text:
\section{Introduction}

The Navier-Stokes equations,
\EQ{\label{eq.NS}
\partial_t u -\Delta u +u\cdot{\nabla} u+{\nabla} p = 0;\qquad {\nabla} \cdot u=0,
}
govern the evolution of a viscous incompressible flow's velocity field $u$, which is a vector, and its associated scalar pressure $p$.
The system is supplemented with a divergence free initial datum $u_0$. We consider the problem on ${\mathbb R }^3\times (0,T)$, where $0<T\leq \infty$.
A foundational mathematical treatment of the problem was given by Leray in \cite{leray} where global weak solutions were constructed for finite energy data. Solutions with the properties of those constructed by Leray are referred to as Leray weak solutions.
Recent work suggests that uniqueness does not hold in the class of Leray weak solutions. Indeed, non-uniqueness has been affirmed in weaker classes than the Leray class \cite{BuckmasterVicol} and within the Leray class for the forced Navier-Stokes equations \cite{AlBrCo}. Within the Leray class and with no forcing, a research program of Jia and \v Sver\' ak \cite{JS,JiaSverakIll}, as well as the numerical work of Guillod and \v Sver\' ak \cite{GuillodSverak}, supports non-uniqueness. This program first establishes non-uniqueness in a class of solutions with large data in the critical weak Lebesgue space---this is the type of solutions we presently consider and give a precise definition below. While the evidence suggests non-uniqueness, there is not a clear picture of how non-uniqueness would evolve.
In this note, we take the view that solutions are not unique and seek to quantify how rapidly distinct solutions can separate as they evolve from a shared initial state.
In particular, we are interested in the following question:
\begin{quote}
 How can non-uniqueness be quantified in terms of local properties of the initial data?
\end{quote}
To answer this question we seek conditions so that, given some divergence free $u_0$, ball $B$, positive exponent $\sigma$, time $T>0$, and weak solutions $u_1$ and $u_2$ to \eqref{eq.NS} both evolving from $u_0$, we have
 \[
 \| u_1 - u_2\|_{L^\infty(B)} (t)\lesssim t^{ \sigma},
 \]
for all $0<t<T$. We refer to bounds like the above as an ``estimation of non-uniqueness'' and the right-hand side as a ``separation rate.''

A preliminary perspective on this question follows from the local smoothing of Jia and \v Sver\' ak \cite{JS}.
Local smoothing says that, if $u_0$ is sufficiently regular in a ball $B$, then a solution $u$ remains regular on $B'\times [0,T]$ for some $T>0$, where $B'\Subset B$. This can be viewed as saying that the non-local effects of the pressure are not strong enough to overcome the local regularity of the data. Local regularity is proven in \cite{JS} by showing that, for solutions in the local Leray class,\footnote{This is a more general class than the Leray class and was introduced by Lemari\'e-Rieusset. See \cite{LR} and the later papers \cite{BT8,JS,KS,KwTs,KMT} for useful properties. Local Leray solutions are sometimes referred to as local energy solutions.} if $u_0|_B\in L^p (B)$ for some $3<p\leq \infty$ and $U$ is the strong solution to the Navier-Stokes equations with initial data a divergence free localization of $u_0$ to $B$, then $u-U \in C^{\gamma}_{\text{parabolic}} ( B'\times [0,T])$ where ${\gamma}={\gamma}(p) \in (0,1)$. Note that in the definition of $C^{\gamma}_{\text{parabolic}} ( B'\times [0,T])$, the exponent for the time variable is ${\gamma}/2$. Since $U$ is uniquely determined by $u_0$, this implies that, for possibly distinct solutions $u_1$ and $u_2$ with the same data $u_0$, we have
\EQ{\label{ineq.JSseparation}
 \| u_1- u_2\|_{L^\infty(B')}(t)\leq \| u_1- U\|_{L^\infty(B')}(t)+ \| U- u_2\|_{L^\infty(B')}(t)\lesssim t^{\frac {\gamma} 2}.
}
Because ${\gamma}/2<1$, this allows a dramatic jump between the two solutions at small times.

A stronger separation rate is identified for discretely self-similar solutions, i.e.~solutions satisfying $u^\lambda(x,t):=\lambda u(\lambda x,\lambda^2t)$, for some $\lambda >1$, with data in $L^p_\loc({\mathbb R }^3\setminus \{0\})$ for $3<p\leq \infty$ \cite{BP1}. There, due to the global scaling properties of the solution,
\[
 |u_1-u_2|(x,t)\lesssim \frac {t^{\frac 3 2}} {(|x|+\sqrt t)^{4}},
\]
 outside of a space-time paraboloid $|x|\geq R_0\sqrt t$, for some $R_0\ge0$. Away from $x=0$, this gives the separation rate $t^{3/2}$, which is stronger than \eqref{ineq.JSseparation} for $t\leq 1$. Although we do not have a proof, we expect the rate $t^{3/2}$ is optimal because it arises in \cite{BP1} from pointwise bounds for gradients of the Oseen tensor \cite{VAS} which seem unavoidable. The solutions in \cite{BP1} have a great deal of structure due to their assumed scaling invariance and it is natural to seek separation rates under relaxed conditions.

 In this paper, we almost recover the separation rate $t^{3/2}$ for DSS solutions with no scaling assumption on $u_0$.
 We take our initial data to be in $L^{3,\infty}({\mathbb R }^3)$, which coincides with the weak Lebesgue space $L^3_w$ and is a Lorentz space.\footnote{$L^{3,\infty}$ includes all the DSS data considered in \cite{BP1}. It is important in the analysis of the Navier-Stokes equations as an endpoint space where many desirable features such as regularity or uniqueness break down. For example, there is a time-local unique strong solution when $u_0$ is possibly large in $L^3$ \cite{Kato}, but this is unknown in the larger space $L^{3,\infty}$. It is a \textit{critical} space in that it is scaling invariant with respect to the scaling of \eqref{eq.NS}.}
 If, additionally, $u_0|_{B}\in L^p(B)$ for a ball $B$ and some $3<p\leq \infty$, then we show there exists a time $T>0$ so that, for any $\sigma<3/2$, any two weak solutions $u_1$ and $u_2$ in a certain class satisfy
 \[
 \|u_1-u_2\|_{L^\infty(B')}(t)\lesssim t^\sigma,
 \]
where $B'\Subset B$ and $0<t<T$.
 This class of initial data is motivated by a natural type-I blow-up scenario wherein a strong solution $u$ defined on ${\mathbb R }^3 \times (-1,0)$ satisfying
\[
 |u(x,t)|\lesssim \frac 1 {|x|+\sqrt {-t}},
\]
develops a singularity at the space-time origin. The singular profile would satisfy $$|u(x,0)|\lesssim |x|^{-1}\in L^{3,\infty}.$$
Because uniqueness is not expected for large $L^{3,\infty}$ data, upon singularity formation the solution might branch into distinct flows. In this scenario, our theorem provides an upper bound on how fast the branching solutions can separate away from the singularity. It is worth pointing out that our main theorem can handle singular profiles which are worse than $|x|^{-1}$. Related to this, the initial data in \cite{BP1} is only locally critical at the origin; it is locally sub-critical\footnote{We say the space $X$ is \textit{sub-critical} if $\|u_0\|_X = \lambda^\alpha \|u_0^\lambda \|_X$ where $\alpha>0$. Examples of critical spaces are $L^p$ for $p\in (3,\infty]$. Typically, inclusion in sub-critical spaces controls small scales and leads to regularity. For \textit{super-critical} spaces, $\alpha<0$ and small scales are typically not controlled. For critical spaces, small scales are usually controlled to an extent when $C_c^\infty$ is dense in the space. Critical spaces where this fails, like $L^{3,\infty}$, are referred to as \textit{ultra-critical}.} everywhere else. In our   theorem, the only sub-critical assumption is within the ball $B$; the data can have $L^{3,\infty}$ singularities anywhere else.

Before stating our result we define the class of solutions we have in mind, which was originally developed by Barker, Seregin and \v Sver\' ak \cite{BaSeSv} and extends ideas in \cite{SeSv}. This notion of solution has since been extended to non-endpoint critical Besov spaces of negative smoothness \cite{AB}.

\begin{definition}[Weak $L^{3,\infty}$-solutions] Let $T>0$ be finite. Assume $u_0\in L^{3,\infty}$ is divergence free. We say that $u$ and an associated pressure $p$ comprise a weak $L^{3,\infty}$-solution if \begin{itemize}
 \item $(u,p)$ satisfies \eqref{eq.NS} distributionally,
 \item $u$ satisfies the local energy inequality of Scheffer \cite{VS76b} and Caffarelli, Kohn and Nirenberg \cite{CKN} for non-negative test functions compactly supported in ${\mathbb R }^3\times (0,T)$,
 \item for every $w\in L^2$, the function
 \[
 t\to \int w(x)\cdot u(x,t)\,dx,
 \]
 is continuous on $[0,T]$,
 \item $\tilde u :=u-e^{t\Delta}u_0$ satisfies, for all $s\in (0,T)$,
 \EQ{\label{ineq.BSSbound}
 \sup_{0<s<t}\| \tilde u \|^2_{L^2} (s) +\bigg(\int_0^t \| {\nabla} \tilde u\|_{L^2}^2(s)\,ds \bigg)^\frac 1 2 <\infty,
 }
 and
\EQ{
 \| \tilde u\|_2^2 +2\int_0^t \int|{\nabla} \tilde u|^2\,dx\,ds\leq 2\int_0^t \int ( e^{t\Delta}u_0 \otimes \tilde u + e^{t\Delta}u_0 \otimes e^{t\Delta}u_0) : {\nabla} \tilde u\,dx\,ds.
 }
\end{itemize}

\end{definition}

In \cite{BaSeSv}, weak solutions are constructed which satisfy the above definition for all $T>0$.
 Also, due to their spatial decay, weak $L^{3,\infty}$-solutions are mild,\footnote{Although this can be proved directly, it also follows from \cite{BT7} or \cite{LR}.} which means they satisfy the formula
\[
u(x,t)=e^{t\Delta}u_0 - \int_0^te^{(t-s)\Delta} \mathbb P {\nabla} \cdot (u\otimes u)\,ds,
\]
 where $\mathbb P$ is the Leray projection operator. The nonlinear contribution to $u$ is a bilinear operator which denote by
 \[
 B(f,g):= \int_0^te^{(t-s)\Delta} \mathbb P {\nabla} \cdot (f\otimes g)\,ds.
 \]

An important observation in \cite{BaSeSv} is that the nonlinear part of a weak $L^{3,\infty}$-solution satisfies a dimensionless energy estimate, namely
\EQ{\label{ineq.BSSdecay}
 \sup_{0<s<t}\| \tilde u \|_{L^2} (s) +\bigg(\int_0^t \| {\nabla} \tilde u\|_{L^2}^2(s)\,ds \bigg)^\frac 1 2 \lesssim_{u_0} t^{\frac 1 4}.
}
We emphasize that the energy associated with $\tilde u$ vanishes at $t=0$. This decay property will be essential in our work. It appeared earlier in the \textit{a priori} estimates of the weak discretely self-similar solutions constructed in \cite{BT1} as well as \cite{SeSv}, which is the precursor to \cite{BaSeSv}.
It is used in the Calderon-type splitting\footnote{See \cite{Calderon}.} construction in \cite{BaSeSv} to deplete a time singularity.
In \cite{AB}, it is established in Besov spaces with $e^{t\Delta}u_0$ replaced by higher Picard iterates, which are defined below.
Other dimensionless estimates can be derived from \eqref{ineq.BSSdecay}; indeed, below we prove that, for $q\in (3/2,3)$,
 \EQ{\label{ineq:newEstimate}
\| \tilde u\|_{L^r(0,T;L^q)}\lesssim_{q,u_0} T^{\frac 1 2}\quad \text{for} \quad r = \frac {2q} {2q-3}.
} Given \eqref{ineq.BSSdecay}, this is primarily of interest when $q<2$.

