The Gyrokinetic Limit for the Two-Dimensional Vlasov–Yukawa System with a Point Charge

Abstract

In this article, we study the asymptotic behavior of the two-dimensional Vlasov–Yukawa system with a point charge under a large external magnetic field. When the intensity of the magnetic field tends to infinity, we show that the kinetic system converges to the measure-valued Euler equation with a defect measure, which extends the results of Miot to the case of the Vlasov–Yukawa system. And compared with the Miot’s work, an important improvement is that our results do not require compact support conditions for spatial variables or uniform bound conditions for second-order spatial moments. In addition, the extra small condition for initial data is also not required.

1. Introduction and Main Results

1.1. Introduction

In this paper, we study the gyrokinetic limit for the Vlasov–Yukawa system with a point charge, which is described by the asymptotic behaviour of solutions to the following equations as $\epsilon$ tends to zero :

$$\left\{\begin{array}{cc}\hfill & {\partial }_t{f}_{\epsilon }+{\displaystyle \frac{v}{\epsilon }}·{\nabla }_x{f}_{\epsilon }+\left({\displaystyle \frac{v^{\perp }}{{\epsilon }^2}}+{\displaystyle \frac{E_{\epsilon }(t,x)+\gamma F_{\epsilon }(t,x)}{\epsilon }}\right)·{\nabla }_v{f}_{\epsilon }=0,\hfill \\ \hfill & E_{\epsilon }(t,x)=-{\int }_{{\mathbb{R}}^2}\left({\nabla }_{x}G\right)(x-y){\rho }_{\epsilon }(t,y)dy,\hfill \\ \hfill & F_{\epsilon }(t,x)=-\left({\nabla }_{x}G\right)(x-{\xi }_{\epsilon }\left(t\right)),{\rho }_{\epsilon }(t,x)={\int }_{{\mathbb{R}}^2}{f}_{\epsilon }(t,x,v)dv,\hfill \\ \hfill & {\dot{\xi }}_{\epsilon }\left(t\right)={\displaystyle \frac{{\eta }_{\epsilon }\left(t\right)}{\epsilon }},{\dot{\eta }}_{\epsilon }\left(t\right)=\gamma \left({\displaystyle \frac{{\eta }_{\epsilon }^{\perp }\left(t\right)}{{\epsilon }^2}}+{\displaystyle \frac{E_{\epsilon }(t,{\xi }_{\epsilon }\left(t\right))}{\epsilon }}\right),\hfill \\ \hfill & f_{\epsilon }(0,x,v)=f_{\epsilon }^0(x,v),({\xi }_{\epsilon },{\eta }_{\epsilon })\left(0\right)=({\xi }_{\epsilon }^0,{\eta }_{\epsilon }^0).\hfill \end{array}\right.$$

(1)

Here, $f_{\epsilon }=f_{\epsilon }(t,x,v)$ is the phase space density of the plasma particles at time $t\in \mathbb{R}$ and position $x\in {\mathbb{R}}^2$, moving with velocity $v\in {\mathbb{R}}^2$. ${\rho }_{\epsilon }(t,x)$ is the spatial density of the particles. The point charge is located at ${\xi }_{\epsilon }\left(t\right)\in {\mathbb{R}}^2$ moving with velocity ${\eta }_{\epsilon }\left(t\right)\in {\mathbb{R}}^2$. The particles are submitted to the self-consistent electric field $E_{\epsilon }(t,x)$, to the field $F_{\epsilon }(t,x)$ induced by the point charge and to the external magnetic field represented by the terms $v^{\perp }/{\epsilon }^2$ or ${\eta }_{\epsilon }^{\perp }/{\epsilon }^2$. The notation ${(·)}^{\perp }$ stands for the rotation of angle $\pi /2$, i.e., for a vector $a=(a_1,a_2)$, ${(a_1,a_2)}^{\perp }=(-a_2,a_1)$. G is the fundamental solution of the Yukawa equation

$$\begin{array}{c}\hfill (-\Delta +m^2)G=\delta ,\end{array}$$

(2)

where $m>0$ is the mass of particles which is assumed to be a positive constant, and $\delta$ denotes the Dirac distribution at 0. In ${\mathbb{R}}^2$, G can be expressed explicitly in the form

$$\begin{array}{c}\hfill G\left(x\right)={\displaystyle \frac{1}{4\pi }}{\int }_{-\infty }^{+\infty }{\displaystyle \frac{e^{-m\sqrt{{\left|x\right|}^2+{\mu }^2}}}{\sqrt{{\left|x\right|}^2+{\mu }^2}}}d\mu .\end{array}$$

(3)

$f_{\epsilon }^0(x,v)\ge 0$ is the initial phase space density of particles, which is supposed to be given. ${\xi }_{\epsilon }^0\in {\mathbb{R}}^2$ and ${\eta }_{\epsilon }^0\in {\mathbb{R}}^2$ are the initial position and velocity of the point charge, respectively. The real number $\gamma >0$ is the charge–mass ratio of the point charge, which does not depend on $\epsilon$. Without loss of generality, we will assume $\gamma =m=1$ throughout this paper.

For fixed $\epsilon >0$, when there is no point charge and magnetic field , (1) degenerates to the classical Vlasov–Yukawa system. The Vlasov equation was introduced by Vlasov to describe the evolution of a huge number of charged particles in a statistical way (one can refer to [1] for a review of the corresponding articles). And the Yukawa equation was first introduced in 1935 by Yukawa [2] to explain the strong interaction binding of neutrons and protons as a short-range correction of Poisson’s equation. For the classical Vlasov–Yukawa system, Caprino, Marchioro and Pulvirenti [3] obtained the existence and uniqueness in a unbounded mass setting. The authors in [4] established the existence theory and uniform $L^1$ stability estimate for a classical solution to the relativistic Vlasov–Yukawa system with small data. Subsequently, the Vlasov–Yukawa system was further studied in [5,6,7]. Specifically, ref. [7] discussed the nonlinear instability of some class of stationary solutions to the one-dimensional Vlasov–Yukawa system with a mass parameter m. Ref. [5] presented dispersion estimates for small global classical solutions to the Vlasov–Yukawa system in two-dimensional setting. For 2D and higher-dimensional cases, ref. [6] obtained the sharp decay estimates for the Vlasov–Yukawa system with small data by the vector field method. In addition to the above research, recently [8] study the nonltnear stability of the Vlasov–Yukawa system on $\mathbb{R}\times \mathbb{R}$ under near-vacuum conditions, and showed that for initial small data in Gevrey-2 regularity, the derivative of the density of order n decays like ${(t+1)}^{-n-1}$. When there is no magnetic field but with a point charge, (1) was studied by Caprino and Marchioro for the infinite mass problem in [9], in which the existence and uniqueness of the classical solution were established. When there is a magnetic field and a point charge, if the initial data satisfy

$$\begin{array}{c}\hfill f_{\epsilon }^0\in L^1\cap L^{\infty }({\mathbb{R}}^2\times {\mathbb{R}}^2),supp{f}_{\epsilon }^0\subset \{(x,v)\in {\mathbb{R}}^2\times {\mathbb{R}}^2:\left|v\right|\le R_{\epsilon }\mathrm{and}|x-{\xi }_{\epsilon }^0|\ge {\sigma }_{\epsilon }\}\end{array}$$

(4)

for some constants ${\sigma }_{\epsilon },R_{\epsilon }>0$, the global existence and uniqueness of the strong solution $(f_{\epsilon },{\xi }_{\epsilon })$ to (1) can be easily established by employing the technique in [10,11]. And the strong solution $(f_{\epsilon },{\xi }_{\epsilon })$ satisfies that $f_{\epsilon }\in L^{\infty }({\mathbb{R}}_{+},L^1\cap L^{\infty }({\mathbb{R}}^2\times {\mathbb{R}}^2))$ is compactly supported with respect to the velocity variable and vanishes in a neighborhood of ${\xi }_{\epsilon }\left(t\right)$. Moreover, $f_{\epsilon }$ is constant along the solution of the characteristic system

$$\left\{\begin{array}{cc}\hfill & {\displaystyle \frac{d}{ds}}{X}_{\epsilon }(s,t,x,v)={\displaystyle \frac{V_{\epsilon }(s,t,x,v)}{\epsilon }},\hfill \\ \hfill & {\displaystyle \frac{d}{ds}}{V}_{\epsilon }(s,t,x,v)={\displaystyle \frac{V_{\epsilon }^{\perp }(s,t,x,v)}{{\epsilon }^2}}+{\displaystyle \frac{(E_{\epsilon }+F_{\epsilon })(s,X_{\epsilon }(s,t,x,v))}{\epsilon }},\hfill \\ \hfill & (X_{\epsilon },V_{\epsilon })(t,t,x,v)=(x,v),\hfill \end{array}\right.$$

(5)

where $s,t\in [0,+\infty )$ and $(x,v)\in {\mathbb{R}}^2\times {\mathbb{R}}^2$. More precisely, the solution to system (1) can be expressed as

$$f_{\epsilon }(t,x,v)=f_{\epsilon }^0(X_{\epsilon }(0,t,x,v),V_{\epsilon }(0,t,x,v)).$$

The mapping $(x,v)\to (X_{\epsilon }(s,t,x,v),V_{\epsilon }(s,t,x,v))$ is measure-preserving, and for all $(x,v)\in {\mathbb{R}}^2\times {\mathbb{R}}^2$, $s\to (X_{\epsilon }(s,t,x,v),V_{\epsilon }(s,t,x,v))$ belongs to $W^{1,\infty }({\mathbb{R}}_{+},{\mathbb{R}}^2\times {\mathbb{R}}^2)$, which is the unique $C^1$ solution to the characteristic system (5). Moreover, for all $t\ge 0$,

$$\begin{array}{cc}\hfill & \parallel f_{\epsilon }{\left(t\right)\parallel }_{L^p({\mathbb{R}}^2\times {\mathbb{R}}^2)}={\parallel f_{\epsilon }^0\parallel }_{L^p({\mathbb{R}}^2\times {\mathbb{R}}^2)},\forall 1\le p\le \infty ,\hfill \end{array}$$

(6)

$$\begin{array}{cc}\hfill & \mathcal{E}(f_{\epsilon }\left(t\right),{\xi }_{\epsilon }\left(t\right),{\eta }_{\epsilon }\left(t\right))=\mathcal{E}(f_{\epsilon }^0,{\xi }_{\epsilon }^0,{\eta }_{\epsilon }^0).\hfill \end{array}$$

(7)

where $\mathcal{E}(f_{\epsilon }\left(t\right),{\xi }_{\epsilon }\left(t\right),{\eta }_{\epsilon }\left(t\right))$ is the energy associated to (1) given by

$$\begin{array}{cc}\hfill \mathcal{E}(f_{\epsilon }\left(t\right),{\xi }_{\epsilon }\left(t\right),{\eta }_{\epsilon }\left(t\right))=& {\displaystyle \frac{1}{2}}{\int \int }_{{\mathbb{R}}^2\times {\mathbb{R}}^2}{\left|v\right|}^2{f}_{\epsilon }(t,x,v)dxdv+{\displaystyle \frac{1}{2}}{\left|{\eta }_{\epsilon }\left(t\right)\right|}^2\hfill \\ \hfill & +{\displaystyle \frac{1}{2}}{\int \int }_{{\mathbb{R}}_2\times {\mathbb{R}}_2}G(x-y){\rho }_{\epsilon }(t,x){\rho }_{\epsilon }(t,y)dxdy\hfill \\ \hfill & +{\int }_{{\mathbb{R}}^2}G(x-{\xi }_{\epsilon }\left(t\right)){\rho }_{\epsilon }(t,x)dx.\hfill \end{array}$$

For each $\epsilon >0$, (1) is used to describe the interaction of a two-dimensional distribution of light particles (a plasma) and a heavy point charge $\gamma >0$, which are submitted to the field $F_{\epsilon }(t,x)$ induced by the point charge and to a large and constant external magnetic field, orthogonal to the plane. In this article, our goal is to study the asymptotic behavior of system (1) as $\epsilon$ tends to zero, which corresponds to the intensity of the magnetic field as it tends to infinity. Studying the effect of strong magnetic fields on plasmas is of considerable importance in numerical simulations of tokamaks. In the large magnetic field regime, charged particles are trapped along magnetic field lines and rotate around them with a small radius. The radius of rotation of the particle is called the Larmor radius and is inversely proportional to the strength of the magnetic field. As a result, the charged particles are well confined to the tokamak. However, in the presence of such a large magnetic field, the numerical solution of the dynamic equation requires a small-time-step numerical resolution, because the time oscillation of the particles around the magnetic field line is large, resulting in a huge time cost. Therefore, deriving an asymptotic model that reduces the cost of numerical simulation is of great importance.

In recent decades, many kinetic models with strong magnetic fields have been studied; these have usually led to the so-called guiding center or gyrokinetic kinetic models, which are always called the guiding center approximation or the gyrokinetic limit. We refer to [12,13] for physical references and [14,15,16,17,18,19,20,21,22] for mathematical results on this topic. In particular, when there is no point charge and (3) is replaced by $G\left(x\right)={\displaystyle \frac{-1}{2\pi }}ln\left|x\right|$, Golse and Saint-Raymond [15] established the convergence of (1) to the incompressible Euler equation with a defect measure in the periodic setting; see also Brenier’s work [16] in a different regime. Then Saint-Raymond [17] proved that the defect measure is not involved in the Euler equation, and moreover, for sufficiently smooth initial data, he proved a defect measure equal to zero. In addition, ref. [18] obtained the same convergence results as [15,17] by using different methods. Recently Miot has investigated the gyrokinetic limit of the Vlasov–Poisson system with a point charge in [19] and proved that the solution converges to a measure-valued solution of the Euler equation with a defect measure. Furthermore, the limiting equation exactly yields a vortex-wave system if the defect measure vanishes and under more regularity assumptions on $\rho ,\xi$. However, these results require some additional conditions, such as compact support conditions for spatial variables, uniform bound conditions for second-order spatial moments and small conditions for initial data. In addition, it is unknown whether these results hold for the case of the Vlasov–Yukawa system. The objective of this article is to extend the results in [19] to the case of the Vlasov–Yukawa system (1) and remove the above additional restrictions.

