.
\end{align}
We note that $f$ is a solution of such equation and that equation $(\ref{poseq})$ can be solved by the method of successive approximations exactly as in the existence theorem. So we consider
$$
\Theta(g):=P_\omega f^0+\int_0^\cdot P_{\cdot-s}^\omega (\overline{Q}(g,g) ds,\ \text{for}\ g\in X_T
$$where $P_\omega(t):=e^{-\omega t}P_t,\ t\geq0$.\\
In the same fashion as Theorem \ref{thm:approx-plm}, $\Theta$ is a contraction in a suitable ball $B_T\subset X_T$, centered in $P_\cdot f^0$, into itself. Taking $T$ small enough we can assume that also $f\in B_T$.
Therefore, since there must be existence and uniqueness of a solution we can find a sequence $g_i$, determined by
$$
g_0=f^0,\ g_{i+1}=\Theta(g_i),\ i\in\mathbb{N},
$$
such that converges to $f$ in $X_T$. This implies that $g_i\rightarrow f$ in $L^2_M$ for $t\in(0,T]$.\\
Since the semigroup $P_t$ preserves positivity, by induction we get that $g_i\in L^2_+$ for $t\in[0,T]$ and $i\in\mathbb{N}$. This means that also $f_t\in L^2_+$ for all $t\leq T$ since the space is closed.\\
Consider now a $\overline{T}\leq T^0$, the maximal time for which $f$ is positive on $[0,\overline{T}]$, then we claim that $\overline{T}=T^0$. Otherwise, we consider now the same equation $\eqref{poseq}$ re-scaled in time
\begin{align*}
    \Dot{g}+\left(\omega+\mathcal{L}\right)g=\overline{Q}(g,g)(t+\overline{T}),\ t\in[0,T^0-\overline{T}],\ g(0)=f_{\overline{T}},
\end{align*}
and this would lead to a contradiction on the maximality of T. Since $T^0$ was arbitrary in the maximal interval of definition we deduce that $f\in L^2_+,\ \forall t\in [0,T^0)$.
\end{proof}
We wish now to understand better the regularity of the truncated solution and the estimate on such computation. To this end we want to use the a priori estimate, but in doing so we first need to establish regularity deriving directly from the semigroup and the nonlinearity.\\
Following from Proposition \ref{ppn:reg-semigp} we can derive some regularity on $f$ and its derivatives in $L^2$ space. In particular, with $f_0\in L^2_M$, we have
$$
f\in L^\infty([0,T],H^2(\mathbb{R}^d))\cap W^{1,2}_{\text{loc}}([0,T]\times\mathbb{R}^d).
$$
This is still not enough and we want to control also the moments of the solution and to this end it is not enough to have $f_0\in L^2_m$, so we first consider an initial condition such that $f_0\in L^{1}_{p,M}:=L^1_M(\brak{v}^pdv)$ for every $p\geq0$ and also in $L^2_{+,M}$, we show that this is enough to get that the solution
$$
f^R\in L^\infty([0,T],L^1_{p,M}\cap L^2_{+,M}(\mathbb{R}^d)),
$$
And this holds for every $p\geq0$.\\
\\
We want to establish a Gronwall relationship, so we integrate the function $f_m$ (we drop the $R$ for readability) for all $t>0$. We note first that following the step of Theorem \ref{thm:approx-plm} it is easy to construct the same function adding the condition that $\int |f|dv$ must be bounded.\\ With this in mind, using the consistency\cite{ConsistSem} of the operator $P_t$, we can write:
\begin{align*}
    \|f_m(t)\|_{1}&\underbrace{=}_{\text{positivity}}\int f_m(t)dv=\int P_tf^0_mdv+\int \int_0^tP_{t-s}\left(df_s+Q_m^R(f,f)_s\right) dsdv\\
    &\underbrace{\leq}_{\text{property\ of}\ P_t} \int f_m^0dv+d\int_0^t \|f_m(s)\|_1ds +T\sup_t \left(\int Q^R_m(f,f)dv\right)
\end{align*}
analogous to the proof of Lemma \ref{lem:conserved} we have $\sup_t \sum_{m=1}^M \int Q^R_m(f,f)dv\leq 0$ and thus summing on $m$ and taking the sup in $t\leq T$ we get the bound of the $L^1_M$ norm, independent on $R$.\\
\\
Establish the $L^1$ bound, we use this results to prove that, depending on $R$ this time, we have bound on  the $p$-th moment for every $p$. To this end consider:
\begin{align*}
    &\int f_m(t)\brak{v}^p dv \leq \iint K(t,w)f_m^0(e^tv-w)\brak{v}^p dwdv+\int_0^t\iint K(s,w)Q^R_m(f,f)(e^sv-w)\brak{v}^pdvdwds\\
    &\underbrace{\leq}_{e^tv-w=z}\int K(t,w)\brak{w}^pdw \int C_t f_m^0(z)\brak{z}^p dz+\int_0^t\int K(s,w)\brak{w}^pdw\int C_s Q^R_m(f,f)(z)\brak{z}^pdzds
\end{align*}
Recall that
\begin{align*}
K(t,w):=\frac{1}{(2\pi(e^{2t}-1))^{\frac{d}{2}}}e^{-\frac{|w|^2}{2\pi(e^{2t}-1)}},
\end{align*}
and that the truncated nonlinearity has a term of the form $\chi_R(z)\chi_R(w)|z-w|$ and as such we get
\begin{align*}
    \int f_m(t)\brak{v}^p dv &\underbrace{\leq}_{\int K(t,w)\brak{w}^pdw<\infty\ \forall p}C_T\|f^0\|_{p,M}^1+C(R)\int_0^tC_s\sum_{n=1}^{m-1}\iint f_n(\varphi(v,w))f_m(w)dwdvds\\
    &\leq C_T\|f^0\|_{p,M}^1+C(R)C_T\sum_{n=1}^{m-1}\sup_t\|f_n\|_1\|f_m\|_1\\
    &\leq C_T\|f^0\|_{p,M}^1+C_T(R,\sup_t\|f\|_{M}^1,M)\sup_t\|f\|^1_{p,M}.
\end{align*}
Summing on $m=1,...,M$, taking the sup over $t\leq T$, we obtain the bound depending on the initial condition, the $L^1$ norm of the solution and the truncation $R$.\\
\\
We show now that a mild solution is indeed a weak solution for our equation and use this to enhance the regularity of the time derivative of the solution.
\begin{lemma}[$L^p$ mild solutions are weak solutions]
Consider $f\in X_T$ a mild solution for equation $(\ref{main-eq-trunc1})$ corresponding to the initial value $f_0$, then $f$ is a weak solution in the sense of Definition \ref{def:weak-sol}.
\end{lemma}
\begin{proof}
Fix $T>0$ an a time $t\in[0,T]$, fix $\phi$ a test function. Consider now the weak formulation for equation $\eqref{main-eq-trunc1}$:
$$
\brak{f_t,\phi}= \brak{f_0,\phi}+\int_0^t\brak{f_s,\mathcal{L}^*\phi}ds+\int_0^t\brak{Q_s(f,f),\phi}ds,
$$
where $\mathcal{L}^*\phi=-v\cdot\nabla\phi+\Delta\phi$. Notice that
$$
\brak{f_t,\mathcal{L}^*\phi}=\brak{\mathcal{L}(f_t)+df_t,\phi}.
$$
Under this definition and observation we consider now $f$ mild solution as before
$$
f_t=P_tf_0+\int_0^tP_{t-s}f_sds+\int_0^tP_{t-s}Q_s(f,f)ds
$$
ans analyze the well defined quanity
\begin{align*}
\int_0^t\brak{f_s,\mathcal{L}^*\phi}ds&=\int_0^t\brak{P_sf_0+\int_0^sP_{s-r}f_rdr+\int_0^sP_{s-r}Q_r(f,f)dr,\mathcal{L}^*\phi}ds\\
&=\underbrace{\int_0^t\brak{P_sf_0,\mathcal{L}^*\phi}ds}_{A}+\underbrace{\int_0^t\brak{\int_0^sP_{s-r}f_rdr,\mathcal{L}^*\phi}ds}_{B}+\underbrace{\int_0^t\brak{\int_0^sP_{s-r}Q_r(f,f)dr,\mathcal{L}^*\phi}ds}_{C}.
\end{align*}
Now we analyze, using the regularity result on $f$ derived in the previous sections and using \cite{MPV}, each of the component on the right hand side.\\
\textbf{A:}
\begin{align*}
    \int_0^t\brak{P_sf_0,\mathcal{L}^*\phi}ds&=\int_0^t\brak{\mathcal{L}P_sf_0,\phi}ds+\int_0^t\brak{P_sf_0,\phi}ds\\
    &=\brak{\int_0^t\frac{d}{ds}P_sf_0ds,\phi}+\int_0^t\brak{P_sf_0,\phi}ds\\
    &=\brak{P_tf_0,\phi}-\brak{f_0,\phi}+\int_0^t\brak{P_sf_0,\phi}ds
\end{align*}
\textbf{B:}
\begin{align*}
\int_0^t\brak{\int_0^sP_{s-r}f_rdr,\mathcal{L}^*\phi}ds&=
\int_0^t\brak{\int_0^s\mathcal{L}P_{s-r}f_rdr,\phi}ds+
\int_0^t\brak{\int_0^sP_{s-r}f_rdr,\phi}ds.\\
&=\int_0^t\brak{\int_r^t\mathcal{L}P_{s-r}f_rds,\phi}dr+
\int_0^t\brak{\int_0^sP_{s-r}f_rdr,\phi}ds\\
&=\int_0^t\brak{\int_r^t\frac{d}{ds}P_{s-r}f_rds,\phi}dr+
\int_0^t\brak{\int_0^sP_{s-r}f_rdr,\phi}ds\\
&=\int_0^t\brak{P_{t-r}f_r,\phi}dr-\int_0^t\brak{f_r,\phi}dr+
\int_0^t\brak{\int_0^sP_{s-r}f_rdr,\phi}ds\\
\end{align*}
\textbf{C:}
\begin{align*}
\int_0^t\brak{\int_0^sP_{s-r}Q_r(f,f)dr,\mathcal{L}^*\phi}ds&=\int_0^t\brak{\int_0^s\mathcal{L}P_{s-r}Q_r(f,f)dr,\phi}ds+\int_0^t\brak{\int_0^sP_{s-r}Q_r(f,f)dr,\phi}ds\\
&=\int_0^t\brak{\int_0^s\mathcal{L}P_{s-r}Q_r(f,f)dr,\phi}ds+\int_0^t\brak{\int_0^sP_{s-r}Q_r(f,f)dr,\phi}ds\\
&=\int_0^t\brak{\int_r^t\mathcal{L}P_{s-r}Q_r(f,f)ds,\phi}dr+\int_0^t\brak{\int_0^sP_{s-r}Q_r(f,f)dr,\phi}ds\\
&=\int_0^t\brak{P_{t-r}Q_r(f,f)-Q_r(f,f),\phi}dr+\int_0^t\brak{\int_0^sP_{s-r}Q_r(f,f)dr,\phi}ds.
\end{align*}
To prove all the following equality we have used the fact that, if $f\in D(\mathcal{L})$, then $P_tf\in D(\mathcal{L})$ and in that case
$$
\frac{d}{dt}P_tf=\mathcal{L}P_tf=P_t\mathcal{L}f.
$$
Thanks to the reguliraty on $f$ we can apply this result, thus putting all together we get:
\begin{align*}
 \int_0^t\brak{f_s,\mathcal{L}^*\phi}ds&=\brak{P_tf_0,\phi}-\brak{f_0,\phi}+\int_0^t\brak{P_sf_0,\phi}ds\\
 &+\int_0^t\brak{P_{t-r}f_r,\phi}dr-\int_0^t\brak{f_r,\phi}dr+
\int_0^t\brak{\int_0^sP_{s-r}f_rdr,\phi}ds\\
&+\int_0^t\brak{P_{t-r}Q_r(f,f),\phi}dr-\int_0^t\brak{Q_r(f,f),\phi}dr+\int_0^t\brak{\int_0^sP_{s-r}Q_r(f,f)dr,\phi}ds.
\end{align*}
For which we derive, using the equality of the mild solution:
\begin{align*}
 \brak{f_0,\phi}&+\int_0^t\brak{f_s,\mathcal{L}^*\phi}ds+\int_0^t\brak{Q_r(f,f),\phi}dr=\underbrace{\brak{P_tf_0,\phi}+\int_0^t\brak{P_{t-r}f_r,\phi}dr+\int_0^t\brak{P_{t-r}Q_r(f,f),\phi}dr}_{:=\brak{P_tf_o+\int_0^tP_{t-r}f_rdr+\int_0^tP_{t-r}Q_r(f,f)dr,\phi}=\brak{f_t,\phi}}\\
 &+\underbrace{\int_0^t\brak{P_sf_0,\phi}ds +\int_0^t\brak{\int_0^sP_{s-r}f_rdr,\phi}ds+\int_0^t\brak{\int_0^sP_{s-r}Q_r(f,f)dr,\phi}ds}_{:=\int_0^t\brak{f_s,\phi}ds}-\int_0^t\brak{f_s,\phi}ds.
\end{align*}
Thus $\brak{f_0,\phi}+\int_0^t\brak{f_s,\mathcal{L}^*\phi}ds+\int_0^t\brak{Q_r(f,f),\phi}dr=\brak{f_t,\phi}$, concluding the proof.
\end{proof}
To conclude we need to extend the regularity on the time derivative and prove that
$$
\partial_t f_m\in L^2([0,T]\times\mathbb{R}^d).
$$
Analyzing the equation solved by $f^R$, proving that $\partial_t f^R_m$ is in $L^2([0,T]\times \mathbb{R}^d)$ boils down to prove that the quantity
$$
v\cdot \nabla f_m\in L^2([0,T]\times\mathbb{R}^d)\text{  i.e. }\ \int_0^T \int |v\cdot \nabla f_m(s)|^2ds<\infty.
$$
In order to to this we use the mild formulation of $f$, the explicit formulation of the nonlinearity $Q^R_m$ and the bound we have previously derived. We reduce again the problem, using Cauchy-Schwartz and the inequality on the weight to prove that
$$
\int_0^t \int|\nabla f_m|^2 \brak{v}^2dvds <\infty.
$$
We consider:
\begin{align*}
    \int_0^t \int|\nabla f_m|^2 \brak{v}^2dvds&\leq \underbrace{\int_0^t\int|\nabla P_tf_0|^2 \brak{v}^2dsdv}_{A}+\underbrace{\int_0^t \int|\nabla \int_0^s P_{s-r}Q_m^R(f,f)dr|^2 \brak{v}^2dvds}_{B}
\end{align*}
we work separately on the two term:
\begin{align*}
    A&:=\int_0^t\int|\nabla \int K(t,w)f_0(e^tv-w)dw|^2\brak{v}^2dvds\\
    &\leq e^Tc_d\int_0^t\int\int K(t,w)|\nabla f_0(e^tv-w)|^2\brak{v}^2dwdvds\\
    &\leq e^Tc_d\int_0^t\int K(t,w)\int|\nabla f_0(e^tv-w)|^2\brak{v}^2dvdwds\\
      &\lesssim  \int_0^t\int K(t,w)\int|\nabla f_0(e^tv-w)|^2(\brak{e^tv}^2+\brak{w}^2)dvdwds\\
    &\lesssim \|\nabla f_0\|_{2,2}+\|\nabla f_0\|_2.
\end{align*}
while for the other terms, we use intensively the form of the nonlinearity $Q^R$. In particular we use the bound obtained with $\chi_R(v)$ and $\chi_R(w)$ to control the weight and conclude.
\begin{align*}
    &B\lesssim\int_0^t \int_0^s \underbrace{\left(\int |\nabla P_{s-r}Q_m^R(f,f)|^2 \brak{v}^2dv\right)}_{C}drds
\end{align*}
and we compute directly on the term $C$.
\begin{align*}
    \left(\int |\nabla P_{s-r}Q_m^R(f,f)|^2 \brak{v}^2dv\right)&\leq e^T\int |P_{s-r}\nabla Q_m^R(f,f)|^2 \brak{v}^2dv\\
    &=e^T\int |P_{s-r}(\left(Q_m^R(\nabla f,f)+Q_m^R(f,\nabla f)\right)|^2 \brak{v}^2dv\\
    &\lesssim \int P_{s-r}|(\left(Q_m^R(\nabla f,f)+Q_m^R(f,\nabla f)\right)|^2 \brak{v}^2dv
\end{align*}
Since the two addenda of the sum are quite similar we focus only on one of them. In particular we focus on each single term one for the "positive" part, the negative part being easier.
\begin{align*}
    &\int \int K(s-r,w)\left|\int \nabla f_n(h(v,w,z) f_{m-n}(z)|z|\chi_R(e^tv-w)\chi_R(z)dz\right|^2dw\brak{v}^2dv\\
    &\leq \int \int K(s-r,w)\left|\int \nabla f_n(h(v,w,z) f_{m-n}(z)|z|\chi_R(e^tv-w)\chi_R(z)dz\right|^2dw\brak{v}^2dv\\
    &\leq C_R^2 \|f\|_1\int \int K(s-r,w)\int |\nabla f_n(h(v,w,z)|^2 f_{m-n}(z)dzdw\brak{v}^2dv\\
    &\underbrace{\lesssim}_{\text{change\ variables}}C_R^2 \|f\|_1\int \int K(s-r,w)\int |\nabla f_n(\gamma)|^2 f_{m-n}(z)\brak{\gamma}^2\brak{w}^2\brak{z}^2d\gamma dzdw\\
    &\lesssim e^T \|\nabla f_n(s)\|_{2,2}^2\|\|f_n\|_{1,2}^2\|_{\infty,T}.
\end{align*}
Putting this in $(B)$ we get
$$
\int_0^t \int_0^s\left(\int |\nabla P_{s-r}Q_m^R(f,f)|^2 \brak{v}^2dv\right)drds\leq C(R,M,T,\|f\|_1,\|f\|_{1,2})\int_0^t \int_0^s\|\nabla f(s)\|_{2,2}^2ds\ dt.
$$
Summing on $M$ we get:
$$
\sum_{m}\int_0^t \int|\nabla f_m|^2 \brak{v}^2dvds \leq C(f_0)+C(R,M,T,\|f\|_1,\|f\|_{1,2})\int_0^t \int_0^s\|\nabla f(s)\|_{2,2}^2ds\ dt
$$
Assuming that $f_0$ is in $H^1_1\cap L^1_2$ we have proven:
$$
f_m\in H^1([0,T]\times\mathbb{R}^d), \forall m=1,...,M.
$$
This extended regularity for the truncated solution $f^R$ make it possible to prove rigorously the a priori estimate for the solutions of the approximated problems, obtaining bound that are independent of $R$.
\iffalse
\begin{lemma}
Consider Solution of theorem $(\cdot))$, $f(t; f^0) \in L_+^p$ for every $t \in [0,T]$, with $T<T(f^0)$ maximal interval. Then the solution has the following properties:
\begin{enumerate}
    \item $|f|\rightarrow 0,\ |v|\rightarrow\infty$;
    \item $|\nabla f|\rightarrow 0,\ |v|\rightarrow\infty$;
    \item $|v||f|, |v||\nabla f|\rightarrow 0,\ |v|\rightarrow\infty$.
\end{enumerate}
\end{lemma}
\begin{proof}
\begin{enumerate}
    \item We can prove it with equivalence of Sobolev space with Bessel space. So we get with Riemann-Lebesgue theorem that $|f|\rightarrow0,\ |v|\rightarrow\infty$;
    \item we can prove, using the representation of the semigruopo $P_t$ that $|v|f\rightarrow 0$;
    \item if we have a little bit more regularity, like $H^{2+\epsilon}$ we could prove also the vanishing of the gradiente $|\nabla f|, |v||\nabla f|$.
\end{enumerate}
\end{proof}
\fi
\begin{corollary}
In the same hypothesis of the last few theorems, the a priori estimate are true for the solution of the approximating problems, with constants independent from $R$, as soon as $f^0$ is regular enough.
\end{corollary}
\begin{proof}
Under the hypothesis, and thanks to the previous lemmas we satisfy the requirement of the a priori estimate and thus conclude the proof. In fact using the established $L^1_k$ bounds and the fact that $f\in H^1([0,T]\times\mathbb{R}^d)$ we can make rigorous the arguments of the a priori estimates lemmas and then proceed to obtain the regularity
$$
f^R\in L^\infty([0,T],H^p_k(\mathbb{R}^d)),\ \forall n\in\mathbb{N}, \forall k\geq0,
$$
with bound independent of $R$.
\end{proof}
We conclude this section with the following:
\begin{theorem}[global existence]
Assume that the initial condition is positive and in $L^2_M$ such that Lemma \ref{lem:H_1-trunc} and Theorem \ref{thm:approx-plm} holds, then the solution $\tilde{f}$ of the approximated problem is global in time, i.e. $T(f^0)=\infty$.
\end{theorem}
\begin{proof}
Thanks to Theorem \ref{thm:approx-plm} we know that the maximal interval of the solution $\tilde{f}$ is open in $R^+$ and of the form $[0,T^0)$.\\
Under the hypothesis of regularity of the initial condition, and thanks to Lemma \ref{lem:H_1-trunc}, we have that
$$
\|\tilde{f}_t\|_{L^2_M}\leq C(T),\ a.e.\ t\in[0,T^0)\cap[0,T].
$$
Thus, since being $T^0$ finite means that $\sup_{t\in[T^0-\epsilon,T^0)}\|\tilde{f}_t\|_{L^2_M}=\infty$, this cannot be happening and this implies that $T^0=\infty$ and the solution is global.\\
This implies that, for each $T>0$, independently of $R$, under the regularity of initial condition, we have a weak solution of equation $\eqref{main-eq-trunc1}$ on $X_T$.
\end{proof}