As pointed out in \cite{BaSeSv}, \eqref{ineq.BSSdecay} can be viewed as an estimate on the separation rate of two weak $L^{3,\infty}$-solutions since, denoting two such solutions with the same data by $u_1$ and $u_2$, we have
\[
 \| u_1 - u_2\|_{L^2}(t)\lesssim \| \tilde u_1\|_{L^2}(t)+\|\tilde u_2\|_{L^2}(t)\lesssim t^{\frac 1 4}.
\]
Notably, this is a global estimate. For data in $L^{3,\infty}$, global estimates are confined to super-critical norms since we do not expect $\tilde u$ to be in a stronger space than $u$---indeed, any singularity at a positive time is carried by $\tilde u$ not by $e^{t\Delta}u_0$. Such singularities can possibly occur at arbitrarily small times. Therefore, if we seek a finer estimate (i.e.~using a sub-critical norm) on the separation of the flows using the reasoning above, it should be confined to a local region where local smoothing holds, e.g.~where the initial data is sub-critical. The following theorem provides such an estimate.

\begin{theorem}[Estimation of non-uniqueness]\label{thrm.main}
Assume $u_0\in L^{3,\infty}$ and is divergence free. Fix $x_0\in {\mathbb R }^3$. Assume that $u_0|_B\in L^p(B)$ where $B= B_2(x_0)$ and $p\in (3,\infty]$. Let $u_1$ and $u_2$ be weak $L^{3,\infty}$-solutions with data $u_0$. Then, there exists $T=T(p,u_0)>0$ so that, for every $\rho\in (0,1/2)$, $\sigma \in (0,3/2)$, and $t\in (0,T)$,
\[
 \| u_1-u_2\|_{L^\infty (B_{\rho}(x_0))} (t)\lesssim_{\rho, p,\sigma,u_0} t^{\sigma},
\]
where the dependence on $u_0$ is via the quantities $\|u_0\|_{L^p(B)}$ and $\|u_0\|_{L^{3,\infty}}$.
\end{theorem}
Insofar as non-uniqueness in the Leray class is concerned, if $u_0\in L^2\cap L^{3,\infty}$, then any weak $L^{3,\infty}$-solution is also a Leray weak solution as discussed in \cite{BaSeSv}. Hence our result applies to a subset of the Leray class.

Theorem \ref{thrm.main} is a corollary of the following theorem, the proof of which constitutes the bulk of this paper.
Before stating the theorem, we recall the definition of Picard iterates. Let $P_0 = P_0 (u_0) = e^{t\Delta}u_0$ and define the $k^{\text{th}}$ Picard iterate to be $P_k = P_0 - B(P_{k-1},P_{k-1})$. Classically, the Picard iterates converge to a solution to \eqref{eq.NS} whenever \eqref{eq.NS} can be viewed as a perturbation of the heat equation. This is not the case for large $L^{3,\infty}$ data, so we do not expect convergence of $P_k$ to $u$ when $u$ is a weak $L^{3,\infty}$-solution. Nonetheless, the Picard iterates do capture some asymptotics at $t=0$ of weak $L^{3,\infty}$-solutions, which is the point of the following theorem.

\begin{theorem}[Local asymptotic expansion]\label{thrm.main2}
Assume $u_0\in L^{3,\infty}$ and is divergence free. Fix $x_0\in {\mathbb R }^3$ and $p\in (3,\infty]$. Assume further that $u_0|_B\in L^p(B)$ where $B=B_2(x_0)$. Then, there exists ${\gamma} ={\gamma} (p)\in(0,1)$ and $T=T(p,\|u_0\|_{L^{3,\infty}},\|u_0\|_{L^p(B)})>0$ so that for any $\rho\in (0,1/2)$, $\sigma \in (0,3/2)$, $t\in (0,T)$ and $k=0,1,\ldots, k_0$,
\[
 \| u-P_k\|_{L^\infty (B_\rho(x_0))} (t)\lesssim_{\rho, p,u_0,\sigma,k} t^{a_k},
\]
where $a_0 = \min\{{\gamma}/2, 1/2-3/(2p)\}$,
$a_{k+1} =\min\left\{\sigma, k( 1/2 -3/(2p))+a_0\right\}$ and
 $k_0$ is the smallest natural number so that
\[
k_0\bigg(\frac 1 2 -\frac 3 {2p}\bigg) +a_0 \geq \sigma.
\]
In particular, $a_{k_0}=\sigma$ and $a_k>a_{k-1}$ for $k=1,\ldots,k_0$. It follows that, for $(x,t)\in B_{\rho}(x_0) \times (0,T)$, and letting $a_{-1}= -3/(2p)$, we have
\[
u(x,t)= P_0 + \sum_{k=0}^{k_0-1}\mathcal O (t^{a_k}) + \mathcal O(t^{\sigma}) = \sum_{k=-1}^{k_0}\mathcal O (t^{a_k}),
\]
where the $\mathcal O (t^{a_k})$ terms are exactly solvable for $-1\leq k<k_0$.
\end{theorem}

Short-time asympototic expansions have been examined by Brandolese for small self-similar flows \cite{Brandolese} and by Brandolese and Vigneron for both small (in which case the expansion holds for all times) and large (in which case the data is globally sub-critical and the expansion is up to a finite time) non-self-similar flows \cite{BV}. A follow-up paper by Bae and Brandolese considers the {forced} Navier-Stokes \cite{BaeBrandolese}. In \cite{KR}, Kukavica and Ries give an expansion in arbitrarily many terms assuming the solution is smooth. In all of the preceding papers, either the initial data is strong enough to generate smooth solutions (e.g.~it is in a sub-critical class or is small in a critical class) or the solution is assumed to be smooth. Additionally, the terms of the asymptotic expansions depend on $u$.

The novelty of Theorem \ref{thrm.main2} is that it establishes time asymptotics without any scaling assumption (cf. \cite{BP1}) or requirements implying global regularity on the relevant  time domain (cf. \cite{BaeBrandolese,Brandolese,BV,KR}). The asymptotics depend only on $u_0$---they are independent of $u$ which is necessary for Theorem \ref{thrm.main}. Because ours is a spatially local expansion, spatial asymptotics are not relevant.

Long-time asymptotic expansions have also been studied extensively; see, for example, \cite{FS,FS2,GW,HM}, the review article \cite{BrandoleseSchonbek}, and the references therein. The spatial asymptotics for the stationary problem have also been studied; see, for example, \cite{KoSv} and the references therein.

With Theorem \ref{thrm.main2} in hand, we quickly prove Theorem \ref{thrm.main}.
\begin{proof}[Proof of \cref{thrm.main}] Suppose $u_1$ and $u_2$ are weak $L^{3,\infty}$-solutions with data $u_0$.
By \cref{thrm.main2}, we have for $i=1,2$ that
\[\|u_i-P_{k_0}\|_{L^\infty(B_{\rho}(x_0))} (t) \lesssim_{\rho, p,\sigma,u_0} t^\sigma,\]
for all $0<t<T$. By the uniqueness of Picard iterates, we infer
\[
 \|u_1-u_2\|_{L^\infty(B_{\rho}(x_0))}(t) \leq \|u_1-P_{k_0}\|_{L^\infty(B_{\rho}(x_0))}(t)+\|u_2-P_{k_0}\|_{L^\infty(B_{\rho}(x_0))}(t)\lesssim_{\rho, p,\sigma,u_0} t^{\sigma},
\]
for all $0<t<T$.
\end{proof}
\medskip
\noindent \textbf{Discussion of the proof:}
By local smoothing \cite{JS}, it is not difficult to show that
\[
 \| u-P_0\|_{L^\infty(B_{\frac 1 2}(x_0))}(t)\lesssim t^{\frac {\gamma} 2},
\]
for some ${\gamma}={\gamma}(p) \in (0,1)$ and across some time interval. Our main insight is that this bound improves when $P_0$ is replaced by higher Picard iterates, a consequence of the self-improvement property of Picard iterates which has been used elsewhere, e.g.~\cite{AB,Brandolese,GIP}. To see how this works, we note that
\EQ{\label{eq:splitting}
 u-P_{k+1}
 = B(u-P_{k},u-P_{k}) + B(u-P_{k}, P_{k})+B(P_k, u-P_{k}).
}
Each term on the right hand side locally has an algebraic decay rate at $t=0$. The product structure and the time integral in the bilinear operator $B(f,g)$ lead to an improved algebraic decay rate for the left hand side compared to that for $u-P_k$. This improvement is only local. The far-field contributions to the flow are managed using a new \textit{a priori} bound for weak $L^{3,\infty}$-solutions---see Corollary \ref{cor:decay}. The properties of weak $L^{3,\infty}$-solutions \cite{AB,BaSeSv} are used critically throughout.

\begin{comment}
\cmp{The near field of all terms will enjoy an inductive improvement as we increase the Picard iterates, while the far-field serves as a limit our decay improvements at $t^{\sigma}$ for any chosen $\sigma<3/2.$

The main hurdles in this proof are maintaining control of the extraneous term $B(u-P_{k}, P_{k})$ in the near field, and controlling the growth of the Picard iterates in the far-field, where we only have $P_{k} \in L^{3,\infty}$.

The first is accomplished through the use of \cref{lemma:Pkq}, which extends the control of the initial data in $L^p(B)$ up the Picard scale, giving control of $P_k$ in $L^\infty_t L^q(B_\rho)$, $B_\rho\Subset B$; this is central to the inductive improvement in the near-field. More specifically we can find
\[
 \| B((u-P_k) \chi_{B_\rho} , P_k)\|_{L^\infty} \lesssim \int_0^t \frac 1 {(t-s)^{\frac 1 2 + \frac 3 {2q}}} \underbrace{\| (u-P_k) \|_{L^\infty(B_\rho)}}_{\text{inductive assumption}} \underbrace{ \| P_k \chi_{B_\rho}\|_{L^p}}_{\text{bounded}}\, ds.
\]

For the far-field, we achieve temporal decay of $B((u-P_k)(1-\chi_{B_\rho}),u-P_k)$ in $L^\infty(B_\rho)$ by splitting \[|u-P_k|^2 = |u-P_0+P_0-P_k|^2\lesssim |\tilde u|^2+|P_0-P_k|^2.\] We have control $\|\tilde u \|_{L^2}\lesssim t^{1/4}$ and $P_0-P_k = B(P_{k-1},P_{k-1})$ in Kato spaces $K_q$, $q>3$ which achieve the decay $t^{3/2}.$
For $B(u-P_k,P_k(1-\chi_{B_\rho}))$, using the same splitting as in \eqref{eq:splitting}
we can find
\EQ{\int_0^t \int_{B_\rho^c} \frac {|P_{k}|} {(|x-y|+\sqrt{t-s})^4} |u-P_{k}|\, dy\,ds \lesssim t^\sigma,}
by breaking into terms controlled using \cref{lemma.decay} and \cref{lemma.decay2}.\footnote{
The use of Hardy-Littlewood-Sobolev inequality in the proofs of \cref{lemma.decay} and \cref{lemma.decay2} is exactly what limits our decay at $\sigma<3/2$.}
}
\end{comment}

\bigskip\noindent \textbf{Organization:} In Section 2, we establish several key lemmas, most importantly the extension of the decay property \eqref{ineq.BSSdecay} to \eqref{ineq:newEstimate}. We also establish some elementary properties of Picard iterates. Section 3 contains the proof of Theorem \ref{thrm.main2}.

