text:
\section{Introduction}

The motion of inviscid incompressible 2D flows is described by the 2D Euler equations,
\begin{equation}
\partial_t \boldsymbol{u} + (\boldsymbol{u} \cdot \nabla) \boldsymbol{u} + \nabla p = 0, \qquad \operatorname{div} \boldsymbol{u} = 0, \qquad \boldsymbol{u}(\boldsymbol{x}, 0) = \boldsymbol{u}_0(\boldsymbol{x}),  \label{Euler}
\end{equation}
where $\boldsymbol{u}=\boldsymbol{u}(\boldsymbol{x}, t)$ is the fluid velocity field and $p=p(\boldsymbol{x}, t)$ is the scalar pressure. Taking the curl of (\ref{Euler}) and defining the vorticity field by $\omega = \operatorname{curl} \boldsymbol{u}$, we obtain the transport equation for $\omega$,
\begin{equation}
\partial_t \omega + (\boldsymbol{u} \cdot \nabla) \omega = 0, \qquad \omega(\boldsymbol{x}, 0) = \omega_0(\boldsymbol{x}),  \label{vEuler}
\end{equation}
and $\omega_0 = \operatorname{curl} \boldsymbol{u}_0$. Then, the velocity $\boldsymbol{u}$ is recovered from the vorticity $\omega$ via the Biot-Savart law,
\begin{equation}
\boldsymbol{u}(\boldsymbol{x}) = \left( \boldsymbol{K} \ast \omega \right)(\boldsymbol{x}) = \int_{\mathbb{R}^2} \boldsymbol{K}(\boldsymbol{x} - \boldsymbol{y}) \omega(\boldsymbol{y}) d \boldsymbol{y}, \label{biot}
\end{equation}
in which $\boldsymbol{K}$ denotes a singular integral kernel defined by
\begin{equation*}
\boldsymbol{K}(\boldsymbol{x}) = \nabla^\perp G(\boldsymbol{x}) = - \frac{1}{2 \pi} \frac{\boldsymbol{x}^\perp}{\left| \boldsymbol{x} \right|^2}, \qquad G(\boldsymbol{x}) = - \frac{1}{2 \pi} \log{|\boldsymbol{x}|},
\end{equation*}
with $\nabla^\perp = (\partial_{x_2}, - \partial_{x_1})$ and $\boldsymbol{x}^\perp = (x_2, - x_1)$. Here, the function $G$ is the fundamental solution to the 2D Laplacian. If a passive particle is initially placed at the position $\boldsymbol{x}$, its evolution is governed by the following equation,
\begin{equation}
\partial_t \boldsymbol{\eta}(\boldsymbol{x}, t) = \boldsymbol{u} \left( \boldsymbol{\eta}(\boldsymbol{x}, t), t \right), \qquad \boldsymbol{\eta}(\boldsymbol{x}, 0) = \boldsymbol{x}, \label{eta}
\end{equation}
where $\boldsymbol{\eta}(\boldsymbol{x}, t)$ is called the Lagrangian flow map describing the position of the passive point $\boldsymbol{x}$ at time $t$. The solution of (\ref{eta}) yields that of (\ref{vEuler}) with the initial vorticity $\omega_0$ since it follows that
\begin{equation}
\omega(\boldsymbol{x}, t) = \omega_0 \left( \boldsymbol{\eta}(\boldsymbol{x}, - t) \right). \label{v-def}
\end{equation}
Hence, finding a solution to (\ref{eta}) with (\ref{biot}) and (\ref{v-def}) is equivalent to solving the problem (\ref{vEuler}) with (\ref{biot}). Regarding the initial value problem of the 2D Euler equations, the global existence of a unique weak solution has been established for $\omega_0 \in L^1(\mathbb{R}^2) \cap L^\infty(\mathbb{R}^2)$ \cite{Marchioro, Yudovich}. We also have global weak solutions with $\omega_0 \in L^1(\mathbb{R}^2) \cap L^p(\mathbb{R}^2)$ and $1 < p < \infty$, though the uniqueness is still open \cite{Diperna}. Moreover, the existence theorem can be extended to the case that $\omega_0$ belongs to the space of finite Radon measure on $\mathbb{R}^2$, denoted by $\mathcal{M}(\mathbb{R}^2)$, with a distinguished sign and its induced velocity $\boldsymbol{u}_0 \in L_{loc}^2(\mathbb{R}^2)$ \cite{Delort, Evans, Majda}. Note that this class includes vortex sheets but not point vortices.

In order to consider the vortex dynamics modeled by singular vortices in the space of Radon measure, such as vortex sheets or point vortices, it is effective to analyze the evolution of them in the regularized Euler flow, since the global existence of a unique solution is guaranteed. Actually, the Euler-$\alpha$ equations, which is a dispersive regularization of the Euler equations, have a unique global weak solution for initial vorticity in $\mathcal{M}(\mathbb{R}^2)$ \cite{Oliver} and point vortex solutions of them are used to analyze a singular vortex dynamics in ideal incompressible flows \cite{G.}. Another important example of the regularization is the vortex blob method that is introduced to compute the evolution of vortex sheets \cite{Chorin, Krasny}. Although the Euler-$\alpha$ equations and the vortex blob method are originally derived in different backgrounds and are usually applied to the different problems, we can derive both of them based on the Euler-Poincar\'{e} framework \cite{Holm(c)}. In the present paper, we investigate the Euler-Poincar\'{e} equations that come through the Euler-Poincar\'{e} framework and a generalization of both the Euler-$\alpha$ equations and the vortex blob method. We prove the existence of a unique global weak solution with its vorticity in the space of Radon measure. Moreover, we show that weak solutions of the Euler-Poincar\'{e} equations converge to those of the Euler equations in the limit of the regularization parameter provided the initial vorticity is integrable and bounded.

This paper is organized as follows. Section~\ref{general} presents the general framework of a regularization of the Euler equations based on the theory of the Euler-Poincar\'{e} equations. Especially, we observe two examples of the regularization, the vortex blob method and the Euler-$\alpha$ equations, in the viewpoint of the Euler-Poincar\'{e} framework in Section~\ref{blob} and \ref{alpha} respectively. In Section~\ref{Main}, we state the main results consisting of the global solvability of the Euler-Poincar\'{e} equations in the space of Radon measure and the convergence of weak solutions to those in the Euler equations for the bounded initial vorticity. The proof of the solvability and the convergence are shown in Section~\ref{proof1} and \ref{proof2} respectively.

\section{The Euler-Poincar\'{e} regularization}
\subsection{General framework}
In this section, we derive a regularized 2D Euler equations based on the framework given in \cite{Foias, Holm(c)}. Let $\boldsymbol{v}$ be an incompressible velocity field that we call {\it singular velocity}. We define the {\it regularized  velocity} by
\begin{equation}
\boldsymbol{u}_h(\boldsymbol{x}) = \left( h \ast \boldsymbol{v} \right) (\boldsymbol{x}) = \int_{\mathbb{R}^2} h\left( \boldsymbol{x} - \boldsymbol{y} \right) \boldsymbol{v}(\boldsymbol{y}) d\boldsymbol{y}, \label{Rvelo}
\end{equation}
where $h$ is a scalar valued function on $\mathbb{R}^2$. In this paper, we assume that $h$ is a integrable function and may have a singularity at the origin. Owing to the convolution with $h$, $\boldsymbol{u}$ is smoother than $\boldsymbol{v}$. Similarly, we define the {\it singular vorticity} $q$ and the {\it regularized vorticity} $\omega_h$ by
\begin{equation*}
q = \operatorname{curl} \boldsymbol{v}, \qquad \omega_h = \operatorname{curl} \boldsymbol{u}_h.
\end{equation*}
Note that it follows that $\operatorname{div} \boldsymbol{u} = 0$ and $\omega_h = h \ast q$ when the convolution commutes with the differential operator.

Now, we consider the Hamiltonian structure with the Hamiltonian,
\begin{equation*}
\mathscr{H} = \frac{1}{2} \int_{\mathbb{R}^2} \boldsymbol{v}(\boldsymbol{x}) \cdot \boldsymbol{u}_h(\boldsymbol{x}) d\boldsymbol{x},
\end{equation*}
and the Lagrangian flow map $\boldsymbol{\eta}_h$ induced by regularized velocity,
\begin{equation}
\partial_t \boldsymbol{\eta}_h(\boldsymbol{x}, t) = \boldsymbol{u}_h \left( \boldsymbol{\eta}_h(\boldsymbol{x}, t), t \right), \qquad  \boldsymbol{\eta}_h(\boldsymbol{x}, 0) = \boldsymbol{x}. \label{h-eta}
\end{equation}
Then, the Euler-Poincar\'{e} equations arise from an application of Hamilton's principle as follows.
\begin{equation}
\partial_t \boldsymbol{v} + (\boldsymbol{u}_h \cdot \nabla) \boldsymbol{v} - (\nabla \boldsymbol{v})^T \cdot \boldsymbol{u}_h - \nabla \Pi = 0, \label{REE}
\end{equation}
where $\Pi$ is a generalized pressure. Taking the curl of (\ref{REE}) with the divergence-free condition, we obtain the transport equation for the singular vorticity convected by the regularized velocity,
\begin{equation*}
\partial_t q + (\boldsymbol{u}_h \cdot \nabla) q = 0.
\end{equation*}
Note that the Biot-Savart law gives $\boldsymbol{u}_h = \boldsymbol{K} \ast \omega_h$. Then, owing to the relation $\omega_h = h \ast q$, we finally obtain the vorticity form of the Euler-Poincar\'{e} equations.
\begin{equation}
\partial_t q + (\boldsymbol{u}_h \cdot \nabla) q = 0, \qquad \boldsymbol{u}_h = \boldsymbol{K}_h \ast q, \qquad \boldsymbol{K}_h = \boldsymbol{K} \ast h. \label{RPDE}
\end{equation}
It is important to remark that the regularized kernel $\boldsymbol{K}_h$ satisfies
\begin{equation}
\boldsymbol{K}_h = \nabla^\perp G_h, \qquad - \Delta G_h = h, \label{KGh}
\end{equation}
where $G_h$ is a solution for the Poisson equation and thus we have $G_h = G \ast h$. Considering the definition of the singular integral kernel, we can see that a feature of the regularized equations (\ref{RPDE}) comes from how to regularize the delta function by the smoothing function $h$.

\label{general}

\subsubsection{The vortex blob method}
The vortex blob method is one of the most common regularizations of the Euler flow. The method was introduced to compute the vortex sheets' evolutions numerically\cite{Anderson, Chorin, Krasny} and has been applied to a variety of fluid dynamics, for instance see \cite{Leonard}. In this method, the regularized integral kernel $\boldsymbol{K}_\delta$ is given by
\begin{equation*}
\boldsymbol{K}^\delta(\boldsymbol{x}) = - \frac{1}{2 \pi} \frac{\boldsymbol{x}^\perp}{|\boldsymbol{x}|^2 + \delta^2}.
\end{equation*}
This vortex blob regularization can be derived from the Euler-Poincar\'{e} framework. Indeed, replacing the function $h$ in (\ref{Rvelo}) by
\begin{equation*}
h^\delta(\boldsymbol{x}) = \frac{1}{2 \pi \delta^2} \psi\left( \frac{|\boldsymbol{x}|}{\delta} \right), \qquad \psi(r) = \frac{2}{(r^2 + 1)^2},
\end{equation*}
we find that $\boldsymbol{K}^\delta$ comes through the solution for the following Poisson equation.
\begin{equation*}
\boldsymbol{K}^\delta = \nabla^\perp G^\delta, \qquad - \Delta G^\delta = h^\delta.
\end{equation*}
It is easily confirmed that the solution $G^\delta$ is given by
\begin{equation*}
G^\delta(\boldsymbol{x}) = G^\delta_r(|\boldsymbol{x}|) = - \frac{1}{2 \pi} \log{\sqrt{|\boldsymbol{x}|^2 + \delta^2}},
\end{equation*}
and $\boldsymbol{K}^\delta$ is rewritten by
\begin{equation*}
\boldsymbol{K}^\delta(\boldsymbol{x}) = \boldsymbol{K}(\boldsymbol{x})\Psi\left( \frac{|\boldsymbol{x}|}{\delta} \right), \qquad \Psi(r) = - 2 \pi r \frac{dG^{\delta=1}_r}{dr}(r) = \frac{r^2}{r^2 + 1}.
\end{equation*}
Note that we have $\Psi'(r) = r \psi(r) $. Hence, both $G^\delta$ and $\boldsymbol{K}^\delta$ have no singularity, which is the essence of the regularization. A remarkable feature of the vortex blob regularization is that the smoothing function $h^\delta$ is bounded and decays algebraically. Regarding the relation with the Euler equations in the $\delta \rightarrow 0$ limit, it has been shown in \cite{Liu} that the vortex blob methods converge to weak solutions of the 2D Euler equations under the conditions that the vorticity is in $\mathcal{M}(\mathbb{R}^2)$ with a distinguished sign and its induced velocity is locally square integrable.