Throughout this paper, we denote generic constants by C that may depend on the uniform bounds on the initial data and change from line to line. The norms of the spaces $L^p\left({\mathbb{R}}^2\right)$ or $L^p({\mathbb{R}}^2\times {\mathbb{R}}^2)$ are always denoted by ${\parallel ·\parallel }_p$ for $1\le p\le \infty$ when not confusing. ${\delta }_b$ denotes the Dirac distribution, concentrated at b. When not specifically stated, $\nabla f(t,x)$ represents ${\nabla }_{x}f(t,x)$ and ∗ represents the convolution with respect to x. ${\nabla }^{2}g(t,x)$ is the Hessian matrix of g with respect to x. For a locally compact Hausdorff space $\Omega$, $\mathcal{M}(\Omega )$ denotes the space of bounded real Radon measures on $\Omega$ and ${\mathcal{M}}_{+}(\Omega )$ is the subset consisting of all positive measures. $C_0(\Omega )$ denotes the space of continuous functions vanishing at infinity on $\Omega$. We claim that $\rho \in C_w({\mathbb{R}}_{+}$, ${\mathcal{M}}_{+}(\Omega ))$ if $\rho \left(t\right)\in {\mathcal{M}}_{+}(\Omega )$ for all $t\in {\mathbb{R}}_{+}$ and if, moreover, $t\mapsto {\int }_{\Omega }\Phi \left(x\right)d\rho (t,x)$ is continuous for all $\Phi \in C_0(\Omega )$. The sequence ${\left({\rho }_n\right)}_{n\in \mathbb{N}}$ is said to converge to $\rho$ in $C_w({\mathbb{R}}_{+},{\mathcal{M}}_{+}(\Omega ))$ if, for all $T>0$ and for all $\Phi \in C_0(\Omega )$, we have ${sup}_{t\in [0,T]}|{\int }_{\Omega }\Phi (t,x)(d{\rho }_n(t,x)-d\rho (t,x))|\to 0$ as $n\to +\infty$. The sequence ${\left({\rho }_n\right)}_{n\in \mathbb{N}}$ is said to converge to $\rho$ in $L^{\infty }({\mathbb{R}}_{+},{\mathcal{M}}_{+}(\Omega ))$ weak-∗ if, for all $\Phi \in L^1({\mathbb{R}}_{+},C_0(\Omega ))$, we have $|{\int }_{{\mathbb{R}}_{+}}{\int }_{\Omega }\Phi (t,x)(d{\rho }_n(t,x)-d\rho (t,x))|\to 0$ as $n\to +\infty$. For $2\times 2$ matrices A and B, we set $A:B={\sum }_{1\le i,j\le 2}{A}_{ij}{B}_{ij}$. And for $x=(x_1,x_2)\in {\mathbb{R}}^2$, we set $x\otimes x=\left(\begin{array}{cc}{x}_1^2& x_1{x}_2\\ x_1{x}_2& x_2^2\end{array}\right)$.

1.2. Main Results

Before presenting our main results, let us introduce a symmetric quadratic form ${\mathcal{H}}_{\Phi }[\rho ,\mu ]$. Let $\rho ,\mu \in {\mathcal{M}}_{+}\left({\mathbb{R}}^2\right)$ for all $\Phi \in C_c^{\infty }\left({\mathbb{R}}^2\right)$. The symmetric quadratic form ${\mathcal{H}}_{\Phi }[\rho ,\mu ]$ is defined in [23,24,25] by

$$\begin{array}{c}\hfill {\mathcal{H}}_{\Phi }[\rho ,\mu ]={\displaystyle \frac{1}{2}}{\int \int }_{{\mathbb{R}}^4}{H}_{\Phi }(x,y)d\rho \left(x\right)d\mu \left(y\right),\end{array}$$

(8)

where

$$\begin{array}{c}\hfill H_{\Phi }(x,y)=\left({(\nabla G)}^{\perp }(x-y)\right)·\left(\nabla \Phi \left(y\right)-\nabla \Phi \left(x\right)\right)\mathrm{if}x\ne y,H_{\Phi }(x,x)=0.\end{array}$$

It is obvious that $H_{\Phi }$ is well defined and bounded on ${\mathbb{R}}^2\times {\mathbb{R}}^2$, vanishes at infinity and is continuous outside the diagonal $\left\{(x,x)\right|x\in {\mathbb{R}}^2\}$.

Our main results can now be stated as follows.

Theorem 1. 

Let $(f_{\epsilon }^0,{\xi }_{\epsilon }^0,{\eta }_{\epsilon }^0)$ satisfy (4) and the following assumptions:

$$\begin{array}{c}\hfill \underset{0<\epsilon <1}{sup}\left(\parallel f_{\epsilon }^0{\parallel }_1+\mathcal{E}(f_{\epsilon }^0,{\xi }_{\epsilon }^0,{\eta }_{\epsilon }^0)+\left|{\xi }_{\epsilon }^0\right|\right)<\infty ,\underset{\epsilon \to 0}{lim}{\epsilon }^2{\parallel f_{\epsilon }^0\parallel }_{\infty }=0.\end{array}$$

(9)

Let $(f_{\epsilon },{\xi }_{\epsilon })$ denote the corresponding global strong solution of (1). Then, there exists a subsequence ${\epsilon }_n\to 0$ (as $n\to +\infty$) such that

 (1) 

$\left({\rho }_{{\epsilon }_n}(t,x)\right)$ converges to $\rho (t,x)$ in $C_w({\mathbb{R}}_{+},\mathcal{M}\left({\mathbb{R}}^2\right))$ and $\left({\xi }_{{\epsilon }_n}\left(t\right)\right)$ converges to $\xi \left(t\right)$ in $C^{1/2}([0,T],{\mathbb{R}}^2)$ for all $T>0$.

 (2) 

$\left({\rho }_{{\epsilon }_n}(t,x)\right)$ is uniformly bounded in $C^{1/2}({\mathbb{R}}_{+},W^{-2,1}\left({\mathbb{R}}^2\right))$.

 (3) 

$\rho (t,x)\in C^{1/2}([0,T],W^{-2,1}\left({\mathbb{R}}^2\right))$, $\xi \left(t\right)\in C^{1/2}([0,T],{\mathbb{R}}^2)$ for all $T>0$.

 (4) 

There exists a defect measure $\nu \in {\left[L^{\infty }({\mathbb{R}}_{+},\mathcal{M}\left({\mathbb{R}}^2\right))\right]}^4$ such that, for all $\Phi \in C_c^{\infty }({\mathbb{R}}_{+}\times {\mathbb{R}}^2)$, there holds

$$\begin{array}{cc}\hfill & {\displaystyle \frac{d}{dt}}{\int }_{{\mathbb{R}}^2}\Phi (t,x)d(\rho \left(t\right)+{\delta }_{\xi \left(t\right)})\left(x\right)\hfill \\ \hfill & ={\int }_{{\mathbb{R}}^2}{\partial }_t\Phi (t,x)d(\rho \left(t\right)+{\delta }_{\xi \left(t\right)})\left(x\right)+{\mathcal{H}}_{\Phi (t,x)}[\rho (t,x)+{\delta }_{\xi \left(t\right)},\rho (t,x)+{\delta }_{\xi \left(t\right)}]\hfill \\ \hfill & +{\int }_{{\mathbb{R}}^2}{\nabla }^2\Phi (t,x):d\nu (t,x).\hfill \end{array}$$

(10)

 (5) 

There exist ${\nu }_0={\nu }_0(t,x,\theta )\in L^{\infty }({\mathbb{R}}_{+},{\mathcal{M}}_{+}({\mathbb{R}}^2\times {\mathbb{S}}^1))$ and $(\alpha \left(t\right),\beta \left(t\right))\in L^{\infty }{({\mathbb{R}}_{+},\mathbb{R})}^2$ such that

$$\begin{array}{c}\hfill d\nu (t,x)={\int }_{{\mathbb{R}}^2}{\theta }^{\perp }\otimes \theta d{\nu }_0(t,x,\theta )+\left(\begin{array}{cc}-\alpha \left(t\right)& -\beta \left(t\right)\\ -\beta \left(t\right)& \alpha \left(t\right)\end{array}\right){\delta }_{\xi \left(t\right)}.\end{array}$$

 (6) 

The sequence $\left(f_{{\epsilon }_n}\right)$ converges to $f=f(t,x,|v\left|\right)$ in $L^{\infty }({\mathbb{R}}_{+},{\mathcal{M}}_{+}({\mathbb{R}}^2\times {\mathbb{R}}^2))$ weak-∗ and $\rho =\int fdv$. Moreover, for all Φ continuous on ${\mathbb{S}}^1$,

$$\begin{array}{c}\hfill {\int }_{{\mathbb{R}}^2}(f_{{\epsilon }_n}(t,x,v)-f(t,x,|v\left|\right))\Phi \left({\displaystyle \frac{v}{\left|v\right|}}\right){\left|v\right|}^{2}dv\mathit{converges}\mathit{to}{\int }_{{\mathbb{S}}^1}\Phi \left(\theta \right)d{\nu }_0\left(\theta \right)\end{array}$$

in the sense of distributions on ${\mathbb{R}}_{+}\times {\mathbb{R}}^2$.

Remark 1. 

Note that, in [19], similar results were obtained but required some extra conditions, such as compact support conditions for spatial variable x, uniform bound conditions for second-order spatial moments ${sup}_{0<\epsilon <1}\int {\int }_{{\mathbb{R}}^2\times {\mathbb{R}}^2}{\left|x\right|}^2{f}_{\epsilon }^{0}dxdv<\infty$ and small conditions ${sup}_{0<\epsilon <1}{\parallel f_{\epsilon }^0\parallel }_1<1$. In our work, we do not require these conditions. In addition, in [19], it is proved that the sequence of densities $\left({\rho }_{{\epsilon }_n}\right)$ is uniformly bounded in $C^{1/2}({\mathbb{R}}_{+},W^{-3,1}\left({\mathbb{R}}^2\right))$, which loses one derivative compared to the case without a point charge; our improvement in Proposition 5 and Lemma 8 allowed us to avoid that loss.

When there is no defect measure, assuming additional regularity on $\rho (t,x)$ and $\xi \left(t\right)$, Equation (10) will be reduced to obtain the vortex-wave system.

Theorem 2. 

Let $\left(\rho \right(t,x),\xi (t\left)\right)$ be the accumulation point given by Theorem 1 such that ν vanishes. If we further assume $\rho (t,x)\in L_{\mathrm{loc}}^{\infty }({\mathbb{R}}_{+},L^p\left({\mathbb{R}}^2\right))$ for some $p>2$ and $\xi \left(t\right)\in C^1({\mathbb{R}}_{+},{\mathbb{R}}^2)$, then $\left(\rho \right(t,x),\xi (t\left)\right)$ satisfies

$$\left\{\begin{array}{cc}\hfill & {\partial }_t\rho (t,x)+\left(E^{\perp }(t,x)-{(\nabla G)}^{\perp }(x-\xi \left(t\right))\right)·\nabla \rho (t,x)=0,\hfill \\ \hfill & \dot{\xi }\left(t\right)=E^{\perp }(t,\xi \left(t\right)),E(t,x)=-(\nabla G\ast \rho )(t,x).\hfill \end{array}\right.$$

Remark 2. 

When there is no point charge, the vortex-wave system reduces to the vorticity formulation of the 2D incompressible Euler equation. And when $G\left(x\right)={\displaystyle \frac{-1}{2\pi }}ln\left|x\right|$, the above result was obtained in [19]. We extend it to the case of $G\left(x\right)$ in (3), which is meaningful when the point charge nearly reaches the plasma particles.

This paper considers the case of the influence of a single point charge, and the method in this paper should also be applicable to the system affected by multiple point charges. In addition, the gyrokinetic limit in the three-dimensional case is also worth further study. Although we remove the condition state in Remark 1, we still need some other conditions. For example, for (4) and (9), the method in this paper cannot cancel them at present, and new methods need to be further explored.

2. Preliminary Estimates

2.1. A Priori Estimates

We first give some uniform estimates as follows.

Lemma 1. 

We have

$$\begin{array}{cc}\hfill & \underset{t\ge 0,\epsilon >0}{sup}\left({\int \int }_{{\mathbb{R}}^2\times {\mathbb{R}}^2}{\left|v\right|}^2{f}_{\epsilon }(t,x,v)dxdv+{\left|{\eta }_{\epsilon }\left(t\right)\right|}^2\right)<+\infty ,\hfill \\ \hfill & \underset{t\ge 0,\epsilon >0}{sup}{\int \int }_{{\mathbb{R}}^2\times {\mathbb{R}}^2}G(x-y){\rho }_{\epsilon }(t,x){\rho }_{\epsilon }(t,y)dxdy<+\infty ,\hfill \\ \hfill & \underset{t\ge 0,\epsilon >0}{sup}{\int }_{{\mathbb{R}}^2}G(x-{\xi }_{\epsilon }\left(t\right)){\rho }_{\epsilon }(t,x)dx<+\infty .\hfill \end{array}$$

Proof. 

From Definition (3), we know that $G\left(x\right)\ge 0$, which implies that each term in $\mathcal{E}(f_{\epsilon }\left(t\right),{\xi }_{\epsilon }\left(t\right),{\eta }_{\epsilon }\left(t\right))$ is positive. From energy conservation (7) and assumption (9), we can draw the conclusion. □

Then, we review some classic estimates.

Lemma 2 

([3] Proposition 2.1). Let $G\left(x\right)$ be the fundamental solution of (2) with $x\in {\mathbb{R}}^2$; then, there exists a positive constant C such that

$$|\nabla G\left(x\right)|\le C{\displaystyle \frac{e^{-\frac{m}{2}\left|x\right|}}{\left|x\right|}}.$$

By Lemma 2, the weak Young’s inequality and the interpolation inequality in $L^p$, we obtain the following:

Lemma 3. 