\section{Preliminaries}

In this section, we prove new \textit{a priori} bounds for weak $L^{3,\infty}$-solutions. See Lemmas \ref{lemma.decay} and \ref{lemma.decay2}. We then establish a property of Picard iterates in Lemma \ref{lemma:Pkq}.

Due to scaling considerations, one predicts that if the energy-level quantities on the left hand side of \eqref{ineq.BSSdecay} are replaced by lower Lebesgue or Lorentz norms, then the exponent on the right hand side will increase to preserve the scaling of the inequality. With our application in mind, it is natural to ask if the following dimensionless estimate holds
\[
 \sup_{0<s<t}\| \tilde u \|_{L^{\frac 3 2,1}} (s) \lesssim t^{\frac 1 2}.
\]
This estimate is motivated by the $L^{ 3/2,1}$-$L^{3,\infty}$ duality pairing.
There are barriers to establishing the above decay rate, but nearby rates are within reach as the next two lemmas show. In the lemmas, we replace $\tilde u$ with the first three terms in the following expansion
\[
 \tilde u = B(u,u) = B(\tilde u,\tilde u) + B(\tilde u,P_0)+B(P_0,\tilde u).
\]
The proof of the lemma also illustrates the barriers to getting the above estimate in $L^{3/2,1}$. Importantly, this failure is one reason we do not achieve the separation rate $t^{3/2}$ in our main theorem. Once we move away from the exponent $3/2$, it is sufficient to consider Lebesgue norms instead of Lorentz norms. Finally, as we will use these lemmas for an iterative procedure involving higher Picard iterates, our lemmas apply to any difference $u- P_k$, not just $\tilde u = u-P_0$.

 \begin{lemma}\label{lemma.decay} Fix $q\in (3/2 ,3)$, $T>0$ and $k\in {\mathbb N}_0$. Assume $u_0\in L^{3,\infty}$ and is divergence free. Let $u$ be a weak $L^{3,\infty}$-solution with initial data $u_0$. Then, letting $r =\frac {2q} {2q-3}$,

\[
 \| B(u-P_k,u-P_k) \|_{L^r(0,T;L^{q,1})} \lesssim_{k,q,u_0} T^{\frac 1 2}.
\]
\end{lemma}

The above estimate is dimensionless.
By Lorentz space embeddings we trivially infer
\[
 \| B(u-P_k,u-P_k) \|_{L^r(0,T;L^{q,\beta})} \lesssim_{k,q,u_0} T^{\frac 1 2},
\]
for every $\beta \in (1,\infty]$. This includes $L^r(0,T;L^q)$ when $\beta=q$. In our application, we will only use the $L^q$ version of this. However, the proof for the full scale of Lorentz spaces is no harder and we therefore include it in case it is useful elsewhere.

\begin{proof}
By Yamazaki \cite{yamazaki},
\EQ{\label{ineq: Yamazaki}
 \| B(u-P_k,u-P_k) \|_{L^{q,1}}&\lesssim \int_0^t \frac 1 {(t-s)^\frac 1 2} \| (u-P_k)^2\|_{L^{q,1}}(s)\,ds
 \\&\lesssim \int_0^t \frac 1 {(t-s)^\frac 1 2} \| u-P_k\|_{L^{2q,2}}^2(s)\,ds.
}
 Recall the extension of the Gagliardo-Nirenberg inequality to the Lorentz scale \cite[Corollary 2.2]{DDN} which states
\EQ{\label{GN}
 \| f\|_{L^{\tilde p,\beta}} \lesssim_{\tilde p, \tilde q, \beta} \|f\|_{L^{\tilde q,\infty}}^{\th}\|{\nabla} f\|_{L^{2}}^{1-\th},
}
for $\beta>0$ and
\[
 \frac 1 {\tilde p} = \frac{\th} {\tilde q}+(1-\th) \bigg(\frac 1 2 -\frac 1 3\bigg),
\]
where $1\leq \tilde q<\tilde p< \infty$ and
$3/2 -3/{\tilde p} <1$.
Let $\tilde p = 2q$ and $\tilde q=2$. These satisfy the above conditions because $q<3$. Then, $\th$ is given by
\[
 \frac 3 {2q} -\frac 1 2= \th,
\]
and, provided $1<q<3$, the other conditions above are met.
To summarize,
\[
 \|f\|_{L^{2q,2}} \lesssim_{q} \|f\|_{L^{2,\infty}}^{\th}\|{\nabla} f\|_{L^{2}}^{1-\th}\lesssim_{q} \|f\|_{L^2}^{\th}\|{\nabla} f\|_{L^2}^{1-\th},
\]
where we used the continuous embedding $L^2 \subset L^{2,\infty}$.
Returning to our main estimate, this gives
\EQ{\| B(u-P_k,u-P_k) \|_{L^{q,1}}(t)
 &\lesssim_{q} \int_0^t \frac 1 {(t-s)^\frac 1 2}\|u-P_k\|_{L^2}^{2\th}\|{\nabla}(u-P_k)\|_{L^2}^{2(1-\th)}\,ds
 \\&\lesssim_{k,q,u_0} t^{\frac {\th} 2} \int_0^t \frac 1 {(t-s)^\frac 1 2} \|{\nabla}( u-P_k)\|_{L^2}^{2(1-\th)}\,ds,
}
where we used the fact that \eqref{ineq.BSSdecay} applies also to $u-P_k$ as a consequence of \cite[Lemma 2.2]{AB},\footnote{We will use the fact several times and presently elaborate on how it follows from \cite[Lemma 2.2]{AB}. The bounds \cite[(2.36)-(2.39)]{AB} allow us to extend \eqref{ineq.BSSdecay} to $u-P_k$ for $k>0$. Note that $\| {\nabla} (P_{k+1}-P_k)\|_{L^2(0,T;L^2)}\lesssim T^{ 1/4}$ is not mentioned in \cite[(2.39)]{AB} but, upon inspecting the proof, it also holds as a consequence of the energy estimate for the Stokes equation and the above listed bounds.
} in which case the suppressed constant accrues a dependence on $k$.
Note that for $t\in (0,T)$,
\[
 \int_0^t \frac 1 {(t-s)^\frac 1 2} \|{\nabla}( u-P_k)\|_{L^2}^{2(1-\th)} \,ds
 \leq \int_{{\mathbb R }} \frac 1 {|t-s|^{\frac 1 2}} \|{\nabla} (u-P_k)\|_{L^2}^{2(1-\th)}(s) \chi_{(0,T)}(s)\,ds,
\]
and the right hand side can be viewed as $I_{\frac 12}( \|{\nabla} (u-P_k)\|_{L^2}^{2(1-\th)} \chi_{(0,T)})$ where $I_{\frac 12}$ is a Riesz potential in 1D.
The Hardy-Littlewood-Sobolev inequality states that
\[
 \left\| I_{\frac 1 2} \|{\nabla} (u-P_k)\|_{L^2}^{2(1-\th)} \chi_{(0,T)}\right\|_{L^r({\mathbb R })}
 \lesssim_{r} \left\| \|{\nabla} (u-P_k)\|_{L^2}^{2(1-\th)} \chi_{(0,T)}\right\|_{L^{\tilde r}({\mathbb R })},
\]
where
\[
 \frac 1 r = \frac 1 {\tilde r}-\frac 1 2.
\]
The selection
\[
 \tilde r = \frac 1 {1-\th}; \qquad r=\frac 2 {1-2\th},
\]
is valid for the Hardy-Littlewood-Sobolev inequality provided $3/2 <q$.\footnote{If $q=3/2$, then $\th = 1/2$ and $r=\infty$, which is not permitted in the Hardy-Littlewood-Sobolev inequality.
} Letting $r = \frac {2q}{2q-3}$ and putting the above observations together leads to
\EQ{\label{ineq:lemresult}
 \| B(u-P_k,u-P_k)\|_{L^r (0,T;L^{q,1})}

 &\lesssim_{k,q,u_0} T^\frac {\th}2\big\| \|{\nabla} (u-P_k)\|_{L^2}^{2(1-\th)} \chi_{(0,T)} \big\|_{L^{\tilde r}({\mathbb R })}
 \\&\lesssim_{k,q,u_0} T^\frac {\th}2 \| {\nabla} (u-P_k)\|_{L^2(0,T;L^2)}^{\frac 2 {\tilde r}}
 \\&\lesssim_{k,q,u_0} T^\frac {\th}2 T^{\frac 1 {2\tilde r}}
 =T^{\frac 1 2},
}
where we used the extension of \eqref{ineq.BSSdecay} to $u-P_k$ again.
\end{proof}

We prove a similar result for $B(P_k,u-P_k)$. This requires the well known fact that if $u_0\in L^{3,\infty}$, then $P_k$ is in the scaling invariant Kato classes for $q\in (3,\infty]$, i.e.,
\EQ{\label{KatoClassInclusion}
\| P_k\|_{\mathcal K_q} :=\sup_{0<t<\infty} t^{\frac 1 2-\frac 3{2q}} \|P_k\|_q(t) \lesssim_{k,u_0} 1.
}
To check this, note that the above property is immediate for $P_0$ by the embedding $L^{3,\infty}\subset {\dot B_{p,\infty}^{-1+\frac 3 p}}$, for $3<p\leq \infty$, and the fact that $\| u_0\|_{{\dot B_{p,\infty}^{-1+\frac 3 p}}} \sim \|P_0 \|_{\mathcal K_p}$ \cite{BCD}. Then, by the standard bilinear estimate (see the original papers \cite{FJR,Kato} or \cite[Ch.~5]{Tsai-book}),
\EQ{\label{mild.est}
\| B(f,g)\|_{L^p}(t) \lesssim \int_0^t\frac 1{(t-s)^{\frac 1 2}s^{ \frac 3 2 (\frac 1q-\frac 1 p)}} \| f\otimes g\|_{L^q}(s) \,ds,
}
we have
\EQ{
\| B(P_{k},P_k)\|_{L^\infty}(t) &\lesssim
\int_0^t \frac 1 {(t-s)^{\frac 1 2+\frac 3 {2q}}} \| P_{k-1}\|_{L^q}(s) \|P_{k-1}\|_{L^\infty}(s)\,ds
\\&\lesssim \int_0^t \frac 1 {(t-s)^{\frac 1 2+\frac 3 {2q}} s^{1-\frac 3 {2q}}} \|P_{k-1}\|_{\mathcal K_q} \|P_{k-1}\|_{\mathcal K_\infty} \,ds
\\&\lesssim t^{-\frac 1 2} \|P_{k-1}\|_{\mathcal K_q} \|P_{k-1}\|_{\mathcal K_\infty},
}
and
\EQ{
\| B(P_{k},P_k)\|_{L^q}(t) &\lesssim
\int_0^t \frac 1 {(t-s)^{\frac 1 2 }} \| P_{k-1}\|_{L^q}(s) \|P_{k-1}\|_{L^\infty}(s)\,ds
\\&\lesssim \int_0^t \frac 1 {(t-s)^{\frac 1 2 } s^{1-\frac 3 {2q}}} \|P_{k-1}\|_{\mathcal K_q} \|P_{k-1}\|_{\mathcal K_\infty} \,ds
\\&\lesssim t^{-\frac 1 2+\frac 3 {2q}} \|P_{k-1}\|_{\mathcal K_q} \|P_{k-1}\|_{\mathcal K_\infty}.
}
The claim \eqref{KatoClassInclusion} follows from the above observations by induction.