\section{Existence and uniqueness for the full System}
Right now we have proven that, fixing $T$ finite, there exists a sequence of function $(f^R)_R$, solutions of $\eqref{main-eq-trunc1}$.\\
Under suitable initial condition $(f^0_m)_{m=1}^M$, we prove now the weak convergence of $f^R$ to a map $f$ in $L^2\cap L^1$ such that solves in the weak sense $\eqref{main-eq1}$.\\
To prove existence for the full problem we begin by recalling the definition of weak solution we are going to use:
\begin{definition}
Fix $T>0$, and $f^0_m\in L^1_2\cap L^2$. A solution for equation $(\ref{main-eq1})$ is a set of functions $f_m\in L^\infty(0,T;L_2^1(\mathbb{R}^d))$ indexed by the masses $m=1,...,M$ such that $\forall m,\ Q_m(f,f)\in L^\infty(0,T;L^1(\mathbb{R}^d))$ and the following holds:
\begin{align*}
    \sum_{m=1}^M\brak{f_m(t),\phi}-\sum_{m=1}^M\brak{f_m^0,\phi}&=\int_0^t \sum_{m=1}^M\brak{f_m(s),\Delta\phi}ds-\int_0^t \sum_{m=1}^M\brak{f_m(s),v\cdot\nabla\phi}ds\\
    &+ \int_0^t \sum_{m=1}^M\brak{Q_m(f(s),f(s)),\phi}ds,
\end{align*}
for a.e. $t\in [0,T]$, and $\forall\phi\in C^\infty_c(\mathbb{R}^d)$. \end{definition}
We note that if a weak solution is sufficiently smooth and satisfy suitable decay for large $|v|$, then it is also a classical solution.
With everything recalled we state the following.
\begin{theorem}
For every $T>0,M>0,\ f_0\geq0$ such that $f_0\in L^1_{2,M}\cap L^2_M$ there exists at least one nonnegative weak solution $f\in L^\infty([0,T],L^1_2(\mathbb{R}^d))$ with initial condition $f(0)=f_0$.\\
If instead $f_0\in L^1_{2,M}\cap H^1_{1,M}$, then for every $t_0>0$ we have $f\in C^1_b([t_0,T],C^{\infty}(\mathbb{R}^d))$ with rapid decay for large $|v|$.
\end{theorem}
We prove here the existence, postponing the uniqueness to the next section.
\begin{proof}(Existence) We start with an initial condition $f_0\in C^\infty_c(\mathbb{R}^d)$. Using the results obtained in the previous section we have establish the $L^1_k$ bounds and the fact that the sequence $f_R\in H^1([0,T]\times\mathbb{R}^d)$. With this we can make rigorous the a priori estimates and get $f_R\in L^\infty([0,T],H^n_k(\mathbb{R}^d)),\ \forall n\in\mathbb{N},k\geq0$, with bounds independent from $R$.\\
$f^R\in L^\infty([0,T],L^2)$, and thanks to the a priori estimate we have that, independently of $R$, $f^R_t\in H^1(\mathbb{R}^d)\cap L^2(\mathbb{R}^d,|v|^2dv)$ and the sequence is equibounded. Analogously we have that, thanks to the regularity of the semi group $P_t$, the solution $f^R$ lies in the space $H^1(0,T;H^{-1})$.\\
Using now the Weighted Aubin-Lions Lemma \ref{lem:aubin} we have that, up to subsequence:
\begin{enumerate}
    \item $f^R\rightharpoonup f$ in $H^1$;
    \item $f^R\rightarrow f$ in $L^2$;
    \item up to subsequence $f^R\rightarrow f$ pointwise a.e.,
\end{enumerate}
and the a priori estimate pass to the limit function $f$.\\
This is enough to allow us to pass to the limit as $R\rightarrow\infty$ in the weak formulation of the system and show that the limit solutions satisfy the equation with the full kernel.\\
Finally we consider $f_0\in L^1_2\cap L^2$ and we take a sequence $f^n_0\in C^\infty_0$ converging to $f_0$ in $L^1_2\cap L^2$, obtaining the bound in $L^1_2$ by Fatou's lemma. Then, since the constants in the moments and energy are independent on $n$ we can pass to the weak $L^1-$limit in the equations obtaining a solution $f$ as expected.\\
To study the regularity of the solution with data in $H^1_1$ we consider that this is enough to get the parabolic regularity $f\in H^1([0,T]\times\mathbb{R}^d)$ for any $T>0$ and using the fact that we have the bound $f\in L^\infty([0,T],L^1_k(\mathbb{R}^d))$ we can make rigorous the argument of the a priori estimates and proceed to obtain the desired regularity.
\end{proof}
\iffalse
\subsection{Weak convergence and weak uniqueness}
Consider now the formulation of the weak solution of each $R>0$, for fixed $T, \kappa>0$ and the initial condition $f^0$ as before. We have:
\begin{align*}
    \sum_{m=1}^M\brak{f^{R,m}_t,\phi}-\sum_{m=1}^M\brak{f_m^0,\phi}&=\int_0^t \sum_{m=1}^M\brak{f^{R,m}_t,\kappa\Delta\phi}ds-\int_0^t \sum_{m=1}^M\brak{f_t^{R,m},v\cdot\nabla\phi}ds\\
    &+ \int_0^t \sum_{m=1}^M\brak{Q^R_m(f^R_t,f^R_t),\phi}ds,
\end{align*}
with equality in $L^2$, for a.e. $t\in [0,T]$, and $\forall\phi\in C^\infty_c(\mathbb{R}^d)$.\\
Thanks to the strong $L^2$ convergence we have, for all the term except the last one, that
\begin{align*}
    \sum_{m=1}^M\brak{f^{,m}_t,\phi}-\sum_{m=1}^M\brak{f_m^0,\phi}&=\int_0^t \sum_{m=1}^M\brak{f^{,m}_t,\kappa\Delta\phi}ds-\int_0^t \sum_{m=1}^M\brak{f_t^{,m},v\cdot\nabla\phi}ds\\
    &+ \lim_{R\rightarrow\infty}\int_0^t \sum_{m=1}^M\brak{Q^R_m(f^R_t,f^R_t),\phi}ds,
\end{align*}
we are left to prove that the nonlinearity converges to the right quantity.\\
To this end we consider again the weak formulation for $\ref{main-eq-trunc1}$ and we write it as:
\begin{align*}
    \sum_{m=1}^M\brak{f^{R,m}_t,\phi}-\sum_{m=1}^M\brak{f_m^0,\phi}&=\int_0^t \sum_{m=1}^M\brak{f^{R,m}_t,\kappa\Delta\phi}ds-\int_0^t \sum_{m=1}^M\brak{f_t^{R,m},v\cdot\nabla\phi}ds\\
    &+ \int_0^t \sum_{m=1}^M\brak{Q_m(f^R_t,f^R_t),\phi}ds\\
    &+ \int_0^t \sum_{m=1}^M\underbrace{\brak{Q^R_m(f^R_t,f^R_t)-Q_m(f^R_t,f^R_t),\phi}}_{=:F^R_\phi(f_t^R)\rightarrow0}ds,
\end{align*}
with equality in $L^2$, for a.e. $t\in [0,T]$, and $\forall\phi\in C^\infty_c(\mathbb{R}^d)$.\\
\subsubsection{nonlinearity $Q$ and $F_\phi$}
Using the a priori estimate for the sequence $(f^R)_R$ we have that all the constant bounding moments and norms of the truncated solutions does not depends on $R$, thus a.e. $v\in\mathbb{R}^d$
\begin{align*}
&|Q_m(f^R,f^R)|\leq\sum_{n=1}^{m-1}K_{m,n} \int |v-w| f^R_n(h(v,w))f^R_{m-n}(w)dw + f_m^R(v)\sum_{m=1}^M\int |v-w| f_n(w) dw\\
&\leq \sum_{n=1}^{m-1} K'_{m,n}\int (|h(v,w)|+|w|) f^R_n(h(v,w))f^R_{m-n}(w)dw + f_m^R(v)\sum_{m=1}^M\int |v-w| f_n(w) dw\\
&\underbrace{\leq}_{Holder} \sum_{n=1}^{m-1} K'_{m,n}\left(\||v|f^R_n(v)\|_2\|f^R_{m-n}\|_2+\|f^R_n\|_2\||v|f^R_{m-n}(v)\|_2\right) + f_m^R(v)\brak{v}\sum_{m=1}^M\int \brak{w} f_n(w) dw\\
&\leq 2K'_{m,n} C_{2,2} C_2 + M C_{1,1} f_m^R(v)\brak{v},
\end{align*}
where $K$ depends only on $m,n$ and the constant come from the application of Holder inequality, translation invariance and the a priori estimate: in particular the estimate on the $L^2_2$ and $L^2$ spaces.\\
So we can split the nonlinearity and get
\begin{align*}
   |Q_m(f^R_t,f^R_t)\phi|&\leq K_m C_{1,2}^\omega C_2|\phi(v)| + M C_1^\omega f_m^R(v)|v| |\phi(v)|
\end{align*}
\begin{align*}
    |\left(Q^R_m(f^R_t,f^R_t)-Q_m(f^R_t,f^R_t)\right)\phi|&\leq K_m C_{1,2}^\omega C_2|\phi(v)| + M C_1^\omega f_m^R(v)|v| |\phi(v)| .
\end{align*}
With this in mind we can consider the positive part of $Q_m$, call it $Q^+_m$ and this part that we can take care with the dominate convergence theorem. We get:
\begin{align*}
    \sum_{m=1}^M\brak{Q^+_m(f^R_t,f^R_t),\phi}\rightarrow \sum_{m=1}^M\brak{Q^+_m(f_t,f_t),\phi}\\
    \sum_{m=1}^M\brak{Q^+_m(f^R_t,f^R_t)-Q^{R,+}_m(f^R_t,f^R_t),\phi}\rightarrow0.
\end{align*}
Where we have used the fact that:
\begin{align*}
    \brak{Q^+_m(f^R_t,f^R_t)-Q^{R,+}_m(f^R_t,f^R_t),\phi}\tilde{=}-\iint\phi(v)|v-w|\left(1-\chi_R(v)\chi_R(w)\right)f^R_n(h(v,w) f^R_{m-n}(w) dwdv
\end{align*}
and the function integrated converges point wise a,e. $v\in\mathbb{R}^d$ to the function $0$.\\
We are left then with the negative part, to this end we find an uniform estimate on $L^\infty$.\\
Here we try to prove for the solution of the approximated problem a $L^\infty$ bound in both velocity and time using the mild formulation and the explicit realization of the semigroup $P_t$. Fix $d=2$:
\begin{align*}
    P_t\phi(v):=\frac{1}{\sqrt2 \pi \left(e^{2t}-1\right)^{1/2}}\int_{\mathbb{R}^d} e^{\frac{-|w|^2}{2\left(e^{2t}-1\right)}}\phi(e^tv-w)dw
\end{align*}
we can use the fact that $\|P_t\phi\|_{\infty}\leq \underbrace{\frac{1}{\sqrt2 \pi \left(e^{2t}-1\right)^{1/2}}\int_{\mathbb{R}^d} e^{\frac{-2|w|^2}{\left(e^{2t}-1\right)}}dw}_{\leq C_T(2\sqrt\pi)}\|\phi\|_{\infty }\lesssim C_T\|\phi\|_{\infty}.$\\
With this in mind, we recall that $f^R$ is $\geq0$, as a nice decay at infinity $|v|f^R\rightarrow0$, and for $f^R$ holds all the a priori estimates. Consider a.e. $v\in\mathbb{R}^d$, we should get:
\begin{align*}
    f_t^R&=P_tf_0+\int_0^tP_{t-s}\left(Q^+(f^R,f^R)-Q^-(f^R,f^R)\right)ds\\
    &\leq P_tf_0+\int_0^tP_{t-s}\left(Q^+(f^R,f^R)\right)ds\\
    &\leq C_0 + dC_T\int_0^t\|f_s^R\|_\infty ds + \sum_{n=1}^{m-1}\int_0^tP_{t-s}\left(\int |\cdot-w|f_n^R(h(\cdot,w))f_{m-n}^R(w)dw\right)ds.
\end{align*}
This should be true since the positivity and the decay. If we found a bound for the last term, passing to the modulus and taking the $\sup_v$ and $\sup_{t\in[0,T]}$ can use Gronwall inequality. Thus fix $n<m$ and we search a bound for the double integral a.e. $v\in\mathbb{R}^2$:
\begin{align*}
    &\int |v-w|f_n^R(h(v,w))f_{m-n}^R(w)dw\leq \int |v|f_n^R(h(v,w))f_{m-n}^R(w)dw+\int |w|f_n^R(h(v,w))f_{m-n}^R(w)dw\\
    &\leq \int \overline{C}_M|h(v,w)|f_n^R(h(v,w))f_{m-n}^R(w)dw+\int \tilde{C}_M|w|f_n^R(h(v,w))f_{m-n}^R(w)dw+\int |w|f_n^R(h(v,w))f_{m-n}^R(w)dw
\end{align*}
using holder inequality, teh traslation invariance of Lebesgue measure, the a priori estimate in the weight space and $L^2$ space we get:
\begin{align*}
    \int |v-w|f_n^R(h(v,w))f_{m-n}^R(w)dw\leq 3C_MC_{1,2}^\omega C_2<\infty,\ a.e. v\in\mathbb{R}^2.
\end{align*}
If the computation is right, then we have, thanks to Holder inequality, on bounded time interval the bound of $L^\infty_v$, independently on $R$, with this we should be able to bound the quantity $|Q_m(f^R_t,f^R_t)\phi|$, both the positive and the negative part, with something that depends on $\phi$, but not on $R$ and is in $L^1$, with:
\begin{align*}
   |Q_m(f^R_t,f^R_t)\phi|&\leq 3C_M C_{1,2}^\omega C_2|\phi(v)| + M C_1^\omega \|f_m^R\||\supp \phi| |\phi(v)|\\
   &\leq  3C_M C_{1,2}^\omega C_2|\phi(v)| + M C_1^\omega 3C_MC_{1,2}^\omega C_2|\supp \phi|;
\end{align*}
\begin{align*}
   &|\left(Q^R_m(f^R_t,f^R_t)-Q_m(f^R_t,f^R_t)\right)\phi|\leq K_m C_{1,2}^\omega C_2|\phi(v)| + M C_1^\omega 3C_MC_{1,2}^\omega C_2|\supp\phi| |\phi(v)|.
\end{align*}
Since $|v|\phi(v)\in L^1$ if $\phi\in C^\infty_c$ we can pass to the limit  for
$$
\brak{Q(f^R,f^R),\phi}\rightarrow\brak{Q(f,f),\phi}
$$
at least for all the test function
\fi
\subsection{Weak Uniqueness}
Consider now two weak solutions $f,g$ of equation $(\ref{main-eq1})$ in the sense of Definition \ref{def:weak-sol}, with initial conditions $f^0_m=g^0_m\in H^1_1\cap L^1_2$ and nonnegative for every $m$. We know they are actually classical solutions. We want to show that, as soon as they start with the same initial, regular condition, then the two solutions must be equal a.e. in $v\in\mathbb{R}^d$ for every fixed time $T\geq0$. \\
Consider for this the set of equation solved by $h_m(t,v):=f_m(t,v)-g_m(t,v),\ \forall m=1,...,M$ and call $H_m(t,v):=f_m(t,v)+g_m(t,v)$.
\begin{align}\label{eq-for-h_m}
    &\begin{cases}
    \partial_t h_m(t,v)=\Delta h_m +\text{div}(vh_m)+\frac{1}{2}\left(Q_m(h,H)+Q_m(H,h)\right)\\[5pt]
    h_m|_{t=0}=0
    \end{cases}
    \end{align}
    where we denote
    \begin{align*}
    Q_m(h,H)&:=\sum_{n=1}^{m-1}\int\left(\frac{m}{n}\right)^2 h_n(\varphi(v,w))H_{m-n}(w)|v-w|dw -2\sum_{m=1}^M h_m(v)\int H_n(w)|v-w|dw,\\
    Q_m(H,h)&:=\sum_{n=1}^{m-1}\int\left(\frac{m}{n}\right)^2  H_n(\varphi(v,w))h_{m-n}(w)|v-w|dw -2\sum_{m=1}^M H_m(v)\int h_n(w)|v-w|dw.
\end{align*}

We consider now the function $\psi_\varepsilon(x)$ an approximation of the function $\text{sgn}(x)$ such that
\begin{align*}
    &\psi_\varepsilon(x):=\begin{cases}
       -1,\ &x\leq-\varepsilon\\
       x/\epsilon,\ &-\varepsilon<x< \varepsilon\\
       1,\ &x\ge \varepsilon
    \end{cases}
\end{align*}
We consider the weight $\brak{v}^2:=1+|v|^2$ and thus the function
$\Psi_\varepsilon(h_m,v):=\psi_\varepsilon(h_m)\brak{v}^2$, multiplying it to the equations \eqref{main-eq1} solved by $h_m, \forall m$ and integrating by part. Defining also
\begin{align*}
\chi_{\varepsilon}(x):=\begin{cases}
    &1,\ \text{if}\ |x|< \varepsilon\\
    &0,\ \text{otherwise},
\end{cases

    \end{align*}
we obtain:
\begin{align}\label{hormander}
    \partial_t\int h_m\psi_\varepsilon(h_m)\brak{v}^2dv&=\int (\partial_th_m)\psi_\varepsilon(h_m)\brak{v}^2dv+\frac{1}{\varepsilon}\int (\partial_th_m)\chi_\varepsilon(h_m)h_m\brak{v}^2dv\\
    &=: A_1+B_1.\nonumber

\end{align}
Now we can work on the quantity $A_1$, using the equation \eqref{main-eq1},
\begin{align*}
    A_1:&=\int\Delta h_m\psi_\varepsilon(h_m)\brak{v}^2+  \div(vh_m)\psi_\varepsilon(h_m)\brak{v}^2+\frac{1}{2}\left(Q_m(h,H)+Q_m(H,h)\right)\psi_\varepsilon(h_m)\brak{v}^2dv\\
    &=-\frac{1}{\varepsilon}\int_{\{|h|<\varepsilon\}}|\nabla h_m|^2\brak{v}^2dv + d\int h_m\psi_\varepsilon(h_m)\brak{v}^2dv+\underbrace{\int \left(\nabla h_m \cdot v\right)\psi_\varepsilon(h_m)\left(|v|^2-1\right)dv}_{A_2}\\
    &+\underbrace{\frac{1}{2}\int\left(Q_m(h,H)+Q_m(H,h)\right)\psi_\varepsilon(h_m)\brak{v}^2dv}_{A_3},
\end{align*}
and we can then drop the first negative term. For $A_2$ we define first the following function:
\begin{align*}
    &\zeta_\varepsilon(x):=\begin{cases}
       -x+\epsilon/2,\ &x\leq-\varepsilon\\
       x^2/(2\epsilon),\ &-\varepsilon<x< \varepsilon\\
       x-\epsilon/2,\ &x\ge\varepsilon
    \end{cases}
\end{align*}
with $\nabla \zeta_\epsilon=\overline\psi_\epsilon$ a.e. and $\zeta_\epsilon\to|x|$ as $\epsilon\to0$, and we can write
\begin{align*}
    A_2=\int \left(\nabla \zeta_\epsilon(h_m) \cdot v\left(|v|^2-1\right)\right)dv&=-\int \zeta_\epsilon(h_m) \div(v\left(|v|^2-1\right))dv\\
    &=-\int \zeta_\varepsilon(h_m)\left(d\left(|v|^2-1\right)+2|v|^2\right)dv.
\end{align*}
Now we get for $A_3$:
\begin{align*}
    A_3:&=\frac{1}{2}\sum_{n=1}^{m-1}\iint\left[h_n(\varphi(v,w))H_{m-n}(w)+h_{m-n}(w)H_{n}(\varphi(v,w))\right]|v-w|\psi_\varepsilon(h_m(v))\brak{v}^2dwdv\\
    &-\sum_{n=1}^{M}\iint  \left[h_m(v)H_{n}(w)+h_{n}(w)H_{m}(v)\right]|v-w|\psi_\varepsilon(h_m(v))\brak{v}^2dwdv

\end{align*}
Next, we turn to the term $B_1$ of \eqref{hormander}. Since $h_m$ appears in the integrand, the set of $v$ such that $h_m(v)=0$ does not contribute to the integral, hence
\begin{align*}
    \left|\frac{1}{\varepsilon}\int (\partial_th_m)\chi_\varepsilon(h_m)h_m\brak{v}^2dv\right| &=\left| \frac{1}{\epsilon}\int_{\{v:\, 0<|h_m(v)|<\epsilon\}}(\partial_th_m)h_m\brak{v}^2dv\right| \\
    &\le\int_{\{v:\, 0<|h_m(v)|<\epsilon\}}|\partial_th_m|\brak{v}^2dv.
\end{align*}
Since $h_m$ is continuous, $|h_m|$ is also continuous sending open sets to open sets, hence $\{0<|h_m(v)|<\epsilon\}\downarrow\emptyset$ as $\epsilon\to0$. We now claim that $\partial_th_m(t)\brak{v}^2\in L^1$ for every $t\in[0,T]$. It will imply that the last integral goes to $0$ as $\epsilon\to0$. By linearity $\partial_th_m=\partial_tf_m-\partial_tg_m$, hence it is enough to verify this claim separately for $f_m$ and $g_m$. Let us take $f_m$ and prove $\partial_t f_m(t)\brak{v}^2\in L^1$ for a.e. $t$. Note that
\begin{align*}
    \int |\partial_t f_m| \brak{v}^2dv &=\int \left|\Delta f_m +\text{div}(vf_m)+Q(f_m,f_m)\right| \brak{v}^2 dv\\
    &=\int \left|\Delta f_m +df_m+v\cdot\nabla f_m+Q(f_m,f_m)\right| \brak{v}^2 dv.
\end{align*}
By Sobolev embedding (which extends to the weighted case), if $f_m\in H^n_k$ for $n,k$ suitably large (in particular, since $f_m$ is a Schwartz function), then the above integral is finite.

Now we sum \eqref{hormander} over $m=1,..,M$ and we get
\begin{align*}
    \partial_t\sum_{m=1}^M\int h_m\psi_\varepsilon(h_m)\brak{v}^2dv&=\sum_{m=1}^M\int (\partial_th_m)\psi_\varepsilon(h_m)\brak{v}^2dv+\sum_{m=1}^M\frac{1}{\varepsilon}\int (\partial_th_m)\chi_\varepsilon(h_m)h_m\brak{v}^2dv\\
    &=:A_1^M+B_1^M.

\end{align*}
We already know that $B_1^M$ is negligible. Further, we have that
\begin{align*}
    A^M_1&=\sum_{m=1}^M\int\Delta h_m\psi_\varepsilon(h_m)\brak{v}^2+ \div(vh_m)\psi_\varepsilon(h_m)\brak{v}^2dv+\frac{1}{2}\int\left(Q_m(h,H)+Q_m(H,h)\right)\psi_\varepsilon(h_m)\brak{v}^2dv\\
    &\leq d\int \sum_{m=1}^M\left(h_m\psi_\varepsilon(h_m)\right)\brak{v}^2dv+\underbrace{\sum_{m=1}^M \int \left(\nabla h_m \cdot v\right)\psi_\varepsilon(h_m)\left(d\left(|v|^2-1\right)+2|v|^2\right)dv}_{A^M_2}\\
    &+\underbrace{\frac{1}{2}\sum_{m=1}^M\int\left(Q_m(h,H)+Q_m(H,h)\right)\psi_\varepsilon(h_m)\brak{v}^2dv}_{A^M_3}
\end{align*}
before estimating the nonlinear part we note that, using the estimate for each $m$, as $\varepsilon\rightarrow0$, under our regularity assumption, we get
\begin{align*}
    &\partial_t\sum_{m=1}^M\int h_m\psi_\varepsilon(h_m)\brak{v}^2dv\rightarrow\partial_t\sum_{m=1}^M\int |h_m|\brak{v}^2dv;\\
    &A^M_2\rightarrow -\sum_{m=1}^M \int |h_m
    \left(d\left(|v|^2-1\right)+2|v|^2\right)dv\leq \sum_{m=1}^M C_d\int |h_m|\brak{v}^2.
\end{align*}
So we miss to estimate the nonlinear term $A^M_3$ and pass to the limit:
\begin{align*}
    A_3^M:&=\frac{1}{2}\sum_{m=1}^M\sum_{n=1}^{m-1}\iint \left(\frac{m}{n}\right)^2\left( h_n(\varphi(v,w))H_{m-n}(w)+h_{m-n}(w)H_{n}(\varphi(v,w))\right)|v-w|\psi_\varepsilon(h_m(v))\brak{v}^2dwdv\\
    &-\sum_{m=1}^M\sum_{n=1}^{M}\iint \left(h_m(v)H_{n}(w)+h_{n}(w)H_{m}(v)\right)|v-w|\psi_\varepsilon(h_m(v))\brak{v}^2dwdv
\end{align*}

By the change of variable
$$
\begin{cases}
    &v'= \varphi(v,w)\\
    & w=w
\end{cases},\ |J(v,w)|=\frac{n}{m}.
$$
and naming the variable $v'$ again by $v$, we have that
\begin{align*}
    A_3^M&= \frac{1}{2}\sum_{m=1}^M\sum_{n=1}^{m-1}\iint  \left(h_n(v)H_{m-n}(w)+h_{m-n}(w)H_{n}(v)\right)|v-w|\psi_\varepsilon(h_m(v))\brak{\frac{nv+(m-n)w}{m}}^2dwdv\\
    &-\sum_{m=1}^M\sum_{n=1}^{M}\iint \left(h_m(v)H_{n}(w)+h_{n}(w)H_{m}(v)\right)|v-w|\psi_\varepsilon(h_m(v))\brak{v}^2dwdv\\
    &\underbrace{\leq}_{\text{Jensen's ineq.}\ \&\ |\psi_\epsilon(h_m)|\leq1}  \frac{1}{2}\sum_{n=1}^M\sum_{\ell=1}^{m-1}\iint \left(|h_n(v)|H_{\ell}(w)+|h_{\ell}|(w)H_{n}(v)\right)|v-w|(\brak{v}^2+\brak{w}^2)dwdv\\
    &-\sum_{m=1}^M\sum_{n=1}^{M}\iint |v-w| \left[h_m(v)H_{n}(w)\psi_\varepsilon(h_m(v))\brak{v}^2+h_{n}(v)H_{m}(w)\psi_\varepsilon(h_m(w))\brak{w}^2\right]dwdv\\
    &\underbrace{\le}_{\text{symmetry}}\sum_{n=1}^{M}\sum_{\ell=1}^M\iint |h_n(v)|H_{\ell}(w)|v-w|(\brak{v}^2+\brak{w}^2)dwdv\\
    &-\sum_{m=1}^M\sum_{n=1}^{M}\iint |v-w|\left[h_m(v)H_{n}(w)\psi_\varepsilon(h_m(v))\brak{v}^2+h_{n}(v)H_{m}(w)\psi_\varepsilon(h_m(w))\brak{w}^2\right]dwdv\\

    &= \iint \sum_{\ell=1}^MH_\ell(w)\sum_{n=1}^M|h_n(v)|\left(\brak{v}^2+\brak{w}^2\right)|v-w|dwdv\\
    &-\iint \sum_{m=1}^MH_m(w)\sum_{n=1}^M\left\{ h_n(v)\psi_\varepsilon(h_n(v))\brak{v}^2+h_{n}(v)\psi_\varepsilon(h_m(w))\brak{w}^2\right\}|v-w|dwdv.
\end{align*}
For the first two integrals we have (renaming $\ell$ by $m$)
\begin{align}\label{cancel-1}
\iint \sum_{m=1}^MH_m(w)\sum_{n=1}^M|h_n(v)|(\brak{v}^2+\brak{w}^2)|v-w|dwdv
\end{align}
while, using the fact that
$$
|h\psi_\epsilon(h)-|h||\leq |h|\chi_\varepsilon(h),
$$
the third integral can be bounded above by
\begin{align}\label{cancel-2}
    \iint \sum_{m=1}^MH_m(w)\sum_{n=1}^M \left[ -|h_n(v)|+|h_{n}|(v)\chi_\varepsilon(h_n(v))\right]\brak{v}^2|v-w|dwdv
\end{align}
while for the fourth, by the elementary inequality $|v-w|\le\langle v\rangle\langle w\rangle$, the moment bound Lemma \ref{lem:moments},  $|\psi_\epsilon(h_m)|\le 1$, is bounded above by
$$
 C_M\iint\sum_{m=1}^MH_m(w)\brak{w}^3\sum_{n=1}^M|h_{n}|(v)\brak{v}dwdv\le C\int \sum_{n=1}^M |h_{n}|(v)\brak{v}^2dv.
$$
We get the upper bound on $A_3^M$ by adding everything together and noting a cancellation in \eqref{cancel-1}-\eqref{cancel-2} that eliminates any higher moments in $v$,
\begin{align*}
    &A_3^M\le C_M\iint \sum_{m=1}^M H_m(w)\sum_{n=1}^M|h_n(v)|\brak{v}\brak{w}^3dvdw\\
    &+ \int \sum_{m=1}^M H_m(w)\sum_{n=1}^M\int_{\{|h_n(v)|<\varepsilon\}}|h_n(v)||v-w|dv\brak{w}^2dw+C\int \sum_{n=1}^M|h_{n}|(v)\brak{v}^2dv.
\end{align*}
Now taking the limit as $\varepsilon\rightarrow0$ we have
$$
\limsup_{\epsilon\to0}A^M_3\le  C\iint \sum_{m=1}^M H_m(w)\sum_{n=1}^M|h_n(v)|\brak{v}\brak{w}^3dvdw+\int \sum_{n=1}^M|h_{n}|(v)\brak{v}^2dv
$$
Consider now all the quantities together after passing to the limit, using again the moment bounds,  we have
\begin{align*}
    \frac{d}{dt}\sum_{m=1}^M\int |h_m(v)|\brak{v}^2d
    &\leq C\int \sum_{m=1}^M|h_m(v)|\brak{v}^2dv

\end{align*}
Now we can apply Gronwall's lemma and get that $\|f-g\|_{L^1_M(\brak{v}^2dv)}=0$, concluding the uniqueness.\qed

\iffalse
\begin{definition}[$L^p$-weighted]
Consider now the quantity $\brak{v}:=\sqrt{1+|v|^2}$ and define the weighted $L^p$ space for $1\leq p \leq\infty$ (with the usual modification for $p=\infty$) as
$$
L^p_1:=\left\{f:\mathbb{R}^d\mapsto\mathbb{R}\ |\ \int_{\mathbb{R}^d} |f|\brak{v}dv<\infty\right\},
$$
with the natural norm $\|f\|_{p}^1:=\int |f|\brak{v}dv$ (and $\sup_v |f\brak{v}|$ if $p=\infty$).\\
In this setting we can explicitly express the duality and the dual space such that
$$
\brak{f,g}_1:=\int fg\brak{v}dv,\ and\ (L^p_1)'=L^{q}_1
$$
where $1\leq p<\infty$ and $q\in (1,\infty]$ satisfy $\frac1p+\frac1q=1$.\\
In particular, with Hahn-Banach and thanks to duality we can express the norm as
$$
\|f\|_p^1:=\sup_{\phi\in L^q_1,\|\phi\|_q^1\leq1}\ \left|\int f\phi\brak{v}dv\right|.
$$
\end{definition}