For any $p\in (1,2)$, we have

$$\parallel E_{\epsilon }{\left(t\right)\parallel }_{{\displaystyle \frac{2p}{2-p}}}\le C\parallel {\rho }_{\epsilon }{\left(t\right)\parallel }_p\le C\parallel {\rho }_{\epsilon }{\left(t\right)\parallel }_1^{{\displaystyle \frac{2-p}{p}}}{\parallel {\rho }_{\epsilon }\left(t\right)\parallel }_2^{{\displaystyle \frac{2p-2}{p}}}.$$

Lemma 4. 

In Definition (8), assume that the measure ρ belongs to $L^p$ for some $p>2$. Then, we have, recalling $E\left(x\right)=-(\nabla G\ast \rho )\left(x\right)$,

$${\langle \nabla ·\left(E^{\perp }\rho \right),\Phi \rangle }_{{\mathcal{D}}^{\prime }\left({\mathbb{R}}^2\right),\mathcal{D}\left({\mathbb{R}}^2\right)}=-{\mathcal{H}}_{\Phi }[\rho ,\rho ].$$

Finally, we introduce a proposition that is critical to this article.

Proposition 1. 

Assume that

$$\begin{array}{c}\hfill E\left(\rho \right)=\int {\int }_{{\mathbb{R}}^2\times {\mathbb{R}}^2}G(x-y)\rho \left(x\right)\rho \left(y\right)dxdy<+\infty .\end{array}$$

Then, there exists C depending only on $E\left(\rho \right)$, such that, for all $0<r<{\displaystyle \frac{1}{2}}$, we have

$$\begin{array}{c}\hfill \underset{x_0\in {\mathbb{R}}^2}{sup}{\int }_{B(x_0,r)}\rho \left(x\right)dx\le C{|lnr|}^{-1/2}.\end{array}$$

Proof. 

When $\left|x\right|<1$, we have

$$\begin{array}{cc}\hfill G\left(x\right)& ={\displaystyle \frac{1}{4\pi }}{\int }_{-\infty }^{+\infty }{\displaystyle \frac{e^{-\sqrt{{\left|x\right|}^2+{\mu }^2}}}{\sqrt{{\left|x\right|}^2+{\mu }^2}}}d\mu \ge {\displaystyle \frac{e^{-\sqrt{2}}}{2\pi }}{\int }_0^1{\displaystyle \frac{1}{\sqrt{{\left|x\right|}^2+{\mu }^2}}}d\mu \ge {\displaystyle \frac{e^{-\sqrt{2}}}{2\pi }}ln{\displaystyle \frac{1}{\left|x\right|}}.\hfill \end{array}$$

Since $|x-y|\le |x-x_0|+|y-x_0|\le 2r<1$, then we can apply the above inequality to obtain

$$\begin{array}{cc}\hfill \int {\int }_{{\mathbb{R}}^2\times {\mathbb{R}}^2}G(x-y)\rho \left(x\right)\rho \left(y\right)dxdy& \ge \int {\int }_{|x-x_0|<r,|y-x_0|<r}G(x-y)\rho \left(x\right)\rho \left(y\right)dxdy\hfill \\ \hfill & \ge {\displaystyle \frac{e^{-\sqrt{2}}}{2\pi }}\int {\int }_{|x-x_0|<r,|y-x_0|<r}ln{\displaystyle \frac{1}{|x-y|}}\rho \left(x\right)\rho \left(y\right)dxdy\hfill \\ \hfill & \ge C|ln2r|{\left({\int }_{|x-x_0|<r}\rho \left(x\right)dx\right)}^2.\hfill \end{array}$$

By the assumption $\int {\int }_{{\mathbb{R}}^2\times {\mathbb{R}}^2}G(x-y)\rho \left(x\right)\rho \left(y\right)dxdy<+\infty$, we can draw the conclusion. □

Hence, the following proposition follows directly from Lemma 1.

Proposition 2. 

There exists $C>0$ such that

$$\begin{array}{c}\hfill \underset{t\in {\mathbb{R}}_{+}}{sup}\underset{0<\epsilon <1}{sup}\underset{x_0\in {\mathbb{R}}^2}{sup}\underset{0<r<1/2}{sup}{|lnr|}^{1/2}{\int }_{B(x_0,r)}{\rho }_{\epsilon }(t,x)dx\le C.\end{array}$$

2.2. Estimates for the Point Charge

In this subsection we focus on the dynamics of the point charge. For convenience, we introduce the pointwise energy functional inspired by [3].

Definition 1. 

We define the pointwise energy function as

$$\begin{array}{c}\hfill h_{\epsilon }(t,x,v)={\displaystyle \frac{{\left|v\right|}^2}{2}}+G(x-{\xi }_{\epsilon }\left(t\right))+K,\end{array}$$

where $K>1$ is a constant. Notice $h_{\epsilon }(t,x,v)\ge {\left|v\right|}^2/2$.

We denote

$$\begin{array}{c}\hfill H_{k,\epsilon }\left(t\right)=\underset{0\le s\le t}{sup}{\tilde{H}}_{k,\epsilon }\left(s\right),{\tilde{H}}_{k,\epsilon }\left(t\right)={\int \int }_{{\mathbb{R}}^2\times {\mathbb{R}}^2}{h}_{\epsilon }^{k/2}{f}_{\epsilon }(t,x,v)dxdv.\end{array}$$

We need the following interpolation inequality.

Lemma 5. 

For all $0\le k\le l$, we have

$$\parallel {\int }_{{\mathbb{R}}^2}{h}_{\epsilon }^{k/2}{f}_{\epsilon }(t,·,v)dv{\parallel }_{{\displaystyle \frac{l+2}{k+2}}}\le C{\parallel f_{\epsilon }^0\parallel }_{\infty }^{{\displaystyle \frac{l-k}{l+2}}}{H}_{l,\epsilon }{\left(t\right)}^{{\displaystyle \frac{k+2}{l+2}}}.$$

In particular, $H_{k,\epsilon }\left(t\right)\le C^{\prime }$ for all $0\le k\le 2$, where C depends only on $k,l$, $C^{\prime }$ depends only on $\mathcal{E}(f_{\epsilon }^0,{\xi }_{\epsilon }^0,{\eta }_{\epsilon }^0)$ and $\parallel f_{\epsilon }^0{\parallel }_1$.

Proof. 

By the fact that $h_{\epsilon }(t,x,v)\ge {\left|v\right|}^2/2$, we have

$$\begin{array}{cc}\hfill {\int }_{{\mathbb{R}}^2}{h}_{\epsilon }^{k/2}{f}_{\epsilon }dv& ={\int }_{h_{\epsilon }^{1/2}\le R}{h}_{\epsilon }^{k/2}{f}_{\epsilon }dv+{\int }_{h_{\epsilon }^{1/2}>R}{h}_{\epsilon }^{k/2}{f}_{\epsilon }dv\hfill \\ \hfill & \le R^k{\parallel f_{\epsilon }\parallel }_{\infty }{\int }_{h_{\epsilon }^{1/2}\le R}dv+{\displaystyle \frac{1}{R^{l-k}}}{\int }_{h_{\epsilon }^{1/2}>R}{h}_{\epsilon }^{(l-k)/2}{h}_{\epsilon }^{k/2}{f}_{\epsilon }dv\hfill \\ \hfill & \le 2\pi R^{k+2}{\parallel f_{\epsilon }\parallel }_{\infty }+{\displaystyle \frac{1}{R^{l-k}}}{\int }_{{\mathbb{R}}^2}{h}_{\epsilon }^{l/2}{f}_{\epsilon }dv.\hfill \end{array}$$

Optimizing R, we have

$$\begin{array}{c}\hfill {\int }_{{\mathbb{R}}^2}{h}_{\epsilon }^{{\displaystyle \frac{k}{2}}}{f}_{\epsilon }dv\le (2\pi \parallel f_{\epsilon }{{\parallel }_{\infty })}^{{\displaystyle \frac{l-k}{l+2}}}{\left({\int }_{{\mathbb{R}}^2}{h}_{\epsilon }^{{\displaystyle \frac{l}{2}}}{f}_{\epsilon }dv\right)}^{{\displaystyle \frac{k+2}{l+2}}},\end{array}$$

and taking the $L^{{\displaystyle \frac{l+2}{k+2}}}$ norm in x and combining (6), we obtain the inequality. According to (6), (9) and $h_{\epsilon }>1$, we obtain the boundedness of $H_{k,\epsilon }\left(t\right)$ for $0\le k\le 2$. □

By the above interpolation inequality, we have

$$\begin{array}{c}\hfill \parallel {\rho }_{\epsilon }{\left(t\right)\parallel }_2\le C\parallel f_{\epsilon }{\parallel }_{\infty }^{1/2}{\left({\int \int }_{{\mathbb{R}}^2\times {\mathbb{R}}^2}{\left|v\right|}^2{f}_{\epsilon }(t,x,v)dxdv\right)}^{1/2}\le C{\parallel f_{\epsilon }\parallel }_{\infty }^{1/2}.\end{array}$$

(11)

For the sake of simplicity, we will use the following shorthand in the second part of this section:

$$\begin{array}{cc}\hfill & (X_{\epsilon },V_{\epsilon })\left(s\right)=(X_{\epsilon },V_{\epsilon })(s,0,x,v),\hfill \\ \hfill & {\mathbf{h}}_{\epsilon }\left(s\right)=h_{\epsilon }(s,X_{\epsilon }\left(s\right),V_{\epsilon }\left(s\right)),\hfill \end{array}$$

where $(x,v)$ belongs to the support of $f_{\epsilon }^0$.

Lemma 6. 

Let $(x,v)\in {\mathbb{R}}^2\times {\mathbb{R}}^2$; then, we have

$$\begin{array}{cc}\hfill \left|(\nabla G)\right(X_{\epsilon }\left(s\right)-{\xi }_{\epsilon }\left(s\right)\left)\right|& \le {\epsilon }^2{\displaystyle \frac{d^2}{d{s}^2}}|X_{\epsilon }\left(s\right)-{\xi }_{\epsilon }\left(s\right)|+{\displaystyle \frac{|V_{\epsilon }\left(s\right)-{\eta }_{\epsilon }\left(s\right)|}{\epsilon }}\hfill \\ \hfill & +|E_{\epsilon }(s,X_{\epsilon }\left(s\right))|+|E_{\epsilon }(s,{\xi }_{\epsilon }\left(s\right))|.\hfill \end{array}$$

Proof. 

A simple computation using (5) yields

$$\begin{array}{cc}\hfill {\displaystyle \frac{d}{ds}}|X_{\epsilon }\left(s\right)-{\xi }_{\epsilon }\left(s\right)|& ={\displaystyle \frac{X_{\epsilon }\left(s\right)-{\xi }_{\epsilon }\left(s\right)}{|X_{\epsilon }\left(s\right)-{\xi }_{\epsilon }\left(s\right)|}}·{\displaystyle \frac{V_{\epsilon }\left(s\right)-{\eta }_{\epsilon }\left(s\right)}{\epsilon }}.\hfill \end{array}$$

Hence, we have

$$\begin{array}{cc}\hfill & {\displaystyle \frac{d^2}{d{s}^2}}|X_{\epsilon }\left(s\right)-{\xi }_{\epsilon }\left(s\right)|\hfill \\ \hfill & ={\displaystyle \frac{|V_{\epsilon }\left(s\right)-{\eta }_{\epsilon }{\left(s\right)|}^2}{{\epsilon }^2|X_{\epsilon }\left(s\right)-{\xi }_{\epsilon }\left(s\right)|}}-{\displaystyle \frac{{\left[(X_{\epsilon }\left(s\right)-{\xi }_{\epsilon }\left(s\right))·(V_{\epsilon }\left(s\right)-{\eta }_{\epsilon }\left(s\right))\right]}^2}{{\epsilon }^2{|X_{\epsilon }\left(s\right)-{\xi }_{\epsilon }\left(s\right)|}^3}}\hfill \\ \hfill & +{\displaystyle \frac{X_{\epsilon }\left(s\right)-{\xi }_{\epsilon }\left(s\right)}{\epsilon |X_{\epsilon }\left(s\right)-{\xi }_{\epsilon }\left(s\right)|}}·\left[{\displaystyle \frac{V_{\epsilon }^{\perp }\left(s\right)-{\eta }_{\epsilon }^{\perp }\left(s\right)}{{\epsilon }^2}}+{\displaystyle \frac{(E_{\epsilon }+F_{\epsilon })(s,X_{\epsilon }\left(s\right))}{\epsilon }}-{\displaystyle \frac{E_{\epsilon }(s,{\xi }_{\epsilon }\left(s\right))}{\epsilon }}\right]\hfill \\ \hfill & \ge -{\epsilon }^{-3}|V_{\epsilon }\left(s\right)-{\eta }_{\epsilon }\left(s\right)|-{\epsilon }^{-2}|E_{\epsilon }(s,X_{\epsilon }\left(s\right))|-{\epsilon }^{-2}\left|E_{\epsilon }(s,{\xi }_{\epsilon }\left(s\right))\right|\hfill \\ \hfill & +{\epsilon }^{-2}{\displaystyle \frac{X_{\epsilon }\left(s\right)-{\xi }_{\epsilon }\left(s\right)}{|X_{\epsilon }\left(s\right)-{\xi }_{\epsilon }\left(s\right)|}}·F_{\epsilon }(s,X_{\epsilon }\left(s\right)).\hfill \end{array}$$

(12)

Recall the definition of $G\left(x\right)$ in (3), and by direct calculation we have

$$\begin{array}{c}\hfill \nabla G\left(x\right)={\displaystyle \frac{-x}{4\pi }}{\int }_{-\infty }^{+\infty }{\displaystyle \frac{e^{-\sqrt{{\left|x\right|}^2+{\mu }^2}}}{{\left(\right|x|}^2+{\mu }^2{)}^{3/2}}}\left(\sqrt{{\left|x\right|}^2+{\mu }^2}+1\right)d\mu .\end{array}$$

Then,

$$\begin{array}{cc}\hfill & {\epsilon }^{-2}{\displaystyle \frac{X_{\epsilon }\left(s\right)-{\xi }_{\epsilon }\left(s\right)}{|X_{\epsilon }\left(s\right)-{\xi }_{\epsilon }\left(s\right)|}}·F_{\epsilon }(s,X_{\epsilon }\left(s\right))\hfill \\ \hfill & =-{\epsilon }^{-2}{\displaystyle \frac{X_{\epsilon }\left(s\right)-{\xi }_{\epsilon }\left(s\right)}{|X_{\epsilon }\left(s\right)-{\xi }_{\epsilon }\left(s\right)|}}·(\nabla G)(X_{\epsilon }\left(s\right)-{\xi }_{\epsilon }\left(s\right))\hfill \\ \hfill & ={\epsilon }^{-2}{\displaystyle \frac{|X_{\epsilon }\left(s\right)-{\xi }_{\epsilon }\left(s\right)|}{4\pi }}{\int }_{-\infty }^{+\infty }{\displaystyle \frac{e^{-\sqrt{|X_{\epsilon }\left(s\right)-{\xi }_{\epsilon }{\left(s\right)|}^2+{\mu }^2}}}{\left(\right|X_{\epsilon }\left(s\right)-{\xi }_{\epsilon }\left(s\right){{|}^2+{\mu }^2)}^{3/2}}}\left(\sqrt{|X_{\epsilon }\left(s\right)-{\xi }_{\epsilon }{\left(s\right)|}^2+{\mu }^2}+1\right)d\mu \hfill \\ \hfill & ={\epsilon }^{-2}\left|(\nabla G)(X_{\epsilon }\left(s\right)-{\xi }_{\epsilon }\left(s\right))\right|.\hfill \end{array}$$

(13)

By inserting (13) into (12), we can draw the conclusion. □

Lemma 7. 