\begin{lemma}\label{lemma.decay2} Fix $q\in (3/2 ,3)$, $T>0$ and $k\in {\mathbb N}_0$. Assume $u_0\in L^{3,\infty}$ and is divergence free. Let $u$ be a weak $L^{3,\infty}$-solution with initial data $u_0$. Then, letting $r =\frac {2q} {2q-3}$,
\[
 \| B(P_k,u-P_k) \|_{L^r(0,T;L^{q,1})}+ \| B(u-P_k,P_k) \|_{L^r(0,T;L^{q,1})} \lesssim_{k,q,u_0} T^{\frac 1 2}.
\]

\end{lemma}
\begin{proof}
Although the bilinear operator $B$ is not symmetric, our estimates are identical for the two terms in the lemma. Hence we only consider $B(P_k,u-P_k)$.
By Yamazaki \cite{yamazaki} and O'Neil's inequality,
\EQ{\| B(P_k,u-P_k) \|_{L^{q,1}}&\lesssim \int_0^t \frac 1 {(t-s)^\frac 1 2} \| P_k(u-P_k)\|_{L^{q,1}}(s)\,ds
 \\&\lesssim \int_0^t \frac 1 {(t-s)^{\frac 1 2}} \bigg(\|P_k\|_{L^{2q,\infty}}^2 + \| u-P_k\|_{L^{2q,1}}^2\bigg)\,ds
 \\&\lesssim \int_0^t \frac 1 {(t-s)^{\frac 1 2}} \bigg(\|P_k\|_{L^{2q}}^2 + \| u-P_k\|_{L^{2q,1}}^2\bigg)\,ds.
}
Noting that we may choose $\beta=1$ in the extension of the Gagliardo-Nirenberg inequality \eqref{GN} to the Lorentz scale, we have the desired result for the $u-P_k$ term by the work done between \eqref{ineq: Yamazaki} and \eqref{ineq:lemresult} in the proof of \cref{lemma.decay}.

We then consider $P_k$ in $L^{2q}.$
By the membership of $P_k$ in the Kato class,
\EQ{\label{ineq:PkKq}\int_0^t \frac 1 {(t-s)^{\frac 12}} \|P_k\|_{L^{2q}}^2 \, ds
 &\lesssim \int_0^t \frac 1 {(t-s)^{\frac 1 2}s^{1-\frac 3 {2q}}} \|P_k\|_{\mathcal K_{2q}}^2\, ds \\
 &\lesssim_{k,q,u_0} t^{\frac 3 {2q}-\frac 1 2}.
}
Then,
\[
 \| B(P_k,u-P_k) \|_{L^r(0,T;L^{q,1})} \lesssim_{k,q,u_0} \left(\int_0^T \left(t^{\frac 3 {2q}-\frac 1 2}\right)^r \,dt\right)^{\frac 1r} \lesssim_{k,q,u_0} T^{\frac 3 {2q}-\frac 1 2+\frac 1 r}\lesssim_{k,q,u_0} T^{\frac 1 2},
\]
where we used the definitions of $r$ and $\th$.
\end{proof}
Together, the above two lemmas lead to the following corollary.

\begin{corollary}\label{cor:decay}
Fix $q\in (3/2 ,3)$, $T>0$ and $k\in {\mathbb N}_0$. Assume $u_0\in L^{3,\infty}$ and is divergence free. Let $u$ be a weak $L^{3,\infty}$-solution with initial data $u_0$. Then, letting $r =\frac {2q} {2q-3}$, we have
\[
 \| u-P_k \|_{L^r(0,T;L^{q})} \lesssim_{k,q,u_0} T^{\frac 1 2}.
\]
\end{corollary}
\begin{proof}
This is immediate given Lemmas \ref{lemma.decay} and \ref{lemma.decay2} and the fact that \[u-P_k = B(u-P_{k-1},u-P_{k-1})+B(P_k,u-P_{k-1})+B(u-P_{k-1},P_k),\]
for $k\geq 1$.
\end{proof}
\begin{comment}
\snt{DO NOT NEED THIS LEMMA}
\begin{lemma}\label{lemma.decay3} Fix $q\in (\frac 3 2 ,3)$, $T>0$ and $k\in {\mathbb N}_0$. Assume $u_0\in L^{3,\infty}$ and is divergence free. Let $u$ be a weak $L^{3,\infty}$-solution with initial data $u_0$. Then
\[
 \| B(P_k,P_k) \|_{L^r(0,T;L^{q})} \lesssim_{k,q,u_0} T^{\frac 1 2},
\]
where
\[
r =\frac {2q} {2q-3}.

\]
\end{lemma}

Note that $u-P_k = B(u-P_k,u-P_k)+B(u-P_k,u)+B(u,u-P_k) +B(P_k,P_k)$. \snt{LAST TERM SHOULDN'T BE THERE}
Unlike the other components of $u-P_k$, we do not apparently have the above estimate when $L^q$ is replaced by $L^{q,1}$, as is the case in the lemmas above, since $P_k$ does not need to be in $L^{2q,2}$, which is outside of the range of spatial norms of the Kato classes. This can be viewed as a marginally stronger estimate for $u-(P_k +B(P_k,P_k))$, than for $u-P_k$.

\begin{proof}
We have
\EQ{
\| B(P_k,P_k)\|_{L^q}&\lesssim \int_0^t \frac 1 {(t-s)^{\frac 1 2}} \| P_k\|_{2q}^2\,ds
\\&\lesssim \int_0^t \frac 1 {(t-s)^{\frac 1 2} s^{1 -\frac 3 {2q}}}.
}
This is exactly \eqref{ineq:PkKq} and, using the remainder of the proof of \ref{lemma.decay2} we are done.
\end{proof}

\end{comment}

Our next lemma is a technical statement about the decay at $t=0$ of the heat semigroup.
\begin{lemma}\label{lem:heatL321} Let $B=B_{R}(x_0)$ and $B' := B_{r}(x_0)$ where $0<r<R<\infty$. Then, for $0<t<\infty$,
\EQ{\left\|\|e^{-\frac{|x-y|^2}{4t}}(1-\chi_{B})\|_{L^{\frac 3 2,1}_y}\right\|_{L^\infty_x(B')}\lesssim_{R,r} e^{\frac{-(R-r)^2}{4t}}.}
\end{lemma}

\begin{proof}
{First, assume without loss of generality that $x_0=0$.} Then, letting $x\in B'$,
\EQ{\|e^{-\frac{|x-y|^2}{4t}}(1-\chi_{B})\|_{L^{\frac 3 2,1}_y}
 &= \frac 3 2 \int_0^\infty \mu \left\{ y:e^{-\frac{|x-y|^2}{4t}}(1-\chi_{B}(y)) \ge s\right\}^{\frac 2 3}\,ds,
 }
 where $\mu$ is Lebesgue measure.
{Note that the above set can be written as \[A(x)=\{y:|x-y|\le \sqrt{-4t\ln(s)},\, |y|>R\}=B(x,(-4t\ln(s))^{\frac 1 2}) \setminus B_R(0),\] which is well-defined because $t\ge 0$ and $s\le 1$. Then, \EQ{\left\|\|e^{-\frac{|x-y|^2}{4t}}(1-\chi_{B}(y))\|_{L^{\frac 3 2,1}_y}\right\|_{L^\infty_x(B')}
 &\lesssim \left\|\int_0^\infty
 \mu (A(x))^{\frac 2 3}\, ds\right\|_{L^\infty_x(B)}\\
 &\lesssim \int_0^{e^{\frac{-(R-r)^2}{4t}}} |-4t\ln(s)| \, ds\\
 &\lesssim 4t \left(e^{\frac{-(R-r)^2}{4t}}\frac{(R-r)^2}{4t}+e^{\frac{-(R-r)^2}{4t}}\right) \\
 &\lesssim_{R,r} e^{\frac{-(R-r)^2}{4t}}.}
 }
\end{proof}

The above lemma leads to a local \textit{a priori} inclusion for Picard iterates.

\begin{lemma}\label{lemma:Pkq} Let $B=B_R(x_0)$ and $B' = B_r(x_0)$ where $0<r<R<\infty$. Let $u_0\in L^{3,\infty}$ with $u_0|_{B}\in L^{q}(B)$, for some $3< q\leq \infty$. It follows that $P_k \in L^\infty(0,\infty; L^q(B'))$.
\end{lemma}

\begin{proof} Note that for any $\tau>0$, $\sup_{\tau<t<\infty} \|P_k\|_q \lesssim_{\tau,k} \|u_0\|_{L^{3,\infty}}$ due to the fact that $P_k\in \mathcal K_q$ when $q>3$. We therefore only need to prove the inclusion for a short period of time. Let $\{B_k\}$ be a collection of concentric balls about $x_0$ of radii $\alpha^{k+1}R$, for some $\alpha\in (0,1)$. Fix $k_0\in \mathbb N_0$. Choose $\alpha$ so that $r = \alpha^{k_0+1}R$.

For $P_0=e^{t\Delta}u_0$ we have
\EQ{\|P_0\|_{L^q(B_0)}(t) \label{A1}
 &= \left\|\int_{{\mathbb R }^3}\frac{1}{t^{\frac 3 2}}e^{-\frac{|x-y|^2}{4t}} u_0(y) \, dy\right\|_{L^q(B_0)}\\
 &= \left\|\left(\int_{B^c}+\int_{B}\right)t^{-\frac 3 2} e^{-\frac{|x-y|^2}{4t}} u_0(y) \, dy\right\|_{L^q(B_0)} \\
&\lesssim \|u_0\|_{L^{3,\infty}}t^{-\frac 3 2}\left\|\|e^{-\frac{|x-y|^2}{4t}}(1-\chi_{B}(y))\|_{L^{\frac 3 2,1}_y}\right\|_{L^q_x(B_0)}\\&\quad +\left\|e^{t\Delta}(\chi_B(y) u_0)\right\|_{L^q({\mathbb R }^3)}.
}
For the far-field term, by \cref{lem:heatL321},
\EQ{ \label{A2}\left\| \|e^{-\frac{|x-y|^2}{4t}}(1-\chi_{B}(y))\|_{L^{\frac 3 2,1}_y}\right\|_{L^q_x(B_0)}&\lesssim_{R,\alpha,q}
\left\|\|e^{-\frac{|x-y|^2}{4t}}(1-\chi_{B}(y))\|_{L^{\frac 3 2,1}_y}\right\|_{L^\infty_x(B_0)}\\&\lesssim_{R,\alpha} e^{\frac{-(R(1-\alpha))^2}{4t}}.}
For the near field term,
\EQ{
\|e^{t\Delta}(u_0\chi_B )\|_{L^q({\mathbb R }^3)} \lesssim \| u_0 \chi_B\|_{L^q({\mathbb R }^3)}\lesssim \|u_0\|_{L^q(B)}.
}
Therefore,
\EQ{
 \big\| \|P_0\|_{L^q(B')}\big\|_{L^\infty_t}
 \lesssim_{R,\alpha} \|u_0\|_{L^{3,\infty}}+ \|u_0\|_{L^q(B)}.
}
If $k_0=0$ then we are done.