With this in mind we prove the following lemma showing us that, under our hypothesis, weak solution of $(\ref{main-eq1})$ are mild solution in the sense of semigroups.
\begin{lemma}[Weak to Mild Solutions]
Consider $f$ weak solution to $(\ref{main-eq1})$ satisfy the bound on momentum and weighted norms, then $f$ is a mild solution of the Non local Smoluchowsky Equation
$$
f_m(t):=P_tf_0(t)+\int_0^t\ P_{t-s}\left[f_m(s)+Q_s(f,f)\right]ds,\ \forall t\in [0,T],\forall m=1,...,M
$$
\end{lemma}
\begin{proof}
\todo[inline]{Fixing the right test function of the form $\varphi:=P_t^*\phi$ we should recover the weak formulation. Since we have enough regularity should be the usual computation. We'll try later if this approach works.}
\end{proof}
We consider now the space $L^1_1:=L^1(\mathbb{R}^d,\brak{v}dv)$ and take the difference of two solutions $\zeta_m(t):=f_m(t)-g_m(t),\ \forall m$.
\begin{align*}
    \sum_{m=1}^M\|\zeta_m(t)\|_1^1&\leq\sum_{m=1}^M\|\zeta_m(0)+\int_0^t\ P_{t-s}\left[\zeta_m(s)+\tilde{Q}_s(\zeta;f,g)\right]ds\|_1^1\\
    &\leq \sum_{m=1}^M\int_0^t\ \|P_{t-s}\left[\zeta_m(s)\right]\|_1^1ds+\int_0^t\ \|P_{t-s}\left[\tilde{Q}_s(\zeta;f,g)\right]\|_1^1ds
\end{align*}
where we have replaced the quantity $\left(Q_s(f,f)-Q_s(g,g)\right)$ with a new collision integral which is greater or equal to the aforementioned quantity:\todo[]{Follows from $(f_1f_2-g_1g_2)\leq(f_1-g_1)(f_2+g_2)$, since $f,g\geq0$.}
\begin{align*}
|\tilde{Q}_s(\zeta;f,g)|&:=|\sum_{n=1}^{m-1}\int \zeta_m(h)(f_m+g_m)(w)|v-w|dw-\sum_{m=1}^{M}\zeta_m(v)\int (f_m+g_m)(w)|v-w|dw|\\
&\leq \sum_{n=1}^{m-1}\int |\zeta_m|(h)(f_m+g_m)(w)|v-w|dw+\sum_{m=1}^{M}|\zeta_m|(v)\int (f_m+g_m)(w)|v-w|dw.
\end{align*}
\begin{lemma}[Semigroup inequality]
Consider a function $\zeta \in L^1_1$ and the semigroup $P_t$ defined in the previous sections for a fixed $t\in[0,T],\ T>0$. Then we have, up to a constant $C<\infty$ depending on $T$, the following inequality:
\begin{align*}
    \|P_{t}\left[\zeta\right]\|_1^1\leq C_T\|\zeta\|_1^1
\end{align*}
\end{lemma}
\begin{proof}
Consider the quantity:
\begin{align*}
    \int\int g_t(w)\zeta(e^tv-w)\brak{v}dwdv&\leq \int\int g_t(w)\zeta(e^tv-w)\brak{e^tv-w}\brak{w}dwdv\\
    &\leq \int g_t(w)\left(\int \zeta(e^tv-w)\brak{e^tv-w}dv\right)\brak{w}dw\\
    &\leq \int g_t(w)(1+|v|)dw\|\zeta\|_1^1\leq (1+K_T)\|\zeta\|_1^1,
\end{align*}
where we have used the translation invariance of usual $L^1$ space and the fact that $\brak{v-w}\leq \brak{v}\brak{w}$ and that $e^t\geq1$.\\
For the last inequality we notice that the kernel of the semigroup is regular enough to obtain $\|g_t\|_1=1$ and that $\|g_t(v)|v|\|_1\sim \sqrt{(e^{2dT})}<\infty.$ Concluding the proof.
\end{proof}

\begin{lemma}[Collision integral inequality]
Consider $\zeta,u\in L^1_1$ and $\phi\in L^\infty_1$, then for each component of the collision integral we have the following estimates:
\begin{align*}
    &\brak{\tilde{Q}_n^+(\zeta,u),\phi}_1=\int\ \int \zeta(h)u(w)|v-w|dw\phi(v)\brak{v}dv\|\leq \|\zeta\|_1^1\|u\|_1^1\|\phi\|^1_{\infty}\\
    &\brak{\tilde{Q}_m^-(\zeta,u),\phi}_1=\int\ \zeta_m(v)\int (u)(w)|v-w|dw\phi(v)\brak{v}dv\leq\|\zeta\|_1^1\|u\|_1^1\|\phi\|^1_{\infty}.
\end{align*}
\end{lemma}
\begin{proof}
The estimate follow easily using the properties of the modulus and the weight
$$
\brak{v-w}\leq \brak{v}\brak{w},\ |v-w|\leq \brak{v}\brak{w}.
$$
In fact we have, taking the first integral as example
\begin{align*}
    \int\ \int \zeta(h)u(w)|v-w|dw\phi(v)\brak{v}dv&\leq \int\ \int \zeta(h)u(w)|h-\alpha w|dw\phi(v)\brak{v}dv\\
    &\lesssim \int\ \int \zeta(h)u(w)\brak{h}\brak{w}dw\phi(v)\brak{v}dv\\
    &\leq \int u(w)\brak{w}\left(\int \zeta(h)\brak{h}dv\right)dw\|\phi(v)\brak{v}\|_{\infty}^1\\
    &=\|\zeta\|_1^1\|u\|_1^1\|\phi\|_1^{\infty}.
\end{align*}
Using the translation invariance of the standard $L^1$ space. Thus concluding the proof.
\end{proof}
Putting all together we have
\begin{align*}
    \sum_{m=1}^M\|\zeta_m(t)\|_1^1&\leq \sum_{m=1}^M\int_0^t\ \|P_{t-s}\left[\zeta_m(s)\right]\|_1^1ds+\int_0^t\ \|P_{t-s}\left[\tilde{Q}_s(\zeta;f,g)\right]\|_1^1ds\\
    &\leq C_T\sum_{m=1}^M\int_0^t\ \|\zeta_m(s)\|_1^1ds+\int_0^t\ \|\tilde{Q}_s(\zeta;f,g)\|_1^1ds\\
    &\leq C_T\sum_{m=1}^M\int_0^t\ \|\zeta_m(s)\|_1^1ds+\int_0^t\ \left(\sup_{\phi\in L^q_1,\|\phi\|_q^1\leq1}\ \int\tilde{Q}_s(\zeta;f,g)\phi(v)\brak{v}dv\right)ds\\
    &\leq\int_0^t\ \leq C_T\sum_{m=1}^M\|\zeta_m(s)\|_1^1ds+\int_0^t\ \left(\|\zeta_m(s)\|_1^1\|f_n+g_n\|_1^1\right)ds\\
    &\leq\int_0^t\ C_T(f,g,M)\sum_{m=1}^M\|\zeta_m(s)\|_1^1ds.
\end{align*}\todo[]{Here we have used the duality of the norm in the weighted space.}
Thus with Gronwall we can conclude and get that $\|f-g\|_1^1=0$, i.e. $f=g\ a.e., \forall t$.
\iffalse
Consider now two solution $f,g$ of the equation $(\cdot)$, in the sense of definition $(\cdot)$, and we compute the derivative in time of the $L^2$ norm of te difference of this two solution with the same initial condition.\\
\begin{align*}
\partial_t \|f_m-g_m\|_2^2&=\int (f_m(t,v)-g_m(t,v))\partial_t (f_m(t,v)-g_m(t,v)) dv\\
&= \int (f_m-g_m) \text{div}_v\left(v(f_m-g_m)\right)dv+\int\kappa (f_m-g_m)\Delta_v (f_m-g_m)dv\\
& +\int  (f_m(t,v)-g_m(t,v))\left(Q(f,f)-Q(g,g)\right)dv\\
&\leq \frac d2\int (f_m-g_m)^2dv +\int  (f_m(t,v)-g_m(t,v))^2dv\int\left(Q(f,f)^+-Q^+(g,g)\right)^2dv\\
&=C_d\|f_m-g_m\|^2_2+\|f_m-g_m\|_2\sqrt{\underbrace{\int\left(Q(f,f)^+-Q^+(g,g)\right)^2dv}_{(A)}}
\end{align*}\todo[]{Aplied Holder Wrong, so it should not work like this. Problem is the quantity $|v-w|$ and the fact that gronwall with square root give us only boundness.}
Where we have eliminated the negative part, since we know that solution is positive and regular to make sense of all the integral. Thus, we just need to estimate $(A)$, but for this quantity, adding and subtracting, using the regularity assumption on the solutions, we get a bound that depends only on $g$ and $f$ on all the interval $[0,T]$. So We get
$$
\partial_t \|f_m-g_m\|_2^2=\left(C(T)C_{1,2}^\omega C_2(f_T,g_T)\right)\|f_m-g_m\|^2_2+\left(C(T)C_{1,2}^\omega C_2(f_T,g_T)\right)\|f_m-g_m\|_2.
$$
but with a generalized Gronwall we obtain only a boundness with the constant of the system:
$$\|f_m-g_m\|_2\leq C_T,\ \forall m=1,...,M, \forall t\in [0,T].$$.
The problem is essentially linked to the quantity:
$$
\|\int |v-w| \left(f_n-g_n\right)(\frac{mv-(m-n)w}{n})f_{m-n}(w)dw\|
$$
whatever standard norm we use, either be $L^\infty$ of $L^2$, we come up with a quantity that goes like
$$
\||v|(f_n-g_n)\|
$$
And even if we try to prove some kind of bound on this quantity to close it and prove Gronwall on a space with norm $L^2,L^2_{weighted}$: $\|f_n-g_n\|+\||v|(f_n-g_n)\|$, we obtain an iterative argument because we get:
$$
\|\int |v||v-w| \left(f_n-g_n\right)(\frac{mv-(m-n)w}{n})f_{m-n}(w)dw\|\sim \|\int |v|^2 \left(f_n-g_n\right)(\frac{mv-(m-n)w}{n})f_{m-n}(w)dw\|
$$
Thus having to deal with $\||v|^2(f_n-g_n)\|$.\\

Even considering the mild formulation seems not to overcome this problem, since the quantity
\begin{align*}
    &\|\int_0^t P_{t-s}\left(\int |v||v-w| \left(f_n-g_n\right)(\frac{mv-(m-n)w}{n})f_{m-n}(w)dw\right)ds\|\sim \\
&\sim\int_0^t C_T\|\left(\int |v||v-w| \left(f_n-g_n\right)(\frac{mv-(m-n)w}{n})f_{m-n}(w)dw\right)\|ds
\end{align*}
Thus having again some constant that does not vanish and does not depend on $\|(f_n-g_n)\|$.\\
If we where on a boudned domain there would have been no problem, I think we should try to use the fact that our function have a fast decay at infinity, but I was not able to use this property in a smart way yet.\\

\textbf{Sketch:} This part try to use the property of vanishing at infinity. We know from the regularity result and definition that a solution of our system is such that:
$$
\|f\|_\infty\leq M,\ \|f\|_1\leq C,\ f(v)|v|\rightarrow0_{|v|\rightarrow\infty}.
$$
This implies that also $\|f|v|\|_\infty\leq M'$. Using the definition of vanishing at infiniry for locally compact topological space, we have that
$$
\forall\epsilon\exists\ K_\varepsilon\ compact\ such\ that\ f(v)|v|\leq\varepsilon\ a.e.\ v\in K^c_\varepsilon.
$$
With this in mind we consider $f_{m-n}$ and $K_\varepsilon:=\left\{v\ :\ f_{m-n}(v)|v|\geq \varepsilon\right\}$ compact set such that $diam(K_\varepsilon)\leq D<\infty$ indepentend of w
$$
\int\int |v-w| \left(f_m-g_m\right)(\frac{mv-(m-n)w}{n})\left(f_m-g_m\right)(v)f_{m-n}(w)dwdv
$$
Non funziona perché $f$ è indipendente da $v$ e siamo punto e a capo.
\fi
\newpage\fi

\iffalse
\section{Enriched coagulation}\label{sec:numerics}
In this section we are going to present numerical results regarding the enrichment of coagulation in the kinetic model proposed.\\
To this end what we have in mind is the understanding of how the small scale turbulence, identified by the parameter $\nu$, interact with the process of coagulation expressed by equation $(\cdot)$.\\
What we found is that, increasing the diffusion parameter related to such turbulence phenomenon at the level of the density of particle, the coagulation is enriched and density of higher order are generated faster.\\
\\
To understand such behavior we have identified a key quantity that can be interpreted as the expelled mass from the finite system, i.e. the mass that coagulate to form particle of higher category.\\
To this end we define the full mass of the system as
\begin{align*}
    \sum_{m=1}^M \iint_{\mathbb{T}^d\times\mathbb{R}^d} mf_m(t,x,v)\, dxdv:=\mathcal{M}_1(t).
\end{align*}
This function represent the total mass present in the system of our consideration at a given time $t$ in our windows of consideration, $t\in[0,T]$.\\
Analyzing the non linearity as in Lemma $(\cdot)$ we notice that
$$
\iint Q^M(f,f)\, dxdv\leq 0, \forall t\in[0,T]
$$
this is enough no understand that our function must be decreasing in time and, since the full system has the total mass conserved (note that $\iint Q^\infty(f,f)\, dxdv=0$, when the type of particle $M$ range in the infinity) then the expelled mass is not lost at all and it is in fact quantity of mass that is sent to higher order of mass-type density.\\ Remember that the negative part of the non linear coagulation operator is of the form, for a fixed $m\leq M$
$$
Q^-_m (f,f)(t,x,v):=\iint f_m(x,t,v)\left(\sum_{n=1}^M f_n(t,x,w)\right) |v-w|\, dxdw
$$
as such, every coagulation at the level of $f_mf_n$, with $m+n>M$, represent a decrease in mass that, ideally, increase the density $f_{m+n}$ that is outside of our system.\\
Therefore, the more (and the fastest) the quantity $\mathcal{M}_1(t)$ decrease over time, the faster and richer the coagulation to higher mass-type is achieved.
\todo[inline]{Here we could explain the numerical setting of the simulation.\\
The restriction on the finite interval thanks to the vanishing at infinity.\\
The result on the faster decay of mass and the interpretation, at least in the case without humidity that we have understood.}

\subsection{Numerical setting and results}
\subsubsection{Smoluchowski Equation with diffusion}
\begin{align*}
    \partial_t f_m(x)=\Delta f_m(x)+&\sum_{n=1}^{m-1}\alpha(n,m-n)
    f_n(x)f_{m-n}(x)\\
    -&2    \sum_{n=1}^\infty \alpha(m,n)
    f_m(x)f_n(x), \quad m\in\mathbb N.
\end{align*}
where $(t,x)\in [0,T]\times \mathbb{T}^d$ and $\kappa>0$. The reason on working on the Torus is linked to the atmospheric problem. The idea is aiming at the study of coagulation inside a cloud formation, thus a bounded domain, that for simplicity reason we model with a torus.
\todo[inline]{Insert graph on the two setting for the standard Smoluchowski equation with diffusion on the torus: Generics initial density, uniform initial density. Showing that things does not work.}
\subsubsection{Smoluchowski with velocity and diffusion}
\begin{align*}
\begin{cases}
\partial_t f_m = -v\cdot \nabla_xf_m+\text{div}_v\left(vf_m\right)+\kappa\Delta_vf_m + Q_m(f,f)\\[5pt]
f_m|_{t=0}=f^0_m
\end{cases}
,\ \quad m=1,...,M
\end{align*}
where $(t,x,v)\in [0,T]\times \mathbb{T}^d\times \mathbb{R}^d$ and $\kappa\ge0$.
\todo[inline]{Insert the equation, explaining the numerical setting and commenting on the picture showing the decay in the uniform initial density in space. Concluding with conjecture that must be the same even with general initial density in space. }
\begin{figure}
     \centering
     \begin{subfigure}[h]{0.4\textwidth}
         \centering
         \includegraphics[width=\textwidth]{SmolVel_CorrezioneIntegrale_DecayPerfettoMassa.png}
     \end{subfigure}
     \hfill
     \begin{subfigure}[h]{0.4\textwidth}
         \centering
         \includegraphics[width=\textwidth]{SmolVel_CorrezioneIntegrale_DifferenzaNoDiffusionDecayPerfettoMassa.png}
     \end{subfigure}
        \caption{Three simple graphs}
        \label{fig:three graphs}
\end{figure}
\begin{figure}
     \centering
     \begin{subfigure}[h]{0.4\textwidth}
         \centering
         \includegraphics[width=\textwidth]{Heatmap_MaximalExpelledMass_EndTime_densXvisc_Correzione.png}
     \end{subfigure}
     \hfill
     \begin{subfigure}[h]{0.4\textwidth}
         \centering
         \includegraphics[width=\textwidth]{Heatmap_MaximalResidualMass_EndTime_densXvisc_Correzione.png}
     \end{subfigure}
        \caption{Three simple graphs}
        \label{fig:three graphs}
\end{figure}
\subsection{Theoretical conjecture}
\todo[inline]{I'm thinking if there is a way to prove some form of theorem on the mass decay depending on the intensity of the laplacian. Maybe using the semigroup theory, since we know explicit formulation of the kernel can help or using some change of variable and look at the total mass. }
\iffalse
We consider the mild formulation for the full problem $\ref{main-eq-trunc1}$ with semigroup: $U_t:=e^tdP_t$ and a regular solution $f_m,\ m=1,...,M$ such that
$$
f^{\mu}_m(t)=U^{\mu}_tf_m^0+\int_0^t U^{\mu}_{t-s}Q_m(f^{\mu},f^{\mu})(s,v)ds
$$
where the viscosity $\mu$ enter in the semigroup generating the solution in the form:
\begin{align*}
    P_t^{\mu}\phi(v)&\sim\frac{1}{\mu^{d/2}}\int e^{-\frac{|w|^2}{\mu}}\phi(e^tv-w)dw\\
    &=\int e^{-|w|^2}\phi(e^tv-\mu w)dw
\end{align*}
Consider the derivative in $\mu$, and a radial initial condition $f^0$ such that $\nabla f^0(-x)\cdot x \leq 0$ and we have
\begin{align*}
    \partial_\mu f^{\mu}_m(t)&=\partial_\mu U^{\mu}_tf_m^0+\int_0^t \partial_\mu U^{\mu}_{t-s}Q_m(f^{\mu},f^{\mu})(s,v)ds\\
    &=-w\cdot \nabla
\end{align*}
\fi
Recall our system of nonlinear integral-differential equations for $\{f_m(t,x,v)\}_{m=1}^M$ defined as:
\begin{align*}
\begin{cases}
\partial_t f_m = -v\cdot \nabla_xf_m+\text{div}_v\left(vf_m\right)+\kappa\Delta_vf_m + Q_m(f,f)\\[5pt]
f_m|_{t=0}=f^0_m
\end{cases}
,\ \quad m=1,...,M
\end{align*}
where $(t,x,v)\in [0,T]\times \mathbb{T}^d\times \mathbb{R}^d$ and $\kappa\ge0$. Since we expect the system to represent density of particles with different masses we take $f^0_m\geq 0,\forall\  m=1,...,M$.\\ The nonlinear term, representing the masses interactions, is:
\begin{align*}
Q_m(f,f)(t,x,v) &:= \sum_{n=1}^{m-1}\iint_{\{nw'+(m-n)w=mv\}} f_n(t,x,w')f_{m-n}(t,x,w)|w-w'|dwdw'\\
&\quad -2\sum_{n=1}^M \int f_m(t,x,v)f_n(t,x,w)|v-w|dw.
\end{align*}
Now, we consider the change of variable $v\rightarrow v/\sqrt{k}$ and call the function $g_m(t,v):=f_m(t,v/\sqrt{k})$. This new function solves the following equation:
\begin{align*}
\begin{cases}
\partial_t g_m = -v\cdot \nabla_xg_m+\text{div}_v\left(vg_m\right)+\Delta_vg_m + \overline{Q}_m(g,g)\\[5pt]
f_m|_{t=0}=f^0_m
\end{cases}
,\ \quad m=1,...,M
\end{align*}
where $(t,x,v)\in [0,T]\times \mathbb{T}^d\times \mathbb{R}^d$ and $\kappa\ge0$. The nonlinear term, representing the masses interactions, is:
\begin{align*}
\overline{Q}_m(g,g)(t,x,v/\sqrt{k}) &:= \sum_{n=1}^{m-1}\iint_{\{nw'+(m-n)w=mv/\sqrt{k}\}} f_n(t,x,w')f_{m-n}(t,x,w)|w-w'|dwdw'\\
&\quad -2\sum_{n=1}^M \int f_m(t,x,v/\sqrt{k})f_n(t,x,w)|v-w|dw\\
&=\sum_{n=1}^{m-1}\int f_n\left(t,\frac{mv/\sqrt{k}-(m-n)w}{n}\right)f_{m-n}(t,w)\left(\frac{m}{n}\right)^2|v/\sqrt{k}-w|dw\\
&\quad -2\sum_{n=1}^M \int f_m(t,v/\sqrt{k})f_n(t,w)|v/\sqrt{k}-w|dw.
\end{align*}
Consider now the total mass associated to this system and we have:
\newpage
\fi
\begin{appendix}
\section{Particle System: Approach and Motivation}\label{appen-particle}

In this section, we provide our original motivation that gives rise naturally to the \abbr{PDE} system we study in this paper. The Smoluchowski type \abbr{SPDE} system we study is conjectured to be the scaling limit of the empirical measure of a system of diffusion particles (idealized rain droplets in the atmosphere) undergoing locally in space coagulation, while subject to an idealized form of turbulence, cf. \cite{FHP} for a discussion. More precisely, for any $d\ge1$ and $N\in\mathbb{N}$, consider a second-order particle system with space variable $x_i^N(t)$ in $\mathbb{T}^d$, velocity variable $v_i^N(t)$ in $\mathbb{R}^d$, mass variable $m_i^N(t)$ in a finite set $\{1,...,M\}$, and initial cardinality $N(0)=N$. Between coagulation events, the motion of an active particle obeys
\begin{equation}\label{particle-sys}
\begin{aligned}
dx^N_i(t)&=v_i^N(t)dt, \quad i\in\mathcal{N}(t)\\
m_i^N(t) dv^N_i(t)&=\alpha (m_i^N(t))^{1/d}\left[\sqrt{2\mu}dB_i(t)+\sum_{k\in K}\sigma_k(x_i^N(t))\,\circ\,dW_t^k-v_i^N(t)dt\right],
\end{aligned}
\end{equation}
(whereas after each coagulation, the velocity will be reset according to the conservation of momentum, to be precised below), where
\begin{itemize}
\item
$\mathcal{N}(t)$ denotes the set of indices of active particles in the system at time $t$, whose cardinality $|\mathcal{N}(t)|\le N$,
\item
$\{B_i(t)\}_{i=1}^\infty$ is a given, countable collection of independent standard Brownian motions in $\mathbb{R}^d$
\item
molecular diffusivity $\mu>0$
\item
$\sigma_k(x):\mathbb{T}^d\to\mathbb{R}^d$, $k\in K$ is a given, finite (or more generally countable, subject to additional assumptions) collection of divergence free vector fields
\item
$\{W_t^k\}_{k\in K}$ is a given, finite (or countable) collection of standard Brownian motions in $\mathbb{R}$
\item
$\circ$ denotes Stratonovich integration.
\end{itemize}
The velocity component of the dynamics obeys  Stokes' law with random force (that depends linearly on the radius $r_i^N=(m_i^N)^{1/d}$) given by an intrinsic noise for each particle, plus a common noise of transport type
\begin{align*}
\dot{\mathcal{W}}(t,x):=\sum_{k\in K}\sigma_k(x)\dot{W}_t^k
\end{align*}
that acts simultaneously on all particles, and as such is seen as an idealized form of turbulent flow in the atmosphere. We denote the $d\times d$ spatial covariance matrix of $\mathcal{W}(t,x)$ by
\begin{align*}
\mathcal{Q}(x,y):=\sum_{k\in K}\sigma_k(x)\otimes\sigma_k(y).
\end{align*}
Moreover, for any fixed $x\in\mathbb{T}^d$ we denote the uniformly elliptic second-order divergence form operator, acting on suitable functions on $\mathbb{R}^d$
\begin{align*}
(\mathcal{L}^{\mathcal{Q},x}_vf)(v):=\left(\mu I+\frac{1}{2}\mathcal{Q}(x,x)\right)\Delta_vf(v).
\end{align*}
Note that $\mathcal{Q}(x,x)$ is nonnegative definite for any $x$. For simplicity, in the sequel we consider the case that
\begin{align}\label{enhanced-diff}
\mathcal{L}^{\mathcal{Q},x}_v\equiv \kappa \Delta_v,
\end{align}
for some constant $\kappa\ge \mu$ and all $x\in\mathbb{T}^d$. In view of diffusion enhancement, we have in mind that $\kappa\gg \mu$.

Each particle $i\in\mathcal{N}(t)$ has a mass $m_i^N(t)\in\{1,2,..,M\}$ which changes over time according a stochastic coagulation rule to be described below. The initial mass $m_i(0)$, $i=1,...,N$, are chosen i.i.d. from $\{1,2,..,M\}$ according to a probability distribution so that $\P(m_1(0)=m)=r(m)$ with $\sum_{m=1}^Mr(m)=1$. We are also given deterministic probability density functions $g_m(x,v):\mathbb{T}^d\times\mathbb{R}^{d}\to\mathbb{R}_+$, $m=1,2,...,M$, satisfying additional assumptions, \footnote{those we impose on the initial condition of the \abbr{PDE} system} such that if $m_i(0)=m$ then the initial distribution of $(x_i(0),v_i(0))$ is chosen with probability density $g_m(x,v)$, independently across $i$. We denote
\begin{align*}
f^0_m(x,v)=r(m)g_m(x,v), \quad (x,v)\in\mathbb{T}^d\times\mathbb{R}^{d},\; m=1,..,M,
\end{align*}
which satisfy the same assumptions as those imposed on $\{g_m(x,v)\}_{m=1}^M$.

The rule of coagulation between pairs of particles is as follows. Let $\theta(x):\mathbb{R}^d\to\mathbb{R}_+$ be a given, $C^\infty$-smooth, symmetric probability density function in $\mathbb{R}^d$ with compact support in $\mathcal{B}(0,1)$ (the unit ball around the origin in $\mathbb{R}^d$) and $\theta(0)=0$. Then, for any $\epsilon\in(0,1)$, denote $\theta^\epsilon(x):\mathbb{T}^d\to\mathbb{R}_+$ by
\begin{align*}
\theta^\epsilon(x):=\epsilon^{-d}\theta(\epsilon^{-1}x), \quad x\in\mathbb{T}^d.
\end{align*}
Suppose the current configuration of the particle system is
\begin{align*}
\eta=(x_1,v_1,m_1,x_2,v_2, m_2,...,x_N,v_N, m_N)\in(\mathbb{T}^d\cup\emptyset)^N\times(\mathbb{R}^d\cup\emptyset)^N\times\{1,...,M,\emptyset\}^N
\end{align*}
where $(x_i,v_i,m_i)$ denotes the position, velocity and mass of particle $i$, by convention if particle $i_0$ is no longer active in the system, we set $x_{i_0}=v_{i_0}=m_{i_0}=\emptyset$. Independently for each pair $(i,j)$ of particles, where $i\neq j$ each running over the index set of active particles in $\eta$, with a rate (recall \eqref{mass-const})
\begin{align}\label{rate}
s(m_i^N,m_j^N)\frac{|v_i-v_j|}{N}\theta^\epsilon(x_i-x_j)
\end{align}
we remove $(x_i,v_i,m_i,x_j,v_j,m_j)$ from the configuration $\eta$, and then add
\begin{align*}
\left(x_i,\frac{m_iv_i+m_jv_j}{m_i+m_j},m_i+m_j,\emptyset,\emptyset,\emptyset\right)
\end{align*}
with probability $\frac{m_i}{m_i+m_j}$, and instead add
\begin{align*}
\left(\emptyset,\emptyset,\emptyset, x_j,\frac{m_iv_i+m_jv_j}{m_i+m_j},m_i+m_j\right)
\end{align*}
with probability $\frac{m_j}{m_i+m_j}$. We call the new configuration obtained this way by $S^1_{ij}\eta$ and $S^2_{ij}\eta$ respectively. In words, if $(i,j)$ coagulate, we decide randomly which of $x_i$ and $x_j$ is the new position of the mass-combined particle. If the position chosen is $x_i$, then we consider $j$ as being eliminated (no longer active) and the new particle has index $i$, whereas if the position chosen is $x_j$, then we consider $i$ as being eliminated and the new particle has index $j$. On the other hand, the velocity of the mass-combined particle is obtained by the conservation of momentum as in perfectly inelastic collisions.

Note that the form of the coagulation rate \eqref{rate} is such that $(i,j)$ can coagulate only if $|x_i-x_j|\le\epsilon$, that is, their spatial positions have to be $\epsilon$-close. We are interested in the case when $\epsilon=\epsilon(N)\to0$ as $N\to\infty$, so that the interaction is not of mean-field type, but rather local, see the statement of our conjectured result below. In particular, if $\epsilon=O(N^{-1/d})$ then each particle typically interacts with a bounded number of others at any given time. The essential feature of our coagulation rate, that of the appearance of $|v_i-v_j|$, is inspired by the coagulation kernels used in the physics literature for describing cloud particles (which in general can depend also on $m_i,m_j$ and other physically relevant quantities), and we believe it is key to demonstrating coagulation enhancement.