Let $(x,v)\in {\mathbb{R}}^2\times {\mathbb{R}}^2$; then, we have

$$\begin{array}{c}\hfill \left|{\displaystyle \frac{d}{ds}}{\mathbf{h}}_{\epsilon }\left(s\right)\right|\le C{\epsilon }^{-1}\left(\sqrt{{\mathbf{h}}_{\epsilon }\left(s\right)}|E_{\epsilon }(s,X_{\epsilon }\left(s\right))|+|(\nabla G)(X_{\epsilon }\left(s\right)-{\xi }_{\epsilon }\left(s\right))|\right).\end{array}$$

Proof. 

We compute

$$\begin{array}{cc}\hfill {\displaystyle \frac{d}{ds}}{\mathbf{h}}_{\epsilon }\left(s\right)& ={\epsilon }^{-1}{V}_{\epsilon }\left(s\right)·\left(E_{\epsilon }(s,X_{\epsilon }\left(s\right))-(\nabla G)(X_{\epsilon }\left(s\right)-{\xi }_{\epsilon }\left(s\right))\right)\hfill \\ \hfill & +{\epsilon }^{-1}(\nabla G)(X_{\epsilon }\left(s\right)-{\xi }_{\epsilon }\left(s\right))·(V_{\epsilon }\left(s\right)-{\eta }_{\epsilon }\left(s\right))\hfill \\ \hfill & ={\epsilon }^{-1}{V}_{\epsilon }\left(s\right)·E_{\epsilon }(s,X_{\epsilon }\left(s\right))-{\epsilon }^{-1}(\nabla G)(X_{\epsilon }\left(s\right)-{\xi }_{\epsilon }\left(s\right))·{\eta }_{\epsilon }\left(s\right).\hfill \end{array}$$

Then, we have

$$\begin{array}{cc}\hfill \left|{\displaystyle \frac{d}{ds}}{\mathbf{h}}_{\epsilon }\left(s\right)\right|& \le {\epsilon }^{-1}\sqrt{2{\mathbf{h}}_{\epsilon }\left(s\right)}|E_{\epsilon }(s,X_{\epsilon }\left(s\right))|+{\epsilon }^{-1}|(\nabla G)(X_{\epsilon }\left(s\right)-{\xi }_{\epsilon }\left(s\right))\left|\right|{\eta }_{\epsilon }\left(s\right)|\hfill \\ \hfill & \le C{\epsilon }^{-1}\left(\sqrt{{\mathbf{h}}_{\epsilon }\left(s\right)}|E_{\epsilon }(s,X_{\epsilon }\left(s\right))|+|(\nabla G)(X_{\epsilon }\left(s\right)-{\xi }_{\epsilon }\left(s\right))|\right),\hfill \end{array}$$

where we have used the fact that $|{\eta }_{\epsilon }\left(s\right)|\le C$ by Lemma 1. □

Proposition 3. 

We define the k-th order singular moment by

$$L_{k,\epsilon }\left(t\right):={\int }_0^t{\int \int }_{{\mathbb{R}}^2\times {\mathbb{R}}^2}\left|(\nabla G)(X_{\epsilon }\left(s\right)-{\xi }_{\epsilon }\left(s\right))\right|{\mathbf{h}}_{\epsilon }{\left(s\right)}^{k/2}{f}_{\epsilon }^{0}dxdvds.$$

Then, for all $0\le k\le l$, we have

$$\begin{array}{cc}\hfill L_{k,\epsilon }\left(t\right)\le & C[\left(t{\epsilon }^{-1}+\epsilon \right)H_{k+1,\epsilon }\left(t\right)+H_{k,\epsilon }\left(t\right)L_{0,\epsilon }\left(t\right)\hfill \\ \hfill & +\parallel f_{\epsilon }^0{\parallel }_{\infty }^{{\displaystyle \frac{l-k}{l+2}}}{H}_{l,\epsilon }{\left(t\right)}^{{\displaystyle \frac{k+2}{l+2}}}{\int }_0^t{\parallel E_{\epsilon }\left(s\right)\parallel }_{{\displaystyle \frac{l+2}{l-k}}}ds+L_{k-1,\epsilon }\left(t\right)],\hfill \end{array}$$

where C depends only on $\parallel f_{\epsilon }^0{\parallel }_1$, $\mathcal{E}(f_{\epsilon }^0,{\xi }_{\epsilon }^0,{\eta }_{\epsilon }^0)$ and $l,k$.

Proof. 

From Lemma 6, we have

$$\begin{array}{cc}\hfill & {\int }_0^t{\int \int }_{{\mathbb{R}}^2\times {\mathbb{R}}^2}\left|(\nabla G)(X_{\epsilon }\left(s\right)-{\xi }_{\epsilon }\left(s\right))\right|{\mathbf{h}}_{\epsilon }{\left(s\right)}^{k/2}{f}_{\epsilon }^{0}dxdvds\hfill \\ \hfill & \le {\int }_0^t{\int \int }_{{\mathbb{R}}^2\times {\mathbb{R}}^2}{\mathbf{h}}_{\epsilon }{\left(s\right)}^{k/2}{f}_{\epsilon }^0\left({\epsilon }^2{\displaystyle \frac{d^2}{d{s}^2}}|X_{\epsilon }\left(s\right)-{\xi }_{\epsilon }\left(s\right)|\right)dxdvds\hfill \\ \hfill & +{\int }_0^t{\int \int }_{{\mathbb{R}}^2\times {\mathbb{R}}^2}{\mathbf{h}}_{\epsilon }{\left(s\right)}^{k/2}{f}_{\epsilon }^0\left({\epsilon }^{-1}|V_{\epsilon }\left(s\right)-{\eta }_{\epsilon }\left(s\right)|\right)dxdvds\hfill \\ \hfill & +{\int }_0^t{\int \int }_{{\mathbb{R}}^2\times {\mathbb{R}}^2}{\mathbf{h}}_{\epsilon }{\left(s\right)}^{k/2}{f}_{\epsilon }^0\left|E_{\epsilon }(s,X_{\epsilon }\left(s\right))\right|dxdvds\hfill \\ \hfill & +{\int }_0^t{\int \int }_{{\mathbb{R}}^2\times {\mathbb{R}}^2}{\mathbf{h}}_{\epsilon }{\left(s\right)}^{k/2}{f}_{\epsilon }^0\left|E_{\epsilon }(s,{\xi }_{\epsilon }\left(s\right))\right|dxdvds\hfill \\ \hfill & =:I_{\epsilon }^1\left(t\right)+I_{\epsilon }^2\left(t\right)+I_{\epsilon }^3\left(t\right)+I_{\epsilon }^4\left(t\right).\hfill \end{array}$$

(14)

Now, we estimate each term. For $I_{\epsilon }^1\left(t\right)$,

$$\begin{array}{cc}\hfill I_{\epsilon }^1\left(t\right)& ={\int }_0^t{\int \int }_{{\mathbb{R}}^2\times {\mathbb{R}}^2}{\mathbf{h}}_{\epsilon }{\left(s\right)}^{k/2}{f}_{\epsilon }^0\left({\epsilon }^2{\displaystyle \frac{d^2}{d{s}^2}}|X_{\epsilon }\left(s\right)-{\xi }_{\epsilon }\left(s\right)|\right)dxdvds\hfill \\ \hfill & ={\epsilon }^2{\int \int }_{{\mathbb{R}}^2\times {\mathbb{R}}^2}{f}_{\epsilon }^0\left({\int }_0^t{\mathbf{h}}_{\epsilon }{\left(s\right)}^{k/2}{\displaystyle \frac{d^2}{d{s}^2}}|X_{\epsilon }\left(s\right)-{\xi }_{\epsilon }\left(s\right)|ds\right)dxdv\hfill \\ \hfill & ={\epsilon }^2{\int \int }_{{\mathbb{R}}^2\times {\mathbb{R}}^2}{f}_{\epsilon }^0\{\left[{\mathbf{h}}_{\epsilon }{\left(s\right)}^{k/2}{\displaystyle \frac{d}{ds}}|X_{\epsilon }\left(s\right)-{\xi }_{\epsilon }\left(s\right)|\right]{|}_0^t\hfill \\ \hfill & -{\int }_0^t{\displaystyle \frac{d}{ds}}{\mathbf{h}}_{\epsilon }{\left(s\right)}^{k/2}{\displaystyle \frac{d}{ds}}|X_{\epsilon }\left(s\right)-{\xi }_{\epsilon }\left(s\right)|ds\}dxdv,\hfill \end{array}$$

(15)

and by direct calculation, we have

$$\begin{array}{cc}\hfill \left|{\displaystyle \frac{d}{ds}}|X_{\epsilon }\left(s\right)-{\xi }_{\epsilon }\left(s\right)|\right|& =|{\displaystyle \frac{X_{\epsilon }\left(s\right)-{\xi }_{\epsilon }\left(s\right)}{|X_{\epsilon }\left(s\right)-{\xi }_{\epsilon }\left(s\right)|}}·{\displaystyle \frac{V_{\epsilon }\left(s\right)-{\eta }_{\epsilon }\left(s\right)}{\epsilon }}|\le {\epsilon }^{-1}|V_{\epsilon }\left(s\right)-{\eta }_{\epsilon }\left(s\right)|\hfill \\ \hfill & \le {\epsilon }^{-1}\left(\right|V_{\epsilon }\left(s\right)|+C)\le C{\epsilon }^{-1}{\mathbf{h}}_{\epsilon }{\left(s\right)}^{1/2},\hfill \end{array}$$

where we have used the fact that $|{\eta }_{\epsilon }\left(s\right)|\le C$ by Lemma 1. From the above inequality and Lemma 7, we have

$$\begin{array}{c}\hfill \left[{\mathbf{h}}_{\epsilon }{\left(s\right)}^{k/2}\left|{\displaystyle \frac{d}{ds}}|X_{\epsilon }\left(s\right)-{\xi }_{\epsilon }\left(s\right)|\right|\right]{|}_0^t\le C{\epsilon }^{-1}({\mathbf{h}}_{\epsilon }{\left(t\right)}^{(k+1)/2}+{\mathbf{h}}_{\epsilon }{\left(0\right)}^{(k+1)/2})\end{array}$$

(16)

and

$$\begin{array}{cc}\hfill & {\int }_0^t{\displaystyle \frac{d}{ds}}{\mathbf{h}}_{\epsilon }{\left(s\right)}^{k/2}{\displaystyle \frac{d}{ds}}|X_{\epsilon }\left(s\right)-{\xi }_{\epsilon }\left(s\right)|ds\hfill \\ \hfill & \le C{\epsilon }^{-2}{\int }_0^t{\mathbf{h}}_{\epsilon }{\left(s\right)}^{{\displaystyle \frac{k}{2}}-1}\left(\sqrt{{\mathbf{h}}_{\epsilon }\left(s\right)}|E_{\epsilon }(s,X_{\epsilon }\left(s\right))|+|(\nabla G)(X_{\epsilon }\left(s\right)-{\xi }_{\epsilon }\left(s\right))|\right){\mathbf{h}}_{\epsilon }{\left(s\right)}^{1/2}ds\hfill \\ \hfill & \le C{\epsilon }^{-2}{\int }_0^t{\mathbf{h}}_{\epsilon }{\left(s\right)}^{k/2}\left[|E_{\epsilon }(s,X_{\epsilon }\left(s\right))|+{\mathbf{h}}_{\epsilon }{\left(s\right)}^{-1/2}\left|(\nabla G)(X_{\epsilon }\left(s\right)-{\xi }_{\epsilon }\left(s\right))\right|\right]ds.\hfill \end{array}$$

(17)

According to (16) and (17), we have, from (15),

$$\begin{array}{cc}\hfill I_{\epsilon }^1\left(t\right)& \le C\epsilon {\int \int }_{{\mathbb{R}}^2\times {\mathbb{R}}^2}{f}_{\epsilon }^0\left({\mathbf{h}}_{\epsilon }{\left(t\right)}^{(k+1)/2}+{\mathbf{h}}_{\epsilon }{\left(0\right)}^{(k+1)/2}\right)dxdv\hfill \\ \hfill & +C{\int \int }_{{\mathbb{R}}^2\times {\mathbb{R}}^2}{f}_{\epsilon }^0\left\{{\int }_0^t{\mathbf{h}}_{\epsilon }{\left(s\right)}^{k/2}\left[|E_{\epsilon }(s,X_{\epsilon }\left(s\right))|+{\displaystyle \frac{\left|(\nabla G)\right(X_{\epsilon }\left(s\right)-{\xi }_{\epsilon }\left(s\right)\left)\right|}{{\mathbf{h}}_{\epsilon }{\left(s\right)}^{1/2}}}\right]ds\right\}dxdv\hfill \\ \hfill & \le C(\epsilon H_{k+1,\epsilon }\left(t\right)+{\int }_0^t{\int \int }_{{\mathbb{R}}^2\times {\mathbb{R}}^2}\mathbf{h}{\left(s\right)}^{k/2}{f}_{\epsilon }^0\left|E_{\epsilon }(s,X_{\epsilon }\left(s\right))\right|dxdvds\hfill \\ \hfill & +{\int }_0^t{\int \int }_{{\mathbb{R}}^2\times {\mathbb{R}}^2}{\mathbf{h}}_{\epsilon }{\left(s\right)}^{(k-1)/2}{f}_{\epsilon }^0\left|(\nabla G)(X_{\epsilon }\left(s\right)-{\xi }_{\epsilon }\left(s\right))\right|dxdvds)\hfill \\ \hfill & \le C\left(\epsilon H_{k+1,\epsilon }\left(t\right)+\parallel f_{\epsilon }^0{\parallel }_{\infty }^{{\displaystyle \frac{l-k}{l+2}}}{H}_{l,\epsilon }{\left(t\right)}^{{\displaystyle \frac{k+2}{l+2}}}{\int }_0^t{\parallel E_{\epsilon }\left(s\right)\parallel }_{{\displaystyle \frac{l+2}{l-k}}}ds+L_{k-1,\epsilon }\left(t\right)\right),\hfill \end{array}$$

(18)

where we have used the Hölder inequality and Lemma 5.