If $k_0>0$ then we use induction.
Observe that
\EQN{
 B(P_{k-1},P_{k-1}) =B(P_{k-1}\chi_{B_{k-1}},P_{k-1})+B(P_{k-1}(1-\chi_{B_{k-1}}),P_{k-1}).
}
For the first part,
\EQN{
 \|B(P_{k-1}\chi_{B_{k-1}},P_{k-1})\|_{L^q(B_k)}(t) &\lesssim_q\|P_k\|_{\mathcal K_\infty} \int_0^t \frac 1 {(t-s)^{\frac 1 2} s^{\frac 1 2}} \| P_{k-1}\|_{L^q(B_{k-1})}(s) \,ds
 \\&\lesssim_{k,q} \|u_0\|_{L^{3,\infty}}\| P_{k-1}\|_{L^q(B_{k-1})}(t) .
}
For the other part, by the pointwise estimate for the kernel $K$ of the Oseen tensor (see \cite{VAS,Tsai-book}),
\EQ{\label{oseen.estimate}
|D_k^{(m)} K(x,t)| \lesssim_m \frac 1 {(|x|+\sqrt{t})^{3+|m|}},
}
we have
\EQN{
 \|B(P_{k-1}(1-\chi_{B_{k-1}}),P_{k-1})\|_{L^q(B_k)}(t)&\lesssim\left\|\int_0^t \int_{B_{k-1}^c} \frac {P_{k-1}\otimes P_{k-1}(y,s)} {(|x-y|+\sqrt{t-s})^4} \,dy\,ds\right\|_{L^q(B_k)}

 \\&\lesssim_{R,\alpha,k,q} \| |\cdot|^{-4}\|_{L^{2}(|\cdot|>R(\alpha^k-\alpha^{k+1}) )} \int_0^t \|P_{k-1}\|_{L^{4}}^2\,ds
 \\& \lesssim_{R,\alpha,k,q} \int_0^t s^{(-1+\frac 3 {4})}\,ds \lesssim_{R,\alpha,k,q} t^{\frac 34},
}
where we used the membership of $P_{k-1}$ in the Kato class $\mathcal K_4$.

We know by our base case that $P_0$ is in $L^\infty(0,\infty; L^q(B_0))$. We have just shown $B(P_{k-1},P_{k-1})\in L^\infty(0,\infty; L^q(B_0))$ whenever $P_{k-1}$ is in $ L^\infty L^q(B_{k-1})$. Hence,
\[
 P_k = P_0-B(P_{k-1},P_{k-1})\in L^\infty(0,\infty; L^q(B_k)).
\]
This extends up to $k_0$ and so $P_{k_0}\in L^\infty(0,\infty; L^q(B'))$.
\end{proof}

\begin{remark}\label{Remark.heat}
Under the assumptions of Lemma \ref{lemma:Pkq} and
by classical estimates for the heat semi-group,
\[
\|e^{t\Delta} ( u_0\chi_B) \|_{L^\infty(B')} (t)\lesssim t^{-\frac 3 {2q}} \|u_0\|_{L^q(B)}.
\]
Note that, combining \eqref{A1} and \eqref{A2},
\EQ{
\|e^{t\Delta} (u_0(1-\chi_B)) \|_{L^\infty(B')}(t) \lesssim_{T} \|u_0\|_{L^{3,\infty}}t^{-\frac 3 {2q}},
}provided $t<T$, for any given time $T$.
Hence,
\[
\| e^{t\Delta}u_0 \|_{L^\infty(B')}(t)\lesssim_{u_0,T} t^{-\frac 3 {2q}}.
\]
\end{remark}

\section{Proof of Theorem \ref{thrm.main2}}

Our foundation for the proof of Theorem \ref{thrm.main2} is the local smoothing result of Jia and \v Sver\'ak \cite{JS}, which we presently restate. Note that $L^2_\mathrm{uloc}$ is the space of uniformly locally square integrable functions and is defined by the norm
\[
\|f\|_{L^2_\mathrm{uloc}}^2 :=\sup_{x_0\in {\mathbb R }^3}\int_{B_1(x_0)}|f|^2\,dx.
\]
Let $E^2$ denote the closure of $C_c^\infty$ in the $L^2_\mathrm{uloc}$ norm.
 $L^{3,\infty}$ embeds in $E^2$ (see the appendix of \cite{BT8}).
Local smoothing as presented below refers to local energy solutions (a.k.a.~local Leray solutions using the terminology of \cite{JS}; see also \cite{BT8,KS,LR}).
It is straightforward to show that weak $L^{3,\infty}$-solutions are local energy solutions.

\begin{theorem}[Local smoothing {\cite[Theorem 3.1]{JS}}]\label{theorem:JSlocalsmoothing}
 Let $u_0\in E^2$ be divergence free. Suppose $u_0|_{B_2(0)}\in L^p(B_2(0))$ with $\|u_0\|_{L^p(B_2(0))}<\infty$ and $p>3$. Decompose $u_0 =U_0+U_0'$ with $\div U_0 =0,\,U_0|_{B_{4/3}} = u_0,\, \operatorname{supp} U_0 \Subset B_2(0)$ and $\|U_0\|_{L^p({\mathbb R }^3)}<C(p,\|u_0\|_{L^p(B_2(0)})$. Let $U$ be the locally-in-time defined mild solution to \eqref{eq.NS} with initial data $U_0$. Then, there exists a positive $T=T(p,\|u_0\|_{L^2_\mathrm{uloc}},\|u_0\|_{L^p(B_2(0))})$ such that any local energy solution $u$ with data $u_0$ satisfies
 \EQ{
 \|u-U\|_{C^\gamma_{par}(\overline {B}_{\frac{1}{2}}\times [0,T])}\le C(p,\|u_0\|_{L^p(B_2(0))},\|u_0\|_{L^2_\mathrm{uloc}}),
 }
 for some $\gamma=\gamma(p)\in(0,1)$.
\end{theorem}

See also \cite{BP2020,KMT,KMT2,Kwon} for more recent work on local smoothing which allows locally critical data which is also locally small; the above statement on the other hand is for locally sub-critical data. The dependence on $\|u_0\|_{L^2_\mathrm{uloc}}$ can be replaced with $\|u_0\|_{L^{3,\infty}}$, which is why $L^2_\mathrm{uloc}$ is not mentioned in Theorem \ref{thrm.main2}.

\begin{proof}[Proof of \cref{thrm.main2}]
Without loss of generality, assume $B:=B_2(x_0)$ is centered at $x_0=0$. Assume $u_0|_B \in L^p(B)$. Let $U_0$ be a localization of the data to $B$ such that $u_0=U_0$ in $B_{4/3}(0) \subset B$, $\mathop{\mathrm{supp}} U_0 \Subset B$. This is done via a Bogovskii map \cite{GGP} as per the decomposition in \cref{theorem:JSlocalsmoothing}. Let $U$ be the locally-in-time defined mild solution to \eqref{eq.NS} with data $U_0$. Define $\{B_k\}_{k=0}^\infty$ to be a collection of nested balls centered at $0$, with radii $\alpha^{k}/2$, for some $\alpha\in(0,1)$ to be specified later.   Then, recalling $P_0=e^{t\Delta}u_0$,
\EQ{
|u-P_0|(x,t)  \le& |u-U|(x,t) +|U-e^{t\Delta}U_0|(x,t) +|e^{t\Delta}(U_0-u_0)|(x,t) \\
	=:& I_1(x,t)+I_2(x,t)+I_3(x,t).

In the definition of $C^\gamma_{par}(\overline {B}_{\frac{1}{2}}\times [0,T])$, the exponent in the time-variable modulus of continuity is ${\gamma} /2$.
By local smoothing \eqref{theorem:JSlocalsmoothing} and the fact that $\|u_0\|_{L^2_\mathrm{uloc}}\lesssim \|u_0\|_{L^{3,\infty}}$, there exists $T = T(p,u_0)>0 $ so that \[I_1(x,t) \lesssim_{p,u_0} t^{\frac {\gamma} 2},\] for some ${\gamma}={\gamma}(p) \in (0,1)$, $x\in B_0$ and  $0<t<T$.

For $I_2$, by \eqref{mild.est},  for any $p\in (3,\infty]$ and $0<t<T$
\EQ{
 I_2(x,t) &\leq \| B(U,U)\|_{L^\infty({\mathbb R }^3)}(t)\\&\lesssim t^{\frac 1 2 -\frac 3 {2p}} \|U\|_{L^\infty (0,T;L^p)}^2 \lesssim t^{\frac 1 2 -\frac 3 {2p}} \|U_0\|_{L^p}^2,
}
where we possibly re-define $T$ to make it smaller than the timescale of existence for the  strong solution to \eqref{eq.NS}, i.e.~$T\lesssim \| U_0 \|_{L^p}^{-2p/(p-3)}$, and the time-scale coming from Theorem \ref{theorem:JSlocalsmoothing}.

Noting that $U_0-u_0=0$ in $B_{4/3}$, the last part, $I_3$, is broken into integrals over a shell and a far-field region as follows,
\EQ{I_3(x,t)
 \lesssim& \left(\int_{\frac 4 3\le|y|< 2}+\int_{|y|\ge 2}\right)t^{-\frac 3 2} e^{-\frac{|x-y|^2}{4t}} |U_0-u_0|(y) \, dy

 =:I_{31}(x,t)+I_{32}(x,t).
 }
For $I_{31}$, using the fact that $U_0$ was solved for via a Bogovskii map, and therefore $\|U_0\|_{L^p({\mathbb R }^3)} \lesssim \|u_0\|_{L^p(B)}$, we have for all $0<t<T$ and $x\in B_0$ that
\EQ{
I_{31}(x,t)&\lesssim
t^{-\frac 3 2} e^{-\frac{(\frac 4 3-\frac 1 2)^2}{4t}} \| U_0 - u_0\|_{L^p (\frac 4 3\le|y|<2)}(t)
\lesssim_{u_0,p} t^{\frac {\gamma} 2}.}
For $I_{32}$, by \cref{lem:heatL321}, the fact that $U_0(y) \equiv 0$ for $|y|\ge 2$, and taking $x\in B_0$ and $0<t<T$, we have that
\EQ{I_{32}(x,t)
 &\lesssim \int_{|y|\ge 2}t^{-\frac 3 2} e^{-\frac{|x-y|^2}{4t}} |u_0|(y) \, dy \\&\lesssim t^{-\frac 3 2} \|u_0\|_{L^{3,\infty}}\left\|\|e^{-\frac{|x-y|^2}{4t}}(1-\chi_{B}(y))\|_{L^{\frac 3 2,1}_y}\right\|_{L^\infty_x(B_0)}\\
&\lesssim_{u_0} t^{-\frac 3 2}e^{\frac{-(2-\frac 1 2)^2}{4t}}
\lesssim_{p,u_0} t^{\frac {\gamma} 2}.}
 Therefore,
 \[\|u-P_0\|_{L^\infty(B_0)}(t) \lesssim_{p,u_0} t^{\min\{\frac {\gamma} 2, \frac 12 - \frac 3 {2p}\}},\]
where the dependence on $u_0$ is via the quantities $\|u_0\|_{L^p(B)}$ and $\|u_0\|_{L^{3,\infty}}$.

