ABSTRACT

Global existence of classical solutions to the relativistic Vlasov-Maxwell system, given sufficiently regular initial data, is a long-standing open problem. The aim of this project is to present in details the results of a paper published in 1986 by Robert Glassey and Walter Strauss. In that paper, a sufficient condition for the global existence of a smooth solution to the relativistic Vlasov-Maxwell system is derived. In the following, the resulting theorem is proved by taking initial data $f_0\in C^2$ , $E_0\mathrm{,}{B}_0\in C^3$ . A small data global existence result is presented as well.

Keywords:

Maxwell Equation, Vlasov Equation, Existence and Uniqueness

1. Introduction and Preliminaries

1.1. Introduction

1.1.1. Collisionless Plasma

Definition 1.1. A plasma is one of the four states of matter, which is a completely ionized gas.

For this work, we assume the following. The plasma is:

• at high temperature.

• at low density.

• collisions are unimportant (i.e. collisions between particles and external forces is negligible).

The plasma is at high temperature implies that

$T>\frac{e^2}{\bar{\gamma }}\cong e^2{N}^{-1/3}$

where N is the total number of charges per unit volume, and $\bar{\gamma }=N^{-1/3}$ is the mean distance between the particles.

Definition 1.2. We call the distance at which the coulomb field of a charge in the plasma is screened a Debye length denoted by a and is defined by:

$a^{-2}=\frac{4\text{π}}{T}{\displaystyle \underset{\alpha }{\sum }}{N}_{\alpha }\left(Z_{\alpha }\right)e^2$

But if we consider only one type of ion, $Z=1$ , then $a={\left(\frac{T}{4\text{π}N{e}^2}\right)}^{1/2}$ . From $T>e^2{N}^{1/3}$ we have that $\frac{e^2{N}^{1/3}}{T}<1,i.e.\frac{e^2{N}^{1/3}}{4\text{π}N{e}^2{a}^2}<1\text{or}\frac{{\bar{\gamma }}^2}{4\text{π}{a}^2}<1$ we can interpret this inequality as, the mean distance between particles is small with respect to the Debye length.

Generally speaking, a plasma is collision-less when the effective collision fre- quency $\nu <\omega$ -that is the frequency of variation of E, B. In this case

$\frac{\partial f}{\partial t}>\text{collisionterm}\text{.}$

1.1.2. The Relativistic Vlasov-Maxwell System

It is a kinetic field model for a collision-less plasma, that is a gas of charged particles which is sufficiently hot and dilute in order to ignore collision effect. Hence the particles are supposed to interact only through electromagnetic forces. In this work let us assume that the plasma is composed of n different particles, (i.e., ions, electrons) with the corresponding masses $m_{\alpha }$ and $e_{\alpha }$ . According to statistical physics the set of the particles of this species is denoted by a distribution function

$f_{\alpha }=f_{\alpha }\left(t,x,p\right)\ge 0$

which is the probability density to find a particle at a time $t>0$ , at a position x with momentum p. Here in the vlasov Maxwell system the motion of the particles is governed by Vlasov’s equation;

${\partial }_t{f}_{\alpha }+v_{\alpha }\cdot {\nabla }_x{f}_{\alpha }+e_{\alpha }\left(E+\frac{v_{\alpha }}{c}\times B\right)\cdot {\nabla }_p{f}_{\alpha }=0$ (1.1)

where $v_{\alpha }$ is the relativistic speed of a particle $\alpha$ , c is the speed of light and E and B are electric and magnetic fields respectively and p is momentum. Here

$v_{\alpha }=\frac{p}{\sqrt{m_{\alpha }^2+\frac{{\left|p\right|}^2}{c^2}}}$ is the relativistic velocity

where $m_{\alpha }$ is the mass of the particle $\alpha$ .

From this we can observe that $\left|v_{\alpha }\right|<c$ (hence relativistic system).

The electric field $E\left(t\mathrm{,}x\right)$ and the magnetic field $B\left(t\mathrm{,}x\right)$ satisfies the fol- lowing Maxwell equations.