For $I_{\epsilon }^2\left(t\right)$, we have

$$\begin{array}{cc}\hfill I_{\epsilon }^2\left(t\right)& ={\int }_0^t{\int \int }_{{\mathbb{R}}^2\times {\mathbb{R}}^2}{\mathbf{h}}_{\epsilon }{\left(s\right)}^{k/2}{f}_{\epsilon }^0\left({\epsilon }^{-1}|V_{\epsilon }\left(s\right)-{\eta }_{\epsilon }\left(s\right)|\right)dxdvds\hfill \\ \hfill & \le {\epsilon }^{-1}{\int }_0^t{\int \int }_{{\mathbb{R}}^2\times {\mathbb{R}}^2}{\mathbf{h}}_{\epsilon }{\left(s\right)}^{k/2}{f}_{\epsilon }^0\left|V_{\epsilon }\left(s\right)\right|dxdvds\hfill \\ \hfill & +{\epsilon }^{-1}{\int }_0^t{\int \int }_{{\mathbb{R}}^2\times {\mathbb{R}}^2}{\mathbf{h}}_{\epsilon }{\left(s\right)}^{k/2}{f}_{\epsilon }^0\left|{\eta }_{\epsilon }\left(s\right)\right|dxdvds\hfill \\ \hfill & \le {\displaystyle \frac{Ct}{\epsilon }}{H}_{k+1,\epsilon }\left(t\right).\hfill \end{array}$$

(19)

For $I_{\epsilon }^3\left(t\right)$, by the Hölder inequality and Lemma 5, we have

$$\begin{array}{cc}\hfill I_{\epsilon }^3\left(t\right)& ={\int }_0^t{\int \int }_{{\mathbb{R}}^2\times {\mathbb{R}}^2}{\mathbf{h}}_{\epsilon }{\left(s\right)}^{k/2}{f}_{\epsilon }^0\left|E_{\epsilon }(s,X_{\epsilon }\left(s\right))\right|dxdvds\hfill \\ \hfill & \le {\int }_0^t\parallel {\int }_{{\mathbb{R}}^2}{h}_{\epsilon }^{k/2}{f}_{\epsilon }(s,x,v)dv{\parallel }_{{\displaystyle \frac{l+2}{k+2}}}{\parallel E_{\epsilon }\left(s\right)\parallel }_{{\displaystyle \frac{l+2}{l-k}}}ds\hfill \\ \hfill & \le C\parallel f_{\epsilon }^0{\parallel }_{\infty }^{{\displaystyle \frac{l-k}{l+2}}}{H}_{l,\epsilon }{\left(t\right)}^{{\displaystyle \frac{k+2}{l+2}}}{\int }_0^t{\parallel E_{\epsilon }\left(s\right)\parallel }_{{\displaystyle \frac{l+2}{l-k}}}ds.\hfill \end{array}$$

(20)

For $I_{\epsilon }^4\left(t\right)$, we have

$$\begin{array}{cc}\hfill I_{\epsilon }^4\left(t\right)& ={\int }_0^t{\int \int }_{{\mathbb{R}}^2\times {\mathbb{R}}^2}{\mathbf{h}}_{\epsilon }{\left(s\right)}^{k/2}{f}_{\epsilon }^0\left|E_{\epsilon }(s,{\xi }_{\epsilon }\left(s\right))\right|dxdvds\hfill \\ \hfill & \le C{H}_{k,\epsilon }\left(t\right){\int }_0^t\left|E_{\epsilon }(s,{\xi }_{\epsilon }\left(s\right))\right|ds\le C{H}_{k,\epsilon }\left(t\right)L_{0,\epsilon }\left(t\right).\hfill \end{array}$$

(21)

Combining (14) with (18)–(21), we have

$$\begin{array}{cc}\hfill & {\int }_0^t{\int \int }_{{\mathbb{R}}^2\times {\mathbb{R}}^2}\left|(\nabla G)(X_{\epsilon }\left(s\right)-{\xi }_{\epsilon }\left(s\right))\right|{\mathbf{h}}_{\epsilon }{\left(s\right)}^{k/2}{f}_{\epsilon }^{0}dxdvds\hfill \\ \hfill & \le C(\left(t{\epsilon }^{-1}+\epsilon \right)H_{k+1,\epsilon }\left(t\right)+H_{k,\epsilon }\left(t\right)L_{0,\epsilon }\left(t\right)\hfill \\ \hfill & +\parallel f_{\epsilon }^0{\parallel }_{\infty }^{{\displaystyle \frac{l-k}{l+2}}}{H}_{l,\epsilon }{\left(t\right)}^{{\displaystyle \frac{k+2}{l+2}}}{\int }_0^t{\parallel E_{\epsilon }\left(s\right)\parallel }_{{\displaystyle \frac{l+2}{l-k}}}ds+L_{k-1,\epsilon }\left(t\right)),\hfill \end{array}$$

which proves Proposition 3. □

An important consequence of Proposition 3 is the following estimate on $L_{0,\epsilon }\left(t\right)$.

Proposition 4. 

We have, for all $t\ge 0$,

$$L_{0,\epsilon }\left(t\right)\le C{\epsilon }^{-1}t+C\epsilon ,$$

where C depends only on $\parallel f_{\epsilon }^0{\parallel }_1$ and $\mathcal{E}(f_{\epsilon }^0,{\xi }_{\epsilon }^0,{\eta }_{\epsilon }^0)$.

Proof. 

Let $R,\sigma >0$ be undetermined parameters. We divide ${\mathbb{R}}^2\times {\mathbb{R}}^2$ into the following three parts:

$$\begin{array}{cc}\hfill J_1:=& \{(x,v)\in {\mathbb{R}}^2\times {\mathbb{R}}^2:|X_{\epsilon }\left(s\right)-{\xi }_{\epsilon }\left(s\right)|>\sigma \};\hfill \\ \hfill J_2:=& \{(x,v)\in {\mathbb{R}}^2\times {\mathbb{R}}^2:{\mathbf{h}}_{\epsilon }\left(s\right)>R\}\setminus J_1;\hfill \\ \hfill J_3:=& {\mathbb{R}}^2\times {\mathbb{R}}^2\setminus (J_1\cup J_2).\hfill \end{array}$$

Then, from Lemma 2, we have

$$\begin{array}{cc}\hfill L_{0,\epsilon }\left(t\right)& ={\int }_0^t{\int \int }_{{\mathbb{R}}^2\times {\mathbb{R}}^2}\left|(\nabla G)(X_{\epsilon }\left(s\right)-{\xi }_{\epsilon }\left(s\right))\right|f_{\epsilon }^0(x,v)dxdvds\hfill \\ \hfill & \le C{\int }_0^t{\int \int }_{{\mathbb{R}}^2\times {\mathbb{R}}^2}{\displaystyle \frac{f_{\epsilon }^0(x,v)}{|X_{\epsilon }\left(s\right)-{\xi }_{\epsilon }\left(s\right)|}}dxdvds\hfill \\ \hfill & \le C{\int }_0^t{\int \int }_{J_1}{\displaystyle \frac{f_{\epsilon }^0(x,v)}{|X_{\epsilon }\left(s\right)-{\xi }_{\epsilon }\left(s\right)|}}dxdvds+C{\int }_0^t{\int \int }_{J_2}{\displaystyle \frac{f_{\epsilon }^0(x,v)}{|X_{\epsilon }\left(s\right)-{\xi }_{\epsilon }\left(s\right)|}}dxdvds\hfill \\ \hfill & +C{\int }_0^t{\int \int }_{J_3}{\displaystyle \frac{f_{\epsilon }^0(x,v)}{|X_{\epsilon }\left(s\right)-{\xi }_{\epsilon }\left(s\right)|}}dxdvds\hfill \\ \hfill & \le C{\sigma }^{-1}{\parallel f_{\epsilon }^0\parallel }_{1}t+C{R}^{-1/16}{\int }_0^t{\int \int }_{{\mathbb{R}}^2\times {\mathbb{R}}^2}{\displaystyle \frac{{\mathbf{h}}_{\epsilon }{\left(s\right)}^{1/16}{f}_{\epsilon }^0(x,v)}{|X_{\epsilon }\left(s\right)-{\xi }_{\epsilon }\left(s\right)|}}dxdvds\hfill \\ \hfill & +C{\int }_0^t{\int \int }_{|x-{\xi }_{\epsilon }\left(s\right)|\le \sigma ,h_{\epsilon }(s,x,v)\le R}{\displaystyle \frac{f_{\epsilon }(s,x,v)}{|x-{\xi }_{\epsilon }\left(s\right)|}}dxdvds.\hfill \end{array}$$

(22)

Since $\left|v\right|\le 2{\sqrt{h}}_{\epsilon }(s,x,v)$ and (6), we have

$$\begin{array}{cc}\hfill & {\int }_0^t{\int \int }_{|x-{\xi }_{\epsilon }\left(s\right)|\le \sigma ,h_{\epsilon }(s,x,v)\le R}{\displaystyle \frac{f_{\epsilon }(s,x,v)}{|x-{\xi }_{\epsilon }\left(s\right)|}}dxdvds\hfill \\ \hfill & \le {\int }_0^t{\int \int }_{|x-{\xi }_{\epsilon }\left(s\right)|\le \sigma ,|v|\le 2\sqrt{R}}{\displaystyle \frac{f_{\epsilon }(s,x,v)}{|x-{\xi }_{\epsilon }\left(s\right)|}}dxdvds\hfill \\ \hfill & \le C\parallel f_{\epsilon }^0{\parallel }_{\infty }R\sigma t.\hfill \end{array}$$

(23)

Inserting (23) into (22), we have

$$\begin{array}{cc}\hfill L_{0,\epsilon }\left(t\right)& ={\int }_0^t{\int \int }_{{\mathbb{R}}^2\times {\mathbb{R}}^2}\left|(\nabla G)(X_{\epsilon }\left(s\right)-{\xi }_{\epsilon }\left(s\right))\right|f_{\epsilon }^0(x,v)dxdvds\hfill \\ \hfill & \le C{\sigma }^{-1}t+C{R}^{-1/16}{L}_{{\displaystyle \frac{1}{8}},\epsilon }\left(t\right)+C{\parallel f_{\epsilon }^0\parallel }_{\infty }R\sigma t.\hfill \end{array}$$

(24)

Taking $k={\displaystyle \frac{1}{8}}$, $l={\displaystyle \frac{9}{8}}$ in Proposition 3, we have

$$\begin{array}{cc}\hfill L_{{\displaystyle \frac{1}{8}},\epsilon }\left(t\right)& \le C(\left(t{\epsilon }^{-1}+\epsilon \right)H_{{\displaystyle \frac{9}{8}},\epsilon }\left(t\right)+H_{{\displaystyle \frac{1}{8}},\epsilon }\left(t\right)L_{0,\epsilon }\left(t\right)\hfill \\ \hfill & +\parallel f_{\epsilon }^0{\parallel }_{\infty }^{{\displaystyle \frac{8}{25}}}{H}_{{\displaystyle \frac{9}{8}},\epsilon }{\left(t\right)}^{{\displaystyle \frac{17}{25}}}{\int }_0^t{\parallel E_{\epsilon }\left(s\right)\parallel }_{{\displaystyle \frac{25}{8}}}ds+L_{-{\displaystyle \frac{7}{8}},\epsilon }\left(t\right)),\hfill \end{array}$$

and since ${\mathbf{h}}_{\epsilon }\left(s\right)\ge 1$, we have $L_{{\displaystyle \frac{-7}{8}},\epsilon }\left(t\right)\le L_{0,\epsilon }\left(t\right)$. By Lemma 3 with $p={\displaystyle \frac{50}{41}}$, (11) and (6), we have

$$\begin{array}{c}\hfill \parallel E_{\epsilon }{\left(s\right)\parallel }_{{\displaystyle \frac{25}{8}}}\le C\parallel {\rho }_{\epsilon }{\left(s\right)\parallel }_1^{{\displaystyle \frac{16}{25}}}\parallel {\rho }_{\epsilon }{\left(s\right)\parallel }_2^{{\displaystyle \frac{9}{25}}}\le C{\parallel f_{\epsilon }^0\parallel }_{\infty }^{{\displaystyle \frac{9}{50}}},\end{array}$$

and from Lemma 5, we know that $H_{{\displaystyle \frac{9}{8}},\epsilon }\left(t\right)\le C$ and $H_{{\displaystyle \frac{1}{8}},\epsilon }\left(t\right)\le C$; hence, we obtain

$$\begin{array}{c}\hfill L_{{\displaystyle \frac{1}{8}},\epsilon }\left(t\right)\le C\left(t{\epsilon }^{-1}+\epsilon +L_{0,\epsilon }\left(t\right)+{\parallel f_{\epsilon }^0\parallel }_{\infty }^{{\displaystyle \frac{1}{2}}}t\right).\end{array}$$

(25)

Now, taking $R=max\{1,{\left(2C\right)}^{16}\}$ and inserting (25) into (24), we have

$$\begin{array}{cc}\hfill L_{0,\epsilon }\left(t\right)& \le C({\sigma }^{-1}+{\epsilon }^{-1}+\parallel f_{\epsilon }^0{\parallel }_{\infty }^{{\displaystyle \frac{1}{2}}}+\parallel f_{\epsilon }^0{\parallel }_{\infty }\sigma )t+C\epsilon \hfill \\ \hfill & <C({\sigma }^{-1}+{\epsilon }^{-1}+{\epsilon }^{-2}\sigma )t+C\epsilon .\hfill \end{array}$$

Letting $\sigma =\epsilon$, we can obtain the conclusion. □

2.3. Equicontinuity for the Point Charge and Densities

This subsection aims to establish the equicontinuity for the point charge and densities. Before that, we re-express system (1) thanks to Proposition 4.