For each $N\in\mathbb{N}$, $T\in(0,\infty)$ and $m\in\{1,..,M\}$, we denote the process of empirical measure on position and velocity of mass-$m$ particles in the system by
\begin{align*}
\mu^{N,m}_t(dx,dv):=\frac{1}{N}\sum_{i\in\mathcal{N}(t)}\delta_{x_i^N(t)}(dx)\delta_{v_i^N(t)}(dv)1_{\{m_i^N(t)=m\}}\in\mathcal{M}_{1,+}(\mathbb{T}^d\times\mathbb{R}^d)
\end{align*}
where $\mathcal{M}_{1,+}:=\mathcal{M}_{1,+}(\mathbb{T}^d\times\mathbb{R}^d)$ denotes the space of subprobability measures on $\mathbb{T}^d\times\mathbb{R}^d$ equipped with weak topology. The choice of the initial conditions for our system implies that $\P$-a.s.
\begin{align*}
\mu_0^N(dx,dv)\to f^0(x,v)dxdv, \quad \text{as }N\to\infty
\end{align*}
where the limit is absolutely continuous. We conjecture that, under the assumption of local interaction, i.e.
\begin{align*}
\lim_{N\to\infty}\epsilon(N)=0, \quad \limsup_{N\to\infty}\frac{\epsilon(N)^{-d}}{N}<\infty.
\end{align*}
for every finite $T$, the collection of empirical measures $\{\mu^N_t(dx,dv):t\in[0,T]\}_{m=1}^M$ converges in probability, as $N\to\infty$, in $\mathcal{D}([0,T],\mathcal{M}_{1,+})^M$ the space of c\`adl\`ag paths taking values in $\mathcal{M}_{1,+}$, equipped with the Skorohod topology, towards an absolutely continuous limit, which is the pathwise unique weak solution $\{f_m(t,x,v):t\in[0,T]\}_{m=1}^M$ of a Smoluchowski-type \abbr{SPDE} system, see \eqref{smol-spde}. The latter \abbr{SPDE} degenerates to the \abbr{PDE} system we study in this paper \eqref{main-eq-intro} when the It\^o term is switched off. While we do not provide a rigorous proof here, we sketch below a heuristic argument and postpone the rigorous derivation of the \abbr{SPDE} from the particle system to a future work. We think that this heuristic argument is sufficient to justify our interest in studying our \abbr{PDE} system. In fact, in the literature there exists specific limiting procedures cf. \cite{Galeati, FGL} that allow, in principle, to obtain the \abbr{PDE} from the \abbr{SPDE} by carefully choosing the vector fields $\sigma_k(x)$.

Let $\phi(x,v)$ be any function of class $C_c^\infty(\mathbb{T}^d\times\mathbb{R}^d)$, for any $m$ we apply It\^o's formula to the process
\begin{align*}
\langle \phi, \mu^{N,m}_t\rangle:=\frac{1}{N}\sum_{i\in\mathcal{N}(t)}\phi(x_i^N(t),v_i^N(t))1_{\{m_i^N(t)=m\}}
\end{align*}
and we get

\begin{align*}
&\langle \phi, \mu^{N,m}_T\rangle=\langle \phi, \mu^{N,m}_0\rangle+\int_0^T\frac{1}{N}\sum_{i\in\mathcal{N}(t)}v_i^N(t)\cdot\nabla_x\phi(x_i^N(t),v_i^N(t))1_{\{m_i^N(t)=m\}}dt\\
&-c(m)\int_0^T\frac{1}{N}\sum_{i\in\mathcal{N}(t)}v_i^N(t)\cdot\nabla_v\phi(x_i^N(t),v_i^N(t))1_{\{m_i^N(t)=m\}}dt\\
&+c(m)\int_0^T\frac{1}{N}\sum_{i\in\mathcal{N}(t)}\nabla_v\phi(x_i^N(t),v_i^N(t))1_{\{m_i^N(t)=m\}}\cdot dB_i(t)\\
&+\mu c(m)^2\int_0^T\frac{1}{N}\sum_{i\in\mathcal{N}(t)}\Delta_v\phi(x_i^N(t),v_i^N(t))1_{\{m_i^N(t)=m\}}dt\\
&+\frac{1}{2}c(m)^2\int_0^T\frac{1}{N}\sum_{i\in\mathcal{N}(t)}\mathcal{Q}(x_i^N(t),x_i^N(t))\Delta_v\phi(x_i^N(t),v_i^N(t))1_{\{m_i^N(t)=m\}}dt\\
&+c(m)\int_0^T\frac{1}{N}\sum_{i\in\mathcal{N}(t)}\sum_{k\in K}\sigma_k(x_i^N(t))\cdot\nabla_v\phi(x_i^N(t),v_i^N(t))1_{\{m_i^N(t)=m\}} dW_t^k\\
&+\int_0^T\frac{1}{N^2}\sum_{n=1}^{m-1}\sum_{\substack{i,j\in\mathcal{N}(t), \\ i\neq j}}s(n,m-n)|v^N_i(t)-v^N_j(t)|\theta^\epsilon(x_i^N(t)-x_j^N(t))\\
&\quad\quad\quad\quad \cdot\phi\left(x_i^N(t),\frac{nv_i^N(t)+(m-n)v^N_j(t)}{m}\right)\frac{n}{m}1_{\{m_i^N(t)=n,\,m_j^N(t)=m-n\}}dt\\
&+\int_0^T\frac{1}{N^2}\sum_{n=1}^{m-1}\sum_{\substack{i,j\in\mathcal{N}(t), \\ i\neq j}}s(n,m-n)|v^N_i(t)-v^N_j(t)|\theta^\epsilon(x_i^N(t)-x_j^N(t))\\
&\quad\quad\quad\quad \cdot \phi\left(x_j^N(t),\frac{nv_i^N(t)+(m-n)v^N_j(t)}{m}\right)\frac{m-n}{m}1_{\{m_i^N(t)=n,\, m_j^N(t)=m-n\}}dt\\
&-\int_0^T\frac{2}{N^2}\sum_{n=1}^{M}\sum_{i,j\in\mathcal{N}(t), \; i\neq j}s(n,m)|v^N_i(t)-v^N_j(t)|\theta^\epsilon(x_i^N(t)-x_j^N(t))\\
&\quad\quad\quad\quad \cdot\phi(x_i^N(t),v_i^N(t))1_{\{m_i^N(t)=m, \, m_j^N(t)=n\}}dt\\
&+ M^{N,J}_T
\end{align*}
where $\{M^{N,J}_t\}_{t\ge0}$ is a martingale associated with coagulation (or jumps) that we do not write explicitly. We can rewrite, using the simplification \eqref{enhanced-diff}, the previous identity more compactly as
\begin{align}\label{iden-emp}
&\langle \phi, \mu^{N,m}_T\rangle=\langle \phi, \mu^{N,m}_0\rangle+\int_0^T\Big\langle v\cdot\nabla_x\phi-c(m)v\cdot \nabla_v+c(m)^2\kappa\Delta_v, \;\mu^{N,m}_t\Big\rangle dt\nonumber\\
&+c(m)\int_0^T\frac{1}{N}\sum_{i\in\mathcal{N}(t)}\nabla_v\phi(x_i^N(t),v_i^N(t))1_{\{m_i^N(t)=m\}}\cdot dB_i(t)\nonumber\\
&+c(m)\int_0^T\sum_{k\in K}\Big\langle \sigma_k(x)\cdot\nabla_v\phi,\;\mu_t^{N,m}\Big\rangle dW_t^k + M_T^{N,J}\nonumber\\
&+\int_0^T\sum_{n=1}^{m-1}\frac{n}{m}\Big\langle s(n,m-n)|v-w|\theta^\epsilon(x-y)\phi\Big(x,\frac{nv+(m-n)w}{m}\Big), \; \mu^{N,n}_t(dx,dv)\mu^{N,m-n}_t(dy,dw)\Big\rangle dt\nonumber\\
&+\int_0^T\sum_{n=1}^{m-1}\frac{m-n}{m}\Big\langle s(n,m-n)|v-w|\theta^\epsilon(x-y)\phi\Big(y,\frac{nv+(m-n)w}{m}\Big), \; \mu^{N,n}_t(dx,dv)\mu^{N,m-n}_t(dy,dw)\Big\rangle dt\nonumber\\
&-\int_0^T2\sum_{n=1}^{M}\Big\langle s(n,m)|v-w|\theta^\epsilon(x-y)\phi(x,v), \; \mu^{N,m}_t(dx,dv)\mu^{N,n}_t(dy,dw)\Big\rangle dt.
\end{align}
We expect that $M_T^{N,J}$ and the stochastic integrals in $dB_i(t)$ vanish in limit as $N\to\infty$ in $L^2(\P)$, whereas the martingale associated with the common noise persists in the limit. Further, suppose that we have proved that the laws of the collection of $\mathcal{D}_T(\mathcal{M}_{1,+})^M$-valued random variables $\{\mu^{N,m}_t:t\in[0,T]\}_{m=1}^M$, $N\in\mathbb{N}$, is tight hence weakly relatively compact. Consider any weak subsequential limit
\begin{align}\label{subseq-conv}
\{\mu^{N_\ell,m}_t:t\in[0,T]\}_{m=1}^M\overset{\ell\to\infty}{\to} \{\overline{\mu}^m_t:t\in[0,T]\}_{m=1}^M.
\end{align}
For the sake of arguments, apply Skorohod's representation theorem and there exists some auxiliary probability space and on which a sequence of random variables having the same laws as the ones in \eqref{subseq-conv} so that the above convergence holds almost surely. By an abuse of notation, below we use the same letters for the variables on the auxiliary space. Assume that we can prove that $\overline{\mu}^m_t$ has a density $f_m(t,x,v)$ with respect to Lebesgue measure for every $m$ and $t$. Then, with minor work the linear part of the identity \eqref{iden-emp} converges as $\ell\to\infty$, i.e.
\begin{align*}
&\langle \phi, \mu^{N_\ell,m}_T\rangle \to \langle \phi, f_m(T,x,v)\rangle,\\
&\langle \phi, \mu^{N_\ell,m}_0\rangle \to \langle \phi,f^0_m(x,v)\rangle, \\
&\int_0^T\left\langle v\cdot\nabla_x\phi-c(m)v\cdot \nabla_v+\kappa c(m)^2\Delta_v, \;\mu^{N_\ell,m}_t\right\rangle dt \\
&\to \int_0^T\left\langle v\cdot\nabla_x\phi-c(m)v\cdot \nabla_v+\kappa c(m)^2\Delta_v, \;f_m(t,x,v)\right\rangle dt,\\
&\int_0^T\sum_{k\in K}\left\langle \sigma_k(x)\cdot\nabla_v\phi,\;\mu_t^{N_\ell,m}\right\rangle dW_t^k \to \int_0^T\sum_{k\in K}\left\langle \sigma_k(x)\cdot\nabla_v\phi,\;f_m(t,x,v)\right\rangle dW_t^k.
\end{align*}
The proof that the nonlinear terms also converge to the corresponding limits is more difficult, hence here we content ourselves with a very heuristic ``two-step argument''. Consider each summand of the last term of \eqref{iden-emp} :
\begin{align*}
\int_0^T\left\langle s(n,m)|v-w|\theta^\epsilon(x-y)\phi(x,v), \; \mu^{N_\ell,m}_t(dx,dv)\mu^{N_\ell,n}_t(dy,dw)\right\rangle dt, \quad 1\le m,n\le M.
\end{align*}
Assume we take $\ell\to\infty$ first, keeping $\epsilon$ fixed, we get
\begin{align*}
&\int_0^T\left\langle s(n,m) |v-w|\theta^\epsilon(x-y)\phi(x,v), \; \mu^{N_\ell,m}_t(dx,dv)\mu^{N_\ell,n}_t(dy,dw)\right\rangle dt\\
&\overset{\ell\to\infty}{\to} \int_0^T\int s(n,m)|v-w|\theta^\epsilon(x-y)\phi(x,v) f_m(t,x,v)f_n(t,y,v)dxdydvdwdt\; ;
\end{align*}
then we take $\epsilon\to0$, and since $\theta^\epsilon(\cdot)$ approximates the delta-Dirac $\delta_0$, we get
\begin{align*}
&\int_0^T\int s(n,m)|v-w|\theta^\epsilon(x-y)\phi(x,v) f_m(t,x,v)f_n(t,y,v)dxdydvdwdt\\
&\overset{\epsilon\to0}{\to}\int_0^T\int s(n,m)|v-w|\phi(x,v)f_m(t,x,v)f_n(t,x,w)dxdvdwdt.
\end{align*}
Similarly, each summand of the second and third terms from the bottom in \eqref{iden-emp} also converge under the two-step argument (using also $n/m+(m-n)/m=1$)
\begin{align*}
&\int_0^T\frac{n}{m}\left\langle s(n,m-n)|v-w|\theta^\epsilon(x-y)\phi\left(x,\frac{nv+(m-n)w}{m}\right), \; \mu^{N_\ell,n}_t(dx,dv)\mu^{N_\ell,m-n}_t(dy,dw)\right\rangle dt\\
&+\int_0^T\frac{m-n}{m}\left\langle s(n,m-n)|v-w|\theta^\epsilon(x-y)\phi\left(y,\frac{nv+(m-n)w}{m}\right), \; \mu^{N_\ell,n}_t(dx,dv)\mu^{N_\ell,m-n}_t(dy,dw)\right\rangle dt\\
&\to\int_0^T\int s(n,m-n)|v-w|\phi\left(x,\frac{nv+(m-n)w}{m}\right)f_n(t,x,v)f_m(t,x,w)dvdwdt.
\end{align*}
Hence, we have (at least under Skorohod's representation) the limit identity satisfied by $\{f_m(t,x,v)\}$:
\begin{align*}
\langle \phi, f_m(T)\rangle =& \langle \phi, f_m^0\rangle +\int_0^T\left\langle v\cdot\nabla_x\phi-c(m)v\cdot \nabla_v+\kappa c(m)^2\Delta_v, \;f_m(t,x,v)\right\rangle dt\\
&+\sum_{n=1}^{m-1}\int_0^T\int s(n,m-n)|v-w|\phi\left(x,\frac{nv+(m-n)w}{m}\right) f_n(t,x,v)f_{m-n}(t,x,w)dxdvdwdt\\
&-2\sum_{n=1}^M\int_0^T\int s(n,m)|v-w|\phi(x,v) f_m(t,x,v)f_{n}(t,x,w)dxdvdwdt\\
&+c(m)\int_0^T\sum_{k\in K}\left\langle \sigma_k(x)\cdot\nabla_v\phi,\;f_m(t,x,v)\right\rangle dW_t^k, \quad m=1,...,M,
\end{align*}
which is the weak formulation of the \abbr{SPDE} system
\begin{align}
\begin{cases}\label{smol-spde}
df_m(t,x,v)=&\left(-v\cdot\nabla_x+c(m)\text{div}_v\left(v\cdot\right)+\kappa c(m)^2\Delta_v\right)f_m(t,x,v)dt \\[5pt]
&-c(m)\sum_{k\in K}\sigma_k(x)\cdot\nabla_vf_m(t,x,v)dW_t^k\\[5pt]
&+\sum_{n=1}^{m-1}\int_{\{nw'+(m-n)w=mv\}}s(n,m-n)|w'-w|f_n(t,x,w')f_{m-n}(t,x,w)dwdw'dt \\[5pt]
&-2\sum_{n=1}^M\int s(n,m)|v-w|f_m(t,x,v)f_n(t,x,w)dwdt \\[10pt]
f_m(\cdot,x,v)|_{t=0}=&f^0_m, \quad m=1,...,M.
\end{cases}
\end{align}
Rewriting the It\^o integral as a Stratonovich integral plus a corrector, we equivalently have that
\begin{align*}
df_m(t,x,v)=&\left(-v\cdot\nabla_x+c(m)\text{div}_v\left(v\cdot\right)+\mu c(m)^2\Delta_v\right)f_m(t,x,v)dt\\
&-c(m)\sum_{k\in K}\sigma_k(x)\cdot\nabla_vf_m(t,x,v)\,\circ\,dW_t^k\\
&+\sum_{n=1}^{m-1}\int_{\{nw'+(m-n)w=mv\}}s(n,m-n)|w'-w|f_n(t,x,w')f_{m-n}(t,x,w)dwdw'dt\\
&-2\sum_{n=1}^M\int s(n,m)|v-w|f_m(t,x,v)f_n(t,x,w)dwdt.
\end{align*}
Thus, we see that the same Stratonovich transport-type noise that acts on the particle system \eqref{particle-sys} also acts on the \abbr{SPDE}.

Lastly, we need to prove that the solution of this \abbr{SPDE} system \eqref{smol-spde} is pathwise unique, which allows to conclude (via nontrivial arguments) that the full sequence of empirical measure converges.
\end{appendix}
\printbibliography

\end{document}

\iffalse
\section{Introduction}
Although clouds play a crucial role in atmospheric phenomena, the effect of turbulence in cloud formation is not well established up to the present.\\
Two processes mutually affecting each other characterize the cloud formation and precipitation development: macro-scale processes such as the fluid motion of air associated with
clouds and the micro-scale processes such as condensation, stochastic
coalescence, and evaporation of water droplets. Thus we can say that cloud formation and precipitation development are typical multiscale-multiphysics phenomena.\\
Motivated by the study of \cite{SuperDrop}, in which the authors propose a novel algorithm to compute the stochastic coalescence of large number of droplet; and \cite{EulLag}, in which the problem of the role of turbulence in the cloud droplet growth is treated with simulation of density equation; we investigate the time of first formation of a raindrop, for the spatial and volume density in a cloud formation, under different settings, to explore the role of turbulent flow in the aggregation of small droplets in clouds.\\
\\
In doing so, we consider the random Smoluchowski coagulation equation for the spatial-volumetric density of the atmosphere
\begin{align}\label{smoleq}
    &\partial_t \rho_t(x,v) + \sum_{j\in H}\left(\sigma_j(x,v)\cdot \nabla_x\rho_t(x,v)\right)\xi^j_t=\varepsilon\Delta_x\rho_t(x,v)+F_v(\rho_t(x,\cdot))\\
    &F_v(\rho_t(x,\cdot)):=\frac{1}{2}\int_0^v\rho_t(x,w)\rho_t(x,v-w)f(w,v-w) dw-\int_{\mathbb{R}_+}\rho_t(x,v)\rho_t(x,w)f(v,w) dw\nonumber
\end{align}
Although Coagulation equations with diffusion are already being studied \cite{aldous1} both in discrete (\cite{hamsurv},\cite{ball1990discrete}) and continuous setting (\cite{hamsurv},\cite{1937-1632_2020_12_3319}), as limit of particle system (\cite{YAGHOUTI20093042},\cite{2006JSP...124..997H,},\cite{ines}) or as regularity of its solution \cite{AIHPC_2010__27_2_639_0}. Little is know about the counterpart $(\ref{smoleq})$ which capture in detail the more complex motion of rain droplet in the atmosphere and is object of this study. Here a diffusion term and a random field act on the cloud density taking account of the macro-scale process, such as fluid motion, acting on the formation.\\
The airflow in which they interact, solution of Navier-Stokes equation, is thus replaced with a random field with finite correlation time, depending on an environmental random noise
produced by Ornstein-Uhlenbeck processes.\\
This choice stems from the idea of Stochastic Model
Reduction \cite{maj} and can be replaced in future work with different approximations.\\
While the nonlinear term $F_v(\rho_t(x,\cdot))$ is the classic coagulation reaction interaction, as used in literature (\cite{10.1214/aoap/1029962598},\cite{aldous1},\cite{amann}).\\
\\
Objective of this paper is the understanding of quantities like
\begin{align}\label{tempcont}
    &\tau_f(\rho):=\inf\{t\geq 0 | \exists\ \eta >0,\ x\in D:\ \rho_t(x,R_{rd})\geq\eta\}
\end{align}
which represent the first time the density put enough mass on effective rain drops.\\
Our goal is to find numerical insight on the decreasing behaviour of such quantity in dependence of the parameter of the system.\\
To this end, we exchange density solution $\rho_t$ of $eq.$\ref{smoleq}, with an approximating particle system in which the aforementioned quantities $\tau_f(\rho)$ is replaced by
\begin{align}
    &\tau_f^N:=\inf\{t\geq 0 | \exists i\in N:\  R^i_t\geq R_{rd}\},
\end{align}
where $N$ represent the particles in the system and $R_i$, $R_{rd}$ are the radius of a particle and a real rain drop respectively.\\
\\
With this in mind, in Section.\ref{psm}, we rigorously define a particle system dynamics that, in addition to being easier to simulate, approximate  our key Smoluchoski equation \ref{smoleq}.\\
Warm clouds and rain consist of large number of water droplets with broad range of radius from micrometers to several millimeters. Several sorts of aerosol particles are also floating in the atmosphere, which can be a seed of a cloud droplet.\\
Let $N$ be the number of such individuals (rain droplets and dust particles) in the considered region, each particle is characterized by its position and volume: $\left(X_t^{i,N},V_t^{i,N}\right),\ i=1, .., N$.\\
To describe the dynamics we see how particle acts on observable $F$ of the system, using the infinitesimal generator $\mathcal{L}$ of the dynamical process.
$$
\mathcal{L}F(\eta)=\mathcal{L}_D F(\eta)+\mathcal{L}_C F(\eta).
$
The first term on the r.h.s. represents the diffusion and random field acting on the particles
$$
\mathcal{L}_D F(\eta):=\sum_i \varepsilon\Delta_{x_i}F(\eta)+\sum_i \mathcal{U}_t(x_i)\cdot\nabla_{x_i}F(\eta).
$$
While the second term, representing the coalescence of the particle, is a mean field counterpart of the classic coagulation interaction term:
\begin{align*}
&\mathcal{L}_C F(\eta):=\sum_{i,j}T_N(i,j)\left[\mathbb{I}_{i,j}F(\eta^{i,j})+\mathbb{I}_{j,i}F(\eta^{j,i})-F(\eta)\right]\\
&\mathbb{I}_{i,j}:=\mathbb{I}_{(v_i>=v_j)},\ \mathbb{I}_{j,i}:=\mathbb{I}_{(v_j>v_i)};\\
&T_N^\delta(i,j):=\frac{1}{2} N^{-1}\delta^{-3}\mathbb{I}_{B_{x_i}(\delta f(v_i,v_j))}(x_j)k(v_i,v_j)
\end{align*}
Defining $\mu_t(x,v):=\frac{1}{N}\sum_{i=1}^N \delta_{(x_t^i,v_t^i)}$ as the empirical measure of the system and coupling it with the random noise $\xi_t^h,\ h\in H$ we are able to obtain convergence $(Thm.\ 2.1)$ of the approximating particle dynamics to a solution of the non local Random Smoluchoski equation, parameterized by $\delta$
\begin{align}\label{dsmoleq}
    &\partial_t \rho^\delta_t(x,v) + \sum_{j\in H}\left(\sigma_j(x,v)\cdot \nabla_x\rho^\delta_t(x,v)\right)\xi^j_t=\varepsilon\Delta_x\rho^\delta_t(x,v)+F^\delta_v(\rho^\delta_t(x,\cdot))
\end{align}
where $F^\delta$ is the non local coagulation kernel closely related to equation \ref{smoleq} as we have shown in \ref{locint}.\\
\\
\iffalse
Although usually $[\cdot,\cdot]$, the position is modelled by a set of second order equation given by
$$
m_i\frac{d}{dt}v_i=m_i g + F_D(v_i,U(x_i),R_i);\ \ \frac{d}{dt}x_i=v_i
$$
Following $[\cdot]$ we suppose that the terminal velocity of the particle is instantaneous and so we replace the second order equation and approximate the velocity with
$$v_t^i=U(X_t^i)-v_{\infty}^i(V_t^i).$$
Here $U$ is the solution to Navier-Stokes equation for the wind, which we approximate with a random field that will be introduced later in the section.
while the relation of the volume is Two real-droplets may collide and coalesce into one
big real-droplet and this process is responsible for precipitation development.
\fi
To this end, Section.\ref{tcseq} is devoted to a formal study of the mean field limit equation. In the section we show that, under suitable hypothesis, stemmed from (\cite{article1},\cite{article2}), we are able to recover a solution of the coagulation-position system (\ref{smoleq}) as show in Proposition $3.1$ as a local convergence in the volume variable
\begin{align*}
    \sup_{t\in[0,T]}\|\rho_t-\rho_t^\delta\|_{L_x^2(\mathbb{R}^d)L^1_{v}(0,R)}\rightarrow 0,\ as\ \delta\rightarrow0,\ \forall\ R>0, \mathbb{P}-a.s.
\end{align*}
Justifying the use of the approximating particle system to compute meaningful quantities of the system of interest and recover information on the effect of the random field on coagulation of droplet in the the region  of space considered.\\
\\
As such, in Section.\ref{numan}, numerical regressions are performed on the particle system to study the mean time formation of rain droplet to recover a decay dependence of $\tau_f^N$ in dependence of the parameter of the random field $|H|$ and $\lambda$: the number of vortex and the intensity of the environmental noise.\\
In particular to analyze how an independent noise and a random turbulent vortex flow could affect the time of rain formation in such a system and the effectiveness of the formation.\\
We studied two different settings: first, as a benchmark, we show numerical results concerning the simulation of
particles subjected only to independent Brownian Motion. We recover the classical conjecture such as the quadratic decay of the time formation with respect to the intensity of the independent noise.\\
The second setting considers the particles under the turbulent random flow and we find an analogous decay of the time of first rain formation, yet with a less efficient rate than the independent noise.\\
In particular, the numerical results show that is expected to have a behavior of such exit time of the form
$$
\mathbb{E}[\tau_f^N]\sim p(|H|,\lambda)^{-1},
$$
with $p(\cdot,\cdot)$ polynomial of at most degree two; showing that although with a less efficient rate that independent theoretical Brownian motion, turbulence produced by the motion of the droplets in the atmosphere under the wind flow enhance the coagulation and the rain formation.
\section{Particle System Model}\label{psm}
The microscopic model that we study in this article consists of a large number of particles, in the space $\mathbb{R}^d,\ d\geq 3$, which move according to independent Brownian motions and a random field acting as the wind on the particle.
\begin{align*}
&dX_t^i=\mathcal{U}_t(X^i_t,\omega)dt+\varepsilon dB^i_t\\
&\mathcal{U}_t(x):=\sum_{k\in K}\sigma_k(x)\xi^k_t\\
&d\xi_t=-\lambda\xi_t dt+\lambda dW^i_t,\ \xi_0=0.
\end{align*}
This velocity field $\mathcal{U}:=\mathcal{U}^{\lambda}, \lambda>1$, with finite correlation time, is of the following form:
\[
\mathcal{U}^{\lambda}\left(  t,x\right)  =\sum_{h\in H}\sigma_{h}\left(  x\right)  \xi_{t}^{h,\lambda}%
\]%
where $H$ is a set of index with finite cardinality, $\{\sigma_h\}_h$ is a suitable family of time independent vector field and $\{\xi_t^{h,\lambda}\}_h$ is a family of stationary i.i.d. Ornstein-Uhlenbeck processes with covariance $cov(\xi_t^{h,\lambda},\xi_s^{h,\lambda})=\frac{\lambda}{2}exp(\lambda |t-s|)$, solution of
\[
d\xi_{t}^{h}  =-\lambda\xi_{t}^{h}dt+\sigma dW_{t}^{h},\ \ \
\xi_{0}^{h}  =0,\ k  =1,...,K.\\
\]%
For a family of i.i.d. Brownian motions $\{W_t^h\}_{h\in H}$ on a probability space $\left(\Omega, \{\mathcal{F}_t\}_{t\in[0,T]}, \mathbb{P}\right)$. Up to changing the filtration, we also have an explicit formula for the solution of such object
$$
\xi_t^{h,\lambda}=\lambda\int_0^t\ e^{-\lambda(t-s)}dW_s^h,\ t\in[0,T]
$$
During our work we select smooth vector fields $\{\sigma_h\}_h$, ideally as smooth approximation of the Biot-Savart Kernel, such that fixing point $x_h\in\mathbb{R}^2$ is defined as
\[
\sigma_{h}\left(  x\right)  \sim\frac{1}{2\pi}\frac{\left(  x-x_{h}\right)
^{\perp}}{\left\vert x-x_{h}\right\vert ^{2}},\ \left(  a,b\right)  ^{\perp}:=\left(  b,-a\right)%
\]%
This choice, admittedly phenomenological, come from the idea that one would like to take
$\mathcal{U}_t(x)\equiv U(x)$, the solution of the Navier-Stokes equations for the wind velocity, or a slight modification, and the use of this "vortex like structure" was the proposed approximation.\\
\\
To each particle we attach a value $v\in (0,\infty)$ which represent the volume of the particle (we assume that mass and volume are around the same and can be recovered from one and another). In the mean field model any pair of particles that approach to within a certain range of interaction are liable to coagulate, at which time they disappear from the system, to be replaced by a particle whose volume is equal to the sum of the volumes of the colliding particles, and whose location is a specific point in the vicinity of the location of the coagulation.\\
\\
As a matter of convenience, we introduce the microscopic model, where the number of particles is initially deterministic. We define a sequence of microscopic models,indexed by a positive integer $N$. A countable set $I:=\mathbb{N},...$ of symbols is provided. A configuration $\eta$ is an $\mathbb{R}^d\times(0,\infty)-valued$ function on a finite subset $I_\eta\subset I$. For any $i\in I_\eta$, the component $\eta(i)$ may be written as $(x_i,v_i)$. The particle labelled by $i$ has volume $v_i$ and location $x_i$.\\
To describe this dynamics we need to see how the particle acts on observable of the system.\\ Let $F:(\mathbb{R}^d\times\mathbb{R}^0_+)^N\rightarrow\mathbb{R}$ denote a function such that $F:(\mathbb{R}^d\times\mathbb{R}_+)^N\rightarrow\mathbb{R}$ is smooth, and $F(x,0)=0,\forall x\in\mathbb{R}^d$. The action on $F$ of the infinitesimal generator $\mathcal{L}$ is given by
$$
\mathcal{L}F(\eta)=\mathcal{L}_D F(\eta)+\mathcal{L}_C F(\eta).
$
The first term on the r.h.s. represents the diffusion and random field acting on the particles
$$
\mathcal{L}_D F(\eta):=\sum_i \varepsilon\Delta_{x_i}F(\eta)+\sum_i \mathcal{U}_t(x_i)\cdot\nabla_{x_i}F(\eta).
$$
While for the second term, representing the coalescence of the particle we have:
\begin{align*}
&\mathcal{L}_C F(\eta):=\sum_{i,j}T_N(i,j)\left[\mathbb{I}_{i,j}F(\eta^{i,j})+\mathbb{I}_{j,i}F(\eta^{j,i})-F(\eta)\right]\\
&\mathbb{I}_{i,j}:=\mathbb{I}_{(v_i>=v_j)},\ \mathbb{I}_{j,i}:=\mathbb{I}_{(v_j>v_i)};\\
&T_N^\delta(i,j):=\frac{1}{2} N^{-1}\delta^{-3}\mathbb{I}_{B_{x_i}(\delta f(v_i,v_j))}(x_j)k(v_i,v_j)
\end{align*}
here we have indicated
$$
\eta^{i,j}_k=\begin{cases}
\eta(k), & \textit{if}\ k\neq i,j\\
(x_i,v_i+v_j), & \textit{if}\ k=i\\
(x_i,0), & \textit{if}\ k=j.
\end{cases}
$$
We note here, as one can see from $\mathcal{L}_C$, that the coalescence generator is as such that two particles interact if they are close in the space depending not only on the radius, but also on some fixed scaling quantities $\delta$.\\
This is done to obtain a scaling limit in the mean field regime for the system. As such, we needed to scale the $T$ interaction Kernel with a factor $1/N$, that take into account the interaction with the particle as $N$ grows.\\
In particular, as $N$ grows the particle are nearer, with mean distance $N^{-1/d}$, and so when we consider the $T$ function we should make a rescale in the position: two particles interact if $|X_t^{i,N}-X^{j,N}_t|\leq N^{-1/d}$, and since in our case the interaction depends also on the volume of each raindrop, we would like to have
$$
|X_t^{i,N}-X^{j,N}_t|\leq N^{-1/d}f(V_t^{i,N},V^{j,N}_t))
$$
and thus making it a local problem. This is out of our capacity for now, so we select a rescaling parameter $\delta$ such that $T$ is substituted whit $T^\delta$ of the form
$$
\delta^{\alpha}\mathbb{I}_{B_x(\delta f(v,w)}g(v,w)
$$
where $\alpha$ depend on the system (in particular the dimension) and $\delta$ depends on $N$.\\
\\
We note that $(x_i(t),v_i(t))\in C([0,T],R^d)\times \mathcal{D}([0,T],R_+)$, we consider now for each $N$ the empirical measure associated to the particles
$$
\mu_t^N:=\frac{1}{N}\sum_{i=1}^N \delta_{(X_i^N(t),V_i^N(t))}\in \mathcal{D}([0,T],\mathcal{M}_1(\mathbb{R}^d\times\mathbb{R}_+)
$$
As in the other setting we consider the pair $(\mu_t^N,\xi_t)\in \mathcal{D}([0,T],\mathcal{M}_1(\mathbb{R}^d\times\mathbb{R}_+))\times C([0,T],\mathbb{R})$. If we consider now $\phi\in C^{1,2}_b$, $\phi(x,0)=0$, non necessarily continuous in $v=0$. we can compute the infinitesimal generator acting on $\brak{\mu_t^N,\phi}$ and we have
\begin{align*}
    \brak{\mu_t^N,\phi}=&\brak{\mu_0^N,\phi}+\int_0^t\varepsilon\brak{\mu_s^N,\Delta\phi}ds+\int_0^t \brak{\mu_s^N,\mathcal{U}_s\cdot\nabla\phi}ds +\\
    &+\int_0^t \brak{\mu_t^N,\brak{\mu_t^N,T^\delta(x,v,y,w)J^\phi(x,v,y,w)}}ds+\tilde{M}_t
\end{align*}
where
\begin{align*}
&T^\delta(x,v,y,w):=\frac{1}{2}\delta^{-3}\mathbb{I}_{B_{x}(\delta f(v,w))}(y)k(v,w);\\
&J^\phi(x,v,y,w):=\left[\mathbb{I}_{(v>=w)}\phi(x,v+w)+\mathbb{I}_{v<w}\phi(y,v+w)-\phi(x,v)-\phi(y,w)\right]
\end{align*}
$\tilde{M}_t$ is a martingale and we can compute is quadratic variation:
$$
\mathbb{E}\left[\tilde{M}_t^2\right]=\mathbb{E}\left[\int_0^t\left(\mathcal{L}\brak{\mu_s^N,\phi}^2-2\brak{\mu_s^N,\phi}\mathcal{L}\brak{\mu_s^N,\phi}ds \right)\right]
$$
which we can compute in two parts $M_D,\ M_C$ such that:
\begin{align*}
& \brak{M_D}:=\mathbb{E}\left[\int_0^t N^{-1}\brak{\mu_t^N,|\nabla\phi|^2}ds\right]\\
& \brak{M_C}:=\mathbb{E}\left[\int_0^t \frac{1}{N^3}\sum_{i,j}T_t^{\delta}(i,j)J_t^{\phi}(i,j)^2ds\right]=\mathbb{E}\left[\int_0^t N^{-1}\brak{\mu_s^N,\brak{\mu_s^N, T^\delta(x,v,y,w)J^\phi(x,v,y,w)^2}}ds\right]
\end{align*}
The last computation is a very tedious one, but not difficult.\\