${\partial }_{t}E=c{\nabla }_x\times B-j;{\nabla }_x\cdot E=\rho$ (1.2)

${\partial }_{t}B=-{\nabla }_x\times E;\nabla \cdot B=0$ (1.3)

where $\rho$ and $j$ are the densities of charge and current respectively, and hence they can be computed by:

$\rho =4\text{π}{\displaystyle \int }{\displaystyle \underset{\alpha }{\sum }}{e}_{\alpha }{f}_{\alpha }\text{d}p,\text{and}j=4\text{π}{\displaystyle \int }{\displaystyle \underset{\alpha }{\sum }}{e}_{\alpha }{f}_{\alpha }{v}_{\alpha }\text{d}p$ (1.4)

The coupled system of Equations (1.1), (1.2), (1.3) and (1.4) is what we call the Vlasov-Maxwell System which is represented as:

$\{\begin{array}{l}{\partial }_t{f}_{\alpha }+v_{\alpha }\cdot {\nabla }_x{f}_{\alpha }+e_{\alpha }\left(E+\frac{v_{\alpha }}{c}\times B\right)\cdot {\nabla }_p{f}_{\alpha }=0\\ \rho =4\text{π}{\displaystyle \int }{\displaystyle \underset{\alpha }{\sum }}{e}_{\alpha }{f}_{\alpha }\text{d}p,\text{and}j=4\text{π}{\displaystyle \int }{\displaystyle \underset{\alpha }{\sum }}{e}_{\alpha }{f}_{\alpha }{v}_{\alpha }\text{d}p\\ {\partial }_{t}E=c\nabla \times B-j;\nabla \cdot E=\rho \\ {\partial }_{t}B=-\nabla \times E;\nabla \cdot B=0\end{array}$ (1.5)

In this system the Vlasov equation governs the motion of the particles and the interaction of the particles are described by the Maxwell equations. So, the aim of this work is to derive a sufficient condition for the global existence of a smooth solution to the system 1.5 with initial data $f_{{\alpha }_0}\left(x\mathrm{,}p\right)\mathrm{,}{E}_0\left(x\right)$ and $B_0\left(x\right)$ , which are supposed to satisfy

${\nabla }_x\cdot E_0={\rho }_0,{\nabla }_x\cdot B_0=0\text{and}{\displaystyle \int }{\rho }_0\text{d}x=0$

In the entire work, we are going to use, the partial derivatives with respect to $x_i\left(i=1,2,3\right)$ will be denoted by ${\partial }_{x_i}$ , while any derivative of order k with respect to t or x or p will be denoted by $D^k$ that is $Df={\partial }_{t}f$ or ${\partial }_{x_i}f$ , $D^{2}f={\partial }_t^a{\partial }_{x_i}^b{\partial }_{x_j}^{c}f$ , $a+b+c=2$ and so on with the convention $D^{o}f=f$ .

1.1.3. A Short Review on the Cauchy Problem for the Vlasov-Maxwell Equations

Now let us provide a short review on the Classical Cauchy problem on the Vlasov Maxwell System.

Robert T.Glassey and Walter A. Strauss [1] , under the title, “In singularity formulation in collision-less plasma could occur at high velocity”, they showed existence and uniqueness of a global smooth solutions in $C^1$ by taking initial data $E_0\mathrm{,}{B}_0$ in $C^2$ and $f_0\in C_0^1$ by assuming existence of a continuous function $\beta \left(t\right)$ such that $f_{\alpha }\left(t,x,p\right)=0$ for $\left|p\right|>\beta \left(t\right)$ . Here our work has many similarities with their work, the only difference is taking $E_0\mathrm{,}{B}_0$ from $C^3$ and $f_0$ is from $C_0^2$ . In another work Robert T. Glassey and Walter A. Strauss [2] , also showed uniqueness and existence of $C^1$ solution by taking sufficiently small $C^2$ initial data. The proof of this result is sketched in the following. Simone Calogero, [3] , investigated global existence for Vlasov-Maxwell equation by modifying the system in which the usual Maxwell systems are replaced by their retarded parts. Sergiu Klainerman and Gigliola Staffilani, [4] showed a new approach to study VM system , that is they showed global existence of unique solution in 3D, under the assumptions of compactly supported particle density by using Fourier transformation of the classical Glassey-Strauss result. Oliver Glass and Daniel Han-Kwan, [5] , explained that existence of classical solutions, from which characteristics are well defined in 2D by using the concept of geometric control condition and strip assumption. Gerhard Rein, [6] investigated the behaviour of classical solutions of the relativistic Vlasov-Maxwell system under small perturbations of the initial data. More recently, Jonathan Luk and Robert Strain 2014, [7] derive a new continuation criterion for the relativistic Vlasov-Maxwell system. But the unconditional global existence in 3D remains an open problem.