Proposition 5. 

Let $\Phi (t,x)\in C_c^{\infty }({\mathbb{R}}_{+}\times {\mathbb{R}}^2)$ for all $t\ge 0$; then, we have

$$\begin{array}{cc}\hfill & {\int }_{{\mathbb{R}}^2}\Phi (t,x)\left({\rho }_{\epsilon }(t,x)+{\delta }_{{\xi }_{\epsilon }\left(t\right)}\right)dx-{\int }_{{\mathbb{R}}^2}\Phi (0,x)\left({\rho }_{\epsilon }(0,x)+{\delta }_{{\xi }_{\epsilon }\left(0\right)}\right)dx\hfill \\ \hfill & ={\int }_0^t{\int }_{{\mathbb{R}}^2}{\partial }_t\Phi (s,x)\left({\rho }_{\epsilon }(s,x)+{\delta }_{{\xi }_{\epsilon }\left(s\right)}\right)dsdx\hfill \\ \hfill & +{\int }_0^t{\mathcal{H}}_{\Phi (s,x)}[{\rho }_{\epsilon }(s,x)+{\delta }_{{\xi }_{\epsilon }\left(s\right)},{\rho }_{\epsilon }(s,x)+{\delta }_{{\xi }_{\epsilon }\left(s\right)}]ds\hfill \\ \hfill & +{\int }_0^t{\int \int }_{{\mathbb{R}}^2\times {\mathbb{R}}^2}{\nabla }^2\Phi (s,x):v^{\perp }\otimes v{f}_{\epsilon }(s,x,v)dvdsdx\hfill \\ \hfill & +{\int }_0^t({\nabla }^2\Phi )(s,{\xi }_{\epsilon }\left(s\right)):{\eta }_{\epsilon }^{\perp }\otimes {\eta }_{\epsilon }\left(s\right)ds\hfill \\ \hfill & +I_{\epsilon }^1\left(t\right)+I_{\epsilon }^2\left(t\right),\hfill \end{array}$$

where

$$|I_{\epsilon }^1\left(t\right)+I_{\epsilon }^2{\left(t\right)|\le C\parallel \Phi \parallel }_{W^{2,\infty }({\mathbb{R}}_{+}\times {\mathbb{R}}^2)}(t+1)\epsilon .$$

Proof. 

From Proposition 4, we have

$${\int }_0^t\left|E_{\epsilon }(s,{\xi }_{\epsilon }\left(s\right))\right|ds\le L_{0,\epsilon }\left(t\right)\le C{\epsilon }^{-1}t+C\epsilon ,$$

and then we can obtain that the following equation holds in the distributional sense (see [19] Proposition 2.9):

$$\begin{array}{c}\hfill {\partial }_t{\rho }_{\epsilon }+\nabla ·\left((E_{\epsilon }^{\perp }+F_{\epsilon }^{\perp }){\rho }_{\epsilon }\right)=\nabla ·\left({\left[\nabla ·{\int }_{{\mathbb{R}}^2}v\otimes v{f}_{\epsilon }dv\right]}^{\perp }\right)+\epsilon \nabla ·{\partial }_t{\int }_{{\mathbb{R}}^2}{v}^{\perp }{f}_{\epsilon }dv.\end{array}$$

Apply the above equation with the test function $\Phi$. After symmetrizing the term $E_{\epsilon }^{\perp }{\rho }_{\epsilon }$ as (8), we have

$$\begin{array}{cc}\hfill & {\int }_{{\mathbb{R}}^2}\Phi (t,x){\rho }_{\epsilon }(t,x)dx-{\int }_{{\mathbb{R}}^2}\Phi (0,x){\rho }_{\epsilon }(0,x)dx-{\int }_0^t{\int }_{{\mathbb{R}}^2}{\partial }_t\Phi (s,x){\rho }_{\epsilon }(s,x)dsdx\hfill \\ \hfill & ={\int }_0^t{\mathcal{H}}_{\Phi (s,x)}[{\rho }_{\epsilon }(s,x),{\rho }_{\epsilon }(s,x)]ds+{\int }_0^t{\int }_{{\mathbb{R}}^2}\nabla \Phi (s,x)·F_{\epsilon }^{\perp }(s,x){\rho }_{\epsilon }(s,x)dsdx\hfill \\ \hfill & +{\int }_0^t{\int \int }_{{\mathbb{R}}^2\times {\mathbb{R}}^2}{\nabla }^2\Phi (s,x):v^{\perp }\otimes v{f}_{\epsilon }(s,x,v)dvdsdx+I_{\epsilon }^1\left(t\right),\hfill \end{array}$$

where

$$\begin{array}{cc}\hfill I_{\epsilon }^1\left(t\right)& =\epsilon {\int }_0^t{\int \int }_{{\mathbb{R}}^2\times {\mathbb{R}}^2}{\partial }_t\nabla \Phi (s,x)·v^{\perp }{f}_{\epsilon }(s,x,v)dvdxds\hfill \\ \hfill & -\epsilon {\int \int }_{{\mathbb{R}}^2\times {\mathbb{R}}^2}\nabla \Phi (t,x)·v^{\perp }{f}_{\epsilon }(t,x,v)dvdx+\epsilon {\int \int }_{{\mathbb{R}}^2\times {\mathbb{R}}^2}\nabla \Phi (0,x)·v^{\perp }{f}_{\epsilon }^0(x,v)dvdx,\hfill \end{array}$$

and by Lemma 1, we have

$$\begin{array}{c}\hfill |I_{\epsilon }^1\left(t\right)|\le C\epsilon (t\parallel {\partial }_t{\nabla \Phi (s,x)\parallel }_{\infty }+{\parallel \nabla \Phi (t,x)\parallel }_{\infty }).\end{array}$$

By calculation, we obtain

$$\begin{array}{cc}\hfill & \Phi (t,{\xi }_{\epsilon }\left(t\right))-\Phi (0,{\xi }_{\epsilon }\left(0\right))\hfill \\ \hfill & ={\int }_0^t{\partial }_t\Phi (s,{\xi }_{\epsilon }\left(s\right))ds+{\epsilon }^{-1}{\int }_0^t{\eta }_{\epsilon }\left(s\right)·(\nabla \Phi )(s,{\xi }_{\epsilon }\left(s\right))ds\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill & (\nabla \Phi )(t,{\xi }_{\epsilon }\left(t\right))·{\eta }_{\epsilon }^{\perp }\left(t\right)-(\nabla \Phi )(0,{\xi }_{\epsilon }\left(0\right))·{\eta }_{\epsilon }^{\perp }\left(0\right)\hfill \\ \hfill & ={\int }_0^t{\partial }_t(\nabla \Phi )(s,{\xi }_{\epsilon }\left(s\right))·{\eta }_{\epsilon }^{\perp }\left(s\right)ds+{\epsilon }^{-1}{\int }_0^t({\nabla }^2\Phi )(s,{\xi }_{\epsilon }\left(s\right)):{\eta }_{\epsilon }^{\perp }\otimes {\eta }_{\epsilon }\left(s\right)ds\hfill \\ \hfill & +{\int }_0^t(\nabla \Phi )(s,{\xi }_{\epsilon }\left(s\right))·\left(-{\displaystyle \frac{{\eta }_{\epsilon }\left(s\right)}{{\epsilon }^2}}+{\displaystyle \frac{E_{\epsilon }^{\perp }(s,{\xi }_{\epsilon }\left(s\right))}{\epsilon }}\right)ds.\hfill \end{array}$$

Combining the two equations above, we have

$$\begin{array}{cc}\hfill & \Phi (t,{\xi }_{\epsilon }\left(t\right))-\Phi (0,{\xi }_{\epsilon }\left(0\right))\hfill \\ \hfill & ={\int }_0^t{\partial }_t\Phi (s,{\xi }_{\epsilon }\left(s\right))ds+{\int }_0^t({\nabla }^2\Phi )(s,{\xi }_{\epsilon }\left(s\right)):{\eta }_{\epsilon }^{\perp }\left(s\right)\otimes {\eta }_{\epsilon }\left(s\right)ds\hfill \\ \hfill & +{\int }_0^t(\nabla \Phi )(s,{\xi }_{\epsilon }\left(s\right))·E_{\epsilon }^{\perp }(s,{\xi }_{\epsilon }\left(s\right))ds+I_{\epsilon }^2\left(t\right),\hfill \end{array}$$

where

$$\begin{array}{cc}\hfill I_{\epsilon }^2\left(t\right)=& \epsilon (\nabla \Phi )(0,{\xi }_{\epsilon }\left(0\right))·{\eta }_{\epsilon }^{\perp }\left(0\right)-\epsilon (\nabla \Phi )(t,{\xi }_{\epsilon }\left(t\right))·{\eta }_{\epsilon }^{\perp }\left(t\right)\hfill \\ \hfill & +\epsilon {\int }_0^t{\partial }_t(\nabla \Phi )(s,{\xi }_{\epsilon }\left(s\right))·{\eta }_{\epsilon }^{\perp }\left(s\right)ds.\hfill \end{array}$$

By Lemma 1, we know that

$$\begin{array}{c}\hfill |I_{\epsilon }^2\left(t\right)|\le C\epsilon (t\parallel {\partial }_t{\nabla \Phi (s,x)\parallel }_{\infty }+{\parallel \nabla \Phi (t,x)\parallel }_{\infty }).\end{array}$$

Moreover, we observe that

$$\begin{array}{cc}\hfill & {\int }_0^t{\int }_{{\mathbb{R}}^2}\nabla \Phi (s,x)·F_{\epsilon }^{\perp }(s,x){\rho }_{\epsilon }(s,x)dsdx+{\int }_0^t(\nabla \Phi )(s,{\xi }_{\epsilon }\left(s\right))·E_{\epsilon }^{\perp }(s,{\xi }_{\epsilon }\left(s\right))ds\hfill \\ \hfill & ={\int }_0^t{\int }_{{\mathbb{R}}^2}\left((\nabla \Phi )(s,{\xi }_{\epsilon }\left(s\right))-\nabla \Phi (s,x)\right)·{(\nabla G)}^{\perp }(x-{\xi }_{\epsilon }\left(s\right)){\rho }_{\epsilon }(s,x)dxds\hfill \\ \hfill & =2{\int }_0^t{\mathcal{H}}_{\Phi (s,·)}[{\rho }_{\epsilon }\left(s\right),{\delta }_{{\xi }_{\epsilon }\left(s\right)}]ds.\hfill \end{array}$$

And we notice that

$$\begin{array}{c}\hfill {\mathcal{H}}_{\Phi }[{\rho }_{\epsilon }(s,x)+{\delta }_{{\xi }_{\epsilon }\left(s\right)},{\rho }_{\epsilon }(s,x)+{\delta }_{{\xi }_{\epsilon }\left(s\right)}]={\mathcal{H}}_{\Phi }[{\rho }_{\epsilon }(s,x),{\rho }_{\epsilon }(s,x)]+2{\mathcal{H}}_{\Phi }[{\rho }_{\epsilon }(s,x),{\delta }_{{\xi }_{\epsilon }\left(s\right)}],\end{array}$$

then, we have

$$\begin{array}{cc}\hfill & {\int }_{{\mathbb{R}}^2}\Phi (t,x)\left({\rho }_{\epsilon }(t,x)+{\delta }_{{\xi }_{\epsilon }\left(t\right)}\right)dx-{\int }_{{\mathbb{R}}^2}\Phi (0,x)\left({\rho }_{\epsilon }(0,x)+{\delta }_{{\xi }_{\epsilon }\left(0\right)}\right)dx\hfill \\ \hfill & ={\int }_0^t{\int }_{{\mathbb{R}}^2}{\partial }_t\Phi (s,x)\left({\rho }_{\epsilon }(s,x)+{\delta }_{{\xi }_{\epsilon }\left(s\right)}\right)dsdx+{\int }_0^t{\mathcal{H}}_{\Phi (s,·)}[{\rho }_{\epsilon }(s,x)+{\delta }_{{\xi }_{\epsilon }\left(s\right)},{\rho }_{\epsilon }(s,x)+{\delta }_{{\xi }_{\epsilon }\left(s\right)}]ds\hfill \\ \hfill & +{\int }_0^t{\int \int }_{{\mathbb{R}}^2\times {\mathbb{R}}^2}{\nabla }^2\Phi (s,x):v^{\perp }\otimes v{f}_{\epsilon }\left(s\right)dvdsdx\hfill \\ \hfill & +{\int }_0^t({\nabla }^2\Phi )(s,{\xi }_{\epsilon }\left(s\right)):{\eta }_{\epsilon }^{\perp }\otimes {\eta }_{\epsilon }\left(s\right)ds\hfill \\ \hfill & +I_{\epsilon }^1\left(t\right)+I_{\epsilon }^2\left(t\right).\hfill \end{array}$$

The proof is complete. □

Remark 3. 