Our result is the following
\begin{theorem}
Assuming that $(\mu,\xi)$ realise a limiting law of the sequence $(\mu^N,\xi)$. Then $\mu$ is a continuous process taking values in $\mathcal{M}_1(\mathbb{R}^{d}\times\mathbb{R}_+)$ and satisfies the limit random \abbr{PDE}:
\begin{align*}
   \brak{\mu_t,\phi}-\brak{\mu_0,\phi}=&\int_0^t\ \brak{\mu_s,\frac{\varepsilon_x^2}{2}\Delta_x\phi}ds+ \int_0^t\ \sum_{j\in J}\brak{\mu_s,\sigma_j(x,v)\cdot\nabla_x\phi}\xi^j_s ds \\
   &+\int_0^t\ \brak{\mu_s,\brak{\mu_s,T^\delta J^\phi}_{y,w}}_{x,v}ds
\end{align*}
for every $t\in[0,T]$ and $\phi\in C_{test}$, with probability 1. Here:
\begin{align*}
&T^\delta(x,v,y,w):=\frac{1}{2}\delta^{-3}\mathbb{I}_{B_{x}(\delta f(v,w))}(y)k(v,w);\\
&J^\phi(x,v,y,w):=\left[\mathbb{I}_{(v>=w)}\phi(x,v+w)+\mathbb{I}_{v<w}\phi(y,v+w)-\phi(x,v)-\phi(y,w)\right].
\end{align*}
Moreover, if $\mu_0$ has finite volume of the particle, also $\mu_t$ does and we have:
$$
\sup_{t\leq T}\int_{\mathbb{R}^d\times\mathbb{R}_+}v\mu_t(dx,dv)\leq \int_{\mathbb{R}^d\times\mathbb{R}_+}v\mu_0(dx,dv)
$$
\end{theorem}
Later we'll show under suitable hypothesis how to pass to the limit as $\delta\rightarrow0$ and retrieve that $\mu_t^\delta\rightarrow\mu_t$ where $\mu_t$ is a solution in the sense of weak formulation for the Random Smoluchoski equation, justifying our particle system approximation scheme:
\begin{definition}
A continuous family of measure $\mu_t\in C([0,T],\mathcal{M}_1(\mathbb{R}^d\times\mathbb{R})$, given $\xi_t$ as before, is a weak solution of the RSPDE if
\begin{align*}
   \brak{\mu_t,\phi}-\brak{\mu_0,\phi}=&\int_0^t\ \brak{\mu_s,\frac{\varepsilon_x^2}{2}\Delta_x\phi}ds+ \int_0^t\ \sum_{j\in J}\brak{\mu_s,\sigma_j(x,v)\cdot\nabla_x\phi}\xi^j_s ds \\
   &+\int_0^t\ \brak{\mu_s,\brak{\mu_s,k(v,w)T(v,w) J_{x,v,w}^\phi}_{x,w}}_{x,v}ds
\end{align*}
for every $t\in[0,T]$ and $\phi\in C_{test}$, with probability 1. Here:
\begin{align*}
&K(v,w)T(v,w):=\frac{\pi}{2}\left(v^{1/3}+w^{1/3}\right)^3E(v,w),\ with\ E\in C^0_b;\\
&J^\phi(x,v,w):=\left[\phi(x,v+w)-\phi(x,v)-\phi(x,w)\right].
\end{align*}
\end{definition}
\iffalse
\begin{theorem}[Existence and uniqueness RSE]
Under the assumption of $Definition$ $14.2$, there exist a solution to the Random Smulochovski Equation and is unique.
\end{theorem}
\begin{proof}
Controllare articoli già letti. Dovrebbe essere una variazione di Alsous con Kernel di questo tipo.
\end{proof}
\begin{corollary}
If $\mu_0$ has density respect the Lebesgue measure on $\mathbb{R}^D\times\mathbb{R}_+$, if $\mu_t$ is a solution in the sense of $Def. 14.2$. Then $\mu_t^N$ has density and solve the RSDE in the weak sense.
\end{corollary}
\begin{proof}
Pensarci bene, dovrebbe seguire dall'unicità e dal fatto che una soluzione funzione debole esiste.
\end{proof}
\fi

\subsection{Scaling Limit: Uniform Estimates and Compactness}
Since $\xi$ does not change in the sequence $(\mu^N,\xi)$, to prove tightness for law $Q^N:=\mathcal{L}(\mu^N,\xi)$ we need to prove it just for the marginal $\mu^N_t$, which is done in the following computation.\\
Suppose that the initial condition
$(X_0^{i,N},V_0^{i,N})$ of the system  are i.i.d from a probability distribution with law $\mu_0$ and $\mathcal{F}_0-measurable$, suppose also that:\\
\\
$a)$ $\sup_{i,N}\mathbb{E}\left[|X_0^{i,N}|+|V_0^{i,N}|\right]<\infty$;\\
$b)$ $\exists\mu_0\in Pr_1(\mathbb{R}^d\times\mathbb{R}_+)$ such that $\brak{\mu_0^N,\phi}\rightarrow_{\mathbb{P}}\brak{\mu_0,\phi}$, for every $\phi\in C^\infty(\mathbb{R}^d\times\mathbb{R}_+)$.\\
$c)$ $\int_{\mathbb{R}^d}v\mu_0(dx,dv)<\infty$;\\
\\
This is true under the particular assumption made in the introduction: $(X_0^{i,N},V_0^{i,N})=(X_0^i,V_0^{i})$, where $\{X_0^i,V_0^i\}_{i\in\mathbb{N}}$ is a sequence of i.i.d. $\mathcal{F}_0-$measurable r.v.'s with common law $\mu_0\in Pr_1(\mathbb{R}^d\times\mathbb{R}_+)$.\\
We also assume the following on system $(3.1)$:\\
\\
$c)$ $\mathcal{U}_t(x,v,\omega)$ is of the form $\sum_{h\in H}\sigma_h(x)D_h(v)\xi_t^h(\omega),\ |H|<\infty$, where $\sigma_h$ is regular  $\forall h, \|\sigma_h\|_\infty\leq K_\sigma<\infty$, and $D_h$ is smooth and bounded with a constant $K_D$;\\
$d)$ the interacting kernel for the volume of the particle is defined as $$T^\delta:=\delta^{-3}\mathbb{I}_{B_x(\delta f(v,w)}(y) k(v,w).$$ Here $k$ represents the efficiency collision and is such that $k\in C^0_b,\ \|k\|_\infty<\infty$, and $f$ is as usual.\\ We'll derive a real interaction kernel at the limit of $\delta\rightarrow0$ later in the next chapter.\\
\\
The crucial step
in the proof of tightness is to establish suitable uniform bounds for the increments of the process $\mu_t^N$.
\begin{proposition}
Let $T\geq 0$ and $\phi\in C^1,2$ be fixed. There exists a constant
$C> 0$ depending only on $T,\phi,\xi,K_\sigma,\delta$ such that for all $N\in\mathbb{N}$, any bounded stopping time
$\tau\leq T$ with respect to the filtration of $\mu^N$, and $\vartheta>0$, it holds
$$
\mathbb{E}\left[|\brak{\mu^N_{\tau+\vartheta}-\mu^N_\tau,\phi}|^2\right]\leq C\vartheta.
$$
\iffalse
Under this assumption, independently on $N$, there exist $C>0$ such that the following condition hold
$$
\mathbb{E}\left[\sup_{t\in[0,T]}\int_{\mathbb{R}^d\times\mathbb{R}_+}\left(\ |x|+|v|\ \right)\mu_t^N(dx,dv)\right]\leq C.
$$
\fi
\end{proposition}
The proof of the latter shall be divided in two parts, corresponding to the generator part and the stochastic (martingale) part of the dynamics of $\mu^N$. Indeed, we can always expand the above increments as
\begin{align*}
    \brak{\mu^N_{\tau+\vartheta}-\mu^N_\tau,\phi}=&\int_\tau^{\tau+\vartheta}\mathcal{L}_s\brak{\mu^N_s,\phi}ds+\left(\tilde{M}_{\tau+\vartheta}^{\phi,N}-\tilde{M}^{N,\phi}_\tau\right).
\end{align*}
where $\tilde{M}$ is a martingale and we know the quadratic variation. Thus we can reduce ourselves to bound separately the first and second summands
in the right-hand side of the expression above.
\iffalse
\begin{proof}
We have, from the definition of empirical measure, that
$$
\int_{\mathbb{R}^d\times\mathbb{R}_+}\left(\ |x|+|v|\ \right)\mu_t^N(dx,dv)=\brak{\mu_t^N,|x|+|v|}=\frac{1}{N}\sum_{i=1}^N\left(|X_t^{i,N}|+|V_t^{i,N}|\right).
$$
Hence, we have (fixing $\omega$):
\begin{align*}
  |X_t^{i,N}|+|V_t^{i,N}|&\leq |X_0^{i,N}|+|V_0^{i,N}|+  \int_0^t|\mathcal{U}_t(X_t^{i,N},V_t^{i,N})|ds\\
  &+ \varepsilon_x|B_t^i|+\varepsilon_v|\tilde{B}_t^i|+ \int_0^t\sum_j|G_N^{\delta}(X_t^{i,N},X_t^{j,N},V_t^{i,N}V_t^{j,N})|ds
\end{align*}
The important terms are the interacting one in the volume and the random field: let's consider them separately.
\begin{align*}
    \int_0^t|\mathcal{U}_t(X_t^{i,N},V_t^{i,N})|ds&\leq \int_0^t\sum_{h\in H}|\sigma_h(X_t^{i,N})D_h(V_t^{i,N})\xi_t^k|ds\\
    &\leq K_D\sum_{h\in H}\int_0^t|\sigma_h(X_t^{i,N})||\xi_t^k|ds
\end{align*}
Here we have used the fact that the functions $D_h$, representing the resistance of the particle to the action of the random field, are smooth and bounded: $\|D_h\|\leq K_D<\infty$.\\
If we also assume that the component of the random field, $\sigma_k$ are smooth and bounded in $L^\infty$ with some constant $K_\sigma$, than we have
\begin{align*}
    \int_0^t|\mathcal{U}_t(X_t^{i,N},V_t^{i,N})|ds&\leq K_D\sum_{h\in H}\int_0^t|\sigma_h(X_t^{i,N})||\xi_t^k|ds\leq T K_D K_\sigma \sum_h \sup_t|\xi_t^h|.
\end{align*}
Concerning the interacting term in the volume, if we suppose that there is a cutoff function $\chi_R(V_t^{i,N})$, we have:
\begin{align*}
     \int_0^t\sum_j|G_N^{\delta}(X_t^{i,N}&,X_t^{j,N},V_t^{i,N}V_t^{j,N})|ds\leq \frac{\delta^{\alpha}}{N}\sum_j\int_0^t |g(V_t^{i,N},V_t^{j,N})\chi_R(V_t^{i,N})|ds\\
     &\leq\frac{\delta^{\alpha}}{N}\sum_j\int_0^t |V_t^{i,N} V_t^{j,N}|\chi_R(V_t^{i,N})ds\leq\delta^{\alpha}R\int_0^t\brak{\mu_s^N,|v|}ds
\end{align*}
And, for the positivity of the quantities under consideration we have
\begin{align*}
    \delta^{\alpha}R\int_0^t\brak{\mu_s^N,|v|}ds&\leq \delta^{\alpha}R\int_0^t\sup_{r\leq s}\brak{\mu_r^N,|v|+|x|}ds
\end{align*}
Summing all together and adding $\frac{1}{N}\sum_{i=1}^N$, we get
\begin{align*}
    \mathbb{E}\left[\sup_{t\leq T}\brak{\mu_t^N,|x|+|v|}\right]&\leq \mathbb{E}\left[|X_0^i|+|V_0^i|\right]+ K_D K_\sigma T |H| \mathbb{E}\left[\sup_{t\leq T}|\xi_t^1|\right]\\
    &+2\max(\varepsilon_x,\varepsilon_v)\mathbb{E}\left[\sup_{t\leq T}|B_t^1|\right]+\delta^{\alpha}R\int_0^T\mathbb{E}\left[\sup_{r\leq s}\brak{\mu_r^N,|v|+|x|}\right]ds
\end{align*}
We remember now, from \textit{S.E.Graversen and G.Peskir}$[\cdot]$, we can find a maximal bound on the Ornstein-Uhlenbeck such that
$$
\mathbb{E}\left[\sup_{t\leq T}|\xi_t^1|\right]\leq C_T\sqrt{\lambda}\log^{1/2}(1+\lambda^{-1}).
$$
And so with Gronwall exist a constant $C_{R,\varepsilon,B,\sigma,D,\mu_0,\xi,T}<\infty$, independent of $N$, such that
$$
\mathbb{E}\left[\sup_{t\leq T}\brak{\mu_t^N,|x|+|v|}\right]\leq C_{R,\varepsilon,B,\sigma,D,\mu_0,\xi,T},
$$
which concludes the proof.
\end{proof}\fi
\begin{lemma}
In the notation above, it holds
$$
\mathbb{E}\left[\big|\int_\tau^{\tau+\vartheta}\mathcal{L}\brak{\mu_t^N,\phi}dt\big|^2\right]\leq C\vartheta
$$
where the constant $C> 0$ only depends on $\xi,\phi,T,K_\sigma$.
\iffalse
Under this assumption there exists $p,\beta,\ and\ C >0$, independently on $N$, such that the following condition hold
$$
\mathbb{E}\left[\mathcal{W}_1(\mu_t^N,\mu_s^N)^p\right]\leq C|t-s|^{1+\beta}.
$$\fi
\end{lemma}
\begin{proof}
Consider the integrating function $\mathcal{L}\brak{\mu_s^N,\phi}$ we can explicitly bound
\begin{align*}
    |\mathcal{L}\brak{\mu_s^N,\phi}|&\leq |\mathcal{L}_D\brak{\mu_s^N,\phi}|+|\mathcal{L}_C\brak{\mu_s^N,\phi}|\leq|\brak{\mu_s^N,\frac{\varepsilon^2}{2}\Delta\phi}|+|\brak{\mu_s^N,\xi_s\sigma\cdot\nabla_x\phi}|+\\
    &+|\brak{\mu_s^N,\brak{\mu_s^N,T^\delta J^\phi}}|\\
    &\leq \frac{\varepsilon^2}{2}\|\phi\|_{1,2}+K_\sigma \sup_{t\in[0,T]}|\xi_t|\|\phi\|_{1,2}+2\delta^{-3}\|k\|_{\infty}\|\phi\|_{\infty}.
\end{align*}
Passing to the square, using Jensen inequality we get
$$
\mathbb{E}\left[\big|\int_\tau^{\tau+\vartheta}\mathcal{L}\brak{\mu_t^N,\phi}dt\big|^2\right]\leq C^2_{\xi,\phi,\sigma,k,\delta}\ \vartheta.
$$
\end{proof}
\iffalse
\begin{proof}
To estimate $\mathcal{W}_1$, we start considering $\phi\in Lip(1)$ and the quantity
\begin{align*}
    \left|\brak{\mu_t^N,\phi}-\brak{\mu_s^N,\phi}\right|&\leq \frac{1}{N}\sum_{i=1}^N|\phi(X_t^{i,N},V_t^{i,N})-\phi(X_s^{i,N},V_s^{i,N})|\\
    &\leq \frac{1}{N}\sum_{i=1}^N|X_t^{i,N}-X_s^{i,N}|+|V_t^{i,N}-V_s^{i,N}|.
\end{align*}
If we remember the definition of Wasserstein distance and using Jensen inequality, we have
$$
\mathbb{E}\left[\mathcal{W}_1(\mu_t^N,\mu_s^N)^p\right]\leq 2^{p-1}\mathbb{E}\left[\frac{1}{N}\sum_{i=1}^N|X_t^{i,N}-X_s^{i,N}|^p+ |V_t^{i,N}-V_s^{i,N}|^p\right].
$$
We fix $i$ and consider the difference one addendum at a time
\begin{align*}
    |X_t^{i,N}-X_s^{i,N}|&\leq \int_s^t |\mathcal{U}_t(X_r^{i,N},V_r^{i,N})|dr + \varepsilon_x|B_t^i-B_s^i|\\
    &\leq |t-s| K_D K_\sigma \sum_h \sup_{r\leq T}|\xi_r^h|+ \varepsilon_x|B_t^i-B_s^i|.
\end{align*}
which implies
\begin{align*}
     \mathbb{E}\left[|X_t^{i,N}-X_s^{i,N}|^p\right]&\lesssim |t-s|^p K^p_D K^p_\sigma |H|^p \mathbb{E}\left[\sup_{r\leq T}|\xi_r^h|^p\right]+ \varepsilon_x\mathbb{E}\left[|B_t^i-B_s^i|^p\right].
\end{align*}
where we have use the property of the sup and the fact that all our function are positive. Again, from \textit{S.E.Graversen and G.Peskir}$[\cdot]$, we can bound the p-th power of the mean of the Ornstein-Uhlenbeck process, and we get for $p>2$
\begin{align}
    \mathbb{E}\left[|X_t^{i,N}-X_s^{i,N}|^p\right]&\lesssim |t-s|^p C(\xi,\sigma,D,H)+C|t-s|^{p/2}\leq C|t-s|^{p/2}.
\end{align}
We now consider the volume component:
\begin{align*}
    |V_t^{i,N}-V_s^{i,N}|&\leq \delta^{\alpha}\int_s^t\frac{1}{N}\sum_{j=1}^N |V_r^{i,N}||V_r^{j,N}|\chi_R(V_r^{i,N})|dr+\varepsilon_v|\tilde{B}_t^i-\tilde{B}_s^i|\\
    &\leq \delta^{\alpha}R\int_s^t \brak{\mu_r^N,|v|}dr+\varepsilon_v|\tilde{B}_t^i-\tilde{B}_s^i|
\end{align*}
we can take now the $p-th$ power and, using property of the supremum and the Jensen inequality we get
\begin{align*}
    |V_t^{i,N}-V_s^{i,N}|^p&\lesssim |t-s|^p \delta^{p\alpha}R^p \sup_{r\leq T}\left(\brak{\mu_r^N,|v|}\right)^p+\varepsilon_v^p|\tilde{B}_t^i-\tilde{B}_s^i|^p
\end{align*}
hence, taking the expectation
\begin{align}
    \mathbb{E}\left[|V_t^{i,N}-V_s^{i,N}|^p\right]&\lesssim |t-s|^p C \mathbb{E}\left[\sup_{r\leq T}\left(\brak{\mu_t^n,|v|}\right)^p\right]+C|t-s|^{p/2}
\end{align}
What we need to do now is estimate the quantity
$$
\mathbb{E}\left[\sup_{r\leq T}\left(\brak{\mu_t^n,|v|}\right)^p\right] \lesssim\mathbb{E}\left[\left(\sup_{r\leq T}\brak{\mu_t^n,|v|^p}\right)\right].
$$
We use Gronwall lemma to prove that is bounded by a constant independent of $N$:
\begin{align*}
    |V_t^i|&\leq\varepsilon_v|\tilde{B}_t^i|+\delta^{\alpha}R\int_0^t\brak{\mu_s^N,|v|}ds
\end{align*}
which we get
\begin{align*}
    |V_t^i|^p&\lesssim\varepsilon_v^p|\tilde{B}_t^i|^p+\delta^{p\alpha}R^p\int_0^t\brak{\mu_s^N,|v|}^p ds\lesssim\varepsilon_v^p|\tilde{B}_t^i|^p+\delta^{p\alpha}R^p\int_0^t\sup_{r\leq s}\brak{\mu_r^N,|v|^p} ds
\end{align*}
summing and passing to the sup, putting all together we get
\begin{align*}
    \mathbb{E}\left[\left(\sup_{r\leq T}\brak{\mu_t^n,|v|^p}\right)\right]&\lesssim\varepsilon_v^p\mathbb{E}\left[\sup_{t\leq T}|\tilde{B}_t^1|^p\right]+\delta^{p\alpha}R^p\int_0^T\mathbb{E}\left[\sup_{r\leq s}\brak{\mu_r^N,|v|^p}\right] ds
\end{align*}
and we can conclude with Gronwall.\\
If we consider now $(3.3),\ (3.4)$ we can finally bound
\begin{align*}
    \mathbb{E}\left[\mathcal{W}_1(\mu_t^N,\mu_s^N)^p\right]\leq C|t-s|^{p/2}
\end{align*}
Since $C>0$ is independent of $N$, choosing $p>2$ and $\beta=\frac{p}{2}-1>0$ we complete the proof.
\end{proof}\fi
\begin{lemma}
In the notation above, it holds
$$
\mathbb{E}\left[\left(\tilde{M}^N_{\tau+\vartheta}-\tilde{M}^N_\tau\right)^2\right]\leq \frac{C\vartheta}{N},
$$
where the constant $C>0$ only depends on $\phi,T,\delta$.
\end{lemma}
\begin{proof}
Using the fact that $\tau$ is a stopping time for $\mu^N$ and $\tilde{M}$ is a martingale, follows
\begin{align*}
\mathbb{E}\left[\left(\tilde{M}^N_{\tau+\vartheta}-\tilde{M}^N_\tau\right)^2\right]&\leq\mathbb{E}\left[\big|\int_\tau^{\tau+\vartheta}\mathcal{L}\brak{\mu_t^N,\phi}^2-2\brak{\mu_t^N,\phi}\mathcal{L}\brak{\mu_t^N,\phi}dt\big|\right]\\
&\leq \vartheta\mathbb{E}\left[\sup_t\left( \mathcal{L}\brak{\mu_t^N,\phi}^2-2\brak{\mu_t^N,\phi}\mathcal{L}\brak{\mu_t^N,\phi}\right)\right].
\end{align*}
From the definition of the two martingale (the diffusive one and coagulation one) we have:
\begin{align*}
& \brak{M_D}:=\mathbb{E}\left[ N^{-1}\brak{\mu_t^N,|\nabla\phi|^2}\right]\leq N^{-1} \|\phi\|_{1,2}\\
& \brak{M_C}:=\mathbb{E}\left[\frac{1}{N^3}\sum_{i,j}T_t^{\delta}(i,j)J_t^{\phi}(i,j)^2\right]\leq N^{-1}2\|\phi\|_{\infty}\delta^{-3}\|k\|_{\infty}.
\end{align*}
Which concludes the proof.
\end{proof}
The combination of Lemma 2.4 and Lemma 2.5 proves Proposition 2.3, from
which we immediately deduce the following tightness result
\begin{proposition}
Under the hypothesis of the previous lemmas, $\{Q^N\}_{N\in\mathbb{N}}$ the law of $\{\mu^N_\cdot\}_{N\in\mathbb{N}}$ on $\mathcal{D}([0,T],\mathcal{M}_1(\mathbb{R}^d\times\mathbb{R}_+))$ is relatively compact.\\
As a consequence the law of $(\mu^N,\xi)\in \mathcal{D}([0,T],\mathcal{M}_1(\mathbb{R}^d\times\mathbb{R}_+))\times C([0,T],\mathbb{R})$ is tight and there exist a sub sequence such that $(\mu^N_k,\xi)\rightarrow_{\mathcal{L}}Q$ a law in $\mathcal{D}([0,T],\mathcal{M}_1(\mathbb{R}^d\times\mathbb{R}_+))\times C([0,T],\mathbb{R})$.
\end{proposition}
\begin{proof}