Organization of the project: Now let us describe how this project is organized. In chapter one, we state some definitions and terms which are related to Vlasov Maxwell system and we try to show the solutions of inhomogeneous wave equations with initial conditions in ${ℝ}^3$ and Gronwall’s lemma is stated and proved. In chapter two, the main theorem is stated and we see representations of the fields and used boundedness to prove the existence and uniqueness of the solution. To prove the theorem, we use an iterative scheme. We construct sequences, then using representations of the fields we showed that these sequences are bounded in $C^1$ , and finally we try to show that the sequences are Cauchy sequences in $C^1$ . In the last chapter, the main theorem is re-stated by changing the hypothesis (taking small initial data conditions) just to show the reader there is at least one case such that the sufficient condition in the main theorem of the latter chapter holds true. In this chapter, only the theorem is stated and the main steps to prove the theorem are described.

1.2. Preliminaries

1.2.1. Inhomogeneous Wave Equations

Theorem 1.3. Kirchhoff’s Formula [8] Suppose $n\ge \mathrm{2,}m\ge 2$ , and $u\in {ℂ}^m\left({ℝ}^n\times \left[\mathrm{0,}\infty \right)\right)$ solves the initial value problem:

$\{\begin{array}{l}{u}_{tt}-\Delta u=0\in {ℝ}^n\times \left(0,\infty \right)\\ u=g,u_t=h\text{on}{ℝ}^n\times \left\{t=0\right\}\end{array}$ (1.6)

Kirchhoff’s states that an explicit formula for u in terms of g and h in three dimensions is:

$u\left(x,t\right)={\displaystyle \underset{\partial B\left(x,t\right)}{\int }}h\left(y\right)t+g\left(y\right)+Dg\left(y\right)\cdot \left(y-x\right)\text{d}S\left(y\right)\left(x\in {ℝ}^3,t>0\right)$

where $\partial B\left(x\mathrm{,}t\right)$ is a sphere in ${ℝ}^3$ centered at x and radius $t>0$ .

Consider the initial value problem for the non homogeneous wave equation

$\{\begin{array}{l}{u}_{tt}-\Delta u=f\in {ℝ}^n\times \left(0,\infty \right)\\ u=0,u_t=0\text{on}{ℝ}^n\times \left\{t=0\right\}\end{array}$

We define $u=u\left(x,t;s\right)$ to be the solution of

$\{\begin{array}{l}{u}_{tt}\left(.;s\right)-\Delta u\left(.;s\right)=0\text{in}{ℝ}^n\times \left(s,\infty \right)\\ u\left(.;s\right)=0,u_t\left(.;s\right)=f\left(.,s\right)\text{on}{ℝ}^n\times \left\{t=s\right\}\end{array}$

Now set

$u\left(x\mathrm{,}t\right)={\displaystyle {\int }_0^t}u\left(x\mathrm{,}t\mathrm{<mo>\; ;\; </mo>}s\right)\text{d}s\left(x\in {ℝ}^n\mathrm{,}t\ge 0\right)$ (1.7)

Duhamel’s principle asserts that this is a solution of

$\{\begin{array}{l}{u}_{tt}-\Delta u=f\in {ℝ}^n\times \left(0,\infty \right)\\ u=0,u_t=0\text{on}{ℝ}^n\times \left\{t=0\right\}\end{array}$