\label{blob}

\subsubsection{The Euler-$\alpha$ model}
The Euler-$\alpha$ model was first introduced in \cite{Holm(a), Holm(b)} as the higher dimensional Camassa-Holm equations. We can also derive it by applying Lagrangian averaging to the Euler equations \cite{Marsden}. A remarkable property of this model is that the regularized velocity $\boldsymbol{u}^\alpha$ is written by $\boldsymbol{u}^\alpha = (1 - \alpha^2 \Delta)^{-1} \boldsymbol{v}$, see \cite{Foias}, and thus the smoothing function $h^\alpha$ in (\ref{Rvelo}) is defined by a fundamental solution for the operator $1 - \alpha^2 \Delta$, namely
\begin{equation*}
h^\alpha(\boldsymbol{x}) = \frac{1}{2\pi\alpha^2} K_0 \left( \frac{|\boldsymbol{x}|}{\alpha} \right),
\end{equation*}
in which $K_0$ denotes the modified Bessel function of the second kind. It is important to remark that $K_0$ has a singularity at the origin like $K_0(r) \sim -\log{r}$ as $r \rightarrow 0$ and decays exponentially. Since the singular velocity $\boldsymbol{v}$ is explicitly expressed by the regularized velocity $\boldsymbol{u}^\alpha$, we find the Euler-Poincar\'{e} equations for $\boldsymbol{u}^\alpha$ as follows.
\begin{equation*}
(1 - \alpha^2 \Delta) \partial_t \boldsymbol{u}^\alpha + \boldsymbol{u}^\alpha \cdot \nabla (1 - \alpha^2 \Delta) \boldsymbol{u}^\alpha + (\nabla \boldsymbol{u}^\alpha)^T \cdot (1 - \alpha^2 \Delta) \boldsymbol{u}^\alpha = \nabla \Pi.
\end{equation*}
This equations are known as the Euler-$\alpha$ equations. Regarding the evolution of the singular vorticity described by the equation corresponding to (\ref{RPDE}), there exist a unique global weak solution for the initial vorticity $q_0 \in \mathcal{M}(\mathbb{R}^2)$ in contrast to the 2D Euler equations \cite{Oliver}. Moreover, it is also shown that a weak solution for $q_0 \in L^\infty(\mathbb{R}^2)\cap L^1(\mathbb{R}^2)$ converges to that of the vorticity equations induced by the 2D Euler equations. In terms of the formulation in (\ref{KGh}), we have
\begin{equation*}
\boldsymbol{K}^\alpha = \nabla^\perp G^\alpha, \qquad - \Delta G^\alpha(\boldsymbol{x}) = h^\alpha(\boldsymbol{x}),
\end{equation*}
and the solution $G^\alpha$ for the Poisson equation is concretely given by
\begin{equation*}
G^\alpha(\boldsymbol{x}) = G^\alpha_r(|\boldsymbol{x}|) = - \frac{1}{2 \pi} \left[ \log{|\boldsymbol{x}|} + K_0\left( \frac{|\boldsymbol{x}|}{\alpha} \right) \right].
\end{equation*}
Hence, we obtain the regularized integral kernel $\boldsymbol{K}^\alpha$ as follows.
\begin{equation*}
\boldsymbol{K}^\alpha(\boldsymbol{x}) = - \frac{1}{2 \pi} \frac{\boldsymbol{x}^\perp}{|\boldsymbol{x}|^2} B_K \left( \frac{|\boldsymbol{x}|}{\alpha} \right) = \boldsymbol{K}(\boldsymbol{x}) B_K \left( \frac{|\boldsymbol{x}|}{\alpha} \right),
\end{equation*}
where $B_K(r) = 1 - r K_1(r)$ and $K_1$ is the first order modified Bessel function of the second kind. Similar to the vortex blob method, we have the relation,
\begin{equation*}
B_K(r) = - 2 \pi r \frac{dG^{\alpha=1}_r}{dr}, \qquad B_K'(r) = r K_0(r).
\end{equation*}
Especially, it follows from the properties of the Bessel function that $G^\alpha$ is bounded and $\boldsymbol{K}^\alpha(0) = 0$, namely they have no singularity.

\label{alpha}

\subsection{Main theorems}
Our main results consist of two theorems, the existence of a unique global weak solution for (\ref{RPDE}) in the space of Radon measure and its convergence to the Euler equations with the integrable and bounded initial vorticity. Recall that $\mathcal{M}(\mathbb{R}^2)$ denotes the space of finite Radon measures on $\mathbb{R}^2$ with the norm,
\begin{equation*}
\| \mu \|_{\mathcal{M}} = \sup \left\{ \left. \int_{\mathbb{R}^2} f d \mu \ \right| \ f  \in C_0(\mathbb{R}^2), \ \| f \|_{L^\infty} \leq 1 \right\},
\end{equation*}
where $C_0(\mathbb{R}^2)$ is the space of continuous functions vanishing at infinity. For later use, we define functions as follows.
\begin{equation*}
\chi_{\log}^{-} (\boldsymbol{x})  = \left\{
\begin{array}{cc}
\displaystyle{ \left( 1 - \log{|\boldsymbol{x}|} \right)^{-1} }  &, \ |\boldsymbol{x}| \leq 1 , \\
\displaystyle{ 0 }  &, \ |\boldsymbol{x}| > 1,
\end{array}
\right. \quad
\chi_{\log}^{+}(\boldsymbol{x})  = \left\{
\begin{array}{cc}
\displaystyle{0}  &,  \  |\boldsymbol{x}| \leq 1, \\
\displaystyle{1 + \log{|\boldsymbol{x}|} } &, \ |\boldsymbol{x}| > 1 ,
\end{array}
\right.
\end{equation*}
and
\begin{equation*}
\chi^{-}_{\alpha}(\boldsymbol{x})  = \left\{
\begin{array}{cc}
|\boldsymbol{x}|^\alpha  &,  \ |\boldsymbol{x}| \leq 1 , \\
0  &,  \ |\boldsymbol{x}| > 1,
\end{array}
\right. \qquad
\chi^{+}_{\alpha}(\boldsymbol{x}) = \left\{
\begin{array}{cc}
0   &,  \ |\boldsymbol{x}| \leq 1, \\
|\boldsymbol{x}|^\alpha  &,  \ |\boldsymbol{x}| > 1 .
\end{array}
\right.
\end{equation*}
We also set $\chi_{\alpha}(\boldsymbol{x}) = \chi_{\alpha}^{-}(\boldsymbol{x}) + \chi_{\alpha}^{+}(\boldsymbol{x}) =  |\boldsymbol{x}|^\alpha$ for $\boldsymbol{x}\in \mathbb{R}^2$. These functions are used to characterize singularities and decay rates of functions. Then, we have the following theorem.

\begin{thm}
Suppose that $h \in C^1(\mathbb{R}^2) \cap W^1_1(\mathbb{R}^2)$ satisfies $\chi_1^{+} h \in L^1(\mathbb{R}^2)$ and
\begin{equation}
\chi_{\log}^{-} h \in L^\infty(\mathbb{R}^2), \qquad  \chi_1 \nabla h \in L^\infty(\mathbb{R}^2). \label{h-asympt}
\end{equation}
Then, for any initial vorticity $q_0 \in \mathcal{M}(\mathbb{R}^2)$, there exists a unique global weak solution of (\ref{RPDE}) such that
\begin{equation*}
\boldsymbol{\eta}_h \in C^1(\mathbb{R};\mathscr{G}), \quad \boldsymbol{u}_h \in C(\mathbb{R}; C (\mathbb{R}^2; \mathbb{R}^2)), \quad q \in C(\mathbb{R};\mathcal{M}(\mathbb{R}^2)),
\end{equation*}
where $\mathscr{G}$ denotes the group of all homeomorphism of $\mathbb{R}^2$ that preserves the Lebesgue measure. \label{well-posed}
\end{thm}

\begin{rem}
The assumption (\ref{h-asympt}) allows the smoothing function $h$ and $\nabla h$ to have singularities and behave
\begin{equation*}
h(\boldsymbol{x}) \sim  \mathcal{O}\left( - \log |\boldsymbol{x}| \right), \qquad \nabla h(\boldsymbol{x}) \sim  \mathcal{O}\left( |\boldsymbol{x}|^{-1} \right),
\end{equation*}
as $|\boldsymbol{x}| \rightarrow 0$. The decay rates of them are estimated by the conditions $\chi_1^{+} h$, $\nabla h \in L^1$.
\end{rem}

In order to consider the convergence of solutions of (\ref{RPDE}) to those in the Euler equations, we consider the parameterized function $h^\varepsilon$ defined by
\begin{equation*}
h^\varepsilon(\boldsymbol{x}) = \frac{1}{2 \pi \varepsilon^2} h\left( \frac{\boldsymbol{y}}{\varepsilon} \right).
\end{equation*}
Moreover, we assume that $h$ is a radial function, namely it depends only on $r = |\boldsymbol{x}|$, and define $h_r(|\boldsymbol{x}|) = h(\boldsymbol{x})$ that is a function on $\mathbb{R}$. Then, we have the following parameterized solutions $(\boldsymbol{\eta}^\varepsilon, \boldsymbol{u}^\varepsilon, q)$ such that
\begin{align*}
\partial_t \boldsymbol{\eta}^\varepsilon(\boldsymbol{x}, t) &= \boldsymbol{u}^\varepsilon \left( \boldsymbol{\eta}^\varepsilon(\boldsymbol{x}, t), t \right), \quad \boldsymbol{\eta}^\varepsilon (\boldsymbol{x}, 0) = \boldsymbol{x}, \\
&q(\boldsymbol{x}, t) = q_0 \left( \boldsymbol{\eta}^\varepsilon(\boldsymbol{x}, -t) \right), \\
\boldsymbol{u}^\varepsilon &=  \boldsymbol{K}^\varepsilon \ast q, \quad \boldsymbol{K}^\varepsilon = \boldsymbol{K} \ast h^\varepsilon,
\end{align*}
and the following theorem about the convergence to the Euler equations in the $\varepsilon \rightarrow 0$ limit holds.

\begin{thm}
In addition to the assumptions in Theorem~\ref{well-posed}, suppose that $h$ is radial and satisfies
\begin{equation*}
\int_0^\infty k h_r(k) dk = 1.
\end{equation*}
Let $q_0 = \omega_0 \in L^1(\mathbb{R}^2) \cap L^\infty(\mathbb{R}^2)$. Then, for any $T > 0$, there exists $C(T) > 0$ such that
\begin{equation*}
\sup_{t\in[0,T]} \sup_{\boldsymbol{x}\in\mathbb{R}^2} \left| \boldsymbol{\eta}^\varepsilon(\boldsymbol{x}, t) - \boldsymbol{\eta}(\boldsymbol{x}, t) \right| \leq C(T) \varepsilon^{e^{-T}}.
\end{equation*}
\label{convergence}
\end{thm}

\begin{rem}
We remark that both the smoothing functions $h^\delta$ and $h^\alpha$ satisfy the assumptions of Theorem~\ref{well-posed} and Theorem~\ref{convergence}. Especially, the Euler-$\alpha$ model is a critical regularization in terms of the singularity since we know $K_0(r) \sim - \log{r}$ as $r \rightarrow 0$ in the smoothing function $h^\alpha$. On the other hand, it is easily confirmed that $h^\delta$ has a slowest algebraic decay to satisfy the assumptions of the theorems.

\end{rem}

\label{Main}

\section{Properties of the integral kernel}
In order to show the main theorems, it is important to investigate the properties of the integral kernel $\boldsymbol{K}_h$ and $\boldsymbol{K}^\varepsilon$ in the regularized Bio-Savart law. Let us first define the function $\varphi(r)$ for $r \geq 0$ by
\begin{equation*}
\varphi(r) = \left\{
\begin{array}{cc}
r(1 - \log{r}) \ &, \quad r < 1 , \\
1 \ &, \quad r \geq 1.
\end{array}
\right.
\end{equation*}
We now show that $\boldsymbol{K}_h$ is well-defined and especially $\boldsymbol{K}_h$ vanishes at the origin. Here, the smoothing function $h$ may have a singularity only at the origin. We actually assume that $h \in L^\infty(\mathbb{R}^2 \backslash B_\delta)$ with sufficiently small $\delta > 0$, where $B_R = \{ \boldsymbol{y}\in \mathbb{R}^2 \ | \ |\boldsymbol{y}| < R \}$.

\begin{lem}
Suppose that $h \in L^1(\mathbb{R}^2) \cap L^\infty(\mathbb{R}^2 \backslash B_\delta)$, $\delta > 0$ and $\chi_\alpha^{-} h \in L^\infty(\mathbb{R}^2)$, $\alpha\in [0,1)$. Then, $\boldsymbol{K}_h$ is uniformly bounded, namely $\boldsymbol{K}_h \in L^\infty(\mathbb{R}^2)$. Moreover, if $h$ is radial, then $\boldsymbol{K}_h(0) = 0$.  \label{K_h-bdd}
\end{lem}

\begin{proof}
We have
\begin{align*}
\left| \boldsymbol{K}_h(\boldsymbol{x}) \right| &\leq \frac{1}{2 \pi} \int_{\mathbb{R}^2} \frac{1}{|\boldsymbol{x} - \boldsymbol{y}|} |h(\boldsymbol{y})| d\boldsymbol{y} \\
& \leq \frac{1}{2 \pi} \int_{|\boldsymbol{x} - \boldsymbol{y}| < \delta } \frac{1}{|\boldsymbol{x} - \boldsymbol{y}|} |h(\boldsymbol{y})| d \boldsymbol{y} + \frac{1}{2 \pi} \int_{|\boldsymbol{x} - \boldsymbol{y}| \geq \delta } \frac{1}{|\boldsymbol{x} - \boldsymbol{y}|} |h(\boldsymbol{y})| d \boldsymbol{y}   \\
& \leq \frac{1}{2 \pi} \| \chi_\alpha^{-} h \|_{L^\infty} \left[ \int_{|\boldsymbol{y}| < |\boldsymbol{x} - \boldsymbol{y}| < \delta }  \frac{1}{|\boldsymbol{y}|^{1+\alpha}} d  \boldsymbol{y} + \int_{ |\boldsymbol{x} - \boldsymbol{y}| \leq |\boldsymbol{y}| < \delta } \frac{1}{|\boldsymbol{x} - \boldsymbol{y}|^{1+\alpha}} d \boldsymbol{y} \right] \\
&\qquad  + \frac{1}{2 \pi} \| h \|_{L^\infty(\mathbb{R}^2 \backslash B_\delta)} \int_{|\boldsymbol{x} - \boldsymbol{y}| < \delta \cap |\boldsymbol{y}| \geq \delta } \frac{1}{|\boldsymbol{x} - \boldsymbol{y}|} d \boldsymbol{y} +  \frac{1}{2 \pi \delta} \int_{\mathbb{R}^2 } |h(\boldsymbol{y})| d \boldsymbol{y}  \\
&\leq \frac{2 \delta^{1-\alpha}}{1 - \alpha} \| \chi_\alpha^{-} h \|_{L^\infty} + \delta \| h \|_{L^\infty(\mathbb{R}^2 \backslash B_\delta)} + \frac{1}{2 \pi \delta} \| h \|_{L^1},
\end{align*}
in which $\| \cdot \|_{L^p} = \| \cdot \|_{L^p(\mathbb{R}^2)}$ and this notation is used throughout this paper. Thus, $\boldsymbol{K}_h$ is uniformly bounded. In terms of the polar coordinates, the value of $\boldsymbol{K}_h$ at the origin is calculated as follows.
\begin{align*}
\boldsymbol{K}_h(0) &= - \frac{1}{2 \pi} \int_{\mathbb{R}^2} \frac{(y_2, - y_1)}{|\boldsymbol{y}|^2} h(\boldsymbol{y}) d\boldsymbol{y} \\
&= - \frac{1}{2 \pi} \int_0^\infty \int_0^{2 \pi}  \frac{(k \sin{\theta}, - k \cos{\theta}) }{k^2} h_r(k) k d\theta dk \\
&= - \frac{1}{2 \pi} \int_0^\infty h_r(k) dk \int_0^{2 \pi}  (\sin{\theta}, - \cos{\theta}) d\theta,
\end{align*}
Since $h_r$ is integrable, we conclude that $\boldsymbol{K}_h(0) = 0$.
\end{proof}

\begin{rem}
When we show that $\boldsymbol{K}_h$ is bounded outside of a neighborhood of the origin, it is enough to suppose that $h \in L^1(\mathbb{R}^2) \cap L^\infty(\mathbb{R}^2 \backslash B_\delta)$.
\end{rem}