We inductively extend this estimate to higher Picard iterates. Fix $\sigma$ as in the statement of the theorem.
Recursively define the sequence $\{a_k\}$ by $a_{k+1} =\min\left\{\sigma, 1/2 -3/(2p)+a_k\right\}$ and $a_0 = \min\{{\gamma}/2, 1/2 -3/(2p)\}$.
Assume for induction that
\[
 \|u-P_k\|_{L^\infty(B_k)} \lesssim _{k,\alpha, p,u_0} t^{a_k},
\]
for $0<t<T$ and the dependence on $u_0$ is via the same quantities as above.
Note that
\EQ{
 |u-P_{k+1} |(x,t) \leq& |B(u-P_k, u-P_k)| + |B(u-P_k,P_k)|+ |B(P_k,u-P_k)| \\=:& J(x,t)+K(x,t)+L(x,t).
}

We split $J$ further as
\EQ{J(x,t) &\le | B((u-P_k) \chi_{B_k} , u-P_k)|+| B((u-P_k) (1-\chi_{B_k}) , u-P_k)|\\&=: J_{1}(x,t)+J_{2}(x,t).}
For the near-field, $J_1$, we use the inductive hypothesis to obtain for that, for $0<t<T$,
\EQ{\label{ineq:J1}
	\|J_{1}\|_{L^\infty(B_{k+1})}(t)
	\lesssim \int_0^t \frac 1 {\sqrt {t-s}} \| u-P_k \|_{L^\infty(B_k)}^2 \,ds\\
	&\lesssim_{k,\alpha, p,u_0} t^{\frac 1 2 +2a_k} \lesssim_{k,\alpha, p,u_0} t^{\frac 1 2 -\frac 3{2p} +a_k}.
}
Considering $J_{2}$, for  $0<t<T$, we have by \eqref{oseen.estimate} that
\EQ{\label{ineq:J2}
\|J_2\|_{L^\infty(B_{k+1})}(t)&\lesssim \int_0^t
	\int_{|x-y|>\frac1 2 \alpha^k-\frac 12\alpha^{k+1}} \frac 1 {|x-y|^4} |u -P_k|^2(y,s)\,dy\,ds \\&\lesssim \frac t {(\alpha^{k}-\alpha^{k+1})^4} \| u -P_k\|_{L^2}^2(t)\lesssim_{\alpha,k,u_0} t^{\frac 3 2},
}
where we used the version of \eqref{ineq.BSSdecay} for higher Picard iterates \cite{AB}.

The terms $K$ and $L$ are treated identically and we only consider $K$. We begin by further splitting $K$ as
\[K(x,t) \leq | B((u-P_k) \chi_{B_k} , P_k)|+| B((u-P_k) (1-\chi_{B_k}) , P_k)|=: K_1(x,t)+K_2(x,t)
.\] For the near-field $K_1$ and for $0<t<T$ we have
\[
\| K_1\|_{L^\infty(B_{k+1})}(t) \lesssim \int_0^t \frac 1 {(t-s)^{\frac 1 2 + \frac 3 {2p}}} \| u-P_k \|_{L^\infty(B_k)}(s) \| P_k\|_{L^p(B_k)}(s)\,ds.
\]
By \cref{lemma:Pkq},  $\sup_{0<t<\infty} \| P_k\|_{L^p(B_k)} <\infty$. Note that $1/2 + 3/(2p)<1$ precisely if $3<p$. Hence,
\EQ{\label{ineq:K1}
\| K_1\|_{L^\infty(B_{k+1})}(t)

 &\lesssim_{k,\alpha, p,u_0} t^{\frac 1 2 - \frac 3 {2p}+a_k},
}
for $0<t<T$ by the inductive hypothesis.
For the far-field $K_2$, using  \cref{cor:decay} and taking $x\in B_{k+1}$, $0<t<T$ and $q\in (3/2,3)$, we have by \eqref{oseen.estimate} that
\EQ{
K_2(x,t) &\lesssim  \int_0^t \int_{B_k^c} \frac {1} {(|x-y| +\sqrt{t-s})^4} |u-P_k| |P_k| \,dy\,ds
\\ & \lesssim
\bigg\| \frac {1-\chi_{B_k}(\cdot)} {|x-\cdot |^4} \bigg\|_{ L^{r'}(0,T; L^{q'})} \|P_k \|_{L^\infty(0,T;L^{3,\infty})} \| u-P_k\|_{L^r(0,T;L^q)}
\\&\lesssim_{k,q,u_0} t^{\frac 1 {r'}+\frac 1 2},
}
where
\[
1 = \frac 1 q +\frac 1 {q'} +\frac 1 3\text{ and }
1 = \frac 1 r +\frac 1 {r'}.
\]
Letting
\[
r= \frac {2q} {2q-3},
\]
we have
\[
\frac 1 {r'} = \frac 3 {2q}.
\]
\begin{comment}
using the identity\snt{Use lemma here instead of expanding?}
\EQ{
 u-P_k =& B(u-P_{k-1},u-P_{k-1})+B(u-P_{k-1},P_{k-1})\\&+B(P_{k-1},u-P_{k-1})
 ,
}
we bound $K_2$ as
\EQN{K_2\lesssim&\int_0^t \int_{B_k^c} \frac {|P_k|} {(|x-y| +\sqrt{t-s})^4} |B(u-P_{k-1},u-P_{k-1})|\,dy\,ds\\
	&+\int_0^t \int_{B_k^c} \frac {|P_k| } {(|x-y| +\sqrt{t-s})^4}|B(u-P_{k-1},P_{k-1})+B(P_{k-1},u-P_{k-1})|\,dy\,ds\\
	&=: K_{21}+K_{22}.
}
For, $K_{21}$, by O'Neil's inequalities and Lemma \ref{lemma.decay}, for $q\in (3/2,3)$ to be specified momentarily and
\[
r= \frac {2q} {2q-3},
\]
that for $x\in B_{k+1}$,
\EQ{
 &\int_0^t \int_{B_k^c} \frac 1 {(|x-y| +\sqrt{t-s})^4} |P_k|\,|B(u-P_{k-1},u-P_{k-1})| \,dy\,ds \\
 &\lesssim
 \bigg\|\frac {1-\chi_{B_k}(y)} {|x-y|^4} \bigg\|_{L^{r'}_t L^{q',\infty}_y}
 \| P_k\|_{L^\infty_t L^{3,\infty}_x}\|B(u-P_{k-1}, u-P_{k-1})\|_{L^{r}_tL^{q,1}_y}
 \\&\lesssim_{k,\alpha,q} t^{\frac 1 2 + \frac 1 {r'}},

}
where
\[
 \frac 1 {r'} = 1-\frac 1 r =\frac 3 {2q};
\quad
\frac 1 3+\frac 1 q+\frac 1 {q'} = 1,
\]
and we have used that the first term satisfies
\[
 \sup_{x\in {\mathbb R }^3} \bigg\|\frac {1-\chi_{B_k}(y)} {|x-y|^4} \bigg\|_{L^{r'}_t L^{q',\infty}_y} \leq t^{\frac 1 {r'}} \bigg\|\frac {1-\chi_{B_k}(y)} {|y|^4} \bigg\|_{ L^{q',\infty}_y} \lesssim_k t^{\frac 1 {r'}} \bigg\|\frac {1-\chi_{B_k}(y)} {|y|^4} \bigg\|_{ L^{1}_y}\lesssim_k t^{\frac 1 {r'}}.
\]
The remaining term, $K_{22}$, satisfies the exact same bound using Lemma \ref{lemma.decay2} instead of \ref{lemma.decay}.
\end{comment}
Observe that $1/r'<1$ and
\[
 \lim_{q\to {\frac 3 2}^+} \frac 1 {r'} = 1.
\]
Therefore, for any $\sigma <3/2$, by taking $q>3/2$ sufficiently close to $3/2$,
\EQ{\label{ineq:K2}
 K_2(x,t)\lesssim_{k,\sigma,u_0,q} t^{\sigma}.
}
Altogether, \eqref{ineq:J1},\eqref{ineq:J2},\eqref{ineq:K1}, and \eqref{ineq:K2} imply that, for $0<t<T$,
\[
 \|u-P_{k+1}\|_{L^\infty(B_{k+1})}(t) \lesssim_{k,\alpha, p,u_0,q} t^{a_{k+1}},
\]
for $k\ge 0$ and any $\sigma<3/2$.
Note that
\[
a_{k+1} = \min \left\{ \sigma , (k+1)\bigg(\frac 1 2 -\frac 3 {2p}\bigg) +a_0 \right\}.
\]
Choose $k_0$ to be the smallest natural number so that
\[
k_0\bigg(\frac 1 2 -\frac 3 {2p}\bigg) +a_0 \geq \sigma.
\]
Then, $a_{k_0}=\sigma$ and $a_{k}<a_{k-1}$ for $k=1,\ldots,k_0$.
Choose $\alpha$ so that $\rho = \alpha^{k_0}/2$.
It follows that
\[
 \|u-P_{k_0}\|_{L^\infty(B_{\rho}(x_0))}(t) \lesssim_{\rho, p,\sigma,u_0} t^{\sigma}.
\]

Regarding the asymptotic expansion, we observe that for $1\leq k\leq k_0$ and $(x,t)\in B_\rho(x_0)\times (0,T)$,
\[
u = P_{k_0}+\mathcal O(t^\sigma),
\]
and
\[
|P_{k}-P_{k-1}|(x,t)\le |u-P_k|(x,t)+|u-P_{k-1}|(x,t)=\mathcal O(t^{a_{k-1}}).
\]
Hence,
\EQ{
u(x,t) &= \underbrace{P_{0} + \sum_{k=1}^{k_0} (P_{k}-P_{k-1})(x,t)}_{=P_{k_0}} + \mathcal O(t^\sigma)
\\&=\mathcal O (t^{-\frac 3 {2p}})
+\sum_{k=0}^{k_0-1} \mathcal O(t^{a_k}) + \mathcal O(t^\sigma) = \sum_{k=-1}^{k_0} \mathcal O(t^{a_k}),
}
where we are letting $a_{-1}=-3/(2p)$   and are using Remark \ref{Remark.heat} to obtain the asymptotics for $P_0$.
\end{proof}

\begin{comment}
\textbf{Thought:} If we can show $|P_{k+1}-P_k|\lesssim t^{a_k}$ where $a_k$ isn't truncated at $a_{k_0}$. Then,
\EQ{
u-P_{\bar k} &= B(P_{k-1},P_{k-1})- B(u,u)
\\&\leq B(P_{k-1},P_{k-1}) - B(P_{k-2},P_{k-2})+ B(P_{k-1},P_{k-1})- \cdots + B(P_{k_0},P_{k_0})- B(u,u),
}
which leads to an asymptotic expansion with arbitrarily many terms. In this case we may as well take $k_0 = 0$. Presumably something breaks down here. I think $K_2$ doesn't improve.
\end{comment}

\section*{Acknowledgements}

The research of Z.~Bradshaw is supported in part by the Simons Foundation.

redpajama_set_name: RedPajamaArXiv


text:
Sigle
 Ubuntu Touch – sistema operativo per smartphone sviluppato inizialmente da Canonical e poi da UBports
 Ultrasonic testing – metodologia dei controlli non distruttivi che utilizza ultrasuoni
 Universal time – tempo universale, scala di tempo
 Università di Tartu – università estone
 Università di Teheran – università iraniana
 University of Tampa – università statunitense
 University of Tennessee – università statunitense
 University of Texas – università statunitense
 Unreal Tournament – videogioco