To find $u\left(x\mathrm{,}t\right)$ explicitly let us consider the case $n=3$ . For $n=3$ , Kirchhoff’s formula implies,

$u\left(x\mathrm{,}t\mathrm{<mo>\; ;\; </mo>}s\right)=\left(t-s\right){\displaystyle {\int }_{\partial B\left(x\mathrm{,}t-s\right)}}f\left(y\mathrm{,}s\right)\text{d}S$

so that

$\begin{array}{c}u\left(x,t\right)={\displaystyle {\int }_0^t}\left(t-s\right)\left({\displaystyle {\int }_{\partial B\left(x\mathrm{,}t-s\right)}}f\left(y,s\right)\text{d}S\right)\text{d}s\\ =\frac{1}{4\text{π}}{\displaystyle {\int }_0^t}{\displaystyle {\int }_{\partial B\left(x\mathrm{,}t-s\right)}}\frac{f\left(y,s\right)}{\left(t-s\right)}\text{d}S\text{d}s\\ =\frac{1}{4\text{π}}{\displaystyle {\int }_0^t}{\displaystyle {\int }_{\partial B\left(x,r\right)}}\frac{f\left(y,t-r\right)}{r}\text{d}S\text{d}r\end{array}$

Therefore

$u\left(x\mathrm{,}t\right)=\frac{1}{\text{4π}}{\displaystyle {\int }_{B\left(x\mathrm{,}t\right)}}\frac{f\left(y\mathrm{,}t-\left|y-x\right|\right)}{\left|y-x\right|}\text{d}y\left(x\in {ℝ}^3\mathrm{,}t\ge 0\right)$

If the initial data is not zero

$u\left(x,t\right)=u_0\left(x,t\right)+\frac{1}{\text{4π}}{\displaystyle {\int }_{B\left(x,t\right)}}\frac{f\left(y\mathrm{,}t-\left|y-x\right|\right)}{\left|y-x\right|}\text{d}y\left(x\in {ℝ}^3,t\ge 0\right)$ (1.8)

where $u_0\left(x,t\right)$ is the solution of the homogeneous equation $u_{tt}-\Delta u=0$ in ${ℝ}^n\times \left(\mathrm{0,}\infty \right)$ , which is in $C^2$ .

1.2.2. Gronwall’s Inequality

In estimating some norm of a solution of a partial differential equation, we are often led to a differential inequality for the norm from which we want to deduce an inequality for the norm itself. Gronwall’s inequality allows one to do this. Roughly speaking, it states that a solution of a differential inequality is bounded by the solution of the corresponding differential equality. There are both linear and non linear versions of Gronwalls’s inequality. We state here only the simplest version of the linear inequality that we are going to use.

Lemma 1.4. Gronwall’s lemma [9]

If $f\mathrm{<mo>\; :\; </mo>}{ℝ}_{+}\to {ℝ}_{+}$ is continuous and bounded above on each closed interval $\left[\mathrm{0,}T\right]$ and satisfies

$f\left(T\right)\le a\left(T\right)+{\displaystyle {\int }_0^T}b\left(t\right)f\left(t\right)\text{d}t$

for increasing function $a\left(t\right)$ and positive (integrable) function $b\left(t\right)$ then

$f\left(T\right)\le a\left(T\right)\exp \left({\displaystyle {\int }_0^T}b\left(t\right)\text{d}t\right)$ (1.9)

In particular if $a\left(T\right)=0$ , then

$f\left(T\right)=0$

Proof: Consider the function

$v\left(t\right)={\text{e}}^{-{\displaystyle {\int }_0^t}b\left(s\right)\text{d}s}{\displaystyle {\int }_0^t}b\left(s\right)f\left(s\right)\text{d}s$

differentiating both sides with respect to t and applying 1.4, we have:

$\begin{array}{c}\frac{\text{d}v\left(t\right)}{\text{d}t}=-b\left(t\right){\text{e}}^{-{\displaystyle {\int }_0^t}b\left(s\right)\text{d}s}{\displaystyle {\int }_0^t}b\left(s\right)f\left(s\right)\text{d}s+b\left(t\right)f\left(t\right){\text{e}}^{-{\displaystyle {\int }_0^t}b\left(s\right)\text{d}s}\\ \le -b\left(t\right){\text{e}}^{-{\displaystyle {\int }_0^t}b\left(s\right)\text{d}s}{\displaystyle {\int }_0^t}b\left(s\right)f\left(s\right)\text{d}s+a\left(t\right)b\left(t\right){\text{e}}^{-{\displaystyle {\int }_0^t}b\left(s\right)\text{d}s}\\ +b\left(t\right){\text{e}}^{-{\displaystyle {\int }_0^t}b\left(s\right)\text{d}s}{\displaystyle {\int }_0^t}b\left(s\right)f\left(s\right)\text{d}s\\ =a\left(t\right)b\left(t\right){\text{e}}^{-{\displaystyle {\int }_0^t}b\left(s\right)\text{d}s}\end{array}$

Integrating both sides and using the increasing property of the functions gives

$\begin{array}{l}{\text{e}}^{-{\displaystyle {\int }_0^T}b\left(t\right)\text{d}t}{\displaystyle {\int }_0^T}b\left(t\right)f\left(t\right)\text{d}t\\ =v\left(T\right)\le {\displaystyle {\int }_0^T}a\left(t\right)b\left(t\right){\text{e}}^{-{\displaystyle {\int }_0}tb\left(s\right)\text{d}s}\\ \le a\left(T\right){\displaystyle {\int }_0^T}b\left(t\right){\text{e}}^{-{\displaystyle {\int }_0^t}b\left(s\right)\text{d}s}\text{d}t=a\left(T\right)\left[1-{\text{e}}^{-{\displaystyle {\int }_0^T}b\left(s\right)\text{d}s}\right]\end{array}$

then using 1.9, and the above bound, we have

$\begin{array}{c}f\left(T\right)\le a\left(T\right)+{\displaystyle {\int }_0^T}b\left(t\right)f\left(t\right)\text{d}t\\ \le a\left(T\right)+{\text{e}}^{{\displaystyle {\int }_0^T}b\left(t\right)\text{d}t}a\left(T\right)\left[1-{\text{e}}^{-{\displaystyle {\int }_0^t}b\left(s\right)\text{d}s}\right]\\ =a\left(T\right)\exp \left({\displaystyle {\int }_0^T}b\left(t\right)\text{d}t\right)\end{array}$

Lemma 1.5 Partial Integration in ${ℝ}^n$ . [8]

When the function $f\left(x\right)\in C_0^1\left({ℝ}^n\right)$ and $g\left(x\right)\in C^1\left({ℝ}^n\right)$ are given, we have

${\displaystyle {\int }_{{ℝ}^n}}g\left(x\right)\frac{\partial }{{\partial }_{x_i}}\left(f\left(x\right)\right)\text{d}x=-{\displaystyle {\int }_{{ℝ}^n}}f\left(x\right)\frac{\partial }{{\partial }_{x_i}}\left(g\left(x\right)\right)\text{d}x,\text{for}i=1,,2,\cdots ,n$ (1.10)

Proof: On account of the property $f\left(x\right)\in C_0^1\left({ℝ}^n\right)$ , we find a radius $r>0$ such that $f\left(x\right)=0$ and $f\left(x\right)g\left(x\right)=0$ is true for all $x\in {ℝ}^n$ with $\left|x_j\right|\ge r$ for at least one index $j\in \left\{\mathrm{1,2,3,}\cdots \mathrm{,}n\right\}$ . Hence the fundamental theorem of differential and integral calculus yields:

$\begin{array}{l}{\displaystyle {\int }_{{ℝ}^n}}\frac{\partial }{{\partial }_{x_i}}\left(f\left(x\right)g\left(x\right)\right)\text{d}x\\ ={\displaystyle {\int }_{-r}^{+r}}\cdots {\displaystyle {\int }_{-r}^{+r}}\left[{\displaystyle {\int }_{-r}^{+r}}\frac{\partial }{{\partial }_{x_i}}\left(f\left(x\right)g\left(x\right)\right)\text{d}{x}_i\right]\text{d}{x}_1\text{d}{x}_2\cdots \text{d}{x}_{i-1}\text{d}{x}_{i+1}\cdots \text{d}{x}_n\mathrm{<mo>\; =\; </mo>0}\end{array}$

where $\text{d}{x}_1\text{d}{x}_2\cdots \text{d}{x}_{i-1}\text{d}{x}_{i+1}\cdots \text{d}{x}_n=\text{d}{S}_{x_i}$ . Therefore,

$0={\displaystyle {\int }_{{ℝ}^n}}\frac{\partial }{{\partial }_{x_i}}\left(f\left(x\right)g\left(x\right)\right)\text{d}x={\displaystyle {\int }_{{ℝ}^n}}g\left(x\right)\frac{\partial }{{\partial }_{x_i}}\left(f\left(x\right)\right)\text{d}x+{\displaystyle {\int }_{{ℝ}^n}}f\left(x\right)\frac{\partial }{{\partial }_{x_i}}\left(g\left(x\right)\right)\text{d}x$

hence the result.

Lemma 1.6, Let $u\left(t\right)$ be a continuous function of t and $\Vert u\left(t\right)\Vert =\underset{0\le s\le t}{{\displaystyle \sup }}{\Vert u\left(s\right)\Vert }_{\infty }$ .

If $\Vert u\left(t\right)\Vert \le c_0+c_1{\Vert u\left(t\right)\Vert }^2$ , then $\Vert u\left(t\right)\Vert$ is bounded provided that either $c_0$ or $c_1$ is sufficiently small.

2. Existence and Uniqueness of Global Smooth Solutions for Vlasov Maxwell Equations

In this chapter, we are going to establish the existence and uniqueness of global smooth solutions for the system in 1.5 under a sufficient condition. To derive the sufficient condition, we shall consider the case of only one species of particles, then at the end we extend the result to the case of a plasma composed of many species. Let us set $c=1,e_{\alpha }=1,m_{\alpha }=1$ and dropping the π factor, the system in 1.5 reduces to:

$\{\begin{array}{l}{\partial }_{t}f+v\cdot {\nabla }_{x}f+\left(E+v\times B\right)\cdot {\nabla }_{p}f=0\\ \rho ={\displaystyle \int }f\text{d}p,\text{and}j={\displaystyle \int }fv\text{d}p\\ {\partial }_{t}E=\nabla \times B-j;\nabla \cdot E=\rho \\ {\partial }_{t}B=-\nabla \times E;\nabla \cdot B=0\end{array}$ (2.1)

The term $E+v\times B$ can be represented by $K$ .That is

$K=E+v\times B$

which we call Lorentz force.

Theorem 2.1. [6] Let $0\le f_0\in C_0^2\mathrm{,}{E}_0\mathrm{,}{B}_0$ in $C^3$ be the initial data which satisfy 2.1 above. Assume there exists a continuous function $\beta \left(t\right)$ such that for all $x\mathrm{,}\alpha$ ;

$f_{\alpha }\left(t,x,p\right)=0\text{for}\left|p\right|>\beta (\; t\; )$

Then there exists a unique $C^1$ solution for all t.

To prove this theorem, we are going to use the concept of representation of the fields and their derivatives. The characteristics equations of the system 1.1 are the solutions of:

$\stackrel{\dot{}}{x}=v,\stackrel{\dot{}}{p}=K,\stackrel{\dot{}}{f}=0$ (2.2)

Hence the solution of this system is:

$\left(X\left(s\mathrm{,}t\mathrm{,}x\mathrm{,}p\right)\mathrm{,}P\left(s\mathrm{,}t\mathrm{,}x\mathrm{,}p\right)\right)$

such that at $s=t,X=x$ and $P=p$ . Therefore

$f\left(t,x,p\right)=f_0\left(X\left(0,t,x,p\right),P\left(0,t,x,p\right)\right)\ge 0$

Since

$0\le f_0\le \max \left(f_0\right)<\infty$

f remains bounded.