Next, we see the asymptotic behavior of $\boldsymbol{K}_h$. The following lemma asserts that it decays at least as fast as the singular kernel $\boldsymbol{K}$.

\begin{lem}
For $h \in L^1(\mathbb{R}^2)$ satisfying $\chi_1^{+} h \in L^1(\mathbb{R}^2) \cap L^\infty(\mathbb{R}^2)$, we have
\begin{equation*}
\limsup_{|\boldsymbol{x}| \rightarrow \infty} \left| \boldsymbol{x} \right| \left|\boldsymbol{K}_h(\boldsymbol{x}) \right| \leq \frac{1}{2 \pi} \| h \|_{L^1}.
\end{equation*}
\label{K_h-decay}
\end{lem}

\begin{proof}
We divide the integral in $\boldsymbol{K}_h$ as follows.
\begin{align*}
\left| \boldsymbol{x} \right| \left| \boldsymbol{K}_h (\boldsymbol{x}) \right| &\leq \frac{1}{2 \pi}\int_{\mathbb{R}^2} \left( 1 + \frac{| \boldsymbol{y} |}{|\boldsymbol{x} - \boldsymbol{y}|} \right) \left| h(\boldsymbol{y}) \right| d\boldsymbol{y} \\
& = \frac{1}{2\pi} \left[ \| h \|_{L^1} + \int_{B_R} \frac{| \boldsymbol{y} |}{|\boldsymbol{x} - \boldsymbol{y}|} \left| h(\boldsymbol{y}) \right| d\boldsymbol{y} + \int_{\mathbb{R}^2 \backslash B_R} \frac{| \boldsymbol{y} |}{|\boldsymbol{x} - \boldsymbol{y}|} \left| h(\boldsymbol{y}) \right| d\boldsymbol{y} \right].
\end{align*}
The second term in the right-hand side vanishes in the $|\boldsymbol{x}| \rightarrow \infty$ limit, since we have
\begin{equation*}
\int_{B_R} \frac{| \boldsymbol{y} |}{|\boldsymbol{x} - \boldsymbol{y}|} \left| h(\boldsymbol{y}) \right| d\boldsymbol{y} \leq \frac{R}{|\boldsymbol{x}| - R} \int_{B_R} \left| h(\boldsymbol{y}) \right| d\boldsymbol{y},
\end{equation*}
for sufficiently large $|\boldsymbol{x}|$. The third term is estimated as follows.
\begin{align*}
\int_{\mathbb{R}^2 \backslash B_R} \frac{| \boldsymbol{y} |}{|\boldsymbol{x} - \boldsymbol{y}|} \left| h(\boldsymbol{y}) \right| d\boldsymbol{y} &= \int_{(\mathbb{R}^2 \backslash B_R) \cap |\boldsymbol{x} - \boldsymbol{y}| < \delta} \frac{| \boldsymbol{y} |}{|\boldsymbol{x} - \boldsymbol{y}|} \left| h(\boldsymbol{y}) \right| d\boldsymbol{y} \\
& \quad  +  \int_{(\mathbb{R}^2 \backslash B_R) \cap |\boldsymbol{x} - \boldsymbol{y}| \geq \delta} \frac{| \boldsymbol{y} |}{|\boldsymbol{x} - \boldsymbol{y}|} \left| h(\boldsymbol{y}) \right| d\boldsymbol{y} \\
&\leq  2 \pi \delta \| \chi_1^{+} h \|_{L^\infty(\mathbb{R}^2 \backslash B_R)} + \frac{1}{\delta} \| \chi_1^{+} h \|_{L^1(\mathbb{R}^2 \backslash B_R)}.
\end{align*}
Summarizing the above estimates, we find
\begin{equation*}
\limsup_{|\boldsymbol{x}| \rightarrow \infty} \left| \boldsymbol{x} \right| \left| \boldsymbol{K}_h (\boldsymbol{x}) \right| \leq  \frac{1}{2 \pi} \| h \|_{L^1} + \delta \| \chi_1^{+} h \|_{L^\infty(\mathbb{R}^2 \backslash B_1)} + \frac{1}{2 \pi \delta} \| \chi_1^{+} h \|_{L^1(\mathbb{R}^2 \backslash B_R)}.
\end{equation*}
Since $\chi_1^{+} h$ is integrable, the third term vanishes in the $R \rightarrow \infty$ limit. Finally, taking the $\delta \rightarrow 0$ limit, we obtain the desired result.
\end{proof}

In the following lemma, we show that $\boldsymbol{K}_h$ is quasi-Lipschitz continuous.

\begin{lem}
Let $h \in C^1(\mathbb{R}^2) \cap W^1_1(\mathbb{R}^2)$ satisfies $\chi_{\log}^{-} h \in L^\infty(\mathbb{R}^2)$ and $\chi_1 \nabla h \in L^\infty(\mathbb{R}^2)$. Then,
\begin{equation*}
\left| \boldsymbol{K}_h (\boldsymbol{x}) - \boldsymbol{K}_h(\boldsymbol{x}') \right| \leq c \varphi\left( \left| \boldsymbol{x} - \boldsymbol{x}' \right| \right).
\end{equation*}
\label{Kh-est}
\end{lem}

\begin{proof}
Let us set $r = |\boldsymbol{x} - \boldsymbol{x}'|$ and note that it is enough to prove the lemma for sufficiently small $r > 0$. We have
\begin{align*}
\left| \boldsymbol{K}_h (\boldsymbol{x}) - \boldsymbol{K}_h(\boldsymbol{x}') \right| & = \left| \int_{\mathbb{R}^2} \boldsymbol{K}(\boldsymbol{y}) \left(  h(\boldsymbol{x} - \boldsymbol{y}) - h(\boldsymbol{x}' - \boldsymbol{y}) \right) d \boldsymbol{y} \right| \\
& \leq \int_{|\boldsymbol{x} - \boldsymbol{y}| < 2 r} \left| \boldsymbol{K}(\boldsymbol{y}) \right| \left(  \left| h(\boldsymbol{x} - \boldsymbol{y}) \right| + \left| h(\boldsymbol{x}' - \boldsymbol{y}) \right| \right) d\boldsymbol{y} \\
&\quad + \int_{2r \leq |\boldsymbol{x} - \boldsymbol{y}| < 2} \left| \boldsymbol{K}(\boldsymbol{y}) \right| \left| h(\boldsymbol{x} - \boldsymbol{y}) - h(\boldsymbol{x}' - \boldsymbol{y}) \right| d\boldsymbol{y}  \\
&\quad + \int_{|\boldsymbol{x} - \boldsymbol{y}| \geq 2} \left| \boldsymbol{K}(\boldsymbol{y}) \right| \left| h(\boldsymbol{x} - \boldsymbol{y}) - h(\boldsymbol{x}' - \boldsymbol{y}) \right| d\boldsymbol{y}  \\
&\equiv I_1 + I_2 + I_3.
\end{align*}
Considering the fact that if $|\boldsymbol{x} - \boldsymbol{y}| < 2 r$, then $|\boldsymbol{x}' - \boldsymbol{y}| < 3 r$, we obtain
\begin{align*}
I_1 &\leq \frac{1}{\pi} \sup_{\boldsymbol{x} \in \mathbb{R}^2} \int_{|\boldsymbol{x} - \boldsymbol{y}| < 3 r} \frac{1}{|\boldsymbol{y}|} \left| h\left( \boldsymbol{x} - \boldsymbol{y} \right) \right| d\boldsymbol{y}  \\
&\leq \frac{1}{\pi} \| \chi_{\log}^{-} h \|_{L^\infty} \sup_{\boldsymbol{x} \in \mathbb{R}^2} \int_{|\boldsymbol{x} - \boldsymbol{y}| < 3 r} \frac{ 1 - \log{|\boldsymbol{x} - \boldsymbol{y}|}}{|\boldsymbol{y}|} d\boldsymbol{y} \\
&\leq \frac{1}{\pi} \| \chi_{\log}^{-} h \|_{L^\infty} \sup_{\boldsymbol{x} \in \mathbb{R}^2} \left[ \int_{|\boldsymbol{y}| < |\boldsymbol{x} - \boldsymbol{y}| < 3 r} \frac{ 1 - \log{|\boldsymbol{y}|}}{|\boldsymbol{y}|} d\boldsymbol{y} \right.\\
& \left. \hspace{40mm} + \int_{ |\boldsymbol{x} - \boldsymbol{y}| \leq |\boldsymbol{y}|\, \cap\, |\boldsymbol{x} - \boldsymbol{y}| < 3 r } \frac{ 1 - \log{|\boldsymbol{x} - \boldsymbol{y}|}}{|\boldsymbol{x} - \boldsymbol{y}|} d\boldsymbol{y} \right] \\
&\leq c_1 \varphi(r) \| \chi_{\log}^{-} h \|_{L^\infty}.
\end{align*}
In order to estimate $I_2$ and $I_3$, we apply the mean value theorem to $h$. Then, we obtain
\begin{equation*}
\left| h(\boldsymbol{x} - \boldsymbol{y}) - h(\boldsymbol{x}' - \boldsymbol{y}) \right| \leq r \int_0^1 \left| (\nabla h)( \boldsymbol{x} - \boldsymbol{y} + \tau (\boldsymbol{x}' - \boldsymbol{x}) ) \right| d\tau.
\end{equation*}
Moreover, it follows from $|\boldsymbol{x} - \boldsymbol{y}| \geq 2r$ that
\begin{equation}
|\boldsymbol{x} - \boldsymbol{y} + \tau (\boldsymbol{x}' - \boldsymbol{x})| \geq \left| |\boldsymbol{x} - \boldsymbol{y}| - \tau r \right| \geq \frac{1}{2}|\boldsymbol{x} - \boldsymbol{y}| + (1 - \tau) r \geq \frac{1}{2}|\boldsymbol{x} - \boldsymbol{y}|. \label{x-y}
\end{equation}
Thus, setting $D = \left\{ \boldsymbol{y}\in\mathbb{R}^2 \ |\  2 r \leq |\boldsymbol{x} - \boldsymbol{y}| < 2 \right\}$, we find
\begin{align*}
I_2 &\leq \frac{r}{2 \pi} \| \chi_1 \nabla h \|_{L^\infty} \int_D \frac{1}{|\boldsymbol{y}|} \int_0^1 \frac{1}{|\boldsymbol{x} - \boldsymbol{y} + \tau (\boldsymbol{x}' - \boldsymbol{x})|} d\tau d\boldsymbol{y} \\
&\leq \frac{r}{\pi} \| \chi_1 \nabla h \|_{L^\infty} \int_D \frac{1}{|\boldsymbol{y}||\boldsymbol{x} - \boldsymbol{y}|} d\boldsymbol{y} \\
&\leq \frac{r}{\pi} \| \chi_1 \nabla h \|_{L^\infty} \left[ \frac{1}{2 r} \int_{D\, \cap\, |\boldsymbol{y}| < 2 r} \frac{1}{|\boldsymbol{y}|} d\boldsymbol{y} + \int_{D\, \cap\, 2 r \leq |\boldsymbol{y}| < |\boldsymbol{x} - \boldsymbol{y}|} \frac{1}{|\boldsymbol{y}|^2} d\boldsymbol{y} \right. \\
&\hspace{65mm}  + \left. \int_{D\, \cap\, |\boldsymbol{x} - \boldsymbol{y}| \leq |\boldsymbol{y}|} \frac{1}{|\boldsymbol{x} - \boldsymbol{y}|^2} d\boldsymbol{y} \right] \\
&\leq 2 r \| \chi_1 \nabla h \|_{L^\infty} \left[ \frac{1}{2 r} \int_0^{2r} dk + 2 \int_{2r}^{2} \frac{1}{k} dk \right] \\
&\leq c_2 \varphi(r) \| \chi_1 \nabla h \|_{L^\infty}.
\end{align*}
Finally, it follows from (\ref{x-y}) that
\begin{align*}
I_3 &\leq \frac{r}{2 \pi} \int_{2 \leq |\boldsymbol{x} - \boldsymbol{y}|} \frac{1}{|\boldsymbol{y}|} \int_0^1 \left| (\nabla h) (\boldsymbol{x} - \boldsymbol{y} + \tau (\boldsymbol{x}' - \boldsymbol{x})) \right| d\tau d\boldsymbol{y} \\
&\leq \frac{r}{2 \pi} \left[  \| \chi_1 \nabla h \|_{L^\infty} \int_{2 \leq |\boldsymbol{x} - \boldsymbol{y}|\, \cap\, |\boldsymbol{y}| < 1} \frac{2}{|\boldsymbol{y}| |\boldsymbol{x} - \boldsymbol{y}|} d\boldsymbol{y} \right. \\
&\hspace{15mm} + \left. \int_{2 \leq |\boldsymbol{x} - \boldsymbol{y}|\, \cap\, |\boldsymbol{y}| \geq 1 } \int_0^1 \left| (\nabla h) (\boldsymbol{x} - \boldsymbol{y} + \tau (\boldsymbol{x}' - \boldsymbol{x})) \right| d\tau d\boldsymbol{y} \right] \\
&\leq r \left[ \| \chi_1 \nabla h \|_{L^\infty} + \frac{1}{2 \pi} \| \nabla h \|_{L^1} \right].
\end{align*}
Combining the above three estimates, we achieve the conclusion.
\end{proof}

\begin{rem}
As a consequence of above lemmas, we find $\boldsymbol{K}_h \in C_0(\mathbb{R}^2)$. This plays an essential role to show the existence of a unique global solution of (\ref{RPDE}) with initial vorticity in $\mathcal{M}(\mathbb{R}^2)$. Indeed, owing to the following estimate,
\begin{equation}
\| \boldsymbol{u}_h \|_{L^\infty} \leq \sup_{\boldsymbol{x} \in \mathbb{R}^2} \left| \int_{\mathbb{R}^2} \boldsymbol{K}_h(\boldsymbol{x} - \boldsymbol{y}) q(\boldsymbol{y}) d\boldsymbol{y} \right| \leq \| \boldsymbol{K}_h \|_{L^\infty} \| q \|_{\mathcal{M}}, \label{unif-bdd}
\end{equation}
and Lemma~\ref{Kh-est}, we find that $\boldsymbol{u}_h$ is uniformly bounded and quasi-Lipschitz continuous , which allows us to apply the classical method to prove Theorem~\ref{well-posed}. \label{C_0}
\end{rem}