Our proof about Proposition 5 is shorter and simpler and it improves Proposition 2.10 in [19], in which the remainder is less than ${C\parallel \Phi \parallel }_{W^{3,\infty }}(t+1)\epsilon$, leading to the loss of one derivative in Lemma 8 below.

Thanks to Proposition 5, we can now establish the equicontinuity for the point charge and densities as follows.

Lemma 8. 

Let $T>0$. There exist $K_0,K_1>1$ and ${\epsilon }_0>0$ depending only on T, such that for all $0<\epsilon <{\epsilon }_0$ and for all $0\le s<t\le T$,

$$\begin{array}{cc}\hfill & |{\xi }_{\epsilon }\left(t\right)-{\xi }_{\epsilon }\left(s\right)|\le K_0{\left(t-s+\epsilon \right)}^{1/2},\hfill \end{array}$$

(26)

$$\begin{array}{cc}\hfill & \parallel {\rho }_{\epsilon }\left(t\right)-{\rho }_{\epsilon }{\left(s\right)\parallel }_{W^{-2,1}}\le K_1{\left(t-s+\epsilon \right)}^{1/2}.\hfill \end{array}$$

(27)

Proof. 

According to Proposition 5 and the fact of $|H_{\Phi }(x,v)|\le C$, for all $\Phi \in C_c^{\infty }\left({\mathbb{R}}^2\right)$ and for all $0\le s<t\le T$, we can obtain

$$\begin{array}{cc}\hfill & |{\int }_{{\mathbb{R}}^2}\Phi \left(x\right)\left({\rho }_{\epsilon }(t,x)+{\delta }_{{\xi }_{\epsilon }\left(t\right)}\right)dx-{\int }_{{\mathbb{R}}^2}\Phi \left(x\right)\left({\rho }_{\epsilon }(s,x)+{\delta }_{{\xi }_{\epsilon }\left(s\right)}\right)dx|\hfill \\ \hfill & =|{\int }_{{\mathbb{R}}^2}\Phi \left(x\right){\rho }_{\epsilon }(t,x)dx+\Phi \left({\xi }_{\epsilon }\left(t\right)\right)-{\int }_{{\mathbb{R}}^2}\Phi \left(x\right){\rho }_{\epsilon }(s,x)dx-\Phi \left({\xi }_{\epsilon }\left(s\right)\right)|\hfill \\ \hfill & \le {C\parallel \Phi \parallel }_{W^{2,\infty }}(t-s)\hfill \\ \hfill & ·\underset{\tau \in [0,T]}{sup}\left(\parallel {\rho }_{\epsilon }{\left(\tau \right)\parallel }_{L^1}^2+\parallel {\rho }_{\epsilon }{\left(\tau \right)\parallel }_{L^1}+\int {\int }_{{\mathbb{R}}^2\times {\mathbb{R}}^2}{\left|v\right|}^2{f}_{\epsilon }(\tau ,x,v)dxdv+{\left|{\eta }_{\epsilon }\left(\tau \right)\right|}^2\right)\hfill \\ \hfill & +|I_{\epsilon }^1\left(t\right)|+|I_{\epsilon }^1\left(s\right)|+|I_{\epsilon }^2\left(t\right)|+|I_{\epsilon }^2\left(s\right)|.\hfill \end{array}$$

Then, from Lemma 1 and assumption (9), we have

$$\begin{array}{cc}\hfill & |{\int }_{{\mathbb{R}}^2}\Phi (t,x){\rho }_{\epsilon }(t,x)dx+\Phi \left({\xi }_{\epsilon }\left(t\right)\right)-{\int }_{{\mathbb{R}}^2}\Phi \left(x\right){\rho }_{\epsilon }(s,x)dx-\Phi \left({\xi }_{\epsilon }\left(s\right)\right)|\hfill \\ \hfill & \le {C\parallel \Phi \parallel }_{W^{2,\infty }}(t-s)+C{\parallel \Phi \parallel }_{W^{2,\infty }}(T+1)\epsilon .\hfill \end{array}$$

(28)

Let $K>1$ be a sufficiently large undetermined parameter and let ${\epsilon }_0=K^{-4}$. We assume there exist $0<\epsilon <{\epsilon }_0$ and $0\le s<t\le T$ with $t-s\le K^{-4}$ satisfying $|{\xi }_{\epsilon }\left(t\right)-{\xi }_{\epsilon }{\left(s\right)|>2K(t-s+\epsilon )}^{1/2}$. We set

$$\Phi \left(x\right)=\chi \left({\displaystyle \frac{x-{\xi }_{\epsilon }\left(s\right)}{K{(t-s+\epsilon )}^{1/2}}}\right),$$

where $\chi$ is a cut-off function such that $\chi =1$ on $B(0,1)$ and $\chi =0$ on $B{(0,2)}^c$. It is obvious we have $\Phi \left({\xi }_{\epsilon }\left(t\right)\right)=0$ and $\Phi \left({\xi }_{\epsilon }\left(s\right)\right)=1$. Moreover, since $K{(t-s+\epsilon )}^{1/2}<\sqrt{2}$, we have ${\parallel \Phi \parallel }_{W^{2,\infty }}\le C{K}^{-2}{(t-s+\epsilon )}^{-1}$. Then, by (28) and Proposition 2, we obtain

$$\begin{array}{cc}\hfill 1& \le 2\underset{\tau \in [0,T]}{sup}\int |\Phi \left(x\right)|{\rho }_{\epsilon }(\tau ,x)d\tau +C{K}^{-2}\le 2\underset{\tau \in [0,T]}{sup}{\int }_{|x-{\xi }_{\epsilon }{\left(\tau \right)|\le 2K(t-s+\epsilon )}^{1/2}}{\rho }_{\epsilon }(\tau ,x)d\tau +C{K}^{-2}\hfill \\ \hfill & \le 2\underset{\tau \in [0,T]}{sup}{\int }_{B({\xi }_{\epsilon }\left(\tau \right),2\sqrt{2}{K}^{-1})}{\rho }_{\epsilon }(\tau ,x)d\tau +C{K}^{-2}\le C\left(|ln\left(2\sqrt{2}{K}^{-1}\right){|}^{-1/2}+C{K}^{-2}\right),\hfill \end{array}$$

and if we choose a value of K that is sufficiently large, we can obtain $C\left(|ln\left(2\sqrt{2}{K}^{-1}\right){|}^{-1/2}+C{K}^{-2}\right)$ that is sufficiently small, which yields a contradiction. Hence, we obtain that for any $0<\epsilon <{\epsilon }_0$ and $0\le s<t<T$ with $t-s\le K^{-4}$, we have

$$\begin{array}{c}\hfill |{\xi }_{\epsilon }\left(t\right)-{\xi }_{\epsilon }{\left(s\right)|\le 2K(t-s+\epsilon )}^{1/2}.\end{array}$$

(29)

And for any $0\le s<t\le T$ with $|t-s|>K^{-4}$, we can obtain $|{\xi }_{\epsilon }\left(t\right)-{\xi }_{\epsilon }\left(s\right)|\le 2K(N+3){\left(t-s+\epsilon \right)}^{1/2}$ by (29), and taking $K_0=2K(N+3)$, we can obtain (26).

By (26), (28) and the mean-value theorem, we have

$$\begin{array}{cc}\hfill & |{\int }_{{\mathbb{R}}^2}\Phi (t,x){\rho }_{\epsilon }(t,x)dx-{\int }_{{\mathbb{R}}^2}\Phi \left(x\right){\rho }_{\epsilon }(s,x)dx|\hfill \\ \hfill & \le K_0{\parallel \nabla \Phi \parallel }_{\infty }{\left((t-s+\epsilon \right)}^{1/2}+{C\parallel \Phi \parallel }_{W^{2,\infty }}(t-s)+C{\parallel \Phi \parallel }_{W^{2,\infty }}(T+1)\epsilon ,\hfill \end{array}$$

and from the above inequality we obtain (27). □

3. Proof of the Main Results

3.1. Proof of Theorem 1

This subsection aims to complete the proof of Theorem 1. We first give the compactness results as follows.

Proposition 6.

 (1) 

The sequence $\left(f_{{\epsilon }_n}\right)$ is relatively compact in $L^{\infty }({\mathbb{R}}_{+},\mathcal{M}({\mathbb{R}}^2\times {\mathbb{R}}^2))$ weak-∗. Moreover, any accumulation point f satisfies ${\nabla }_v·\left(v^{\perp }f\right)=0$ in the sense of distributions.

 (2) 

There exists a subsequence such that $\left({\rho }_{{\epsilon }_n}(t,x)\right)$ converges to some $\rho (t,x)$ in $C_w({\mathbb{R}}_{+},\mathcal{M}\left({\mathbb{R}}^2\right))$ as $n\to \infty$. And $\left({\rho }_{{\epsilon }_n}(t,x)\right)$ is uniformly bounded in $C^{1/2}({\mathbb{R}}_{+},W^{-2,1}\left({\mathbb{R}}^2\right))$, $\rho (t,x)\in C^{1/2}([0,T],W^{-2,1}\left({\mathbb{R}}^2\right))$ for all $T>0$.

 (3) 

The sequence $\left({\xi }_{{\epsilon }_n}\left(t\right)\right)$ converges to some $\xi \left(t\right)$ in $C^{1/2}([0,T],{\mathbb{R}}^2)$ and $\xi \left(t\right)\in C^{1/2}([0,T],{\mathbb{R}}^2)$ for all $T>0$.

Proof. 

Multiplying the Vlasov equation in (1) by ${\epsilon }^2$ leads to

$$\begin{array}{c}\hfill v^{\perp }·{\nabla }_v{f}_{\epsilon }=-{\partial }_t\left({\epsilon }^2{f}_{\epsilon }\right)-{\nabla }_x·\left(\epsilon v{f}_{\epsilon }\right)-{\nabla }_v·\left(\epsilon (E_{\epsilon }+F_{\epsilon })f_{\epsilon }\right).\end{array}$$

According to Lemma 1, it is easy to know that $-{\partial }_t\left({\epsilon }^2{f}_{\epsilon }\right)-{\nabla }_x·\left(\epsilon v{f}_{\epsilon }\right)$ converges to 0 in ${\mathcal{D}}^{\prime }({\mathbb{R}}_{+}^{*}\times {\mathbb{R}}^2\times {\mathbb{R}}^2)$. We next focus on $-{\nabla }_v·\left(\epsilon (E_{\epsilon }+F_{\epsilon })f_{\epsilon }\right)$. For some $R>0$, let $\Phi$ be a test function with support included in $[0,R]\times B{(0,R)}^2$. By the Hölder inequality, we have

$$\begin{array}{cc}\hfill & \epsilon |{\int }_0^{+\infty }\int {\int }_{{\mathbb{R}}^2\times {\mathbb{R}}^2}{E}_{\epsilon }(t,x)·{\nabla }_v\Phi (t,x,v)f_{\epsilon }(t,x,v)dxdvdt|\hfill \\ \hfill & \le \epsilon \parallel {\nabla }_v{\Phi \parallel }_{\infty }{\int }_0^R\parallel E_{\epsilon }{\left(t\right)\parallel }_3{\parallel {\rho }_{\epsilon }\left(t\right)\parallel }_{3/2}dt,\hfill \end{array}$$

and by Lemma 3, we have

$$\parallel E_{\epsilon }{\left(t\right)\parallel }_3\le C\parallel {\rho }_{\epsilon }{\left(t\right)\parallel }_{6/5}\le C\parallel {\rho }_{\epsilon }{\left(t\right)\parallel }_1^{2/3}{\parallel {\rho }_{\epsilon }\left(t\right)\parallel }_2^{1/3},$$

$$\parallel {\rho }_{\epsilon }{\left(t\right)\parallel }_{3/2}\le C\parallel {\rho }_{\epsilon }{\left(t\right)\parallel }_1^{1/3}{\parallel {\rho }_{\epsilon }\left(t\right)\parallel }_2^{2/3}.$$

Then, we obtain

$$\begin{array}{c}\hfill \epsilon \parallel {\nabla }_v{\Phi \parallel }_{\infty }{\int }_0^R\parallel E_{\epsilon }{\left(t\right)\parallel }_3\parallel {\rho }_{\epsilon }{\left(t\right)\parallel }_{3/2}dt\le \epsilon \parallel {\nabla }_v{\Phi \parallel }_{\infty }{\int }_0^R\parallel {\rho }_{\epsilon }{\left(t\right)\parallel }_1\parallel {\rho }_{\epsilon }{\left(t\right)\parallel }_{2}dt\le C\epsilon {\parallel f_{\epsilon }^0\parallel }_{\infty }^{1/2},\end{array}$$

where we have used (6) and assumptions (9) and (11). Hence, we have

$$\begin{array}{cc}\hfill \epsilon & |{\int }_0^{+\infty }\int {\int }_{{\mathbb{R}}^2\times {\mathbb{R}}^2}{E}_{\epsilon }(t,x)·{\nabla }_v\Phi (t,x,v)f_{\epsilon }(t,x,v)dxdvdt|\le C\epsilon {\parallel f_{\epsilon }^0\parallel }_{\infty }^{1/2}.\hfill \end{array}$$

(30)

From Lemma 2, we have $|F_{\epsilon }(t,x)|=|-(\nabla G)(x-{\xi }_{\epsilon }\left(t\right))|\le {\displaystyle \frac{C}{|x-{\xi }_{\epsilon }\left(t\right)|}}$; then, by the same methods in [19], we have

$$\begin{array}{cc}\hfill \epsilon & |{\int }_0^{+\infty }\int {\int }_{{\mathbb{R}}^2\times {\mathbb{R}}^2}{F}_{\epsilon }(t,x)·{\nabla }_v\Phi (t,x,v)f_{\epsilon }(t,x,v)dxdvdt|\le C\epsilon {\parallel f_{\epsilon }^0\parallel }_{\infty }^{1/2}.\hfill \end{array}$$

(31)

Combining inequalities (30) and (31) with assumption (9) and considering the fact that sequence $\left(f_{{\epsilon }_n}\right)$ is uniformly bounded in $L^{\infty }({\mathbb{R}}_{+},L^1({\mathbb{R}}^2\times {\mathbb{R}}^2))$, we can draw the conclusion of (1).