it suffices to show that the laws of couplings$\brak{\mu_t^N,\phi}$ is tight on $\mathcal{D}([0,T],\mathbb{R})$ for any fixed $\phi$ in a dense subset of $C^0$
including the constant $\phi \equiv 1$. Such tightness is in turn verified if:
\begin{itemize}
    \item for any $t\in[0,T]$ the sequence of random variables $\brak{\mu_t^N,\phi}$ is tight;
    \item for any $\overline{\delta} > 0$ it holds
    $$
    \lim_{\varepsilon\rightarrow0}\limsup_{N\in\mathbb{N}}\sup_{\tau\in \mathcal{T}_T^N}\sup_{\vartheta\leq \varepsilon} Q^N(|\brak{\mu^N_{\tau+\vartheta}-\mu^N_\tau,\phi}|>\delta)=0
    $$
    where $\mathcal{T}_T^N$ denotes the family of stopping times with respect to the filtration
of $\mu_t^N$ bounded by $T$, that is $\tau\leq T$ almost surely;
\end{itemize}
The former condition is an easy consequence since $|\mu_t^N|\leq 1$, the collection $(\brak{\phi,\mu_t^N})_{N\in\mathbb{N}}$ for fixed $t$ is automatically tight because $\brak{\phi,\mu_t^N}\leq \|\phi\|_\infty$. The second condition, follows immediately from  the previous proposition and Markov inequality.
\end{proof}
\subsection{Passing to the Limit}
As a consequence of proposition $(2.6)$ the law of $(\mu^N,\xi)\in C([0,T],Pr_1(\mathbb{R}^d\times\mathbb{R}_+))\times C([0,T],\mathbb{R})$ is tight and we can extract a subsequence, whcich we still denote $Q^N$, such that converges weakly to a law $Q$ in $C([0,T],Pr_1(\mathbb{R}^d\times\mathbb{R}_+))\times C([0,T],\mathbb{R})$.\\
Let us now consider $(\mu_t,\xi)$, a process such that its law is equal to $Q$ and such that
$$
(\mu_t^{N_k},\xi)\rightarrow_{\mathcal{L}}(\mu_t,\xi).
$$

\begin{theorem}
If $(\mu,\xi)$ realise the law of a weak limit point of a subsequence of $\{Q^N\}_{N\in\mathbb{N}}$, then $\mu$ solves the RPDE with probability 1 over test function.
\end{theorem}
\begin{proof}
We consider the functional
\begin{align*}
\Psi_{\phi}(\nu_\cdot,f_\cdot):=&\sup_{t\in [0,T]}\big|\brak{\nu_t,\phi}-\brak{\nu_0,\phi}-\int_0^t\ \brak{\nu_s,\frac{\varepsilon_x^2}{2}\Delta_x\phi}ds\\
   &-\int_0^t\ \brak{\nu_s,\sigma(x,v)\cdot\nabla_x\phi}f_s\ ds -\int_0^t\ \brak{\nu_s,\brak{\nu_s,T^\delta J^\phi}}ds\ \big|
\end{align*}
defined for every $\phi\in C^{1,2}_b(\mathbb{R}^d\times\mathbb{R})$.\\
It is continuous on $\mathcal{D}([0.T],Pr_1(\mathbb{R}^d\times\mathbb{R}))\times C([0,T],\mathbb{R})$.\\
Hence, if $(\mu_t^{N_k},\xi)\rightarrow_{\mathcal{L}}(\mu_t,\xi)$ is a subsequence which weakly converges to $(\mu,\xi)$, by Portmanteau theorem we have
\begin{align*}
    \mathbb{P}(\Psi_{\phi}(\mu_\cdot,\xi_\cdot)>\delta)\leq \liminf_{k\rightarrow\infty}\mathbb{P}\left(\Psi_{\phi}(\mu^{N_k}_\cdot,\xi_\cdot)>\tilde{\delta}\right).
\end{align*}
Using the definition of the functional and the identity satisfied by $(\mu_t^N,\xi)$ in $lemma\ (3.3)$, the r.h.s. is equal to
$$
=\mathbb{P}\left(\sup_{t\in[0,T]}|\brak{\mu_0^{N_k},\phi}-\brak{\mu_0,\phi}+M_t^{\phi,N_k,x}+M_t^{\phi,N_k,v}|>\tilde{\delta}\right).
$$
Using the hypothesis on the initial condition, regularity of the velocity field and interaction kernel, standard computation for the martingale part gives us
$$
    \mathbb{P}(\Psi_{\phi}(\mu_\cdot,\xi_\cdot)>\tilde{\delta})\leq\liminf_{k\rightarrow\infty}\mathbb{P}\left(\Psi_{\phi}(\mu^{N_k}_\cdot,\xi_\cdot)>\tilde{\delta}\right)=0.
$$
Since this hold true for every $\tilde{\delta}>0$, we deduce that
$$
\mathbb{P}(\Psi_{\phi}(\mu_\cdot,\xi_\cdot)>\tilde{\delta})=1.
$$
Hence $\forall\phi\in C^{1,2}_b(\mathbb{R}^d\times\mathbb{R})$:
\begin{align*}
   \brak{\mu_t,\phi}-\brak{\mu_0,\phi}=&\int_0^t\ \brak{\mu_s,\frac{\varepsilon_x^2}{2}\Delta_x\phi}ds \\
   &+\int_0^t\ \brak{\mu_s,\sigma(x,v)\cdot\nabla_x\phi}\xi_s ds +\int_0^t\ \brak{\nu_s,\brak{\nu_s,T^\delta J^\phi}}ds
\end{align*}
$\forall t\in [0,T]$, with probability 1.
\end{proof}
We now can conclude the proof of Theorem (2.1)
\begin{proof}
What we need to prove to conclude the theorem is that we can swap the quantifiers on $\phi\in \mathcal{D}(R^{d+1})$.\\ Since this space is separable we can work on a infinite countable dense subset and we have
$$
\mathbb{P}(\forall\phi\ \Psi_\phi(\mu,\xi)=0)=\mathbb{P}(\forall\phi_n\ \Psi_{\phi_n}(\mu,\xi)=0)=1.
$$
And so we have that:
\begin{align*}
   \brak{\mu_t,\phi}-\brak{\mu_0,\phi}=&\int_0^t\ \brak{\mu_s,\frac{\varepsilon_x^2}{2}\Delta_x\phi}ds \\
   &+\int_0^t\ \brak{\mu_s,\sigma(x,v)\cdot\nabla_x\phi}\xi_s ds +\int_0^t\ \brak{\mu_s,\brak{\mu_s,T^\delta J^\phi}}ds
\end{align*}
for every $t\in[0,T]$ and $\phi\in C_{test}$, with probability 1.\\
The continuity of $\mu$ follows from the fact that every element in the r.h.s. is continuous.
\end{proof}
What we have now is that any process realizing the limiting law is a solution of the random evolution equation in the sense above. What we need to conclude now is:
\begin{enumerate}
    \item Prove uniqueness of solutions for this equation.
    \item Conclude all limiting laws agree, and hence that we have convergence in law.
\end{enumerate}
\begin{theorem}
If $(\mu,\xi),\ (\nu,\xi)$ are two processes such that $\mu,\nu$ satisfy the weak formulation as in $Thm.\ 8.3$, $\mathbb{P}-a.s.$, then:
$$\|\mu_t-\nu_t\|=0,\ \forall t\in[0,T],\ \mathbb{P}-a.s.$$
\end{theorem}
\begin{proof}
Existence in the weak sense has been proved above. Let us prove uniqueness. To
shorten the expressions we set $\varepsilon_x=\varepsilon_v=1$ and $|H|=1$.\\
Let $\mu$ and $\nu$ be two solutions. Using as test
functions the heat kernel multiplied by a test function we have that both satisfy
\begin{align*}
    \brak{\mu_t-\nu_t,\phi}=&\int_0^t\ \brak{\mu_s-\nu_s,\sigma(x,v)\cdot\nabla_x(e^{(t-s)\Delta}\phi)}\xi_s ds\\
    &+\int_0^t\ \brak{\mu_s,\brak{\mu_s,T^\delta J^{(e^{(t-s)\Delta}\phi)}}}ds\\
    &-\int_0^t\ \brak{\nu_s,\brak{\nu_s,T^\delta J^{(e^{(t-s)\Delta}\phi)}}}ds
\end{align*}
We pass to the module and we estimate for $\mathbb{P}-a.s.$
\begin{align*}
    |\brak{\mu_t-\nu_t,\phi}|\leq&\int_0^t\ |\brak{\mu_s-\nu_s,\sigma(x,v)\cdot\nabla_x(e^{(t-s)\Delta}\phi)}|ds\|\xi\|_{L_t^\infty}\\
    &+\int_0^t\ |\brak{\mu_s-\nu_s,\brak{\mu_s,T^\delta J^{(e^{(t-s)\Delta}\phi)}}}|ds\\
    &+\int_0^t\ |\brak{\nu_s,\brak{\mu_s-\nu_s,T^\delta J^{(e^{(t-s)\Delta}\phi)}}}|ds
\end{align*}
We the first term in the r.h.s. we have $\mathbb{P}-a.s.$
\begin{align*}
\int_0^t\ |\brak{\mu_s-\nu_s,\sigma(x,v)\cdot\nabla_x(e^{(t-s)\Delta}\phi)}|ds\|\xi\|_{L_t^\infty}&\leq \int_0^tK_{\sigma} \|\phi\|_\infty\|\nabla e^{(t-s)\Delta}\|\|\mu_s-\nu_s\|ds \|\xi\|_{L_t^\infty}\\
&\leq\int_0^t\frac{C(\xi, \phi,\sigma)}{\sqrt{t-s}}\|\mu_s-\nu_s\|ds.
\end{align*}
For the second term in the r.h.s we have
\begin{align*}
\int_0^t\ |\brak{\mu_s-\nu_s,\brak{\mu_s,T^\delta J^{(e^{(t-s)\Delta}\phi)}}}|ds&\leq \int_0^t\ \brak{\mu_s-\nu_s,2\delta^{-3}\|\phi\|_{\infty}\|K\|_{\infty}}ds
\end{align*}
where we have used the fact that
\begin{align*}
|T^\delta J^{e^{(t-s)\Delta}\phi}|&=|\int \frac{1}{2}\delta^{-3}\mathbb{I}_{B_x(\delta f(v,w))}k(v,w)4|e^{(t-s)\Delta}\phi|d\mu^N_s|\\
&\leq 2\delta^{-3}\int d\mu_s^N(dx,dv)\|\phi\|_\infty\|k\|_{\infty}\leq 2\delta^{-3}\|\phi\|_{\infty}\|K\|_{\infty}<\infty.
\end{align*}
putting all together we have
$$
\int_0^t\|\brak{\mu_s-\nu_s,\brak{\mu_s,T^\delta J^{(e^{(t-s)\Delta}\phi)}}}|ds\leq\int_0^t C(\delta,k,\phi)\|\mu_s-\nu_s\|ds.
$$
For the last term we have
\begin{align*}
\int_0^t\ |\brak{\nu_s,\brak{\mu_s-\nu_s,T^\delta J^{(e^{(t-s)\Delta}\phi)}}}|ds&\leq C(k,\delta,\phi)\int_0^t \|\mu_s-\nu_s\|ds
\end{align*}
If we combine all the inequality we get, over the sup on the test function with $\|\phi\|=1$
$$
\|\mu_t-\nu_t\|\leq C(\xi,k,\sigma,T,\delta)\int_0^t \frac{\sqrt{t-s}+1}{\sqrt{t-s}}\|\mu_s-\nu_s\|ds
$$
and with a modification
of a standard lemma we are able to conclude that\\
$\|\mu_t-\nu_t\|=0.$
\end{proof}
\begin{corollary}
The family $\{Q^N\}_{N\in\mathbb{N}}$
of laws of the processes $(\mu_\cdot^N,\xi_\cdot)$ weakly converge
to $Q$ where $\mu\in C([0,T],\mathcal{M}_1(\mathbb{R}^d\times\mathbb{R}_+))$.
\end{corollary}
\begin{proof}[Proof.]
What we have now is that every subsequence of $\{Q^N\}_{N\in\mathbb{N}}$ admit a limit measure $Q$ such that every $(\mu,\xi)$ realizing the law solves $\mathbb{P}-a.s.$ the weak formulation of $(2.2)$.\\
From the previous theorem we now that any process $\nu_t$ that solves the same identity is such that $\mu_t=\nu_t,\ \mathbb{P}-a.s.$.\\
So any process realizing any limiting law are $\mathbb{P}-a.s.$ equal and so they have the same law.
\end{proof}
\section{Conjecture on \abbr{PDE}-PDE Limit: Local Interaction}\label{tcseq}
Before analyzing numerically our particle system and retrieve useful insight on turbulence and coagulation formation; In this section we show formally, how we can relate the limiting equation of our particle system to the Smoluchoski Coagulation random \abbr{PDE} (1.1), objective of this paper.\\
We also conjecture the convergence of a sequence of solution $\rho_t^\delta$ of the random Smulochovski \abbr{PDE}, as $\delta\rightarrow0$, in the interaction kernel for the volume.\\%, $\mathbb{P}-a.s.$
First of all, following the result of (\cite{amann},\cite{1937-1632_2020_12_3319},\cite{article2}), we suppose that our system admits a measure valued solution with  density $\rho_t(x,v)\in C([0,T],L^2(\mathbb{R}^d\times \mathbb{R}_+))\cap L^1(\Omega,C([0,T],L^2(\mathbb{R}^d\times\mathbb{R}_+))$.
\subsection{Approximation of the Local Interaction Kernel}\label{locint}
The rescaled interaction Kernel considered for the volume exchange between particles is of the form
$$
T^{\delta}(x,v,y,w):= \delta^{-3}\mathbb{I}_{B_x(\delta f(v,w))}\ k(v,w),
$$
where we have defined:
\begin{align*}
    &B_x(\delta f(v,w)):=\{y\in\mathbb{R}^d\ :\ |x-y|_d\leq \delta f(v,w)\}, \forall\delta,v,w\in\mathbb{R};\\
    &f(v,w):=\left(v^{1/3}+w^{1/3}\right)\left(\frac{3}{4\pi}\right)^{1/3};\\
    &k(v,w)\in C^0_b,\ collision\ coefficient.
\end{align*}
Fix $d=3$, we consider now the definition of the mean field equation in weak form, when $\mu_t$ has a density $\rho_t$, for a fixed $\omega,\phi$:
\begin{align}
   \brak{\rho_t,\phi}-\brak{\rho_0,\phi}=&\int_0^t\ \brak{\rho_s,\frac{\varepsilon_x^2}{2}\Delta_x\phi}ds+ \int_0^t\ \brak{\rho_s,\sigma(x,v)\cdot\nabla_x\phi}\xi_s ds \\
   &+\int_0^t\ \brak{\rho_s,\brak{\rho_s,T^\delta J^\phi}_{y,w}}_{x,v}ds\nonumber
\end{align}
where the bracket are to be intend as a $L^2$ product. We see that the only complicated term is the one involving $T^{\delta}$
\begin{align*}
&\left(  \brak{\rho_s,T^\delta J^\phi}_{y,w}\right)  \left(  x,v\right):=\\
&=\int \delta^{-3}\mathbb{I}_{B_x(\delta f(v,w))}\ k(v,w)\left(  \int \underbrace{\left[\mathbb{I}_{(v>=w)}\phi(x,v+w)+\mathbb{I}_{v<w}\phi(y,v+w)-\phi(x,v)-\phi(y,w)\right]}_{J^\phi_{x,v,y,w}}\rho\left(  y,w\right)  dw\right)  dy\\
&=\int_{\mathbb{R}_+} \left(\delta^{-3}\int_{B_x(\delta f(v,w))}J^\phi_{x,v,y,w}\rho_t(y,w)dy\right)k(v,w)dw
\end{align*}
Heuristically, if $\rho$ is regular enough in $w$, since $\phi$ is a test function and therefor regular, we can use a Lebesgue differentiation theorem, and if we take the mean of the integral over the ball as $\delta\rightarrow0$, we have:
$$
a.e.\ x\in\mathbb{R}^d\, \left(\left(\delta f(v,w)\right)^{-3}\int_{B_x(\delta f(v,w))}J^\phi_{x,v,y,w}\rho_t(y,w)dy\right)\rightarrow J^\phi_{x,v,w}\rho_t(x,w),\ as\ \delta\rightarrow0,\ \forall\ w\in\mathbb{R}.
$$
where
$$
J^\phi(x,v,w)=\left[\phi(x,v+w)-\phi(x,v)-\phi(x,w)\right],
$$
and so the convergence of ``$T^{\delta}\ast\rho$" to the new kernel:
\begin{align*}
\left(  \brak{\rho_s,T^\delta J^\phi}_{y,w} \right)  \left(  x,v\right)    &
:=\int_\mathbb{R} \left(f(v,w)\right)^3\left(\left(\delta f(v,w)\right)^{-3}\int_{B_x(\delta f(v,w))}J^\phi_{x,v,y,w}\rho_t(y,w)dy\right)k(v,w)dw\\
&\rightarrow \int_\mathbb{R}\ f(v,w)^3\  k(v,w) J^\phi(x,v,w)\rho_t(x,w) dw,\ a.e. x\in\mathbb{R}^d,\ \forall\ v\in\mathbb{R}_+.
\end{align*}
Resulting in the Random Smoluchoski Coagulation \abbr{PDE} with a turbulence field as in definition $2.2$:
\begin{align}
   \brak{\rho_t,\phi}-\brak{\rho_0,\phi}=&\int_0^t\ \brak{\rho_s,\frac{\varepsilon_x^2}{2}\Delta_x\phi}ds+ \int_0^t\ \brak{\rho_s,\sigma(x,v)\cdot\nabla_x\phi}\xi_s ds \\
   &+\int_0^t\ \brak{\rho_s,\brak{\rho_s,k(v,w)(f(v,w))^3 J^\phi(x,v,w)}_{x,w}}_{x,v}ds\nonumber
\end{align}
\subsection{Conjecture: Converge to Smoluchoski Random Equation}
To conclude, in this section we give a brief idea of a possible convergence of sequence of solution under some additional assumption on the regularity of the random \abbr{PDE}.\\
Consider again the candidate limiting equation in weak form:
\begin{align}
   \brak{\rho_t,\phi}-\brak{\rho_0,\phi}=&\int_0^t\ \brak{\rho_s,\frac{\varepsilon_x^2}{2}\Delta_x\phi}ds+ \int_0^t\ \brak{\rho_s,\sigma(x,v)\cdot\nabla_x\phi}\xi_s ds \\
   &+\int_0^t\ \brak{\rho_s,\brak{\rho_s,k(v,w)(f(v,w))^3 J^\phi(x,v,w)}_{x,w}}_{x,v}ds\nonumber
\end{align}
And the approximate sequence of solutions $\rho_t^\delta$ that solves in weak sense:
\begin{align}
   \brak{\rho_t,\phi}-\brak{\rho_0,\phi}=&\int_0^t\ \brak{\rho_s,\frac{\varepsilon_x^2}{2}\Delta_x\phi}ds+ \int_0^t\ \brak{\rho_s,\sigma(x,v)\cdot\nabla_x\phi}\xi_s ds \\
   &+\int_0^t\ \brak{\rho_s,\brak{\rho_s,T^\delta J^\phi}_{y,w}}_{x,v}ds\nonumber
\end{align}

\begin{proposition}[Limit $\delta$]
Let $\rho_t(x,v)\in C([0,T],L^1\cap L^2(\mathbb{R}^d\times \mathbb{R}_+))\cap L^1(\Omega,C([0,T],L^1\cap L^2(\mathbb{R}^d\times\mathbb{R}_+))$ be a weak solution of $(1.1)$. Suppose that $\mu_0$ has density $\rho_0$ that satisfy hypothesis $(a,b,c)$ as in section.\ref{psm}. Suppose also that the kernel is symmetric and $K(v,w)f(v,w)^3\leq v^\alpha$, with $\alpha\leq 1$.\\
Let $\{\rho_t^\delta\}_\delta$ solutions of the equation with kernel $T^\delta$, then we have
\begin{align*}
    \sup_{t\in[0,T]}\|\rho_t-\rho_t^\delta\|_{L^2(\Omega\times(0,R))}\rightarrow 0,\ as\ \delta\rightarrow0,\ \forall\ R>0,\ \forall \Omega \subset\subset \mathbb{R},\ \mathbb{P}-a.s.
\end{align*}
\iffalse
\begin{align*}
    \sup_{t\in[0,T]}\|\rho_t-\rho_t^\delta\|_{L_x^2(\mathbb{R}^d)L^1_{v}(0,R)}\rightarrow 0,\ as\ \delta\rightarrow0,\ \forall\ R>0,\ \forall \Omega \subset\subset \mathbb{R},\ \mathbb{P}-a.s.
\end{align*}
\textbf{N.B.}\todo[]{SISTEMARE CONTO} During computation it seems that we need a bound like:
$$
\sup_t\|\rho_t^\delta\|_\infty\leq C,\ independent\ from\ \delta.
$$
But is a bit too much to ask.
\fi
\end{proposition}
\begin{proof}
Let's consider a test function $\phi$ and the difference between $\rho_t^\delta,\ \rho_t$ in the weak formulation:
\begin{align*}
\brak{\rho_t-\rho_t^\delta,\phi}&= \int_0^t\brak{\rho_t-\rho_s^\delta,\mathcal{U}_s(x,v)\cdot \nabla_x\left(\phi\right)}ds+\int_0^t\brak{\rho_t-\rho_s^\delta,\Delta_x\left(\phi\right)}ds\\
&\ \ \ +F_{x,v}^{\phi,\delta}(\rho_t^\delta)-F_{x,v}^\delta(\rho_t).
\end{align*}
where the nonlinear term are
\begin{align*}
&F_{x,v}^\phi(\rho_t)=\int_0^t\brak{\rho_s,\brak{\rho_s,T(v,w)J^{\left(e^{(t-s)A}\phi\right)}}_x}ds\\
&F_{x,v}^{\phi,\delta}(\rho_t^\delta)=\int_0^t\brak{\rho^\delta_s,\brak{\rho^\delta_s,T^\delta J_y^{\left(e^{(t-s)A}\phi\right)}}}ds.
\end{align*}
Consider now the modulus and then, term by term, $\mathbb{P}-a.s.$
\begin{align*}
    &\int_0^t|\brak{\rho_t-\rho_s^\delta,\mathcal{U}_s(x,v)\cdot \nabla_x\left(\phi\right)}|ds\leq \|\xi\|_{L^\infty}\int_0^tC^\phi_{\varepsilon,T}K_\sigma\|\rho_t-\rho_s^\delta\|^2_{2}ds\\
    &\int_0^t|\brak{\rho_t-\rho_s^\delta,\Delta_x\left(\phi\right)}|ds\leq \int_0^t C^\phi_{T}\|\rho_t-\rho_s^\delta\|^2_{2}ds
\end{align*}
while for the other terms we need to add an subtract to get