Finally, we investigate the relation between $\boldsymbol{K}^\varepsilon$ and $\boldsymbol{K}$ to prove Theorem~\ref{convergence}. Before that, we show the following lemma.
\begin{lem}
Suppose that $h \in L^1(\mathbb{R}^2) \cap L^\infty(\mathbb{R}^2 \backslash B_\delta)$, $\delta > 0$ satisfies $\chi_\alpha^{-} h \in L^\infty(\mathbb{R}^2)$, $\alpha\in[0,1]$, and $\chi_{\log}^{+} h \in L^1(\mathbb{R}^2)$. Then, the scalar function
\begin{equation*}
G_h(\boldsymbol{x}) = - \frac{1}{2 \pi} \int_{\mathbb{R}^2} \left( \log{|\boldsymbol{x} - \boldsymbol{y}|} \right) h\left( \boldsymbol{y} \right) d\boldsymbol{y},
\end{equation*}
is well-defined for any $\boldsymbol{x}\in\mathbb{R}^2$ and satisfies the Poisson equation, $- \Delta G_h = h$, in the weak sense that $\langle \nabla G_h, \nabla\phi \rangle = - \langle h, \phi \rangle$ for any $\phi \in C_0^\infty(\mathbb{R}^2)$. Moreover, we have
\begin{equation}
\lim_{|\boldsymbol{x}| \rightarrow \infty} \frac{G_h(\boldsymbol{x})}{\log{|\boldsymbol{x}|}} =  - \frac{1}{2 \pi} \int_{\mathbb{R}^2} h(\boldsymbol{y}) d\boldsymbol{y}. \label{E_h-lim}
\end{equation}
Especially, if $h$ is a radial function, $G_h$ is also radial, namely $G_h(\boldsymbol{x}) = G_{h,r}(|\boldsymbol{x}|)$.

Regarding the derivative of $G_{h,r}$, suppose that the radial function $h \in L^1(\mathbb{R}^2)$ satisfies $\chi_\alpha^{-} h \in L^\infty(\mathbb{R}^2)$, $\alpha\in[0,1)$ and $\chi_1^{+} h \in L^1(\mathbb{R}^2) \cap L^\infty(\mathbb{R}^2)$. Then, we have $G'_{h,r}(0) = 0$ and
\begin{equation}
\lim_{r \rightarrow \infty} r G'_{h,r}(r) = - \frac{1}{2 \pi} \int_{\mathbb{R}^2} h(\boldsymbol{y}) d\boldsymbol{y}. \label{G'_h-lim}
\end{equation}
\label{K_eps-lem}
\end{lem}

\begin{proof}
We first show that $G_h$ is well-defined. For any fixed $\boldsymbol{x}\in\mathbb{R}^2$, we divide the integral into two parts,
\begin{equation*}
\left| G_h(\boldsymbol{x}) \right| \leq \frac{1}{2 \pi} \left[ \int_{|\boldsymbol{x} - \boldsymbol{y}| < 1} \left| \log{|\boldsymbol{x} - \boldsymbol{y}|} \right| | h\left( \boldsymbol{y} \right) | d\boldsymbol{y} + \int_{|\boldsymbol{x} - \boldsymbol{y}| \geq 1} \left| \log{|\boldsymbol{x} - \boldsymbol{y}|} \right| | h\left( \boldsymbol{y} \right) | d\boldsymbol{y} \right].
\end{equation*}
Each integral is estimated as follows.
\begin{align*}
\int_{|\boldsymbol{x} - \boldsymbol{y}| < 1} \left| \log{|\boldsymbol{x} - \boldsymbol{y}|} \right| | h\left( \boldsymbol{y} \right) | d\boldsymbol{y} & \leq \| \chi_{\alpha}^{-} h \|_{L^\infty} \left[ \int_{ |\boldsymbol{y}| < |\boldsymbol{x} - \boldsymbol{y}| < 1 \, \cap\, |\boldsymbol{y}| < \delta} \frac{ - \log{|\boldsymbol{y}|} }{|\boldsymbol{y}|^\alpha} d\boldsymbol{y} \right. \\
& \hspace{25mm} \left. + \int_{ |\boldsymbol{x} - \boldsymbol{y}| \leq |\boldsymbol{y}| < \delta} \frac{ - \log{|\boldsymbol{x} - \boldsymbol{y}|} }{|\boldsymbol{x} - \boldsymbol{y}|^\alpha} d\boldsymbol{y}\right] \\
& \quad + \| h \|_{L^\infty(\mathbb{R}^2 \backslash B_\delta)} \int_{|\boldsymbol{x} - \boldsymbol{y}| < 1 \, \cap\, |\boldsymbol{y}| \geq \delta} \left( - \log{|\boldsymbol{x} - \boldsymbol{y}|} \right) d\boldsymbol{y} \\
&\leq 4 \pi \| \chi_{\alpha}^{-} h \|_{L^\infty} \int_0^\delta k^{1-\alpha} ( - \log{k}) dk \\
& \quad + 2 \pi \| h \|_{L^\infty(\mathbb{R}^2 \backslash B_\delta)} \int_0^1 k ( - \log{k}) dk\\
&\leq c \left(  \| \chi_{\alpha}^{-} h \|_{L^\infty} + \| h \|_{L^\infty(\mathbb{R}^2 \backslash B_\delta)} \right),
\end{align*}
and
\begin{align*}
\int_{|\boldsymbol{x} - \boldsymbol{y}| \geq 1} \left| \log{|\boldsymbol{x} - \boldsymbol{y}|} \right| | h\left( \boldsymbol{y} \right) | d\boldsymbol{y} & \leq \int_{|\boldsymbol{x} - \boldsymbol{y}| \geq 1} \log{( |\boldsymbol{x}| + |\boldsymbol{y}|)} | h\left( \boldsymbol{y} \right) | d\boldsymbol{y} \\
&\leq \chi_{\{|\boldsymbol{x}| \geq 1/2 \}}\int_{|\boldsymbol{y}| < |\boldsymbol{x}|} \log{(2|\boldsymbol{x}|)} | h\left( \boldsymbol{y} \right) | d\boldsymbol{y} \\
&\qquad  + \int_{|\boldsymbol{y}| \geq |\boldsymbol{x}|\, \cap\, |\boldsymbol{y}| \geq 1/2} \log{(2|\boldsymbol{y}|)} | h\left( \boldsymbol{y} \right) | d\boldsymbol{y} \\
&\leq \chi_{\log}^{+}(\boldsymbol{x}) \| h \|_{L^1} + c\left(  \| h \|_{L^1} +  \| \chi_{\log}^{+} h \|_{L^1} \right).
\end{align*}
Thus, we find
\begin{equation}
\left| G_h(\boldsymbol{x}) \right| \leq  \chi_{\log}^{+}(\boldsymbol{x}) \| h \|_{L^1} + c \left(  \| h \|_{L^1} +  \| \chi_{\log}^{+} h \|_{L^1} + \| \chi_{\alpha}^{-} h \|_{L^\infty} + \| h \|_{L^\infty(\mathbb{R}^2 \backslash B_\delta)} \right). \label{E_h-WD}
\end{equation}
In order to see an asymptotic behavior of $G_h$, we decompose the integral as follows.
\begin{align*}
\frac{G_h(\boldsymbol{x})}{\log{|\boldsymbol{x}|}} = - \frac{1}{2 \pi} \left[ \int_{B_R} \frac{\log{|\boldsymbol{x} - \boldsymbol{y}|}}{\log{|\boldsymbol{x}|}} h(\boldsymbol{y}) d\boldsymbol{y} + \int_{\mathbb{R}^2 \backslash B_R} \frac{\log{|\boldsymbol{x} - \boldsymbol{y}|}}{\log{|\boldsymbol{x}|}} h(\boldsymbol{y}) d\boldsymbol{y} \right]
\end{align*}
As for the first term, since we have the pointwise convergence, $\log{|\boldsymbol{x} - \boldsymbol{y}|} / \log{|\boldsymbol{x}|} \rightarrow 1$ as $| \boldsymbol{x} | \rightarrow \infty$, and
\begin{equation*}
\frac{\log{|\boldsymbol{x} - \boldsymbol{y}|}}{\log{|\boldsymbol{x}|}} |h(\boldsymbol{y})| \leq c |h(\boldsymbol{y})|,
\end{equation*}
for $\boldsymbol{y} \in B_{R}$ and sufficiently large $|\boldsymbol{x}|$, the dominated convergence theorem gives
\begin{equation*}
 \lim_{|\boldsymbol{x}| \rightarrow \infty} \int_{B_R} \frac{\log{|\boldsymbol{x} - \boldsymbol{y}|}}{\log{|\boldsymbol{x}|}} h(\boldsymbol{y}) d\boldsymbol{y} = \int_{B_R} h(\boldsymbol{y}) d\boldsymbol{y}.
\end{equation*}
Based on the calculations in (\ref{E_h-WD}), we find the following estimate for sufficiently large $|\boldsymbol{x}|$.
\begin{align*}
\left| \int_{\mathbb{R}^2 \backslash B_R} \frac{\log{|\boldsymbol{x} - \boldsymbol{y}|}}{\log{|\boldsymbol{x}|}} h(\boldsymbol{y}) d\boldsymbol{y} \right| & \leq \| h \|_{L^1(\mathbb{R}^2 \backslash B_R)} \\
& \quad + \frac{c}{\log{|\boldsymbol{x}|}} \left(  \| h \|_{L^1} +  \| \chi_{\log}^{+} h \|_{L^1} + \| h \|_{L^\infty(\mathbb{R}^2 \backslash B_\delta)} \right).
\end{align*}
Thus, taking the $|\boldsymbol{x}| \rightarrow \infty$ limit and then the $R \rightarrow \infty$ limit, we obtain the desired result (\ref{E_h-lim}). According to the classical theory, we readily confirm that $G_h$ is a weak solution of the Poisson equation, see \cite{Masaki}. Next, we rewrite $G_h$ in the polar coordinates,
\begin{equation*}
G_{h,r}(r, \theta) = - \frac{1}{2 \pi} \int_{\mathbb{R}^2} \left( \log{\sqrt{(r \cos{\theta} - y_1)^2 + (r \sin{\theta} - y_2)^2}} \right) h(\boldsymbol{y}) d\boldsymbol{y}.
\end{equation*}
Then, for a radial function $h$, we have
\begin{align*}
G_{h,r}(r, \theta) &= - \frac{1}{2 \pi} \int_0^\infty \int_0^{2 \pi} \left( \log{ \sqrt{ (r \cos{\theta} - \tilde{r}\cos{\tilde{\theta}})^2 + (r \sin{\theta} - \tilde{r}\sin{\tilde{\theta}})^2 } } \right) \tilde{r} h_r(\tilde{r}) d\tilde{\theta} d\tilde{r} \\
&= - \frac{1}{2 \pi} \int_0^\infty \int_0^{2 \pi} \left( \log{\sqrt{ r^2 + \tilde{r}^2 - 2 r \tilde{r}\cos{(\theta - \tilde{\theta})} }} \right) \tilde{r} h_r(\tilde{r}) d\tilde{\theta} d\tilde{r} \\
&= - \frac{1}{2 \pi} \int_0^\infty \int_0^{2 \pi} \left( \log{\sqrt{ r^2 + \tilde{r}^2 - 2 r \tilde{r}\cos{\tilde{\theta}} }} \right) \tilde{r} h_r(\tilde{r}) d\tilde{\theta} d\tilde{r}.
\end{align*}
Thus, $G_{h,r}$ is independent of $\theta$ and represented by
\begin{equation*}
G_{h,r}(r) = G_{h,r}(r, 0) = - \frac{1}{2 \pi} \int_{\mathbb{R}^2} \left( \log{\sqrt{(r - y_1)^2 + y_2^2}} \right) h(\boldsymbol{y}) d\boldsymbol{y}.
\end{equation*}
Its derivative is given by
\begin{equation*}
G'_{h,r}(r) = - \frac{1}{2 \pi} \int_{\mathbb{R}^2} \frac{r - y_1}{(r - y_1)^2 + y_2^2} h(\boldsymbol{y}) d\boldsymbol{y},
\end{equation*}
and we easily find that $G'_{h,r}$ is uniformly bounded and $G'_{h,r}(0) = 0$ based on the calculation in Lemma~\ref{K_h-bdd}. Finally, we see the asymptotic behavior of $G'_{h,r}$. Note that
\begin{equation*}
r G'_{h,r}(r) = - \frac{1}{2 \pi} \int_{\mathbb{R}^2} h(\boldsymbol{y}) d\boldsymbol{y} + \frac{1}{2 \pi} \int_{\mathbb{R}^2} \frac{ - y_1(r - y_1) + y_2^2}{(r - y_1)^2 + y_2^2} h(\boldsymbol{y}) d\boldsymbol{y}.
\end{equation*}
Then, setting $\boldsymbol{x}_r = (r, 0)$, we obtain
\begin{equation*}
\left| \int_{\mathbb{R}^2} \frac{- y_1(r - y_1) + y_2^2}{(r - y_1)^2 + y_2^2} h(\boldsymbol{y}) d\boldsymbol{y} \right| \leq \int_{\mathbb{R}^2} \frac{|\boldsymbol{y}|}{| \boldsymbol{x}_r - \boldsymbol{y} |} |h(\boldsymbol{y})| d\boldsymbol{y} \quad \rightarrow \quad 0,
\end{equation*}
as $r \rightarrow \infty$ in the same way as the proof in Lemma~\ref{K_h-decay}. Hence, we obtain (\ref{G'_h-lim}) and complete the proof.
\end{proof}