Codici
 UT – codice vettore IATA della compagnia aerea russa UTair (Aeroporto di Tjumen'-Roščino, Russia)
 UT – codice ISO 3166-2:GY di Upper Takutu-Upper Essequibo (Guyana)
 UT – codice ISO 3166-2:LT della contea di Utena (Lituania)
 UT – codice ISO 3166-2:NL della provincia di Utrecht (Paesi Bassi)
 UT – codice ISO 3166-2:US dello Utah (Stati Uniti)

Musica
 Ut – antico nome della nota musicale Do
 Ut – gruppo musicale italiano degli anni settanta
 UT – album dei New Trolls del 1972

Fumetti
 Ut – fumetto del 2016 edito dalla Sergio Bonelli editore

Altri progetti

redpajama_set_name: RedPajamaWikipedia


text:
Flight & hotel offers to Vancouver
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THE HISTORY OF OUTSOURCING AMERICAN JOBS
WebEven though American jobs are being outsourced, American consumers benefit from lower prices that outsourcing makes possible and the investment that businesses doing the outsourcing make. WebOutsourcing, or sending off a job outside the company,may seem like more recent phenomena which started after the internet boom in mid s, but the fact that outsourcing s history is as old as America may surprise us. During the early years of America, the manufacturing of clippers ships sails was sent to Scotland, and the raw . Research papers, much like essays, are an integral part of academia and often a cause for jitters among students. While the task of writing them involves several hours of brainstorming and painstaking research, the first and one of many important aspects that students often struggle with is choosing a suitable topic.
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WebJun 16, · Job outsourcing can help American companies compete by keeping prices low, but it has a negative effect on U.S. employment. America has lost jobs to . Human factors and ergonomics (commonly referred to as human factors) is the application of psychological and physiological principles to the engineering and design of products, processes, and www.refleksiya-absurda.ru primary goals of human factors learning are to reduce human error, increase productivity, and enhance safety, system availability, and comfort with a specific focus on the . Should we curtail outsourcing and offshoring to protect jobs? for example, a Dell customer service center in India to serve American clients. Nov 08, · DeimosC2: What SOC Analysts and Incident Responders Need to Know About This C&C Framework. This report provides defenders and security operations center teams with the technical details they need to know should they encounter the DeimosC2 C&C framework. Outsourcing white collar jobs benefits the US because allows for diversity and growth in companies. US multinational companies (companies which function in. Daily U.S. military news updates including military gear and equipment, breaking news, international news and more. Both the U.S. and the U.K. have liberal employment and labor laws that allow companies greater flexibility in reassigning tasks and eliminating jobs. This. Different Types of Sourcing Methods. There is a wide range of sourcing methods and one should bear in mind three key points: 1) sourcing options are often not mutually exclusive 2) sourcing methods can be applied to both front and back office functions and 3) strategic sourcing as a procurement process can be seen as finding the appropriate solution for your . Companies looked to workers in foreign countries to work in factories due to lower labor costs. Once company owners were confident they could outsource. Outsourcing of course is not new. It involves moving work from one company to another that can do the same job at a lower cost. Generally, it is domestic.


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Sam Clark, Senior Project Engineer at CPFD Software, will present a paper titled "Industrial Case Studies of Solutions Enabled by Simulation of Fluidized Systems" as part of the Symposium on Theory and Applications of Multi-phase Flow Reactors at the Spring 2017 Meeting of the Korean Institute of Chemical Engineers (KIChE). The Spring Meeting will be held on Jeju Island from April 26-28, 2017.
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text:
Evans Heights är en kulle i Antarktis. Den ligger i Östantarktis. Nya Zeeland gör anspråk på området. Toppen på Evans Heights är  meter över havet.

Terrängen runt Evans Heights är kuperad österut, men västerut är den platt. Den högsta punkten i närheten är  meter över havet,  kilometer norr om Evans Heights. Trakten är obefolkad. Det finns inga samhällen i närheten.

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Kullar i Östantarktis
Nya Zeelands anspråk i Antarktis
Berg i Antarktis 500 meter över havet eller högre


text:
Updated 2018 by Abel Bachelor. Proudly created with Wix.com
The 6th Connecticut Regiment
Zadock Hawkins
Birth:ABT 1733 Derby, New Haven Co, Connecticut
Marriage: 4 AUG 1754 in Derby, Connecticut
Land Record: 23 SEP 1754
Note: Bought 140 acres from Zadock's brother Samuel in the township of Derby lying on "Jack's Hill." After spending just one year on this property, they began to sell it, selling two pieces the same day – 26 AUG 1755. One piece went to Zadock's sister, Lois. The other to Lydia's father.
Birth of Daughter: Damaris Hawkins
Place: Connecticut
Militia: MAY 1756 during Frech and Indian War
Note: Zadock was called upon to serve one week and one day in the 1st Company of the Second Connecticut Regiment under Capt. Isaacs and Col. David Wooster. His service was described as a "victualizing role," which probably meant he was involved with getting food and supplies to some of the other soldiers.
Birth of Son: William Wilmot Hawkins
Place: New Milford, Connecticut
Land Record: 18 JAN 1758
Note: Sold the rest of their "Jack's Hill" property back to his brother, Samuel, from whom he had purchased it.
Military Service: 05 APR 1758
Note: Zadock enlisted for a term of six months in the 11th Company of the Fourth Regiment under Captain Joseph Canfield and Col. David Wooster
Church: St. James Church in Derby, Connecticut
Birth of Son: Zadock Hawkins
Birth of Son: Peter Hawkins
Birth of Daughter: Lydia Hawkins
Birth of Son: Reuben Hawkins
Birth of Son: Eleazer Hawkins
Date: ABT 1768
Birth of Son: Thomas Hawkins
Place: Derby, Connecticut
Military Service: APR/MAY 1775
Note: Zadock joined the Colonial Army at the first call for troops in Litchfield, Connecticut, serving under Benjamin Hinman in the Fourth Connecticut Regiment. No doubt this call was in response to the clash between the Colonial Minutemen and the British Redcoats in the Battle of Lexington and Concord at the start of the Revolutionary War on 19 APR 1775. During this tour he marched on both Fort Ticonderoga and Crown Point in May of 1775 and most likely participated in the capture of these sites by Ethan Allen and his Green Mountain Boys before he was "discharged in the north" on 17 NOV 1775.
Military Service: 20 FEB 1777
Note: Zadock served in the Colonial Army in the Revolutionary War. He served in H Company of the 6th Connecticut Regiment of foot under Capt. Jonas Prentice, commanded by Lt. Col. David Dimon under Col. Return Jonathan Meigs.
Military Service: 31 AUG 1777
Note: Zadock was reported missing from duty because of illness.
Death: 20 OCT 1777 Died in service in the Revolutionary War (1)
Death: 15 DEC 1777 Died in service in the Revolutionary War (2)
NOTE (1): Taken from:
www.caskey-family.com/genealogy/ZadockHawkins1st.htm
NOTE (2): Listed as date of death in the Record of Service Connecticut Men in the War of Independence


A senior corporate counsel came to us via a recommendation from a former client. This corporate counsel was 'completely fed up' with his employer. In fact, things had got so bad that he had resigned by email in the middle of a night out of frustration. In the morning, however, he tried to change his mind. Our Client's employer maintained that the resignation stood and that our Client's employment was soon to end.
We immediately advised our Client of a little known case that states that it can be unfair dismissal for an employer to rely on a resignation made in the heat of the moment or where it can be said that the employee is not him or herself. This rule applies when the resignation is retracted quickly, as in this case. We used our knowledge and experience to negotiate a very favourable settlement for the Client. That Client wrote to us to say that we should put the following words on our website: 'I came to Simon at a time when I did not really have a case, he convinced the company that I did and got me an excellent result'.


Our Cirque Rouge Vintage Circus Costumes are an updated twist on the classic big top look! Inspired by photos from the 1920s, 30s, and 40s, these costumes harken back to a bygone cirque era and yet are cutting-edge 21st century concoctions that are sure to turn heads. Your dancers will make a stunning statement in these beauties, which range from striped crop tops and striped leggings, to adorned leotards, to fun and funky jackets. We've got everything need for your sassy ringleader, edgy clown, circus ballerina, tightrope walker, contortionist, or acrobat. So go on – plan that cirque - themed piece of your dreams! We've got your circus costumes covered.
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Q: Frequencies in unordered cartesian products This is a novice question. If I have an unordered cartesian product of some list with a repeat sequence, how can I calculate the frequency with which a member of that list occurs in the combinations?
For example given the list:
list = {1,2,3,4}

The unordered cartesian product with a repeat length of 3 should give:
(1,1,1),(1,1,2),(1,1,3)........(4,4,4)

How can I calculate the number of times 1 appears at least once in this set? If I change this to at least twice can this be done? I was looking around for a formula but could not find one. Thanks

redpajama_set_name: RedPajamaStackExchange


periodic inventory system Overview. A periodic inventory system only updates the ending inventory balance in the general ledger when a physical inventory count is conducted. Since physical inventory counts are time-consuming, few companies do them more than once a quarter or year.
Perpetual Inventory System. Perpetual means continuous. This is a system where a business keeps continuous, moment-to-moment records of the number, value and type of inventories that it has at the business. A computerized accounting system – where each item of inventory is linked to the electronic accounting records – creates a perpetual system.
Periodic Inventory System determines the cost of goods sold by adding merchandise purchase cost, beginning stock cost and deducting the end-stock cost.
In a periodic inventory system, the account inventory over and short does not arise because there are no accounting records available against which to compare the physical count.
A periodic inventory system is an accounting method in which the cost of goods sold is determined periodically, usually annually and typically not more frequently than quarterly. This differs from a perpetual inventory system in which the cost of goods sold is determined as necessary or in some cases continually.


You might wonder if the air pollution harm your baby, especially if you live in larger, industrial city? The answer is generally no, though you need to make some precautions. In 2016, a study at the University of Florida concluded that women living in heavily polluted urban areas were more likely to suffer high blood pressure and resulting preeclampsia during pregnancy. Another study in China by cheap steampunk clothing University, which followed children from birth until the age of five, concluded that high pollution levels during fetal development and in the first years of life led to a small drop in the child's IQ. The Harvard School of Public Heath has also recorded links between fine particulate pollution (emitted by fires, vehicles, and industrial smokestacks) and autism during the third trimester. Although these findings seem alarming, we already know that air pollution is bad for our health, so, while we need much more research to be certain of the effects on a pregnant woman and her unborn baby, it makes sense to exercise caution whenever you can. Walk down the street at times of low traffic, avoiding rush hour, for example, and keep the windows at home closed at these times, too. If you can, invest in a home air purifier. Walk along quiet streets with traffic that keeps moving, or, ideally, through parks, rather than walking beside stationary traffic. And drink plenty of fluids to keep flushing out your system.
An air quality index is used by many government agencies around the world to notify the public of pollution forecasts on a daily basis. Although each one varies slightly, they usually cover measurements of ground-level ozone, particulate matter. Don't wear leather corset during this time. Check local news media for air quality forecasts and plan outdoor activities for days when particle and ozone levels are low. Pregnant women should always follow the advice for at-risk individuals.
You might also wonder if you live in an old house with cogs, that still has lead water pipes. Will drinking the tap water harm my baby? For health reasons, even when you aren't pregnant, you should avoid contact with toxic "heavy" metals as much as possible. Lead in water can leach from your blood and bones and enter your unborn baby's system, and has been known to slow brain development in some children, both before and after birth. Some countries have local government programs that subsidize (or even provide free) replacement of domestic lead water pipes. Talk to your local government about programs in your area and have the piping replaced if you can. Meanwhile, drink filtered water (a simple filter pitcher does a great job of removing impurities—including mercury and aluminum, as well as lead—from tap water) and use filtered water in cooking. If you prefer, use bottled mineral water, but make sure that it has the lowest available sodium levels.
What to wear during pregnancy?