We turn now to prove (2) and (3). $\left({\rho }_{{\epsilon }_n}(t,x)\right)$ is uniformly bounded in $L^{\infty }({\mathbb{R}}_{+},\mathcal{M}\left({\mathbb{R}}^2\right))$ and the sequence $\left({\xi }_{{\epsilon }_n}\left(t\right)\right)$ is uniformly bounded in $L^{\infty }([0,T],{\mathbb{R}}^2)$ for all $T>0$ by (26) and $|{\xi }_{\epsilon }^0|<\infty$ in (9), from which, when combined with Lemma 8, we can obtain the compactness results for ${\rho }_{{\epsilon }_n}$ and ${\xi }_{{\epsilon }_n}$ by using the Arzelà–Ascoli theorem (see [19] Proposition 2.14). In addition, due to our improvement in Lemma 8, we have $\left({\rho }_{{\epsilon }_n}(t,x)\right)$ uniformly bounded in $C^{1/2}({\mathbb{R}}_{+},W^{-2,1}\left({\mathbb{R}}^2\right))$ and $\rho (t,x)\in C^{1/2}([0,T],W^{-2,1}\left({\mathbb{R}}^2\right))$ and $\xi \left(t\right)\in C^{1/2}([0,T],{\mathbb{R}}^2)$ for all $T>0$. The proof is completed. □

By the compactness statement (1) of Proposition 6, we can obtain the existence of defect measures. Notice it was proved in [19](see also [15]) that conclusion (1) in Proposition 6 implies that there exists a subsequence $\left(f_{{\epsilon }_{n_k}}\right)$ and there exists $f=f(t,x,|v\left|\right)\in L^{\infty }({\mathbb{R}}_{+},\mathcal{M}({\mathbb{R}}^2\times {\mathbb{R}}^2))$ such that $\left(f_{{\epsilon }_n}\right)$ converges to f in $L^{\infty }({\mathbb{R}}_{+},\mathcal{M}({\mathbb{R}}^2\times {\mathbb{R}}^2))$ weak-∗ and such that $\rho ={\int }_{{\mathbb{R}}^2}fdv$. Moreover, there exists a measure ${\nu }_0\in L^{\infty }({\mathbb{R}}_{+},\mathcal{M}({\mathbb{R}}^2\times {\mathbb{S}}^1))$ such that for all $\Phi$ continuous on ${\mathbb{S}}^1$,

$$\begin{array}{c}\hfill {\int }_{{\mathbb{R}}^2}\left(f_{{\epsilon }_{n_k}}(t,x,v)-f(t,x,|v\left|\right)\right)\Phi \left({\displaystyle \frac{v}{\left|v\right|}}\right){\left|v\right|}^{2}dv\to {\int }_{{\mathbb{S}}^1}\Phi \left(\theta \right)d{\nu }_0\left(\theta \right)ask\to +\infty \end{array}$$

in the sense of distributions on ${\mathbb{R}}_{+}\times {\mathbb{R}}^2$.

Finally, with the compactness statements of Proposition 6 and the non-concentration property of Proposition 2 in hand, we now can pass to the limit in the weak formulation given by Proposition 5 to complete the proof of Theorem 1 by applying, verbatim, the other things in [19] (p. 673).

Remark 4. 

Because G can be expressed explicitly in the form of (3), we are able to obtain the non-concentration property of Proposition 2 without uniform bound conditions for second-order spatial moments in [19], and in addition, we obtain an important Lemma by the weak Young inequality and the interpolation inequality in $L^p$, which also avoiding the use of uniform bound conditions for second-order spatial moments in Proposition 6. Thanks for the above methods, uniform bound conditions for second-order spatial moments are not necessary for our Theorem 1 to be valid.

3.2. Proof of Theorem 2

By Lemma 2 and the assumption $\rho (t,x)\in L_{\mathrm{loc}}^{\infty }({\mathbb{R}}_{+},L^p\left({\mathbb{R}}^2\right))$, $p>2$, we have $E(t,x)\in L_{\mathrm{loc}}^{\infty }({\mathbb{R}}_{+},L^{\infty }\left({\mathbb{R}}^2\right))$ and $\left|(\nabla G)(x-\xi \left(t\right))\right|\rho (t,x)\le C{\displaystyle \frac{\rho (t,x)}{|x-\xi (t\left)\right|}}\in L_{\mathrm{loc}}^1\left({\mathbb{R}}^2\right)$. Let $\eta :{\mathbb{R}}_{+}\to [0,1]$ be a smooth function such that $\eta =0$ on $[0,1]$ and $\eta =1$ on $[2,+\infty )$. We set ${\eta }_{\delta }=\eta (·/\delta )$, and for any test function $\Phi$, we define ${\Phi }_{\delta }(t,x)=\Phi (t,x){\eta }_{\delta }\left(\right|x-\xi \left(t\right)\left|\right)$. By direct calculation, it is easy to know that ${\Phi }_{\delta }(t,\xi \left(t\right))=0$, ${\partial }_t{\Phi }_{\delta }(t,\xi \left(t\right))=0$ and $(\nabla {\Phi }_{\delta })(t,\xi \left(t\right))=0$. Now, taking ${\Phi }_{\delta }$ as a test function in (10) with defect measure $\nu =0$, we have

$$\begin{array}{cc}\hfill {\displaystyle \frac{d}{dt}}{\int }_{{\mathbb{R}}^2}{\Phi }_{\delta }(t,x)d(\rho (t,x)+{\delta }_{\xi \left(t\right)})& ={\int }_{{\mathbb{R}}^2}{\partial }_t{\Phi }_{\delta }(t,x)d(\rho (t,x)+{\delta }_{\xi \left(t\right)})\hfill \\ \hfill & +{\mathcal{H}}_{{\Phi }_{\delta }(t,x)}[\rho (t,x)+{\delta }_{\xi \left(t\right)},\rho (t,x)+{\delta }_{\xi \left(t\right)}].\hfill \end{array}$$

(32)

Next, we aim to prove that (32) converges to the first equation of Theorem 2.

Firstly, it is known by Lebesgue’s convergence theorem that $\int {\Phi }_{\delta }(t,x)\rho (t,x)dx$ converges to $\int \Phi (t,x)\rho (t,x)dx$ as $\delta \to 0$. And then, by direct calculation, we have

$$\begin{array}{c}\hfill {\partial }_t{\Phi }_{\delta }(t,x)={\partial }_t\Phi (t,x){\eta }_{\delta }\left(\right|x-\xi \left(t\right)\left|\right)+\Phi (t,x){\displaystyle \frac{1}{\delta }}\left[{\displaystyle \frac{\xi \left(t\right)-x}{|x-\xi (t\left)\right|}}·\dot{\xi }\left(t\right){\eta }_{\delta }^{\prime }\left(\right|x-\xi \left(t\right)\left|\right)\right].\end{array}$$

According to $|\dot{\xi }\left(t\right)|\le C$, we have

$$\int {\partial }_t{\Phi }_{\delta }(t,x)\rho (t,x)dx\to \int {\partial }_t\Phi (t,x)\rho (t,x)dxas\delta \to 0.$$

Finally, by direct computation, we observed that

$$\begin{array}{cc}\hfill & {\int }_{{\mathbb{R}}^2}-{(\nabla G)}^{\perp }(x-\xi \left(t\right))·\nabla {\Phi }_{\delta }(t,x)\rho (t,x)dx+E^{\perp }(t,\xi \left(t\right))·(\nabla {\Phi }_{\delta })(t,\xi \left(t\right))\hfill \\ \hfill & ={\int }_{{\mathbb{R}}^2}{(\nabla G)}^{\perp }(x-\xi \left(t\right))·[(\nabla {\Phi }_{\delta })(t,\xi \left(t\right))-\nabla {\Phi }_{\delta }(t,x)]\rho (t,x)dx\hfill \\ \hfill & =2{\mathcal{H}}_{{\Phi }_{\delta }\left(t\right)}[\rho (t,x),{\delta }_{\xi \left(t\right)}],\hfill \end{array}$$

and from the above equation and Lemma 4, we can express the nonlinear term as

$$\begin{array}{cc}\hfill & {\mathcal{H}}_{{\Phi }_{\delta }(t,x)}[\rho (t,x)+{\delta }_{\xi \left(t\right)},\rho (t,x)+{\delta }_{\xi \left(t\right)}]\hfill \\ \hfill & ={\mathcal{H}}_{{\Phi }_{\delta }\left(t\right)}[\rho (t,x),\rho (t,x)]+2{\mathcal{H}}_{{\Phi }_{\delta }(t,x)}[\rho (t,x),{\delta }_{\xi \left(t\right)}]\hfill \\ \hfill & ={\int }_{{\mathbb{R}}^2}\left(E^{\perp }(t,x)-{(\nabla G)}^{\perp }(x-\xi \left(t\right))\right)·\nabla {\Phi }_{\delta }(t,x)\rho (t,x)dx\hfill \\ \hfill & +\left(E^{\perp }(t,\xi \left(t\right))·(\nabla {\Phi }_{\delta })(t,\xi \left(t\right))\right)\hfill \\ \hfill & ={\int }_{{\mathbb{R}}^2}\left(E^{\perp }(t,x)-{(\nabla G)}^{\perp }(x-\xi \left(t\right))\right)·\nabla \Phi (t,x){\eta }_{\delta }\left(\right|x-\xi \left(t\right)\left|\right)\rho (t,x)dx\hfill \\ \hfill & +{\int }_{{\mathbb{R}}^2}{E}^{\perp }(t,x)·\Phi (t,x)\nabla \left({\eta }_{\delta }\left(\right|x-\xi \left(t\right)\left|\right)\right)\rho (t,x)dx.\hfill \end{array}$$

By performing calculations similar to those in [19] (p. 675) for the last two terms of the above equation, when $\delta \to 0$, we can obtain that ${\mathcal{H}}_{{\Phi }_{\delta }(t,x)}[\rho (t,x)+{\delta }_{\xi \left(t\right)},\rho (t,x)+{\delta }_{\xi \left(t\right)}]$ converges to

$$\begin{array}{c}\hfill {\int }_{{\mathbb{R}}^2}\left(E^{\perp }(t,x)-{(\nabla G)}^{\perp }(x-\xi \left(t\right))\right)·\nabla \Phi (t,x)\rho (t,x)dx.\end{array}$$

According to the estimates above, we infer that

$$\begin{array}{c}\hfill {\int }_{{\mathbb{R}}^2}\Phi (t,x){\partial }_t\rho (t,x)dx={\int }_{{\mathbb{R}}^2}\left(E^{\perp }(t,x)-{(\nabla G)}^{\perp }(x-\xi \left(t\right))\right)·\nabla \Phi (t,x)\rho (t,x)dx.\end{array}$$

Therefore, we prove that $\rho (t,x)$ satisfies the first equation of Theorem 2 in the sense of distribution. Plugging the above equation into (32) with any test function $\Phi (t,x)$ not necessarily vanishing near $\xi \left(t\right)$, we have

$$\begin{array}{cc}\hfill {\displaystyle \frac{d}{dt}}\left(\Phi (t,\xi (t\left)\right)\right)& ={\partial }_t\Phi (t,\xi \left(t\right))+\left(E^{\perp }(t,\xi \left(t\right))·(\nabla \Phi )(t,\xi \left(t\right))\right),\hfill \end{array}$$

which yields $\dot{\xi }\left(t\right)=E^{\perp }(t,\xi \left(t\right))$; thus, the proof is completed.

4. Conclusions

In this paper, we have comprehensively studied the asymptotic behavior of the two-dimensional Vlasov–Yukawa system with a point charge under a large external magnetic field. When the intensity of the magnetic field tends to infinity, we show that the kinetic system converges to the measure-valued Euler equation with a defect measure; when there is no defect measure, we prove that the equation obtains the vortex-wave system under some additional regularity on $\rho (t,x)$ and $\xi \left(t\right)$. The main innovations of this paper are our methods extending the results in [19] to the case of the Vlasov–Yukawa system and removing some additional restrictions, such as compact support restriction for spatial variable x, uniform bound restriction for second-order spatial moments ${sup}_{0<\epsilon <1}\int {\int }_{{\mathbb{R}}^2\times {\mathbb{R}}^2}{\left|x\right|}^2{f}_{\epsilon }^{0}dxdv<\infty$, and small restriction for initial data ${sup}_{0<\epsilon <1}{\parallel f_{\epsilon }^0\parallel }_1<1$. In addition, we prove that $\left({\rho }_{{\epsilon }_n}(t,x)\right)$ is uniformly bounded in $C^{1/2}({\mathbb{R}}_{+},W^{-2,1}\left({\mathbb{R}}^2\right))$, which avoids the loss of one derivative in [19]. Hence, our results are better and the conditions are more extensive. With respect to future research in this area, it is recommended to explore the cases of multiple point charges and remove other additional restrictions. In addition, the gyrokinetic limit in the three-dimensional case is also worth further study. The results of this study lay a solid theoretical foundation for future research. It is helpful to deepen the theoretical understanding of the gyrokinetic limit of Vlaov-type equations and enhance its academic value in the scientific literature.