\begin{rem}
Under the condition of Theorem~\ref{convergence}, it follows from Lemma~\ref{K_eps-lem} that $G^\varepsilon = G \ast h^\varepsilon$ is radial, namely $G^\varepsilon(\boldsymbol{x}) = G_r^\varepsilon(|\boldsymbol{x}|)$ and
\begin{equation}
\lim_{r \rightarrow \infty} r \frac{d G_r^1}{dr}(r) = - \frac{1}{2 \pi}\int_0^\infty k h_r(k)dk = - \frac{1}{2 \pi}. \label{G-lim}
\end{equation}
Note that the relation,
\begin{equation*}
G_r^\varepsilon(|\boldsymbol{x}|) = G_r^1\left( \frac{|\boldsymbol{x}|}{\varepsilon} \right) - \frac{1}{2 \pi}\log{\varepsilon}.
\end{equation*}
Then, we find
\begin{equation*}
\boldsymbol{K}^\varepsilon(\boldsymbol{x}) = \nabla^\perp G_r^\varepsilon(|\boldsymbol{x}|)  =  \boldsymbol{K}(\boldsymbol{x}) \mathcal{G}\left( \frac{|\boldsymbol{x}|}{\varepsilon} \right) , \qquad \mathcal{G}(r) =  - 2 \pi r \frac{dG_r^1}{dr} (r).
\end{equation*}
Moreover, owing to (\ref{G-lim}) and $\frac{dG_r^1}{dr}(0) = 0$, we have
\begin{equation*}
\mathcal{G}(0) = 0, \qquad \lim_{r \rightarrow \infty} \mathcal{G}(r) = 1.
\end{equation*}
Hence, if $\boldsymbol{x} \neq 0$, then $\boldsymbol{K}^\varepsilon(\boldsymbol{x})$ converges pointwise to $\boldsymbol{K}(\boldsymbol{x})$.
\end{rem}

\begin{prop}
Let $h \in L^1(\mathbb{R}^2)$ be a radial function that satisfies $\chi_\alpha^{-} h \in L^\infty(\mathbb{R}^2)$, $\alpha\in[0,1)$, $\chi_1^{+} h \in L^1(\mathbb{R}^2) \cap L^\infty(\mathbb{R}^2)$ and $\int_0^\infty k h_r(k) dk = 1$. Then, we have
\begin{equation*}
\int_{\mathbb{R}^2} \left| \boldsymbol{K}^\varepsilon(\boldsymbol{x}) - \boldsymbol{K}(\boldsymbol{x}) \right| d\boldsymbol{x} \leq  \frac{\varepsilon}{2 \pi} \| \chi_1 h \|_{L^1}
\end{equation*}
\label{K_con}
\end{prop}

\begin{proof}
We first note that $h \in L^1(\mathbb{R}^2)$ and $\chi_1^{+} h \in L^1(\mathbb{R}^2)$ yields $\chi_1 h \in L^1(\mathbb{R}^2)$. We have
\begin{align*}
\int_{\mathbb{R}^2} \left| \boldsymbol{K}^\varepsilon(\boldsymbol{x}) - \boldsymbol{K}(\boldsymbol{x}) \right| d\boldsymbol{x} &= \int_{\mathbb{R}^2} \left| \int_{\mathbb{R}^2} \boldsymbol{K}(\boldsymbol{x} - \boldsymbol{y}) \frac{1}{2\pi\varepsilon^2} h\left( \frac{ \boldsymbol{y} }{\varepsilon} \right) d\boldsymbol{y} - \boldsymbol{K}(\boldsymbol{x}) \right| d\boldsymbol{x} \\
&= \varepsilon \int_{\mathbb{R}^2} \left| \frac{1}{2\pi} \int_{\mathbb{R}^2} \boldsymbol{K}(\boldsymbol{x} - \boldsymbol{y}) h\left( \boldsymbol{y} \right) d\boldsymbol{y} - \boldsymbol{K}(\boldsymbol{x}) \right| d\boldsymbol{x} \\
&= \frac{\varepsilon}{2 \pi} \int_{\mathbb{R}^2} \left| \nabla^\perp \left( G_h(\boldsymbol{x}) + \log{\left| \boldsymbol{x} \right|}  \right) \right| d\boldsymbol{x}.
\end{align*}
Owing to Lemma~\ref{K_eps-lem}, $G_h$ is a radial function, namely $G_h(\boldsymbol{x}) = G_{h,r}(|\boldsymbol{x}|)$. Then, we find
\begin{equation*}
\nabla^\perp \left( G_h(\boldsymbol{x}) + \log{\left| \boldsymbol{x} \right|}  \right)  =  \frac{\boldsymbol{x}^\perp}{|\boldsymbol{x}|} \left( G'_{h,r}(|\boldsymbol{x}|) + \frac{1}{|\boldsymbol{x}|} \right),
\end{equation*}
and thus
\begin{align*}
\int_{\mathbb{R}^2} \left| \boldsymbol{K}^\varepsilon(\boldsymbol{x}) - \boldsymbol{K}(\boldsymbol{x}) \right| d\boldsymbol{x} &= \frac{\varepsilon}{2 \pi} \int_{\mathbb{R}^2} \left| G'_{h,r}(|\boldsymbol{x}|) + \frac{1}{|\boldsymbol{x}|} \right| d\boldsymbol{x} \\
&= \varepsilon \int_0^\infty \left| r G'_{h,r}(r) + 1 \right| dr.
\end{align*}
Since $G_h$ is a solution to the Poisson equation,
\begin{equation*}
- \Delta G_h(\boldsymbol{x}) = - \frac{1}{r}\frac{d}{dr}\left( r G'_{h,r}(r) \right) = h_r(r),
\end{equation*}
and (\ref{G'_h-lim}) gives $r G'_{h,r}(r) \rightarrow - 1$ as $r \rightarrow \infty$, we obtain
\begin{equation*}
1 + r G'_{h,r}(r) = \int_r^\infty k h_r(k) dk.
\end{equation*}
Hence, it follows that
\begin{align*}
\int_0^\infty \left| r G'_{h,r}(r) + 1 \right| dr &\leq  \int_0^\infty \int_r^\infty k \left| h_r(k) \right| dk  dr \\
& = \left[ r \int_r^\infty k \left| h_r(k) \right| dk \right]_0^\infty + \int_0^\infty r^2 \left| h_r(r) \right| dr \\
& = \int_0^\infty r^2 \left| h_r(r) \right| dr \\
& = \frac{1}{2 \pi} \| \chi_1 h \|_{L^1}.
\end{align*}
Here, we have used the fact that $h \in L^1(\mathbb{R}^2)$ and $\chi_1 h \in L^1(\mathbb{R}^2)$ yield
\begin{equation*}
\left[ r \int_r^\infty k \left| h_r(k) \right| dk \right]_0^\infty = 0.
\end{equation*}
The proof of the proposition is now complete.
\end{proof}

\section{Proofs of main theorems}
\subsection{Global solvability in the space of Radon measure}
We prove Theorem~\ref{well-posed} by a classical iterative method. In this method, a uniform boundedness and a quasi-Lipschitz continuity of the velocity field play an essential role. Actually, the following lemma holds.

\begin{lem}
Consider the initial value problem for $\boldsymbol{x}(t) \in \mathbb{R}^n$, $t \in [0, \infty)$.
\begin{equation}
\frac{d}{dt} \boldsymbol{x}(t) = \boldsymbol{b} (\boldsymbol{x}(t), t), \qquad \boldsymbol{x}(0) = \boldsymbol{x}_0,  \label{ode}
\end{equation}
where $\boldsymbol{b} \in C\left( \mathbb{R}^n \times [0,\infty); \mathbb{R}^n \right)$ is uniformly bounded and quasi-Lipschitz continuous in space, namely
\begin{equation*}
\left| \boldsymbol{b}(\boldsymbol{x}, t) - \boldsymbol{b}(\boldsymbol{x}', t) \right| \leq c\varphi\left( |\boldsymbol{x} - \boldsymbol{x}'| \right),
\end{equation*}
and $c$ is independent of $t$. Then, (\ref{ode}) has a global unique solution $\boldsymbol{x}(\cdot) \in C^1[0, \infty)$.
\label{ODE}
\end{lem}
The proof of the lemma is given in \cite{Marchioro}. Our proof of Theorem~\ref{well-posed} is also based on that in \cite{Marchioro, Oliver}. We begin with introducing a sequence of approximate solutions for $n \in \mathbb{N}$, which are given by
\begin{align*}
q^0(\boldsymbol{x}, t) &= q_0(\boldsymbol{x}), \\
\boldsymbol{u}^n(\boldsymbol{x}, t)& = \left( \boldsymbol{K}_h \ast q^{n-1} \right)(\boldsymbol{x}, t),  \\
\partial_t\boldsymbol{\eta}^n(\boldsymbol{x},t) &= \boldsymbol{u}^n(\boldsymbol{\eta}^n(\boldsymbol{x},t), t),  \\
\boldsymbol{\eta}^n(\boldsymbol{x},0) &= \boldsymbol{x},  \\
q^n(\boldsymbol{x}, t) &= q_0(\boldsymbol{\eta}^n(\boldsymbol{x},-t) ).
\end{align*}
As the first step, we prove that $\boldsymbol{\eta}^n \in C^1([0,\infty) ; \mathscr{G})$ for $n \in \mathbb{N}$ by induction. For $n = 1$, it follows from (\ref{unif-bdd}) and Lemma~\ref{Kh-est} that $\| \boldsymbol{u}^1 \|_{L^\infty} \leq \| \boldsymbol{K}_h \|_{L^\infty} \| q_0 \|_{\mathcal{M}}$ and
\begin{align*}
\left| \boldsymbol{u}^1(\boldsymbol{x}) - \boldsymbol{u}^1(\boldsymbol{x}') \right| &\leq \sup_{\boldsymbol{y} \in \mathbb{R}^2} \left| \boldsymbol{K}_h(\boldsymbol{x} - \boldsymbol{y}) - \boldsymbol{K}_h(\boldsymbol{x}' - \boldsymbol{y}) \right| \| q_0 \|_{\mathcal{M}}  \\
& \leq c \varphi\left( |\boldsymbol{x} - \boldsymbol{x}'| \right) \| q_0 \|_{\mathcal{M}}.
\end{align*}
Thus, $\boldsymbol{u}^1$ is uniformly bounded on $\mathbb{R}^2 \times[0,\infty)$, quasi-Lipschitz in space and obviously continuous in time. Using Lemma~\ref{ODE}, we obtain $\boldsymbol{\eta}^1 \in C^1([0,\infty) ; \mathscr{G})$. For $\boldsymbol{\eta}^{n-1} \in C^1([0,\infty) ; \mathscr{G})$, considering $\| q^{n-1}(t) \|_{\mathcal{M}} = \| q_0 \|_{\mathcal{M}}$, we readily find that $\boldsymbol{u}^n$ is uniformly bounded and quasi-Lipschitz in space as well. Regarding the continuity in time, we have
\begin{align*}
|\boldsymbol{u}^n(\boldsymbol{x}, t) - \boldsymbol{u}^n(\boldsymbol{x}, t')| & \leq \sup_{\boldsymbol{y} \in \mathbb{R}^2} \left| \boldsymbol{K}_h(\boldsymbol{x} - \boldsymbol{\eta}^{n-1}(\boldsymbol{y},t)) - \boldsymbol{K}_h(\boldsymbol{x} - \boldsymbol{\eta}^{n-1}(\boldsymbol{y},t'))\right| \| q_0 \|_{\mathcal{M}} \\
& \leq c \sup_{\boldsymbol{y} \in \mathbb{R}^2} \varphi \left( \left| \boldsymbol{\eta}^{n-1}(\boldsymbol{y},t) - \boldsymbol{\eta}^{n-1}(\boldsymbol{y},t') \right| \right) \| q_0 \|_{\mathcal{M}} \\
& \leq c \sup_{\boldsymbol{y} \in \mathbb{R}^2} \sup_{s \in [t, t']} \varphi \left( \left| \partial_t\boldsymbol{\eta}^{n-1}(\boldsymbol{y},s) \right| |t - t'| \right) \| q_0 \|_{\mathcal{M}} \\
& \leq c \sup_{\boldsymbol{y} \in \mathbb{R}^2} \sup_{s \in [t, t']} \varphi \left( \left| \boldsymbol{u}^{n-1}(\boldsymbol{y},s) \right| |t - t'| \right) \| q_0 \|_{\mathcal{M}}.
\end{align*}
Since $\boldsymbol{u}^{n}$ is uniformly bounded, a uniform continuity in time holds. Hence, it follows from Lemma~\ref{ODE} that $\boldsymbol{\eta}^n \in C^1([0,\infty) ; \mathscr{G})$.