\begin{align*}
    &F_{x,v}^{\phi,\delta}(\rho_t^\delta-\rho_t)-(F_{x,v}^{\phi,\delta}-F_{x,v}^\phi)(\rho_t):=\\
    &=|\int_0^t\brak{\rho^\delta_s-\rho_s,\brak{\rho^\delta_s,T^\delta(v,w)J^{\left(\phi\right)}}}ds+\int_0^t\brak{\rho_s,\brak{\rho^\delta_s-\rho_s,T^\delta(v,w)J^{\left(\phi\right)}}}ds-(F_{x,v}^{\phi,\delta}-F_{x,v}^\phi)(\rho_t)|\\
    &\leq\left(\sup_t\|\rho_s^\delta\|_1+\sup_t\|\rho_s\|_1\right)\int_0^tC^\phi_{\Omega,R}\|\rho^\delta_s-\rho_s\|_2^2ds+|(F_{x,v}^{\phi,\delta}-F_{x,v}^\phi)(\rho_t)|
\end{align*}
For the last term we have
\begin{align*}
    &(F_{x,v}^{\phi,\delta}-F_{x,v}^\phi)(\rho_t)=\\
    &=\int_0^t|\brak{\rho_s,\brak{\rho_s,T^\delta\ J^{\left(\phi\right)}}-\brak{\rho_s,T J^{\left(\phi\right)}}_x}|ds\leq\\
    &\int_0^t|\brak{\rho_s,\int_0^R k(v,w)\left(\int_\Omega \rho_t T^\delta \tilde{J}^\phi dy-\rho_sT J^{\left(\phi\right)}\right)dw}ds\leq C^\phi\int_0^t|\brak{\rho_s,\int_0^R k(v,w)Ld^\delta(\rho_t T J^\phi)dw}ds
\end{align*}
Where
$$
Ld^\delta(\rho_tTJ^\phi):=\left(\int_\Omega \rho_t T^\delta \tilde{J}^\phi dy-\rho_sT J^{\left(\phi\right)}\right)\rightarrow0,\ in L^\infty_vL^1_x,\ as \delta\rightarrow0
$$
thanks to Lebesgue differentiation theorem and bound on the volume component of the density. Putting all together we have:
\begin{align*}
    & C^\phi\int_0^t|\brak{\rho_s,\int_0^R k(v,w)Ld^\delta(\rho_t T J^\phi)dw}ds\\
    &\leq \int_0^tc^\phi M_1(v\rho_t)\sup_{w\in(0,R)}\|Ld^\delta(\rho_tT)\|\leq c^\phi M_1(v\rho_0)\sup_{w\in(0,R)}\|Ld^\delta(\rho_tT)\|ds:=\int_0^tA_\delta C^\phi_{R,\rho_0,\Omega}ds
\end{align*}
where the last inequality follows from the bound on the microscopic kernel $k$. If we put all the estimate together, taking the sup over all test functions with $L^2$ norm equal 1, we obtain with Gronwall the following
$$
\sup_t\|\rho_t^\delta-\rho_t\|_2^2\leq \left(\sup_t (A_\delta) T\right)\exp(C^\phi_{R,\Omega,\rho_0})\rightarrow0,\ as\ \delta\rightarrow0.
$$
With this we conclude the proof.
\end{proof}
\iffalse
Let's consider a test function $\phi$ and the heat semigroup $e^{tA},\ A=\frac{\varepsilon^2}{2}\Delta$, and compute the weak formulation of the equation we get:
\begin{align*}
\brak{\rho_t-\rho_t^\delta,\phi}&= \int_0^t\brak{\rho_t-\rho_s^\delta,\mathcal{U}_s(x,v)\cdot \nabla_x\left(e^{(t-s)A}\phi\right)}ds\\
&+\int_0^t\brak{\rho_s,\brak{\rho_s,T(v,w)J^{\left(e^{(t-s)A}\phi\right)}}_x}ds-\int_0^t\brak{\rho^\delta_s,\brak{\rho^\delta_s,T^\delta J_y^{\left(e^{(t-s)A}\phi\right)}}_y}ds.
\end{align*}
Consider now the modulus and then, term by term, $\mathbb{P}-a.s.$
\begin{align*}
    \int_0^t|\brak{\rho_t-\rho_s^\delta,\mathcal{U}_s(x,v)\cdot \nabla_x\left(e^{(t-s)A}\phi\right)}|ds\leq \|\xi\|_{L^\infty}\int_0^t\frac{C_{\varepsilon,T}K_\sigma}{\sqrt{t-s}}\|\rho_t-\rho_s^\delta\|_{L^1\cap L^2}ds
\end{align*}
while for the other terms we need to add an subtract to get
\begin{align*}
    &|\int_0^t\brak{\rho_s,\brak{\rho_s,T(v,w)J^{\left(e^{(t-s)A}\phi\right)}}_x}ds-\int_0^t\brak{\rho^\delta_s,\brak{\rho^\delta_s,T^\delta J^{\left(e^{(t-s)A}\phi\right)}}_y}ds|=\\
    &\leq|\int_0^t\brak{\rho_s-\rho_s^\delta,\brak{\rho_s,T(v,w)J^{\left(e^{(t-s)A}\phi\right)}}_x}ds|+|\int_0^t\brak{\rho^\delta_s,\brak{\rho_s,T\ J^{\left(e^{(t-s)A}\phi\right)}}_x-\brak{\rho_s,T^\delta J^{\left(e^{(t-s)A}\phi\right)}}_y}ds|+\\
    &+|\int_0^t\brak{\rho^\delta_s,\brak{\rho_s,T^\delta\ J^{\left(e^{(t-s)A}\phi\right)}}_y-\brak{\rho^\delta_s,T^\delta J^{\left(e^{(t-s)A}\phi\right)}}_y}ds|
\end{align*}
and estimate each different quantities:
\begin{align*}
   |\int_0^t\brak{\rho_s-\rho_s^\delta,\brak{\rho_s,T(v,w)J^{\left(e^{(t-s)A}\phi\right)}}_x}ds|\leq \int_0^t \brak{\|\rho_s-\rho_s^\delta\|_2, ||M_1(\rho_s)||_2}ds
\end{align*}

Similarly, under the (\textbf{N.B.}) assumption:
\begin{align*}
    \int_0^t|\brak{\rho^\delta_s,\brak{\rho_s-\rho_s^\delta,T^\delta\ J^{\left(e^{(t-s)A}\phi\right)}}_y}|ds&\leq
    C_\phi\int_0^t|\brak{\rho^\delta_s,\brak{\rho_s-\rho_s^\delta,T^\delta }_y}|ds\\
    &\leq C_\phi\int_0^t|\brak{\rho^\delta_s, v}|\brak{|\rho_s-\rho_s^\delta|,\mathbb{I}^\delta }ds\\
    &\leq C_\phi\int_0^t|\brak{\rho^\delta_s, v}|ds\sup_t\{\brak{|\rho_s-\rho_s^\delta|,\mathbb{I}^\delta }\}\\
    &\leq C_\phi M_T(\rho_0)\sup_t\{\brak{|\rho_s-\rho_s^\delta|,\mathbb{I}^\delta }\}
\end{align*}
For the last term we have
\begin{align*}
    &\int_0^t|\brak{\rho^\delta_s,\brak{\rho_s,T^\delta\ J^{\left(e^{(t-s)A}\phi\right)}}_y-\brak{\rho^\delta_s,T^\delta J^{\left(e^{(t-s)A}\phi\right)}}_y}|ds\leq\\
    &\leq\int_0^t\sup_t\|\rho^\delta_t\|_\infty\ C \int_\mathbb{R}|f(v,w)^3\left(J_x^\phi\rho_t(x,w)-|(B_x(\delta f(v,w))|\int_{B_x(\delta f(v,w)}J_{y,x}^\phi \rho_t(y,w)dy\right)||K(v,w)| dwds\\
    &\leq \int_0^t\mathbb{C}(\rho_T^\delta,T,k)\|\rho_sJ_x^\phi-\vartheta_\delta\ast J_{y,x}\rho^y_s\|_1ds.
\end{align*}
Thus taking the sup over $\phi\in C^{\infty}$ under the $L^2\cap L^1$ norm,using the Lebesgue differentiation theorem and Gronwall lemma, we conclude the proof.
\fi

\iffalse
\section{Appendix: Idea in the case of Environmental Noise}
Let's consider the same system as before, but now the particle are under the action of a common noise of the form $\sum_{i=1}^K \int_0^t\sigma^i(x,v)dW^i_s$. We select $\sigma$ uniformly bounded and such that $\sigma\sigma^T=\kappa Id$. So we can define the infinitesimal generator:
$$
\mathcal{L}\brak{\mu_t^N,\phi}= \brak{\mu_t^N,(\varepsilon +\kappa)\Delta\phi}+\brak{\mu_t^N,\brak{\mu_t^N, T^\delta J^\phi_{x,v,y,w}}}
$$
where in general the contribution of the common noise should be $\frac{1}{2}Tr\left( \sigma\sigma^T\ Hess(\phi)\right)$. Against this kind of test function we can compute with Ito formula:
\begin{align*}
    \brak{\mu_t^N,\phi}=&\brak{\mu_0^N,\phi}+\int_0^t\brak{\mu_s^N,(\varepsilon +\kappa)\Delta\phi}ds+\int_0^t \brak{\mu_s^N,\sigma(x,v)\cdot\nabla_x\phi}dW_s +\\
    &+\int_0^t \brak{\mu_t^N,\brak{\mu_t^N,T^\delta(x,v,y,w)J^\phi(x,v,y,w)}}ds+\tilde{M}_t
\end{align*}
where
\begin{align*}
&T^\delta(x,v,y,w):=\frac{1}{2}\delta^{-3}\mathbb{I}_{B_{x}(\delta f(v,w))}(y)k(v,w);\\
&J^\phi(x,v,y,w):=\left[\mathbb{I}_{(v>=w)}\phi(x,v+w)+\mathbb{I}_{v<w}\phi(y,v+w)-\phi(x,v)-\phi(y,w)\right]
\end{align*}
$\tilde{M}_t$ is a martingale and we can compute is quadratic variation:
$$
\mathbb{E}\left[\tilde{M}_t^2\right]=\mathbb{E}\left[\int_0^t\left(\mathcal{L}\brak{\mu_s^N,\phi}^2-2\brak{\mu_s^N,\phi}\mathcal{L}\brak{\mu_s^N,\phi}ds \right)\right]
$$
which we can compute in two parts $M_D,\ M_C$ such that:
\begin{align*}
& \brak{M_D}:=\mathbb{E}\left[\int_0^t N^{-1}\brak{\mu_t^N,|\nabla\phi|^2}ds\right]\\
& \brak{M_C}:=\mathbb{E}\left[\int_0^t \frac{1}{N^3}\sum_{i,j}T_t^{\delta}(i,j)J_t^{\phi}(i,j)^2ds\right]=\mathbb{E}\left[\int_0^t N^{-1}\brak{\mu_s^N,\brak{\mu_s^N, T^\delta(x,v,y,w)J^\phi(x,v,y,w)^2}}ds\right]
\end{align*}
Roughly we can do the same computation as before in the mean field case, the only thing we need to check carefully is, taking the law of $(\mu,W)$ limit of our sequence $(\mu^{N_k},W)$ in law, then we have the convergence of the ito integral of the common noise in the limiting equation.\\
\\
If the detail can be filled correctly out, than we should focus on the \abbr{SPDE}-to-SPDE limit in $\delta$ giving us a Smoluchovski diffusion \abbr{SPDE} with Environmental Noise.\\
\\
Notare che nel kernel ci vuole funzione $E(v,w)$ tale che $E(v,w)=\mathcal{E}(\max(\frac{v}{w}, \frac{w}{v}))$ con $\mathcal{E}(x)\rightarrow0$ as $x\rightarrow0$ faster that $(v+w)$ goes to infinity.
\fi
\section{Numerical Simulation: First Time Rain Generation}\label{numan}
We conclude this work with a numerical simulation, with a subsequent analysis, of the particle system in two different setting to better understand the role of turbulence in the generation of rain inside a cloud formation.\\
In detail, what we are interested in the study of the first formation time
$$\tau_f:=\inf\{t\geq 0 | \exists i\in N_t,\  R^i_t>=R_{rd}\},$$
where $R_{rd}:=0.0004$ is the radius of the typical raindrop.\\
What we obtain is a decay dependence of the aforementioned exit time in different setting: a pure Brownian case, and a random field driven particles.\\
\\
In the general setting we consider the following positional equation for the particle system
\begin{align}
    &dX^i_t=\varepsilon_{rf}\sum_{k\in K}\sigma_k(X_t^i)\xi_t^k dt-\hat{e}_2f(V_t^i)dt+\varepsilon_{bm} \sigma dB_t^i,
\end{align}
for $i=1,...,N$, which we have simulated with an Eulero-Maruyama scheme.\\
$\varepsilon_{rf},\varepsilon_{bm}\in\{0,1\}$ select the setting in which we investigate the formation time $\tau_f$.\\
The domain in which we have done our simulation is $[-2,2]^2$ (that is intended to be measured in $cm$) with periodic condition on the border, so that it represent a zoom inside the cloud and we assume that as the particles move they are replaced with probability one with particles of almost the same size.\\
Concerning the velocity random field in which the particles moves, it is defined as the stochastic process $\mathcal{U}\left(  t,x\right)  $%
\[
\mathcal{U}\left(  t,x\right)  =\sum_{k\in K}\sigma_{k}\left(  x\right)  \xi_{t}^{k}%
\]%
\[
\sigma_{k}\left(  x\right)  =\frac{1}{2\pi}\frac{\left(  x-x_{k}\right)
^{\perp}}{\left\vert x-x_{k}\right\vert ^{2}},\ \left(  a,b\right)  ^{\perp}=\left(  b,-a\right)%
\]%
where points $x_{k}$ are fixed and selected with a uniform distribution over the considered domain, representing vortex generated by the turbulent flow in which the droplet are submerged. We have used the notation such that $\sigma_k(x_k)=0$.\\
The real-valued stochastic processes $\xi_t^k$ are Ornstein-Uhlenbech that satisfy:
\[
d\xi_{t}^{k}  =-\lambda\xi_{t}^{k}dt+\lambda dB_{t}^{k},\ \ \
\xi_{0}^{k}  =0,\ k  =1,...,K.\\
\]%
which we have also simulated with an Eulero-Murayama scheme and used the result to compute $(4.1)$.\\
The initial condition of the particle, $X_0^i$, are sampled from a uniform distribution over the domain, while the initial volumes $V_0^i>0$ are selected uniformly in the range of a large raindroplet ($\sim 10^3m)$.\\
We specify also that we assume the raindrops are spherical and as such the volume can be represented as
$$
\forall i=1,...,N,\ V_t^i\sim\frac{4}{3}\pi(R_t^i)^3.
$$
While the function $f$ representing the limit velocity induced by the mass of the raindrop was simulated with a standard logistic function such that the limit is exactly the terminal velocity as in $[\cdot]$.
\subsubsection{Stochastic coalescence:}
Two real-droplets may collide and coalesce completely into one big real-droplet and this process is responsible for precipitation development.\\
Our primary postulate, which is commonly used in many previous studies, is that the coalescence process can be described in a probabilistic way.\\
So, what we expect is that our $V_t^{i,N}$, of the $i-th$ particle, changes not in a continuous way, but with jumps proportional to the particle that $i$ interact with.\\
With this we mean that, when two particle interact with each other there is a chance that the volume of the $i-th$ particle absorb or get absorbed by the other particle. So that the volume of a particle can increase with a certain probability, only when collide with other rain droplets.\\
\\
For the sake of
completeness we need to define a $P_{mean}$
that represent a threshold that encapsulate the possibility that two rain droplets can interact or bounce from each other, and if extracting a random number we surpass this threshold that coalescence can happen, otherwise the particles get along with their motion.\\
To state it clearly this is how our coalescence works:\\
\paragraph{\textbf{Algorithm for stochastic coalescence}}
\begin{enumerate}
    \item Let $x_i,\ i=1,...,N$ the particles of the system, and denote $R_i$ their radius;
    \item For every couple $(i,j)$  such that $|x_i-x_j|\leq R_i+R_j$, we generate a number $\phi_i^j\in (0,1)$ with a uniform distribution;
    \item If $\phi_i^j>P_{mean}$, then the coalescence happens in the following way:
    \begin{itemize}
        \item If $V_t^{i,N}>=V_t^{j,N}$ then
        \begin{itemize}
            \item $V_{t}^{i,N}= V_t^{i,N} + V_t^{j,N}$;
            \item $V_{t}^{j,N}=0$ and the particle $j$ is removed from the system.
        \end{itemize}
        \item Vice versa if  $V_t^{j,N}>V_t^{i,N}$;
    \end{itemize}
    \item If $\phi_i^j<P_{mean}$, then the raindroplet bounced of from each other.
\end{enumerate}
\subsection{Pure Brownian Movement}
In this small section we analyze the particle system under the action of indendent brownian motion attachted to the droplets.\\
We set $\varepsilon_{rf}=0,\varepsilon_{bm}=1$ in $(4.1)$ and we reduce the model to the motion equation given by:
\begin{align}
    &dX^i_t=-\hat{e}_2f(V_t^i)dt+\sigma dB_t^i,
\end{align}
where $X_0^i$ are drawn from a uniform distribution over the domain $[-2,2]^2$.\\
The radius of the particles are selected in such a way that a raindrop is created as soon as the value it the threshold
$$
R^i_t\sim 0.004 m
$$
and the radius $R_0^i$ of the starting rain droplets  are drawn from a uniform distribution between values $10^{-1},\ 10^{-2}$ smaller than a real raindrop.\\
Finding the right time step was challenging, since the position of the particle is discrete and as such we could lose interaction between particle in the motion of the rain droplets, causing numerical error and inconsistency in the behaviour of the formation time.\\
While the code can run on a large number of particle, to analyze efficiently the result we set our parameters as follows:
$$
dt=0.0001,\ \ N=1000,\ \ T_{MaxIt}=3000,
$$
where $N$ represents the initial number of particle and $T_{MaxIt}$ the maximal iteration of the system.\\
To simplify the analysis we postpone the investigation of the coefficient $P_{mean}$ assuming it is independent from the fact that the motion is subjected under the action of a random field or independent Brownian Motion.\\
In such a way we can assume without loss of generality that $P_{mean}=0$, and the coalescence algorithm is as follows:
\begin{enumerate}
    \item Let $x_i,\ i=1,...,N$ the particles of the system, and denote $R_i$ their radius;
    \item For every couple $(i,j)$  such that $|x_i-x_j|\leq R_i+R_j$:
    \begin{itemize}
        \item If $V_t^{i,N}>=V_t^{j,N}$ then
        \begin{itemize}
            \item $V_{t}^{i,N}= V_t^{i,N} + V_t^{j,N}$;
            \item $V_{t}^{j,N}=0$ and the particle $j$ is removed from the system.
        \end{itemize}
        \item Vice versa if  $V_t^{j,N}>V_t^{i,N}$;
    \end{itemize}
\end{enumerate}
We wan to analyze the formation time $\tau_{f}$ dependent on the intensity $\sigma$ of the independent Brownian motions.\\
To do so, for each parameter, we have simulated ten times the particle system, collecting the time formation in a time series from which we extract the mean time. In detail we have, calling $N_r$ the number of repetition, $T^i_f$ the formation time of the $i-th$ simulation, the following:
$$
N_r=10,\ \ T_{mean}=\mathbb{E}[T_f]\sim\frac{1}{N_r}\sum_{i=1}^{N_{r}} T_f^i, \forall \sigma \in \Sigma.
$$
Where $\Sigma:=\{0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9,1\}$, is a partition of the reasonable interval, $[0,1]$, for the intensity of a realistic Brownian motion.
\subsubsection{A simplified theoretical reasoning}
Before showing the numerical result, we show a theoretical reasoning on a simplified model that can explain the expected result in such a model.\\
Let us consider two particles moving subjected to independent Brownian motion with different initial condition: $\sigma W^1_t+x_1$, $\sigma W_t^2+x_2$ on the unitary Torus.\\
Call $x_0 = x_1-x_2$ and define the random time
\[
\tau_{\sigma,x_0}:=\inf\{t\geq0: |\sigma W_t^1-\sigma W_t^2 +x_0|\leq \varepsilon\}
\]
which is linked to our formation time, since coalescence happens when two raindrops are near depending on a setted threshold.\\
Then we ask the question: how $\sigma$ impact the collision time of particles?\\
This quantity cannot be independent from $\sigma$, indeed call $B_t = W_t^1- W^2_t$, this is a new Brownian motion, with intensity dependent from the dimension (but fixed), so without loss of generality we fixed it at 1.\\
Thus we consider the quantity $\tau_{\sigma,x_0}$:
\begin{align*}
\tau_{\sigma,x_0}:&=\inf\{t\geq0: |\sigma B_t +x_0|\leq \varepsilon\}\\
&=\inf\{t\geq0: |\sigma B_{\sigma^2 t/\sigma^2} +x_0|\leq \varepsilon\}\\
&=\inf\{t\geq0: |W_{\sigma^2t} +x_0|\leq \varepsilon\}\\
&=\tau_{1,x_0}/\sigma^2.
\end{align*}
Here we have used the auto-similarity of the Brownian motion.\\
What we recover is that there is an inverse quadratic dependence of the formation time with the intensity of the Brownian motion moving the particles. That we can express in a sintetic form as
$$
\mathbb{E}\left[\tau_{\sigma,x_0}\right]\sim \sigma^{-2}.
$$
\subsubsection{Results on Regression}
Let's start showing the result table of the mean formation time and the standard deviation, in dependence of Brownian intensity.\\ (Note that to obtain a time measure we need to multiply the epoch with the step time $dt$).\\
Following the previous reasoning we expect that our formation time depend with a square inverse power with the intensity of the diffusion
$$
\mathbb{E}\left[\tau_{f}(\sigma)\right]\sim \sigma^{-2},
$$
where we have highlighted the dependence on $\sigma$.\\
\begin{center}
\begin{tabular}{ |c|c|c| }
 \hline
 Brownian intensity &  Mean time & Standard Deviation \\
 1 & 636 & 114.9043 \\
 0.9 & 641.4 & 119.8025\\
 0.8 & 691.5 & 93.29014 \\
 0.7 & 734.9 & 170.0514\\
 0.6 & 808 & 165.7607 \\
 0.5 & 917.6 & 234.5115 \\
 0.4 & 1086.9 & 312.3816 \\
 0.3 &  1569.8 & 260.9294\\
 0.2 &  2896.5 &  387.9338;\\
 0.1 &  4567 & 434.2\\
 \hline
\end{tabular}
\end{center
\begin{figure}[!tbp]
  \centering
  \begin{minipage}[b]{0.7\textwidth}
\includegraphics[width=0.7\textwidth]{SoloBrown1.png}
  \end{minipage}
\hfill
  \begin{minipage}[b]{0.7\textwidth}
    \includegraphics[width=0.7\textwidth]{QuadraticSoloBrown1.png}
  \end{minipage}
  \hfill
  \begin{minipage}[b]{0.7\textwidth}
\includegraphics[width=0.7\textwidth]{LogSoloBrown1.png}
  \end{minipage}
  \caption{From top to bottom: (a) Plot of Brownian Intensity/Formation time that shows the time decay in dependence with increase in $\sigma$, intensity; (b) Linear-log plot: quadratic regression, in red the interpolated numerical data and in blue the regression curve;(c) log-log plot: logarithmic regression, in red the interpolated numerical data and in blue the regression curve.}
  \label{f1}
\end{figure}
As we can see from the table, there is a significance decay in the mean time as the intensity grows, and, as expected, the standard deviation became smaller.\\
As $\sigma$ grows the initial condition of the system became less and less important for the mean formation time, as we can see as the standard deviation decrease as $sigma$ increases.\\
To understand the decay we proceed our analysis with a quadratic and logarithmic regression on the model.\\
In $Figure\ 1$ we can see plotted, $(a)$, the decay of mean formation time that shows the time decay in dependence with the increase in $\sigma$, intensity of the Brownian motions. Again we can appreciate the stabilization of the randomness in the standard deviation, plotted as black bar in figure.\\
\\
First we performed a quadratic regression, for which we show the results in fig.\ref{f1}(b) as a Lin-Log plot. We can already notice how the behavior of the numerical data, interpolated in red, fit the regression line in blue and the most significant error are in the very small regime of $\sigma\sim 0.1$, since the particle are almost still in their motion and the dependence from the randomly generated data is stronger.\\
The regression show a residual plot without any particular structure, with a residual standard error equal to 0.5232 and the R-square equal to 0.992 and a summary that agrees with our conjecture on the quadratic decreasing of time respect to the intensity of the noise.\\
\\
\textbf{Quadratic Regression Summary:}
\begin{verbatim}
Residual standard error: 0.5232 on 7 degrees of freedom
Multiple R-squared:  0.992,	Adjusted R-squared:  0.9897
F-statistic: 431.5 on 2 and 7 DF,  p-value: 4.673e-08.
\end{verbatim}

To validate our results we concluded the analysis with a logarithmic regression to capture lower order power and fluctuation in the intensity dependence of the exit time. The results in fig.\ref{f1}(c) as a Log-Log plot confirm how the behavior of the numerical data, interpolated in red, fit the regression line, in blue, in accordance to the quadratic regression. Small fluctuations around the regression line can be appreciated, showing lower order dependencies on the intensity, but the residual plot showing no structures with a standard error of 0.1 and a R-squared equal to 0.979, agrees with the hypothesis $\mathbb{E}[\tau_f^\sigma]\sim\sigma^{-2}$.\\
\\
\textbf{Logarithmic Regression Summary:}
\begin{verbatim}
Residual standard error: 0.1061 on 8 degrees of freedom
Multiple R-squared:  0.979,	Adjusted R-squared:  0.9764
F-statistic: 373.6 on 1 and 8 DF,  p-value: 5.329e-08
\end{verbatim}
\iffalse
\begin{Nota}Qui le regressioni in accordo:\\
\textbf{Regressione logaritmica:}
\begin{verbatim}
Call:
lm(formula = log(reg) ~ log(sigma), data = data)

Residuals:
     Min       1Q   Median       3Q      Max
-0.18634 -0.06982  0.01082  0.06145  0.14747

Coefficients:
            Estimate Std. Error t value Pr(>|t|)
(Intercept)  2.92623    0.05086   57.54 9.24e-12 ***
log(sigma)   0.93251    0.04825   19.33 5.33e-08 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 0.1061 on 8 degrees of freedom
Multiple R-squared:  0.979,	Adjusted R-squared:  0.9764
F-statistic: 373.6 on 1 and 8 DF,  p-value: 5.329e-08
\end{verbatim}
\iffalse
Call:
lm(formula = reg ~ sigma + sigma2, data = data)

Residuals:
     Min       1Q   Median       3Q      Max
-0.82114 -0.33775  0.04893  0.40605  0.54079

Coefficients:
            Estimate Std. Error t value Pr(>|t|)
(Intercept)  -0.9470     0.6154  -1.539  0.16774
sigma        28.1888     2.5701  10.968 1.16e-05 ***
sigma2      -10.4304     2.2770  -4.581  0.00254 **
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
\fi
\end{Nota}
\fi
\iffalse
\subsection{PDE deterministica}
Una piccola digressione sulla \abbr{PDE} deterministica.\\
Possiamo solo vagamente immaginare come sarà la \abbr{PDE} nel caso in cui il campo aleatorio sia assente (dunque solo browniani nelle particelle), ma ci sia presenza di coalescenza totale: in questo caso infatti le nostre particelle non avrebbero in $V_t^i$ una formulazione continua, ma sarebbero dei processi stocastici con salti, probabilmente dettati da una qualche sorta di poisson non omogeneo in cui compaglia la codifica del volume stesso e della posizione.\\