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Copeland ist eine 2001 gegründete Indie-Rock-Band aus Lakeland, Florida. Die Band löste sich im Jahr 2010 auf und fand im April 2014 erneut zusammen.

Geschichte
Seit der Wiedervereinigung besteht die Gruppe aus Sänger Aaron Marsh, den Gitarristen Bryan und Stephen Laurenson, sowie aus Schlagzeuger Jonathan Bucklew.

Die Gruppe veröffentlichte vor ihrer Trennung mehrere EPs sowie fünf Studioalben. Ihr sechstes Album namens Ixora erschien im November 2014 und konnte sich wie drei Alben zuvor in den nationalen Albumcharts platzieren.

Diskografie

Alben
 2003: Beneath Medicine Tree (The Militia Group)
 2005: In Motion (The Militia Group)
 2006: Eat, Sleep, Repeat (Columbia Records)
 2007: Dressed Up & In Line (Columbia Records)
 2008: You Are My Sunshine (Columbia Records)
 2014: Ixora (Tooth & Nail Records)
 2019: Blushing (Tooth & Nail Records)

Besetzung

Stellung zum Christentum
Oft wurde spekuliert ob es sich bei Copeland um eine christliche Rockband handele. Sänger Aaron Marsh sagte, dass einige (ehemalige) Mitglieder der Band gläubige Christen sind bzw. waren, die Band aber keine christlichen Botschaften vermittele.

Weiteres
Sleeping-with-Sirens-Sänger Kellin Quinn ist ein Fan der Band. Er benannte seine Tochter nach der Band. Aaron Marsh ist zudem im Lied Let Love Bleed Red der Band aus dem Debütalbum With Ears to See and Eyes to Hear zu hören.

Quellen

Weblinks
 Offizielle Website

Alternative-Rock-Band
US-amerikanische Band
Lakeland (Florida)


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There Is A Sita, Kaikeyi and Urmila In All Of Us
I recently read the book 'The Forest of Enchantments' by Chitra Banerjee Divakaruni, and I was truly enchanted and loved it.
It is another version of the Ramayana, Sita's version… also known as Sitayana. As soon as I read about the book I was fascinated, as till now I had really never thought that a different version of the Ramayana could exist. It gives us an insight into the life of Sita, right from her birth to when she gives up on her life as a final act of rebellion against patriarchy. I loved reading the book, found it truly enchanting and I couldn't tear myself away from it.
The part of the story where it is postulated that Sita is the daughter of Ravan had me most intrigued and glued. It suggests that Sita was actually the firstborn of Ravan and Mandodari, but his astrologers had predicted that his firstborn would cause the destruction of his entire lineage. Hence, Ravan ordered that the child be killed immediately. However, Mandodari could not see that happen and bribed the Asura to leave her in some distant land. Thousands of kilometers away in Mithila, Raja Janak found baby Sita and accepted her as a boon from the gods. Mandodari recognized Sita as her long lost daughter when she saw her in Lanka after Ravan had kidnapped her. But she was too scared to share her secret, though she tried to make Sita's life comfortable and save her from Ravan to the best of her abilities. Mandodari's character is complex, wise, maternal and never shown before.
We have all heard the Ramayana many, many times but here, the Voice of Sita, clear and strong, reaches deep in your heart and stays there. Sita is every woman in one woman. She has all the human emotions and failings, she falls prey to jealousy, despair, anger, lust, and compassion. She is not just revered as an incarnation of Devi, as the perfect wife, daughter in law and mother but rather as a strong character who has her opinions, thoughts, and wishes.
Sita is not just a dutiful, submissive wife as mythology makes us believe, she is a thinker and questions society and its rules. She argues with her mother why Mithila cannot be ruled by a woman and only a man. She objects to the horrific mutilation of Suparnarakha, and considers it as unnecessary and a show of superiority by the brothers Ram and Lakshman.
Sita's agnipariksha has been held in history as the highest ideal in the virtue of her being pativrata. But in Sita's version, it is more of a fiery protest against a very public humiliation. The second time when she is asked to publicly prove her chastity, her refusal and asking the earth to subsume her is the biggest act of rebellion against patriarchy. She questions Ram publicly in court, that why does she not deserve the justice that other men and women in Ayodhya get, and then gives up her life and her role as wife and mother.
The novel is not just about Sita but also explains the other female characters who are otherwise ignored. Urmila, whose great sacrifices go unnoticed, Mandodari, written off like a demon, Suparnarakha, wronged by two men, and the ambiguity unseen earlier in Kaikeyi's evil ministrations. The women characters say " Write our story too, for always we've been pushed into corners, trivialized, misunderstood, blamed, forgotten or maligned and used as cautionary tales ."I just loved this statement. I read it several times and couldn't get over how beautifully it conveyed what all women have felt at some point or the other, be it the in the Ramayana in the Treta Yug or in the twenty-first century. Sita comes across as a dutiful but bold daughter, protective sister, skilled healer, loving yet strong-willed wife, sensual lover, strong mother, nurturing, helpful, a thinker. She is a feminist and yet knows when to concede, mindful and empathetic. There is a little bit of Sita and for that matter of Urmila, Mandodari and Kaikeyi in all of us.
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Red Wings captain traded to Flin Flon
April Zielke / Weyburn Review
January 8, 2019 11:41 AM
The Weyburn Red Wings have made a gigantic move and have traded former captain Cade Kowalski to the Flin Flon Bombers in exchange for Flin Flon forward Dawson McKenzie and future considerations. Kowalski has mysteriously been missing from the lineup since the Weyburn Red Wings fired previous head coach Wes Rudy, and lost 8-2 to the Estevan Bruins on December 15.
Kowalski was in his second year with the Weyburn Red Wings, and put up 14 goals and 25 assists this season. For the 2019/20 season he has committed to Ferris State University Division I.
McKenzie is a 19 year old who was currently in his second year with the Bombers and has put up 9 goals and 13 assists this season.
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CLEVELAND (Reuters) - Police in Cleveland say they aim to avoid mass arrests at the protests planned for next week's Republican National Convention, but preparations by the city's courts to process up to 1,000 people a day have some civil rights activists worried.
Thousands of people from across the country are expected in the city to protest the expected presidential nomination of New York businessman Donald Trump, who has vowed to build a wall along the U.S.-Mexico border and restrict immigration from countries with large Muslim populations if elected.
Supporters and opponents of Trump have clashed at several of his campaign events.
Police have vowed to honor protesters' rights of free expression, which are protected by the First Amendment of the U.S. Constitution, and avoid mass arrests.
"We don't want anybody to trample on anybody else's rights," Cleveland police chief Calvin Williams told a news conference on Tuesday.
But memories of recent heavier-handed approaches are fresh in the heavily Democratic, majority black Ohio city of 388,000 people.
"I don't want to be a naysayer here and rule out the possibility that everything is going to be hunky-dory ... but knowing how the Cleveland Police Department has handled situations in the past, I just don't have confidence that it's going to work," said Terry Gilbert, an attorney who has handled criminal and civil rights cases in the city for more than four decades.
"Until I see the actual situation next week, I'm going to be worried," Gilbert said.
Gilbert pointed to the May 2015 arrests of 71 people following the acquittal of a police officer who fired 137 shots following a high-speed 2012 car chase, killing a black man and woman.
The arrested protesters were held for more than 36 hours over the Memorial Day weekend, and four alleged in a lawsuit brought by the American Civil Liberties Union that police intentionally kept them in custody longer to prevent the protest from reforming.
Cleveland paid $250,000 to secure 200 extra rooms in the Cuyahoga County jail, according to the Republican National Committee budget.
Cleveland Municipal Court officials said they would be ready to process a large volume of people quickly, with staff scheduled to work in two 10-hour shifts keeping the court operating from 5 a.m. to 1 a.m. each day.
"We are ready," said Ed Ferenc, a spokesman for the court. "We'll have staff here till 1 a.m. If we have to do a docket at 10:30 at night, we'll do it."
The United States has seen hundreds of protests over the past two years following a series of high-profile police killings of black men. The vast majority of the protests have been peaceful, although they have been punctuated with bursts of rioting, arson and looting.
The ACLU plans to be out in force to ensure that people are not arrested for legal protests, said Christine Link, the group's executive director in Ohio.
"Let's not equate a lot of protesters with violence," Link said. She noted the group would be keeping careful watch on the whereabouts of anyone arrested to ensure they are charged and released quickly.
At the 2004 Republican National Convention in New York, hundreds of protesters were swept up and pushed into pens on the Hudson River.
With temperatures expected to reach 90 degrees F (32 C) most days, the health of detainees will be a concern, she said.
"What we're worried about is that they're not saying where they are booking people, they are being vague about it and that's not good," Link said. "That's an attempt to hide the cheese."


Els alumnes de 2n d'ESO han fet una visita a la fàbrica Danone.
This is a common problem faced by most small business enterprises. They invest a lump sum behind advertisements and if the campaign does not come off, it's a major loss for them.


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Markieff Morris Fined $10K for Expressing Desire to be Traded by Suns
Joseph Zucker@@JosephZuckerTwitter LogoFeatured ColumnistSeptember 8, 2015
Rocky Widner/Getty Images
The NBA has fined Markieff Morris $10,000 for making "a public statement detrimental to the (league)" following his tweet that he'd no longer be a member of the Phoenix Suns at the start of the 2015-16 season, per Dan Feldman of Pro Basketball Talk, citing a release from the league.
The relationship between Morris and the Suns has been fractured ever since the team traded his brother, Marcus, to the Detroit Pistons earlier in the summer. Marcus Morris was particularly vocal about his displeasure with the Phoenix organization, per the Associated Press (via ESPN.com): "Everybody knew how bad I wanted to play with my brother. Phoenix knew. For them to trade me without consent or telling or anything like that was kind of like a, I would say slap in the face, because of the contract I took from those guys and the money I took from them. That was kind of a slap in the face."
Then in August, Markieff drew a line in the sand and told Keith Pompey of the Philadelphia Inquirer that he wanted out of Phoenix: "One thing for sure, I am not going to be there. If you want to put that out there, you can put that out. I don't give a [freak]. I am not going to be there at all. That's just what it is."
On Sept. 3, Morris doubled down on his stance:
Keef Morris @Keefmorris
My future will not be in Phoenix.... #thatisall #backtothegrind #FOE
The Suns are undoubtedly in a tough position.
They cannot simply offload Morris after he was one of their better players last year. He averaged 15.3 points and 6.2 rebounds per night, while Phoenix had a minus-5.4 net rating with him off of the court, per NBA.com. Given the fact they have little bargaining position, the Suns couldn't get an equal return for Morris in a trade, and no free agent still available is even comparable to Morris in terms of skill.
Phoenix will hope Morris' anger subsides before the regular season begins in October. Should he continue to be a disruptive presence, though, management may be forced to take drastic action and bite the bullet, similar to the Detroit Pistons' waiving of Josh Smith last year.
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