Next, we show the existence of a limit flow $\overline{\boldsymbol{\eta}} \in C^1([0,\infty) ; \mathscr{G})$. The sequence $\left\{ \boldsymbol{\eta}^n \right\}_{n \in \mathbb{N}}$ is Cauchy in $C([0,T) ; \mathscr{G})$ for some $T > 0$. Indeed, for $n \geq N$, we have
\begin{align*}
&\left| \boldsymbol{u}^n \left( \boldsymbol{\eta}^n(\boldsymbol{x}) \right) - \boldsymbol{u}^{n-1} \left( \boldsymbol{\eta}^{n-1}(\boldsymbol{x}) \right) \right|  \\
&\leq \int_{\mathbb{R}^2} \left| \boldsymbol{K}_h \left( \boldsymbol{\eta}^n(\boldsymbol{x}) - \boldsymbol{\eta}^{n-1}(\boldsymbol{y}) \right) - \boldsymbol{K}_h \left( \boldsymbol{\eta}^{n-1}(\boldsymbol{x}) - \boldsymbol{\eta}^{n-1}(\boldsymbol{y}) \right) \right| |q_0(\boldsymbol{y})| d\boldsymbol{y} \\
& \quad + \int_{\mathbb{R}^2} \left| \boldsymbol{K}_h \left( \boldsymbol{\eta}^{n-1}(\boldsymbol{x}) - \boldsymbol{\eta}^{n-1}(\boldsymbol{y}) \right) - \boldsymbol{K}_h \left( \boldsymbol{\eta}^{n-1}(\boldsymbol{x}) - \boldsymbol{\eta}^{n-2}(\boldsymbol{y}) \right) \right| |q_0(\boldsymbol{y})| d\boldsymbol{y} \\
&\leq c \| q_0 \|_{\mathcal{M}} \varphi\left( \left| \boldsymbol{\eta}^n(\boldsymbol{x}) - \boldsymbol{\eta}^{n-1}(\boldsymbol{x}) \right| \right) + c \int_{\mathbb{R}^2} \varphi\left( \left| \boldsymbol{\eta}^{n-1}(\boldsymbol{y}) - \boldsymbol{\eta}^{n-2}(\boldsymbol{y}) \right| \right) | q_0(\boldsymbol{y}) | d\boldsymbol{y} \\
&\leq c \| q_0 \|_{\mathcal{M}} \varphi\left( \sup_{n \geq N-1} \sup_{\boldsymbol{x} \in \mathbb{R}^2} \left| \boldsymbol{\eta}^n(\boldsymbol{x}) - \boldsymbol{\eta}^{n-1}(\boldsymbol{x}) \right| \right),
\end{align*}
in which we drop the explicit time dependence. Defining
\begin{equation*}
\rho^N(t) \equiv \sup_{n \geq N} \sup_{\boldsymbol{x} \in \mathbb{R}^2} \left| \boldsymbol{\eta}^n(\boldsymbol{x}, t) - \boldsymbol{\eta}^{n-1}(\boldsymbol{x}, t) \right|,
\end{equation*}
we obtain
\begin{align*}
\rho^N(t) &\leq \sup_{n \geq N} \sup_{\boldsymbol{x} \in \mathbb{R}^2}\int_0^t \left| \boldsymbol{u}^n \left( \boldsymbol{\eta}^n(\boldsymbol{x}, s),s  \right) - \boldsymbol{u}^{n-1} \left( \boldsymbol{\eta}^{n-1}(\boldsymbol{x}, s),s  \right) \right| ds \\
&\leq c \| q_0 \|_{\mathcal{M}} \int_0^t \varphi\left( \rho^{N-1}(s) \right) ds.
\end{align*}
This implies $\rho^N \rightarrow 0$ uniformly on $[0, T]$ as $N \rightarrow + \infty$ for sufficiently small $T$. Since $T$ depends only on $\| q_0 \|_{\mathcal{M}}$, the convergence holds for arbitrary times and thus the limit flow $\overline{\boldsymbol{\eta}} \in C([0,\infty) ; \mathscr{G})$ exits.

In turn, we see the convergences of the vorticity and the velocity. Let us define $\overline{q}$ and $\overline{\boldsymbol{u}}$ by
\begin{equation}
\overline{q}(\boldsymbol{x}, t) = q_0\left( \overline{\boldsymbol{\eta}}(\boldsymbol{x}, -t) \right), \qquad \overline{\boldsymbol{u}}(\boldsymbol{x}, t) = \int_{\mathbb{R}^2} \boldsymbol{K}_h(\boldsymbol{x} - \boldsymbol{y}) \overline{q}(\boldsymbol{y}, t) d\boldsymbol{y}. \label{limit_q-u}
\end{equation}
Then, we find that $q^n \rightharpoonup \overline{q}$ weakly in $\mathcal{M}(\mathbb{R}^2)$ and $\boldsymbol{u}^n \rightarrow \overline{\boldsymbol{u}}$ in $C(\mathbb{R}^2)$. Note that both converge uniformly on any finite time interval. Indeed, for any $\psi \in C_0^1(\mathbb{R}^2)$, it follows that
\begin{align*}
\left| \int_{\mathbb{R}^2} \left( q^n(\boldsymbol{x}, t) - \overline{q}(\boldsymbol{x}, t) \right) \psi(\boldsymbol{x}) d\boldsymbol{x} \right| &\leq \int_{\mathbb{R}^2} |q_0(\boldsymbol{x})| \left| \psi\left( \boldsymbol{\eta}^n(\boldsymbol{x}, t) \right) - \psi\left( \overline{\boldsymbol{\eta}}(\boldsymbol{x}, t) \right) \right| d\boldsymbol{x} \\
& \leq \| q_0 \|_{\mathcal{M}} \| \nabla \psi \|_{L^\infty} \sup_{\boldsymbol{x} \in \mathbb{R}^2} \left| \boldsymbol{\eta}^n(\boldsymbol{x}, t) - \overline{\boldsymbol{\eta}}(\boldsymbol{x}, t) \right|,
\end{align*}
and
\begin{align*}
\left| \boldsymbol{u}^n(\boldsymbol{x},t) - \overline{\boldsymbol{u}}(\boldsymbol{x},t) \right| & \leq c \| q_0 \|_{\mathcal{M}} \varphi\left( \sup_{\boldsymbol{y} \in \mathbb{R}^2} \left| \boldsymbol{\eta}^n(\boldsymbol{y}, t) - \overline{\boldsymbol{\eta}}(\boldsymbol{y}, t) \right| \right).
\end{align*}
Hence, the convergence of $\boldsymbol{\eta}^n$ and the density argument yield the above assertions.

Finally, we show that $(\overline{\boldsymbol{\eta}}, \overline{q}, \overline{\boldsymbol{u}})$ is the solution to (\ref{h-eta}). Consider the integral form of (\ref{h-eta}),
\begin{align*}
& \left| \overline{\boldsymbol{\eta}}(\boldsymbol{x}, t) - \boldsymbol{x} - \int_0^t \overline{\boldsymbol{u}}\left( \overline{\boldsymbol{\eta}}(\boldsymbol{x}, s), s \right) ds  \right| \\
& = \left| \overline{\boldsymbol{\eta}}(\boldsymbol{x}, t) - \boldsymbol{x} - \int_0^t \overline{\boldsymbol{u}}\left( \overline{\boldsymbol{\eta}}(\boldsymbol{x}, s), s \right) ds - \left( \boldsymbol{\eta}^n(\boldsymbol{x}, t) - \boldsymbol{x} - \int_0^t \boldsymbol{u}^n\left( \boldsymbol{\eta}^n(\boldsymbol{x}, s), s \right) ds \right)  \right| \\
& \leq \left| \boldsymbol{\eta}^n(\boldsymbol{x}, t) - \overline{\boldsymbol{\eta}}(\boldsymbol{x}, t) \right| + \int_0^t \left| \boldsymbol{u}^n\left( \boldsymbol{\eta}^n(\boldsymbol{x}, s), s \right) - \overline{\boldsymbol{u}}\left( \boldsymbol{\eta}^n(\boldsymbol{x}, s), s \right) \right| ds \\
& \hspace{40mm} + \int_0^t \left| \overline{\boldsymbol{u}}\left( \boldsymbol{\eta}^n(\boldsymbol{x}, s), s \right) - \overline{\boldsymbol{u}}\left( \overline{\boldsymbol{\eta}}(\boldsymbol{x}, s), s \right) \right| ds \\
& \leq \sup_{\boldsymbol{x} \in \mathbb{R}^2} \left| \boldsymbol{\eta}^n(\boldsymbol{x}, t) - \overline{\boldsymbol{\eta}}(\boldsymbol{x}, t) \right| + c \| q_0 \|_{\mathcal{M}} \int_0^t \varphi\left( \sup_{\boldsymbol{y} \in \mathbb{R}^2} \left| \boldsymbol{\eta}^{n-1}(\boldsymbol{y}, s) - \overline{\boldsymbol{\eta}}(\boldsymbol{y}, s) \right| \right) ds \\
& \hspace{48mm} + c \| q_0 \|_{\mathcal{M}} \int_0^t \varphi\left( \sup_{\boldsymbol{x} \in \mathbb{R}^2} \left| \boldsymbol{\eta}^n(\boldsymbol{x}, s) - \overline{\boldsymbol{\eta}}(\boldsymbol{x}, s) \right| \right) ds.
\end{align*}
Then, the right side converges zero uniformly in space as $n \rightarrow + \infty$. Since $\overline{\boldsymbol{u}}\left( \overline{\boldsymbol{\eta}}(\boldsymbol{x}, t), t \right)$ is continuous in time, $\overline{\boldsymbol{\eta}}$ satisfies (\ref{h-eta}) and $\partial_t \overline{\boldsymbol{\eta}}$ is continuous in time. Considering the time reversibility of (\ref{h-eta}), we find that the solutions are extended to negative times. Moreover, we can see that the solutions $\overline{q}$ and $\overline{\boldsymbol{u}}$ satisfy
\begin{equation*}
\int_{\mathbb{R}} \int_{\mathbb{R}^2} \left( \partial_t \psi + \overline{\boldsymbol{u}} \cdot \nabla \psi \right) \overline{q} d\boldsymbol{x} dt = 0,
\end{equation*}
for any $\psi \in C_0^\infty(\mathbb{R}\times\mathbb{R}^2)$. The uniqueness of the solutions can be proven based on the same estimates in the existence part.