\textbf{Pertanto possiamo fare qualche ipotesi sul modello continuo}\\
\textbf{1.} Consideriamo l'equazione candidata limite nel nostro caso, assumiamo che $\sigma_v=\sigma_x$, si ha:
\begin{align}
&\partial_{t}\rho +\operatorname{div}_{v}\left(  \rho G\ast\rho\right) =\frac{\sigma^{2}}{2}\Delta\rho, \ \rho=\rho_t(x,v)\\
&\left(  G^{\left(  \delta\right)  }\ast\rho\right)  \left(  x,v\right):=\int_R\ f(v,w)^d\ vw\ \overline{sgn}(v-w) \rho_t(x,w) dw.\nonumber
\end{align}
Consideriamo il cambio di variabili $\tilde{\rho}(t,x,v)=\rho(t/\sigma^2,x,v)$, si ha:
\begin{align*}
    &\partial_{t}\tilde{\rho}_t(x,v)=\frac{1}{\sigma^2}\partial_s\rho(s,x,v)|_{s=t/\sigma^2}= \frac{1}{2}\Delta\rho(t/\sigma^2,x,v)-\frac{1}{\sigma^2}\operatorname{div}_{v}\left(  \rho(t/\sigma^2,x,v) G\ast\rho\right) ,\\
    &\left(  G^{\left(  \delta\right)  }\ast\rho\right)  \left( t/\sigma^2, x,v\right):=\int_R\ f(v,w)^d\ vw\ \overline{sgn}(v-w) \rho(t/\sigma^2,x,w) dw.\nonumber
\end{align*}
Cioè, otteniamo l'equazioni
$$
\partial_{t}\tilde{\rho}_t(x,v)=\frac{1}{2}\Delta\tilde{\rho}_t(x,v)-\frac{1}{\sigma^2}\operatorname{div}_{v}\left(  \tilde{\rho} G\ast\tilde{\rho}\right).
$$
Dunque,supponendo questa come equazione limite del modello continuo, quando $\sigma\rightarrow\infty$, si ottiene una somiglianza con l'equazione del calore
$$
\partial_{t}\tilde{\rho}_t(x,v)\sim\frac{1}{2}\Delta\tilde{\rho}_t(x,v).
$$
E dunque si dovrebbe verificare una accelerazione che va come $C/\sigma^2$, circa.\\
\\
\textbf{2.} Una cosa che si potrebbe provare a pensare è la seguente:\\
Quando analizziamo l'equazione delle particelle, anche aiutati dalle analisi numeriche che avevamo fatto all'inizio di questo percorso, potevamo introdurre una "costante di assorbimento", $M\in\mathbb{R}^+$, al cui crescere implicava un maggior trasferimento di massa, cioè:
$$
dV_{t}^{i}   =M\sum_{j\neq i}G\left(
X_{t}^{i},V_{t}^{i},X_{t}^{j},V_{t}^{j}\right)  dt +\sigma d\tilde{B}_t^i
$$
E si poteva "congetturare numericamente" che, per $M\rightarrow\infty$ il trasferimento di massa fosse istantaneo. In questo caso l'equazione limite è la stessa con $G_M\ast\rho=M*(G\ast\rho)$.\\
Ripetendo il cambio di variabili si otterrebbe:
$$
\partial_{t}\tilde{\rho}_t(x,v)=\frac{1}{2}\Delta\tilde{\rho}_t(x,v)-\frac{M}{\sigma^2}\operatorname{div}_{v}\left(  \tilde{\rho} G\ast\tilde{\rho}\right).
$$
Ed in questo caso per capire il comportamento dell'equazione bisognerebbe capire
$$
\lim_{M\rightarrow\infty,\ \sigma\rightarrow\infty}\left(\frac{M}{\sigma^2}\right).
$$
In ogni caso il problema di capire qualche risultato numerico dal modello di \abbr{PDE} teorico si fa molto complesso a causa della differenza di coalescenza: in uno continua e nell'altro completamente stocastica.
\fi
\iffalse
\subsection{Comparison between pure noise and random field}
In this section we show result for the time of the first raindrop formation.\\

In questo momento, come eravamo rimasti in accordo ho concentrato gli sforzi sull'analisi del tempo di formazione della prima goccia.\\
Mi sono ridotto, scalando il volume, a 1000 particelle (in questo modo ogni simulazione singola ha bisogno di una 20ina di secondimassimo).\\
Da qui sto provando a controllare la media dei tempi di prima formazione sotto diverse ipotesi, ad esempio:
\begin{itemize}
    \item Solo moti browniani con parametri piccolissimo o molto grandi (da $0.00001$ a $10^9$);
    \item Fissare lambda del processo di O-U a livelli ragionevoli (scelto tramite altre prove precedenti) e analizzato la media dei tempi di caduta su 500 iterazioni al variare del numero di vortici presenti;
    \item Analisi, con numero di vortici ottimizzato, di diversi lambda e della media dei tempi di prima caduta.
\end{itemize}
Giusto per fare qualche esempio e chiarire alcune cose che si possono notare dagli esperimenti.
Prendiamo per esempio l'analisi eseguita con l'intensita del processo di O.-U. fissata ($\lambda=150$) e il numero di vortici variabili.\\
Come si può notare dalla figura, che rappresenta l'andamento della media del tempo di prima caduta (ognuna calcolata su 500 iterazioni, con condizioni iniziali randomizzate) al variare del numero di vortici presenti nel dominio, l'aumentare dei vortici, che in un certo senso aumente la sparsità delle particelle produce tempi via via più lenti non appena si supera una soglia ben precisa (che nel nostro caso è tra i 25 e i 30 vortici).\\
\includegraphics[width=\textwidth]{vortexmeantime.png}
In un qualche senso questo sembrerebbe collegato ai risultati che ho ottenuto analizzando il problema solo con i moti browniani, senza la presenza del campo aleatorio.\\
In questo caso (solo browniano) abbiamo ottenuto un tempo di formazione della prima goccia molto più elevato
\begin{center}
\begin{tabular}{ |c|c|c| }
 \hline
 $\sigma$, Brownian Intensity & Mean time first raindrop formation  \\
 0.00001 & 5590  \\
 0.1 & 5500\\
 0.3 & 5433\\
 0.4 & 5230\\
 0.6 & 5199\\
 150 &  5037\\
 10000 &  4434 \\
 $10^9$ & 4200\\
 \hline
\end{tabular}
\end{center}
In particolare sembrerebbe che il problema giace nel fatto che le gocce, cose che accade quantomento guardando le simulazioni di turbolenza al computer nei vari paper, tendono a formare spostaneamente dei cluster sospinte dal fluido in cui sono immerse, cosa che invece non accade quando il moto è solo browniano e, nonstante le particelle si muovano randomicamente ovunque le collisioni sembrerebbero più distribuite e quindi il tempo effettivo affinchè una goccia si formi aumenta.\\
Questo fenomeno accade, anche se con tempi minori nel caso di $\lambda$ troppo grandi ($\lambda$ prima dell'istabilità numerica), ad esempio:
\begin{center}
\begin{tabular}{ |c|c|c| }
 \hline
 $\lambda$, RF intensity & best numebr of vortex & Mean time first raindrop formation  \\
 100 & 35 &  59.082\\
 150 & 25 & 36.662\\
 200 & 25 & 91.166\\
 300 & NA & INSTABILE\\
 \hline
\end{tabular}
\end{center}
Quello che si può riscontrare a livello qualitativo è che, dopo le prime iterazioni i cluster con $\lambda$ troppo grande si dissolvono e il movimento delle particelle somiglia sempre più a quello di un movimento browniano ed il tempo di caduta aumenta in base a quanto poco sono durati i cluster.\\
In figura riporto due immagini qualitative, la prima figura con dinamica solo brownianan e la seconda figura con $\lambda$ molto elevato (250), si può notare come apprezzabilmente le due dinamiche siano molto simili, mentre (terza figura) il caso con $\lambda=150$ mostra cluster che simuovono all'unisono aumentando la velocità di caduta.\\
\begin{figure}[h]
\caption{Brownian motion $\sigma=10^9$, $\lambda=0$}
\centering
\includegraphics[width=\textwidth]{browniano.png}
\end{figure}
\begin{figure}[h]
\caption{NO Brownian motion $\sigma\sim 0$, $\lambda=250$}
\centering
\includegraphics[width=\textwidth]{lambda250.png}
\end{figure}
\begin{figure}[h]
\caption{No Brownian motion $\sigma\sim 0$, $\lambda=150$}
\centering
\includegraphics[width=\textwidth]{lambda150.png}
\end{figure}
In questa tabella abbiamo iniziato a raccogliere i primi risultati in cui
\begin{center}
\begin{tabular}{ |c|c|c|c| }
 \hline
 $\lambda$, RF intensity & Brwonian intesity & best numbers of vortex & Mean time first raindrop formation  \\
 150 & 0 & 25 & 40.561\\
 150 & $10^{-6}$ & 25 & media=36.662, sqm=13.17\\
 150 & 0.1  & 25 & media=149,36, sqm=83.127\\
 150 & 0.2 & 25 & media=227,497, sqm=129.495\\
 \hline
\end{tabular}
\end{center}
In tutti gli esperimenti una cosa che ho notato molto è la varianza campionaria molto grande nelle varie serie temporali. Credo che questo sia dovuto alla dipendenza dalla condizioni iniziali, dal numero di sample non grandissimo (500 per ogni serie temporale) e dalle approssimazioni numeriche dovute alle operazioni.
\fi

\subsection{Random Field Movement}
In this section we are going to analyze the particle system under the action of the Velocity Random Field.\\
Here we present a brief numerical study for the particle system under theoretical study, to motivate future work on the stochastic coalescence and turbulence formation in this interacting system.\\
We considered the following system for the position of the particles
\begin{align}
    &dX^i_t=\sum_{k\in K}\sigma_k(X_t^i)\xi_t^k dt-\hat{e}_2f(V_t^i)dt+\varepsilon dB_t^i
\end{align}
for $i=1,...,N$, which we have simulated with an Eulero-Maruyama scheme.\\
The domain in which we have done our simulation is $[-2,2]^2$ (that is intended to be measured in $cm$) with periodic condition on the border, so that it represent a zoom inside the cloud and we assume that since the particle moves they are replaced with probability one with particles of almost the same size.\\
Concerning the velocity random field in which the particles moves, it is defined as the stochastic process $\mathcal{U}\left(  t,x\right)  $%
\[
\mathcal{U}\left(  t,x\right)  =\sum_{k\in K}\sigma_{k}\left(  x\right)  \xi_{t}^{k}%
\]%
\[
\sigma_{k}\left(  x\right)  =\frac{1}{2\pi}\frac{\left(  x-x_{k}\right)
^{\perp}}{\left\vert x-x_{k}\right\vert ^{2}},\ \left(  a,b\right)  ^{\perp}=\left(  b,-a\right)%
\]%
where points $x_{k}$ are fixed and selected with a uniform distribution over the considered domain. We have used the notation such that $\sigma_k(x_k)=0$.\\
The real-valued stochastic processes $\xi_t^k$ are Ornstein-Uhlenbech that satisfy:
\[
d\xi_{t}^{k}  =-\lambda\xi_{t}^{k}dt+\lambda dB_{t}^{k},\ \ \
\xi_{0}^{k}  =0,\ k  =1,...,K.\\
\]%
which we have also simulated with an Eulero-Murayama scheme and used the result to compute $(5.1)$.\\
The initial condition of the particle, $X_0^i$, are sampled from a uniform distribution over the domain, while the initial volumes $V_0^i>0$ are selected uniformly in the range of a large raindroplet ($\sim 10^3m)$.\\
We specify also that we assume the raindrop are spherical and as such the volume can be represented as
$$
\forall i=1,...,N,\ V_t^i\sim\frac{4}{3}\pi(R_t^i)^3.
$$
While the function $f$ representing the limit velocity induced by the mass of the raindrop was simulated with a standard logistic function such that the limit is exactly the terminal velocity as in $[\cdot]$.
\subsubsection{Stochastic coalescence:}
Two real-droplets may collide and coalesce completely into one big real-droplet and this process is responsible for precipitation development.\\
Our primary postulate, which is commonly used in many previous studies, is that the coalescence process can be described in a probabilistic way.\\
So, what we expect is that our $V_t^{i,N}$, of the $i-th$ particle, changes not in a continuous way, but with jumps proportional to the particle that $i$ interact with.\\
With this we mean that, when two particle interact with each other there is a chance that the volume of the $i-th$ particle absorb or get absorbed by the other particle. So that the volume of a particle can increase with a certain probability, only when collide with other rain droplets.\\
\\
For the sake of
completeness we need to define a $P_{mean}
that represent a threshold that encapsulate the possibility that two rain droplets can interact or bounce from each other, and if extracting a random number we surpass this threshold that coalescence can happen, otherwise the particles get along with their motion.\\
To state it clearly this is how our coalescence works:\\
\paragraph{\textbf{Algorithm for stochastic coalescence}}
\begin{enumerate}
    \item Let $x_i,\ i=1,...,N$ the particles of the system, and denote $R_i$ their radius;
    \item For every couple $(i,j)$  such that $|x_i-x_j|\leq R_i+R_j$, we generate a number $\phi_i^j\in (0,1)$ with a uniform distribution;
    \item If $\phi_i^j>P_{mean}$, then the coalescence happens in the following way:
    \begin{itemize}
        \item If $V_t^{i,N}>=V_t^{j,N}$ then
        \begin{itemize}
            \item $V_{t}^{i,N}= V_t^{i,N} + V_t^{j,N}$;
            \item $V_{t}^{j,N}=0$ and the particle $j$ is removed from the system.
        \end{itemize}
        \item Vice versa if  $V_t^{j,N}>V_t^{i,N}$;
    \end{itemize}
    \item If $\phi_i^j<P_{mean}$, then the raindroplet bounced of from each other.
\end{enumerate}
Since our objective is study the mean time of such rain formation, we fixed\\ $P_{mean}=0$ as we already expect some fluctuation in the results given by the initial condition, and postpone to later studies the understanding of such physical quantity.
\subsubsection{Random Field:}\textit{Theoretical Conjecture}.
Before moving to the numerical results, we briefly illustrate theoretical conjecture on what we expect to find in the next section.\\
Following the idea from (\cite{f1},\cite{flandoli2021mixing},\cite{f2}) in which a fluid dynamical equations with transport noise converges to deterministic viscous equations such that the viscosity is enhanced by the turbulent random field.\\
We expect that, albeit less impactful than independent Brownian motion attach to each particles, the first exit time depends on the random field parameters as an inverse power $\alpha\in(-2,0)$.\\
\\
Consider our Smoluchoski equation with random field
$$
\partial_t \rho_t+ \sum_{k\in K}\left(\sigma_k(x,v)\cdot \nabla_x\rho_t\right)\xi_t=F_v(\rho_t)
$$
We expect that, under suitable assumption, the solution of such coagulation \abbr{PDE} is close to the solution of the diffusion Smoluchoski equation
$$
\partial_t \overline{\rho}_t+ \frac{\kappa^2}{2}\Delta\overline{\rho}_t=F_v(\overline{\rho}_t),
$$
where $\kappa:=\kappa(K,\lambda)$ depends on the parameters of the system in study.\\
From the theory on the Brownian motion case we expect that
$$
\mathbb{E}\left[\tau_f(\kappa)\right]\sim p(\kappa)^{-1}
$$
where $p(\kappa)$ is at most a polynomial of degree $2$.
\begin{figure}[h]
\centering
\includegraphics[width=\textwidth]{01ParticelleCluster.png}
\caption{Snapshot of particle system dynamics: we can observe the cluster formation obtained from the presence of the vortex. We expect that such cluster, and the randomness of the Ornstein–Uhlenbeck process, are the reason for the decrease in time of rain formation.}
\end{figure}

\subsection{Random Field Analysis: nonlinear Regression}
We divide the analysis in two main parts. We start fixing the intensity of the Ornstein-Ulhembeck processes and varying the number of vortex in the cloud; later we fix the vortex number and move the intensity of the environmental stochastic processes.\\
Following the previous reasoning we expect that our formation time depend with at most a square inverse power with the number of vortex and intensity of the random processes
$$
\mathbb{E}\left[\tau_{f}(|H|,\lambda)\right]\sim p(|H|,\lambda)^{-1},
$$
where we have highlighted the dependence on the parameter.\\
From the theoretical conjecture in the previous subsection, we expect that as vortex increase, so does linearly the covariance matrix of the noise, resulting in a square decay for the time.\\ While $\lambda$ model the  temporal structure of the phenomena, thus we expect that the effect is smaller on the vector fields $\sigma$ with a inverse decay trend $\sim \sqrt{\lambda}$.
\subsubsection{Fixed $\lambda$, Vortex Moving}
We analyze here the case in which the intensity of the noise is fixed and the number of vortex in the field changes. We'll see, as claimed at the start of the section that the decay is visible when we increase the number of field $\sigma_j$ interacting with the particles. We show here regression that support the conjecture of a polynomial decay of order $2$.\\ We note that the initial condition produce a very fluctuating system when the vortex number is small and the particles moves very slowly. This is expected, since the collision are reduced and the random initial condition characterize the system a lot. This is show in the high standard deviation that decreases as soon as the vortex number increase.\\
We fix $\lambda$ at 1500, to have enough speed in the particle and make sure that the collisions of droplet happen fast enough to be appreciated, in particular the choice is made to obtain a fast enough system for the required simulation to be made even with very few vortex-like structure.\\
We first performed a quadratic regression from which the residual plot showed no sign of residual structure. The analysis show a great degree of understanding of the data, as such R-squared is equal to $\sim0.98$ and the p-value suggest that our hypothesis on the decay is validated. We report here the complete summary of the regression performed in Fig.4 (left).\\
\\
\textbf{Quadratic Regression Summary Moving Vortex Number:}
\begin{verbatim}
Residual standard error: 0.4133 on 8 degrees of freedom
Multiple R-squared:  0.9798,	Adjusted R-squared:  0.9748
F-statistic: 194.2 on 2 and 8 DF,  p-value: 1.658e-07
\end{verbatim}
\begin{figure}[!tbp]
  \centering
  \begin{minipage}[c]{0.45\textwidth}
    \includegraphics[width=\textwidth]{1TabellaVortici.png}
  \end{minipage}
  \hfill
  \begin{minipage}[c]{0.45\textwidth}
    \includegraphics[width=\textwidth]{plotvortex.png}

  \end{minipage}
  \caption{Left: table of time decay as vortex number increase and as such the number of vortex in the system grows. Mean time and Standard deviation decays accordingly to the growth of $\#Vortex$; Right: The black line represent the mean time varying the number of vortex, while the red strip are the upper bound and lower bound of the standard deviation.}
\end{figure}

To conclude the analysis we perform a logarithmic regression, that we plotted in Fig.3 (right) in a log-log scale. The analysis is consistent with the quadratic regression performed and showed a good R-squared value. No significant structures were found in the residual plot and the value of the regression supported our hypothesis of a quadratic decay in the time, with small fluctuation due to the randomness of the initial condition that stabilize at higher vortex number.\\
\\
\textbf{Logarithmic Regression Summary Moving Vortex Number:}
\begin{verbatim}
Residual standard error: 0.4133 on 8 degrees of freedom
Multiple R-squared:  0.9798,	Adjusted R-squared:  0.9748
F-statistic: 194.2 on 2 and 8 DF,  p-value: 1.658e-07
\end{verbatim}

\begin{figure}[!tbp]
  \centering
  \begin{minipage}[b]{0.45\textwidth}
    \includegraphics[width=\textwidth]{quadVortex.png}
  \end{minipage}
  \hfill
  \begin{minipage}[b]{0.45\textwidth}
    \includegraphics[width=\textwidth]{logvortex.png}

  \end{minipage}
  \caption{Left:Lin-Log plot of quadratic regression with fixed intensity and varying vortex number; Right: Log-Log plot for Logarithmic regression with fixed intensity and varying vortex number.}
\end{figure}

\subsubsection{Fixed Vortex, $\lambda$ Moving}
We analyze here the case in which vortex are fixed and the intensity of the Random field changes. We'll see, as claimed at the start of the section that the quadratic decay is no longer visible and the initial condition produce a very unstable numerical system.\\
We fix the vortex number at 200, the choice is made to obtain a fast enough system for the required simulation to be made even with very slow random field intensity, but selecting also a number of vortex that already showed a nice decay in time.\\

\begin{figure}[!tbp]
  \centering
  \begin{minipage}[b]{0.5\textwidth}
    \includegraphics[width=\textwidth]{ValoriAltiIntensity.png}
  \end{minipage}
  \hfill
  \begin{minipage}[b]{0.45\textwidth}
    \includegraphics[width=\textwidth]{NoQuadIntensity.png}

  \end{minipage}
  \caption{Left: numerical plot of mean time with standard deviation. It is shown a decay in time when the intensity increase, with also a decrease in the standard deviation; Right: Lin-Log plot quadratic regression of mean time with varying $\lambda$, fixed vortex number.}
\end{figure}
\iffalse
\begin{figure}[h]
\caption{Andamento dei tempi medi al variare di $\lambda$, con vortici fissi a 200}
\centering
\includegraphics[width=0.8\textwidth]{0D-Vortici200-Lambda25a1000-ErroriNumericiValoriAlti.png}
\end{figure}
\fi
\iffalse
\begin{figure}[h]
\caption{Regressione Quadratica, tempi medi al variare di $\lambda$, con vortici fissi a 200}
\centering
\includegraphics[width=0.7\textwidth]{0D-Vortici200-Lambda25a1000-ErroriNumericiForse-NoQuadratica.png}
\end{figure}
\fi

We first performed a quadratic regression from which the residual plot showed no sign of residual structure. The residual error and the standard deviation was overall higher, even with high intensity parameter that should make coagulation faster. This suggested, as we expected from the theoretical conjecture, that the influence of the intensity in the time structure is less strong than the number of vortex-like structures in the field.\\
The analysis show a less degree of understanding of the data, and while R-squared is still high, we expect that power of lower order describe the decay curves in a more robust way. We report here the complete summary of the regression performed in Fig.5 (right).\\
\textbf{Quadratic Regression Summary Moving $\lambda$}
\begin{verbatim}
Residual standard error: 2.747 on 21 degrees of freedom
Multiple R-squared:  0.8392,	Adjusted R-squared:  0.8239
F-statistic:  54.8 on 2 and 21 DF,  p-value: 4.633e-09
\end{verbatim}

\iffalse
\begin{figure}[h]
\caption{Regressione Cubica, tempi medi al variare di $\lambda$, con vortici fissi a 200}
\centering
\end{figure}
\fi

To see if a regression with lower power of the parameters can explains better the decay of time formation, we performed a logarithmic regression, plotted in Fig.6. From the summary below, we see that more structure is captured when this type of analysis is performed, suggesting that a nonlinear regression of the form $\sim a/(1+b\cdot\lambda)$ is the right expected decay for the time as a function of the intensity parameter.\\
\textbf{Logarithmic Regression Summary Moving $\lambda$}
\begin{verbatim}
Residual standard error: 0.1084 on 22 degrees of freedom
Multiple R-squared:  0.9145,	Adjusted R-squared:  0.9106
F-statistic: 235.3 on 1 and 22 DF,  p-value: 3.135e-13
\end{verbatim}

\iffalse
Residual standard error: 2.013 on 20 degrees of freedom
Multiple R-squared:  0.9178,	Adjusted R-squared:  0.9054
F-statistic:  74.4 on 3 and 20 DF,  p-value: 5.033e-11
Figura 13 - Regressione Cubica - call:

lm(formula = reg ~ sigma + sigma2 + sigma3, data = data)

Residuals:
    Min      1Q  Median      3Q     Max
-2.6230 -1.4261 -0.3232  1.0057  4.6588

Coefficients:
              Estimate Std. Error t value Pr(>|t|)
(Intercept)  6.802e+00  1.492e+00   4.560 0.000190 ***
sigma        9.121e-02  1.396e-02   6.534 2.28e-06 ***
sigma2      -1.569e-04  3.252e-05  -4.824 0.000103 ***
sigma3       9.238e-08  2.113e-08   4.371 0.000295 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 2.013 on 20 degrees of freedom
Multiple R-squared:  0.9178,	Adjusted R-squared:  0.9054
F-statistic:  74.4 on 3 and 20 DF,  p-value: 5.033e-11
\fi

\iffalse
\begin{figure}[h]
\caption{Regressione Quartica, tempi medi al variare di $\lambda$, con vortici fissi a 200}
\centering
\includegraphics[width=0.5\textwidth]{0D-Vortici200-Lambda25a1000-ErroriNumericiForse-ForseQuadrica.png}
\end{figure}
\begin{verbatim}
figura 14- Regressione y=a^4

Call:
lm(formula = reg ~ sigma + sigma2 + sigma3 + sigma4, data = data)

Residuals:
    Min      1Q  Median      3Q     Max
-2.5372 -0.9136 -0.4303  0.7264  4.1374

Coefficients:
              Estimate Std. Error t value Pr(>|t|)
(Intercept)  4.558e+00  1.873e+00   2.434  0.02497 *
sigma        1.332e-01  2.654e-02   5.018 7.64e-05 ***
sigma2      -3.434e-04  1.068e-04  -3.215  0.00456 **
sigma3       3.793e-07  1.586e-07   2.391  0.02731 *
sigma4      -1.415e-10  7.764e-11  -1.823  0.08405 .
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 1.906 on 19 degrees of freedom
Multiple R-squared:   0.93,	Adjusted R-squared:  0.9153
F-statistic: 63.12 on 4 and 19 DF,  p-value: 1.049e-10
\end{verbatim}
\fi

\begin{figure}[h]
\centering
\includegraphics[width=0.5\textwidth]{LogLogIntensity.png}
\caption{Logarithmic Regression: Log-Log plot regression mean time with varying $\lambda$, fixed vortex number.}
\end{figure}
\iffalse
\begin{verbatim}
Concludiamo con una regressione Logaritmica, che mostra una ragionevole sicurezza
nel supporre che l'andamento dei tempi decada all'aumentare dell'intensità.

Call:
lm(formula = log(reg) ~ log(sigma), data = data)

Residuals:
      Min        1Q    Median        3Q       Max
-0.286313 -0.046006  0.006676  0.065507  0.215377

Coefficients:
            Estimate Std. Error t value Pr(>|t|)
(Intercept) -1.08432    0.12989  -8.348 2.89e-08 ***
log(sigma)  -0.34114    0.02224 -15.338 3.13e-13 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 0.1084 on 22 degrees of freedom
Multiple R-squared:  0.9145,	Adjusted R-squared:  0.9106
F-statistic: 235.3 on 1 and 22 DF,  p-value: 3.135e-13
\end{verbatim}
\fi

To this end, to get a more precise estimation on the behaviour of the first formation time $\tau_f(\lambda)$, with fixed vortex, we performed a nonlinear regression on a more refined sample of intensity.\\
As show in Fig.7 we iterate the system for intensity $\lambda\in[100,1100]$ with step of $10$.\\
After the logarithmic regression, we expect that we have a dependence on the parameter of the form $\sim \frac{a}{(b+\lambda)}$. For this reason we set $a_{in}=0.08,\ b_{in}=0.048$ starting parameter and we perform a regression that shown in Fig.7.\\
As we can see from the summary below the residual error equal to 0.0029 with a correlation with the numerical data equal to 0.79. As we see no structure is present in the residual plot and the decay is captured by the regression curve.\\
In conclusion when the intensity of the noise increase we still see a decay in time, but since the noise interact only in the time structure the decay has lower degree.\\
\\
\textbf{nonlinear Regression $\sim a/(b+\lambda)$ Summary:}
\begin{verbatim}
residual sum-of-squares: 0.002907
Achieved convergence tolerance: 1.489e-06
cor(NumericalData,NonLinearModel)=0.7997127
\end{verbatim}
\iffalse
Number of iterations to convergence: 11
\fi
\begin{figure}[!tbp]
  \centering
  \begin{minipage}[b]{0.5\textwidth}
    \includegraphics[width=\textwidth]{6.png}
  \end{minipage}
  \hfill
  \begin{minipage}[b]{0.45\textwidth}
    \includegraphics[width=\textwidth]{7.png}

  \end{minipage}
  \caption{Left:nonlinear Regression for moving $\lambda$ and fixed vortex. Expected decay as $\sim \lambda^{-1}$; Right: Residual plot. There is no presence of structure in the residual.}
\end{figure}

\printbibliography

\end{document}
\fi