For the next two sections, the reader can consult the material [10] for more details.

2.1. Representation of Electric and Magnetic Fields

Theorem 2.2. Assume that the function $\beta \left(t\right)$ is as in Theorem 2.1 above. Let

$S={\partial }_t+{\displaystyle \underset{k=1}{\overset{3}{\sum }}}{v}_k{\partial }_{x_k}$

Then for $i=1,2,3$ , $E^i$ and $B^i$ are represented by:

$4\text{π}{E}^i\left(t,x\right)={\left(E^i\right)}_0\left(t,x\right)+E_T^i\left(t,x\right)+E_S^i\left(t,x\right)$

$4\text{π}{B}^i\left(t,x\right)={\left(B^i\right)}_0\left(t,x\right)+B_T^i\left(t,x\right)+B_S^i\left(t,x\right)$

where

$E_T^i\left(t,x\right)=-{\displaystyle \underset{\left|y-x\right|\le t}{\iint }}\frac{\left(w_i+v_i\right)\left(1-{\left|v\right|}^2\right)}{{\left(1+v\cdot w\right)}^2}f\left(t-\left|y-x\right|,y,p\right)\text{d}p\frac{\text{d}y}{{\left|y-x\right|}^2}$

$E_S^i\left(t,x\right)=-{\displaystyle \underset{\left|y-x\right|\le t}{\iint }}\frac{\left(w_i+v_i\right)}{\left(1+v\cdot w\right)}\left(Sf\right)\left(t-\left|y-x\right|,y,p\right)\text{d}p\frac{\text{d}y}{\left|y-x\right|}$

where

$w=\frac{y-x}{\left|y-x\right|}$

Just replacing $w_i+v_i$ , in each expression above by ${\left(w\times v\right)}_i$ , we can represent $B^i\left(t\mathrm{,}x\right)$ in a similar way.

Proof:

Using chain rule

${\partial }_{y_i}\left(f\left(t-\left|y-x\right|,y,p\right)\right)={\partial }_{y_i}f-w_i{\partial }_{t}f$

Hence, let us denote this by $T_{i}f$ . That is $T_{i}f={\partial }_{y_i}f-w_i{\partial }_{t}f$ .

Here, $T_i$ is the tangential derivative along the surface of a backward cha- racteristic cone.

Now let us replace the usual operators ${\partial }_t$ and ${\partial }_i$ by $T_i$ and $S$ . From

$T_{i}f={\partial }_{y_i}f-w_i{\partial }_{t}f\text{and}S={\partial }_t+{\displaystyle \underset{k=1}{\overset{3}{\sum }}}{v}_k\partial x_k,\text{we}\text{get},$

${\partial }_t=\frac{S-v\cdot T}{1+v\cdot w}\text{and}$

$\begin{array}{l}{\partial }_i=T_i+\frac{w_i}{\left(1+v\cdot w\right)}\left(S-v\cdot T\right),v\cdot T=v_j\cdot T_j\\ =\frac{w_{i}S}{1+v\cdot w}+T_i-\frac{w_{i}v\cdot T}{1+v\cdot w}\\ =\frac{w_{i}S}{1+v\cdot w}+\left({\delta }_{ij}-\frac{w_i{v}_j}{1+v\cdot w}\right)T_j\end{array}$ (2.3)

For relativistic Vlasov Maxwell system, the fields satisfy the inhomogeneous wave equation:

$\left({\partial }_t^2-\Delta \right)E^i=-\left(\nabla \rho +{\partial }_t{j}^i\right)=-\left({\partial }_i\rho +{\partial }_t{j}^i\right)$

But from

$\rho ={\displaystyle \int }f\text{d}p\text{and}{j}^i={\displaystyle \int }f{v}_i\text{d}p$

we have

$\left({\partial }_t^2-\Delta \right)E^i=-\left({\partial }_i\rho +{\partial }_t{j}^i\right)=-{\displaystyle \int }\left({\partial }_{i}f+v_i{\partial }_{t}f\right)\text{d}p$ (2.4)