\label{proof1}

\subsection{Convergence to the Euler equations}
In this section, we prove Theorem~\ref{convergence}. The proof proceeds in the same way as \cite{Oliver}. Before that, let us review the following quasi-Lipschitz estimates for the singular kernel $\boldsymbol{K}$.
\begin{lem}
For $\omega \in L^1(\mathbb{R}^2) \cap L^\infty(\mathbb{R}^2)$,
\begin{equation*}
\int_{\mathbb{R}^2} \left| \boldsymbol{K}(\boldsymbol{x} - \boldsymbol{y}) - \boldsymbol{K}(\boldsymbol{x}' - \boldsymbol{y}) \right| \left| \omega(\boldsymbol{y}) \right| d\boldsymbol{y} \leq c \varphi\left( \left| \boldsymbol{x} - \boldsymbol{x}' \right| \right) \left( \|\omega\|_{L^1} + \|\omega\|_{L^\infty} \right).
\end{equation*}
\label{K-esti1}
\end{lem}
The proof of this lemma is straightforward and we find it, for example, in \cite{Benedetto, Marchioro, McGrath}. The following lemma is a modified estimate of the singular kernel, which was proven in \cite{Benedetto, Oliver}.
\begin{lem}
Let $\omega \in L^1(\mathbb{R}^2) \cap L^\infty(\mathbb{R}^2)$ and $\phi$ be an area preserving measurable transformation on $\mathbb{R}^2$. Then
\begin{equation*}
\int_{\mathbb{R}^2} \left| \boldsymbol{K}(\boldsymbol{x} - \boldsymbol{y}) - \boldsymbol{K}(\boldsymbol{x} - \phi( \boldsymbol{y} )) \right| \left| \omega(\boldsymbol{y}) \right| d\boldsymbol{y} \leq c \sup_{x \in \mathbb{R}^2} \varphi\left( \left| \boldsymbol{x} - \phi(\boldsymbol{x}) \right| \right) \left( \|\omega\|_{L^1} + \|\omega\|_{L^\infty} \right).
\end{equation*}
\label{K-esti2}
\end{lem}
We estimate the difference of the flow maps.
\begin{align*}
&\left| \boldsymbol{\eta}^\varepsilon(\boldsymbol{x}, t) - \boldsymbol{\eta}(\boldsymbol{x}, t) \right| \leq \int_0^t \left| \boldsymbol{u}^\varepsilon\left( \boldsymbol{\eta}^\varepsilon(\boldsymbol{x}, s), s \right) - \boldsymbol{u}\left( \boldsymbol{\eta}(\boldsymbol{x}, s), s \right) \right| ds  \\
&\leq \int_0^t \int_{\mathbb{R}^2} \left| \boldsymbol{K}^\varepsilon\left( \boldsymbol{\eta}^\varepsilon(\boldsymbol{x}, s) - \boldsymbol{\eta}^\varepsilon(\boldsymbol{y}, s) \right) - \boldsymbol{K} \left( \boldsymbol{\eta}^\varepsilon(\boldsymbol{x}, s) - \boldsymbol{\eta}^\varepsilon(\boldsymbol{y}, s) \right) \right| \left| \omega_0(\boldsymbol{y}) \right| d\boldsymbol{y} ds \\
&\quad + \int_0^t \int_{\mathbb{R}^2} \left| \boldsymbol{K}\left( \boldsymbol{\eta}^\varepsilon(\boldsymbol{x}, s) - \boldsymbol{\eta}^\varepsilon(\boldsymbol{y}, s) \right) - \boldsymbol{K} \left( \boldsymbol{\eta}(\boldsymbol{x}, s) - \boldsymbol{\eta}^\varepsilon(\boldsymbol{y}, s) \right) \right| \left| \omega_0(\boldsymbol{y}) \right| d\boldsymbol{y} ds \\
&\quad + \int_0^t \int_{\mathbb{R}^2} \left| \boldsymbol{K}\left( \boldsymbol{\eta}(\boldsymbol{x}, s) - \boldsymbol{\eta}^\varepsilon(\boldsymbol{y}, s) \right) - \boldsymbol{K} \left( \boldsymbol{\eta}(\boldsymbol{x}, s) - \boldsymbol{\eta}(\boldsymbol{y}, s) \right) \right| \left| \omega_0(\boldsymbol{y}) \right|  d\boldsymbol{y} ds \\
&\equiv \int_0^t J_1 + J_2 + J_3 ds.
\end{align*}
As for $J_1$, it follows from Lemma~\ref{K_con} that
\begin{equation*}
J_1 \leq \| \omega_0 \|_{L^\infty} \int_{\mathbb{R}^2} \left| \boldsymbol{K}^\varepsilon(\boldsymbol{y}) - \boldsymbol{K}(\boldsymbol{y}) \right| d\boldsymbol{y} \leq \frac{\varepsilon}{2 \pi} \| \omega_0 \|_{L^\infty} \| \chi_1 h \|_{L^1} .
\end{equation*}
The other two terms are estimated by using Lemma~\ref{K-esti1} and Lemma~\ref{K-esti2} respectively. We find
\begin{align*}
J_2 &= \int_{\mathbb{R}^2}  \left| \boldsymbol{K}\left( \boldsymbol{\eta}^\varepsilon(\boldsymbol{x}, s) - \boldsymbol{y} \right) - \boldsymbol{K} \left( \boldsymbol{\eta}(\boldsymbol{x}, s) - \boldsymbol{y} \right)  \right| \left| q(\boldsymbol{y}, s) \right|  d\boldsymbol{y} \\
&\leq c_2 \varphi \left( \left| \boldsymbol{\eta}^\varepsilon(\boldsymbol{x}, s) - \boldsymbol{\eta}(\boldsymbol{x}, s) \right| \right) \left( \| \omega_0 \|_{L^1} + \| \omega_0 \|_{L^\infty} \right),
\end{align*}
and
\begin{align*}
J_3 &= \int_{\mathbb{R}^2} \left| \boldsymbol{K}\left( \boldsymbol{\eta}(\boldsymbol{x}, s) - \boldsymbol{\eta}^\varepsilon \left( \boldsymbol{\eta}^{-1}(\boldsymbol{y}, s), s \right) \right) - \boldsymbol{K} \left( \boldsymbol{\eta}(\boldsymbol{x}, s) - \boldsymbol{y} \right) \right| \left| \omega(\boldsymbol{y}, s) \right|  d\boldsymbol{y} \\
&\leq c_3 \sup_{\boldsymbol{x}\in\mathbb{R}^2} \varphi \left( \left| \boldsymbol{\eta}^\varepsilon(\boldsymbol{x}, s)  - \boldsymbol{\eta}(\boldsymbol{x}, s)  \right| \right) \left( \| \omega_0 \|_{L^1} + \| \omega_0 \|_{L^\infty} \right).
\end{align*}
Combining above estimates and setting $\rho(t) = \sup_{\boldsymbol{x}\in\mathbb{R}^2} \left| \boldsymbol{\eta}^\varepsilon(\boldsymbol{x}, t)  - \boldsymbol{\eta}(\boldsymbol{x}, t) \right|$, we obtain
\begin{equation*}
\rho(t) \leq c \int_0^t \left[ \varepsilon + \varphi\left( \rho(s) \right) \right] ds.
\end{equation*}
This yields $\rho(t) \leq C(T) \varepsilon^{\exp{(-T)}}$ for $t\in [0, T]$, see the calculation in \cite{Oliver}. The proof is now complete.

\label{proof2}

redpajama_set_name: RedPajamaArXiv


It's a busy week here on Mostly Homemade Mom and I'm loving it! All week long we're celebrating yummy soups perfect for this chilly fall weather we have now. So far I've brought you Simple Spinach Tortellini Soup and Crockpot Loaded Baked Potato Soup. Today I'm bringing you another yummy soup from the slow cooker.
To the Crockpot, add tomatoes (with juice), carrots, celery, chicken broth, onion, oregano, and basil.
Cook on low for 5 hours, test carrots and celery to make sure they're tender. Now it's roux time! In a medium bowl, melt butter and whisk in flour.
Drizzle in 1 cup hot soup, stirring the entire time. Add another 3 cups soup, again continuing to stir. Add roux back into the Crockpot along with the parmesan cheese and warmed half and half. Cover and continue cooking on low for another hour.
Have some great soup recipes to share? Head on over to our special Soup Week edition of Show & Share Wednesday.
I printed this baby out. Sounds great!
That looks delicious! I love tomato soup and can't wait to try this version!
Tomato Basil is one of our favorites, we would love this soup. Hope you have a fantastic weekend and thank you so much for sharing with Full Plate Thursday.
YUM! Pinned it and following now!
This looks yummy! Perfect for having on a cool fall evening. Thanks for sharing on the SPARK link party. I am co-hosting this month.
So I'm smitten with this soup. Do you mind if I copy your picture and feature this soup in a holiday line up and link back to the recipe?
Another great ecipe from your website that I am adding to mine! looks YUMMY! Thanks for sharing. There is nothing like hot soup on a cold day..or when you have a sore throat!


text:
Region's museum exhibits shine
Topics range from dinosaurs and Golden Books to holidays and the honeybee
Region's museum exhibits shine Topics range from dinosaurs and Golden Books to holidays and the honeybee Check out this story on stargazette.com: http://stargaz.tt/1qLAsC1
Correspondent Published 12:00 a.m. ET Nov. 16, 2014 | Updated 12:24 a.m. ET Nov. 16, 2014
Richard Scarry's illustration for "I Am a Bunny" (1963) is part of the exhibit "Golden Legacy: 65 Years of Golden Books" at the Memorial Art Gallery of the University of Rochester.(Photo: Provided photo)
Winter weather may be looming, but that doesn't mean hunkering down until spring returns. Museums around New York state offer plenty of opportunities to learn about art, history and culture — and many of them will make a worthwhile daytrip in the coming months.
For example, "Dinomania" has returned to Syracuse's Milton J. Rubenstein Museum of Science & Technology (MOST) for the sixth time since 1989 — its last appearance was in 2011 — but the latest version is the most complex to date.
Three separate scenes spotlight a variety of animatronic dinosaurs created by Kokoro Exhibits, including Pachycephalosaurs, Maiasaurs and a Tyrannosaurus rex.
New technology was required to make sure all the lights, video and sound work together in synchronicity for "living wall" behind the T-rex — it features a video that mimics a two-day, two-night cycle of the environment in which dinosaurs lived. The goal of the show is to scientifically accurate as possible depict what life was like for dinosaurs in the Mesozoic Era.
There's also an interactive game called "Dino Dodgeball" that uses Xbox Kinect sensors to track the movements of visitors and transform them into "dinosaurs" that will be projected on a screen — there, they can throw flaming asteroids at each other dodgeball.
Below is a look at several exhibitions around upstate New York that are currently on display, including some that have opened in recent weeks.
What: "Dorothea Lange's America"
Where: Fenimore Art Museum, 5798 State Route 80, Cooperstown
When: Through Dec. 31
Info: www.fenimoreartmuseum.org
The Great Depression was the catalyst for a tremendous outburst of creative energy in America's photographic community. The devastation wreaked upon the country inspired a host of socially conscious photographers to capture the painful stories of the time. This exhibition features the work of 13 of these artists, including Dorothea Lange (1895-1965), whose empathetic images of migrant workers, suffering families and tortured landscapes have seared the imagery of the Depression into America's consciousness.
Hours are from 10 a.m. to 4 p.m. Tuesday-Sunday through Dec. 31. Admission is $12, $10.50 for seniors, free for children 12 and under.
Looking at Books
What: "Printed Editions in the Sixties and Seventies: LeWitt, Roth, Ruscha"
Where: Albright-Knox Art Gallery, 1285 Elmwood Avenue, Buffalo
When: Through Jan. 4, 2015
Info: www.albrightknox.org
A selection of artists' books and prints from the Albright-Knox's Collection by three pioneering artists who re-envisioned what a book could be: Sol LeWitt, Dieter Roth and Ed Ruscha. This exhibition highlights some of the best-known artists' books in the Albright-Knox's Collection and offers an opportunity for visitors to rediscover the form of the book through the hands of artists who understood the poetry of small things.
Also on display: "Lucas Samaras: Reflections," which comprises prints, sculpture, photographs alongside Samaras's famously disorienting "Room No. 2, 1966" (through Nov. 16); Millie Chen's "The Miseries & Vengeance Wallpapers" (though Nov. 16); and "Tacita Dean: The Friar's Doodle" (through Jan. 11, 2015).
Hours are from 10 a.m. to 5 p.m. daily; admission is $12 for adults, $8 for students and seniors, $5 for children 6-12.
Dinosaurs in Armory Square
What: Dinomania 2014
Where: Milton J. Rubenstein Museum of Science & Technology, 500 S, Franklin St., Syracuse
Info: www.most.org
Dinomania returns to downtown Syracuse, as the popular animatronic dinosaurs, which were created by Kokoro Exhibits with input from paleontologists to be as scientifically accurate as possible, are shown in scenes that depict what life was like for them in the Mesozoic Era. The exhibit also includes a remote-control dinosaur frame to explain the technology behind Kokoro's dinosaurs, a dino dig box, rub stations and hands-on activities from MOST educators. The MOST is also bringing back a favorite movie, "T-Rex: Back to the Cretaceous," to its Bristol IMAX Omnitheater, and will present "The Changing Universe" in its planetarium.
The MOST is open from 10 a.m. to 5 p.m. Wednesday-Sunday, with additional hours during holiday weeks. Admission is $10 for adults, $8 for children 2-11 for exhibit hall or the Omni Theatre; combined passes are available for $14 for adults and $12 for children 2-11.
Castles made of LEGOs
What: LEGO Castle Adventure
Where: The Strong National Museum of Play, One Manhattan Square, Rochester
When: Through Feb. 1, 2015
Info: www.museumofplay.org
Learn how castles were planned and built by examining LEGO models of real-world castles Blarney (Ireland), Arundel (England), and Neuschwanstein (Germany) created by LEGO® Master Builders. Then put your newfound knowledge to the test and join the Builder's Guild to plan the ideal defenses and construct your own castle. Three age-appropriate building areas encourage young builders to create a medieval masterpiece.
The Strong is open from 10 a.m. to 5 p.m. Monday-Thursday; from 10 a.m. to 8 p.m. Friday and Saturday, and from noon to 5 p.m. Sunday. Admission is $13.50 for age 2 and older; children under age 2 and members are admitted free of charge.
Cowboy Art in Corning
What: "Harmless Hunter: The Wildlife Work of Charles M. Russell"
Where: The Rockwell Museum, 111 Cedar St., Corning
Info: rockwellmuseum.org
"Harmless Hunter," a new exhibit of artwork by famed cowboy artist Charles M. Russell, showcases a lesser-known strength of his portfolio — wildlife art. This touring exhibit, which features more than 40 works and a companion publication, premiered at the National Museum of Wildlife Art before coming to Corning.
Russell painted, drew and sculpted various Western animals, among them the grizzly bear, the deer and the buffalo, throughout his long and storied career. Russell's works also address more problematic environmental themes like the rapid changes wrought by the onset of settlement, economic development and near decimation of wild game in the American West.
The Rockwell Museum is open from 9 a.m. to 5 p.m. daily. Admission is $5-$9; children 19 and under are admitted free of charge.
Silk Road Inspiration
What: Daniel Buckingham's "Secret Invitation"
Where: Munson-Williams-Proctor Arts Institute, 310 Genesee St., Utica
When: Through May 31, 2015
Info: www.mwpai.org
Daniel Buckingham's visual language is inspired by his solo journey along the ancient Silk Road — a 4,000-mile long network of trade routes that connected China to Europe, the Middle East and North Africa — where, by bicycle, he explored 60 countries over an 8-year span. His towering three-arch structure "reflects the ancient Caravansary architecture and functions as a metaphorical space for dialogue and contemplation. The arch becomes a portal for our passage through time; an invitation to travel inside and outside history to open a dialogue in the present."
Also on display: "In Company with Angels: Seven Rediscovered Tiffany Windows" and "A Century of Olmsted: Utica and Beyond," both through Jan. 4, 2015.
Hours are from 10 a.m. to 5 p.m. Tuesday-Thursday & Saturday (to 8 p.m. Friday) and 1-5 p.m. Sunday. Admission is free for most exhibitions. ($5-$10 for Tiffany exhibit.)
Surrealistic Influences
What: "Surrealism and Magic"
Where: Herbert F. Johnson Museum of Art, Cornell University, 114 Central Ave., Ithaca
Info: museum.cornell.edu
Inspired by the magic-themed library of Kurt Seligmann (1900-1962), acquired by Cornell upon his death, this exhibition explores the surrealists' interest in magic, the occult and indigenous spirituality. The show includes works of art by Seligmann along with drawings, collages and paintings by other surrealists and artists including André Breton, Victor Brauner, Max Ernst, René Magritte, Leonora Carrington, Dorothea Tanning, Roberto Matta, Jean Dubuffet, Enrico Donati and Wifredo Lam.
Also on display: "Jie (Boundaries): Contemporary Art from Taiwan," in which contemporary artists from Taiwan examine the effects of globalization on the island's transformation (through Dec. 21); "An Eye for Detail: Dutch Painting from the Leiden Collection," which presents 12 17th-century Dutch paintings by Pieter Brueghel the Younger and others from the New York-based Leiden Collection. (through June 21, 2015). Hours are from 10 a.m. to 5 p.m. Tuesday-Sunday; admission is free, but donations are welcome.
Familiar Favorites
What: "Golden Legacy: 65 Years of Golden Books"
Where: Memorial Art Gallery of the University of Rochester, 500 University Ave., Rochester
Info: mag.rochester.edu
This exhibition presents the most extensive public showing ever of original illustration art from American publishing's best-loved and most important picture-book series, Little Golden Books. Launched in 1942, the series made high-quality illustrated books available at affordable prices for the first time to millions of young children and their parents. Fifty-seven masterpieces of original illustration art will be featured in the exhibition, including examples from such picture-book classics as "Tootle," "Home for a Bunny," "Five Little Firemen" and "I Can Fly."
Also on display: Matt Elson's "Infinity Boxes" by Matt Elson, which use color, light and mirrors to create optical illusions that envelop viewers who peer inside. (through Jan. 4, 2015).Hours are from 11 a.m. to 5 p.m. Wednesday-Sunday (until 9 p.m. Thursdays). Admission is $12, $8 for seniors and $5 for college students and children 6-18.
Glass Master
What: "René Lalique: Enchanted by Glass"
Where: Corning Museum of Glass, One Museum Way, Corning
Info: cmog.org
This exhibition brings together glass, jewelry, production molds and design drawings by René Lalique (French, 1860-1945). His aesthetic choices in his designs informed the styles of Art Nouveau and Art Deco in France, and the objects he created have become iconic reflections of these periods. Lalique also embraced industrial innovations, like mass production, allowing luxury glass to be placed in more and more households around the world. Shown in conjunction with "Designing for a New Century: Works on Paper by Lalique and his Contemporaries."
Hours are 9 a.m. to 5 p.m. daily, except for holidays. Admission is $16 for adults, $13.50 for college students, free for 19 and under.
Casts of Characters
What: "Firing the Canon," an exhibit of Cornell plaster casts
Where: Weinhold Chilled Water Plant, 101 Forest Home Drive, Cornell University, Ithaca
When: Weekdays through December
Info: as.cornell.edu/150/
"Firing the Canon," a College of Arts and Sciences Sesquicentennial exhibit, explores how Cornell's prized collection of plaster casts was "embraced, defaced, and dethroned." Cornell used to have one of the largest collections of plaster casts in North America, attracting many tourists to Ithaca. But over time, casts fell out of favor, and the collection deteriorated. About twenty casts from Cornell's remaining 500 or so pieces will be on display among the aging mechanics of the water plant. Hours are from 10 a.m. to 4 p.m.; admission is free.
Making Electronic Music
What: "Switched-On: The Birth of the Moog Synthesizer"
Where: The History Center of Tompkins County, 401 E. State St., Ithaca
When: Through May 2015
Info: www.thehistorycenter.net
This show tells about Bob Moog and the development of the groundbreaking electronic instruments bearing his name, many of which were made in his Trumansburg factory 1963-1971. Several vintage instruments are on display, and interactive exhibits allow visitors to create their own electronic sounds. Also on display through August 2015 is "Captains, Commerce, and Community: The Impact of the Erie Canal on Tompkins County," which explores the lasting effects of the Erie Canal on the area. Hours are from 11 a.m. to 5 p.m. Tuesdays, Thursdays and Saturdays, and by appointment.
Roving the Galaxy
What: "Mars and Stars" exhibit
Where: The Sciencenter, 601 1st St., Ithaca
Info: sciencenter.org
Explore our galaxy and beyond by taking an interactive flight over the moon using high-resolution data of the lunar surface. Visit astronomical objects such as planets, stars, and nebulae, see the constellations in our local night sky, and explore a Martian Volcano. A recently added remote-controlled Mars Rover exhibit allows guests to direct a model rover in real time to drill into rocks and collect dust as the rover moves over simulated Marian terrain. Hours are from 10 a.m. to 5 p.m. and noon to 5 p.m. Sunday. Admission is $8 for adults, $7 for seniors, $6 for children (3-17 years), and free to members and toddlers (0-2 years).
What: "To Do No Harm: Medicine in Chemung County"
Where: The Chemung County Historical Society, 415 E. Water St., Elmira
When: Opens Nov. 20
Info: www.chemungvalleymuseum.org
Explore the changes in medical practices and health care throughout the history of Chemung County through stories of the men and women who worked as physicians, nurses, dentists, and other health care professionals in the community. Learn about the many public and private hospitals that operated around the turn of the 20th century, explore advances in medical technology and discover why some pharmaceuticals are no longer prescribed by doctors.
Hours are from 10 a.m. to 5 p.m. Monday-Saturday. Admission is $3 for adults, $2 for seniors, $1 for students 6-18, free to children under 5.
Holiday Happening
What: O Tannenbaum Holiday Showcase
Where: Tioga County Historical Society Museum, 110 Front St., Owego
Info: www.tiogahistory.org
The annual holiday showcase features trees, wreaths, and other items donated by the community; they're displayed throughout the museum as part of a silent auction. The event culminates on Final Bid Night on Dec. 13 — the museum is open from 10 a.m. to 8 p.m., with bidding ending at 7 p.m. All fundraising proceeds directly benefit museum. Regular hours are from 10 a.m. to 4 p.m. Wednesday-Saturday.
Bees, If You Please
What: "What's The Buzz? A Honey of an Exhibit"
Where: The Discovery Center of the Southern Tier, 60 Morgan Road, Binghamton
When: Ongoing
Info: www.thediscoverycenter.org
The center's 30th anniversary exhibit tells all about the honeybee and its impact on life as a worldwide pollinator. Climb into a 14-foot beehive, take a seat on the Queen Bee's throne, try a hand at bee mythology with puppets, and explore the history of bees in math, literature and the arts. Hours are from 10 a.m. to 3 p.m. Monday (toddler only), 10 a.m. to 4 p.m. Tuesday-Friday, 10 a.m. to 5 p.m. Saturday, and noon to 5 p.m. Sunday. Admission is $7 (children under 1 free).
Spirit of the Season
What: "Home for the Holidays" returns
Where: Roberson Museum and Science Center, 30 Front St., Binghamton
Info: www.roberson.org
The Roberson Museum and Science Center is celebrating 60 Years in 2014 — and 60 years of a treasured tradition of showcasing holiday traditions and celebrations from around the world. "Home for the Holidays" hours are from 11 a.m. to 5 p.m. Saturday-Wednesday; 11 a.m. to 5 p.m. Thursday-Friday; visit the website for a complete schedule of events. The center's 60th anniversary also features a new natural history exhibit, "Journey From Our Prehistoric Past," comprising items loaned by the New York State Museum in Albany and the Museum of the Earth in Ithaca.
Coastal Tales
What: "Under the Isles of Shoals: Archaeology on Smuttynose Island"
Where: Museum of the Earth, 1259 Trumansburg Road, Ithaca
When: Through Monday, Nov. 17
Info: www.museumoftheearth.org
Explore more than 6,000 years of history through The Isles of Shoals, a cluster of rocky islands in the Gulf of Maine, about 10 miles east of Portsmouth, N.H. More than 250,000 artifacts have been unearthed in recent years, and these bones, stones, and fragments of human occupation tell volumes about the people who have tried to make their living on these rocks. Hours are from 10 a.m. to 5 p.m. Monday, Thursday-Saturday, and from 11 a.m. to 5 p.m. Sunday (closed Tuesday and Wednesday). Admission is $5-$8.
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redpajama_set_name: RedPajamaCommonCrawl


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Understated Accessibility
It's a common dilemma for designers: Pleasing the clients who are remodeling or building to accommodate small children or aging parents - or even themselves as they age - but don't want to base decisions on accessibility. "Most of my clients who are aging really don't want to talk about it," says Jessica Williamson, an interior designer based in Millburn, N.J. "I have to [make things safe] without telling them."
The good news? Several popular kitchen products offer significant benefits for multigenerational households, even though they may not be marketed that way. Here are a few such products that any client can get excited about.
Induction cooktops. The fact that you can touch an induction cooktop without getting burned (it reacts only to magnetic cookware) makes it great for homes with young children or older adults. Roberta Kravette, founder and managing director of Nieuw Amsterdam Kitchens in Manhattan, says she wishes her aging father had one when he was living alone. He tended to lean on the front of the range and turn on the gas by accident or not turn it off completely.
Microwave drawers. It's safer to lift a hot dish from a drawer than to reach for it overhead. The microwave drawer also frees up counter space and is easily accessed by someone in a wheelchair. On the downside, liquids can slosh around when the drawer is opened, something to consider if the client will use the microwave primarily for heating beverages.
Refrigerators with bottom-mounted freezers. The refrigerator gets opened an average of 14 times more often than the freezer, says Carm Ricotta, an interior designer based in Brantford, Ontario. Putting the freezer on the bottom raises the refrigerated items nearer to eye level. Kravette likes French-door refrigerators with double-deck freezer drawers, such as those made by LG: "It's easy access for kids. It's also wheelchair accessible," she says.
Pull-down faucets. One benefit offered by the pull-down faucet is that it can fill a pot that's sitting on the counter. That's great for someone who has trouble lifting a pot of water out of the sink. Moen's Edwyn Spot Resist Stainless Spring Pulldown Kitchen Faucet offers flexible water delivery and retracts with ease after use.
Sinks with varying depths. Deep sinks are not optimal for people with back problems because the user must lean over further to use them. Plus, children like to drop things - or throw them - into the sink. That's why designers recommend two sinks in a multigenerational household, one of which is shallow.
Rollout shelves. Besides making good use of cabinet space, rollout shelves are extremely helpful to someone in a wheelchair, or to someone who has trouble bending - whether an expectant mother or a teen on crutches.
Drawer cabinets. Kravette and Williamson recommend that plates be stored in base cabinets with drawers. "The best thing is to be able to pick up something heavy rather than pull it down," Kravette notes. Drawer storage makes great sense if the kids normally set the table.


Andrea collection from Swissmar has spectacle mills designed with meticulous grinding mechanism and charming crystal clear acrylic body for checking refill level easily. The brine mate with sober white tinted wood top is a loyal chef's mate to abet culinary experiences on a splendid note. It is a versatile addition to pro-bar counters as well as in kitchens marked for their specialty continental cuisines.
Swissmar Andrea Salt Mill in White – 20 cm is neatly packed and shipped in a corrugated box.


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Lisinski may refer to:

Vatroslav Lisinski (1819–1854), Croatian composer
Vatroslav Lisinski Concert Hall, concert hall and convention center in Zagreb
Vatroslav Lisinski School of Music, secondary-level music school in Zagreb
Lisinski (film), 1944 film about Vatroslav Lisinski directed by Oktavijan Miletić

redpajama_set_name: RedPajamaWikipedia


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January 2023 Today
April Today June
Tech Lab Consults
Alzheimer's Café
Memory Loss: Guide to Next Steps
Tuesday, May 2, 2023 (all day)
Sign up for a free 30-minute one-on-one consult with Dr. Carolyn Parsey to answer questions about your current devices (cellphones, tablets, laptops, etc) or devices you are considering purchasing. Bring your questions and devices, and explore how apps and other technologies can help solve daily life challenges - from remembering where you parked your car, to reminding you to take your medications.
Tuesday, May 2, 2023 at 10:00am
Tuesday, May 9, 2023 at 2:30pm
Join the Frye Art Museum and the UW Memory and Brain Wellness Center for an uplifting time of social connection! A unique opportunity for people living with dementia, their care partners, family, and friends to enjoy companionship, live music, and fun in a relaxed environment. (Registration Required)
Thursday, May 11, 2023 at 2:00pm
Join the UW Memory and Brain Wellness Center for this free, introductory class for people diagnosed with memory loss, and their families.
Tuesday, May 16, 2023 (all day)
Tuesday, May 16, 2023 at 10:00am


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Game Creator
Picking up coins.
Pandar 3 months ago in Game Creator • updated by Marti (Lead Developer) 3 months ago • 1
Hey. I want to make a simple 3D walker. The character collects coins (their number should be visible in the UI and in the in-game shop), then: Esс - Main Menu - Shop - and buy something. How difficult is it to organize such a structure, taking into account the preservation of data during the transition to the next level? Coins should be Global Variables? How to organize text in UI and in the shop panel? Need additional modules, like inventory?
Unity version:
Game Creator version:
Marti (Lead Developer) 3 months ago
Hi Pandar; The Inventory module would be ideal in this case. Specially if you plan to have multiple purchasable items. The Inventory would greatly simplify this. However, let me give you a brief explanation about how to do this without the Inventory module.
Coins, as you said, should be made as a Numeric Global Variable, marked as "savable". To pick up coins you'd need to first need to create a game object (let's say a sphere with the isTrigger option enabled) and add a Trigger "On Player Enter". This Trigger will be executed when the Player touches it. The Actions sequence when picking a coin could be:
Add Variable +5 to "Coins" Global Variable
Play a nice sound effect & some particles
Destroy the game object
In order to reuse this coin, you can create a prefab out of this and create multiple instances across your scenes. This will allow you to modify the prefab and affect all coins in your levels without having to modify them one by one.
Great! Now we have the picking mechanic. To display the amount of coins on a UI panel, all you need to do is use the Text Variable component that comes with Game Creator. This looks exactly the same as the Unity Text component, but allows you to select a variable (Global Variable "Coins" in your case) and display its value on screen. The value will be automatically updated when it changes, so you don't need to worry about anything.
As for purchasing power-ups in a store, this is a bit trickier. You'll need to use a Button to detect when the user clicks on a particular power-up from the store and launch a Conditions object, which checks if the variable "Coins" is greater or equal to the price of the power-up. If so, you can enable the power-up (through a boolean variable, for example). Otherwise, you can play an "error" sound effect and print a message that says: You don't have enough coins.
With the Inventory module, however, this last section is really easy. You just need to define the item, its price and create a Merchant object, which allows you to define which objects are on sale though a pre-made UI, with discounts, reselling prices, etc...
I'd suggest you try the approach without the Inventory if you don't have it. If it's enough, great! If not, switching to the Inventory shouldn't be much of a hassle.
GOOD, I'M SATISFIED
Satisfaction mark
Satisfaction mark by Pandar 3 months ago