Using 2.3 above

$\begin{array}{c}{\partial }_i+v_i{\partial }_t=\frac{w_{i}S}{1+v\cdot w}+\left({\delta }_{ij}-\frac{w_i{v}_j}{1+v\cdot w}\right)T_j+v_i\left(\frac{S-v\cdot T}{1+v\cdot w}\right)\\ =\frac{w_{i}S}{1+v\cdot w}+\frac{v_{i}S}{1+v\cdot w}+\left({\delta }_{ij}-\frac{w_i{v}_j}{1+v\cdot w}\right)T_j-\frac{v_i{v}_j}{1+v\cdot w}\\ =\frac{\left(w_i+v_i\right)S}{1+v\cdot w}+\left({\delta }_{ij}-\frac{w_i{v}_j}{1+v\cdot w}\right)T_j-\frac{v_i{v}_j}{1+v\cdot w}{T}_j\\ =\frac{\left(w_i+v_i\right)S}{1+v\cdot w}+\left({\delta }_{ij}-\frac{\left(w_i+v_i\right)v_j}{1+v\cdot w}\right)T_j\end{array}$ (2.5)

Hence, substituting 2.5 in to 2.4, we have

$\left({\partial }_t^2-\Delta \right)E^i=-{\displaystyle \int }\left[\frac{\left(w_i+v_i\right)Sf}{1+v\cdot w}+\left({\delta }_{ij}-\frac{\left(w_i+v_i\right)v_j}{1+v\cdot w}\right)T_{j}f\right]\text{d}p$ (2.6)

By applying Equation (1.8) in chapter one, we have

$\begin{array}{c}{E}^i\left(t,x\right)={\left(E^i\right)}_0\left(t,x\right)-\frac{1}{4\text{π}}{\displaystyle {\int }_{\left|y-x\right|\le t}}{\displaystyle \int }\frac{\left(w_i+v_i\right)Sf}{1+v\cdot w}\left(t-\left|y-x\right|,y,p\right)\text{d}p\frac{\text{d}y}{\left|y-x\right|}\\ -{\displaystyle {\int }_{\left|y-x\right|\le t}}{\displaystyle \int }\left({\delta }_{ij}-\frac{\left(w_i+v_i\right)v_j}{1+v\cdot w}\right)\left(T_{j}f\right)\left(t-\left|x-y\right|,y,p\right)\text{d}p\frac{\text{d}y}{\left|y-x\right|}\end{array}$

This implies,

$\begin{array}{c}4\text{π}{E}^i\left(t,x\right)={\left(E^i\right)}_0\left(t,x\right)-{\displaystyle {\int }_{\left|y-x\right|\le t}}{\displaystyle \int }\frac{\left(w_i+v_i\right)Sf}{1+v\cdot w}\left(t-\left|y-x\right|,y,p\right)\text{d}p\frac{\text{d}y}{\left|y-x\right|}\\ -{\displaystyle {\int }_{\left|y-x\right|\le t}}{\displaystyle \int }\left({\delta }_{ij}-\frac{\left(w_i+v_i\right)v_j}{1+v\cdot w}\right)\left(T_{j}f\right)\left(t-\left|x-y\right|,y,p\right)\text{d}p\frac{\text{d}y}{\left|y-x\right|}\end{array}$ (2.7)

where ${\left(E^i\right)}_0\left(t\mathrm{,}x\right)$ is the solution of the homogeneous wave equation.

From this we can easily see that the second term is $E_S^i$ . Let

$a_j=\left({\delta }_{ij}-\frac{\left(w_i+v_i\right)v_j}{1+v\cdot w}\right)\text{and}\gamma =\left|y-x\right|$

Hence, we can re-write the last integral as:

$-{\displaystyle {\int }_{\left|y-x\right|\le t}}{\displaystyle \int }\frac{a_j}{\gamma }\frac{\partial }{{\partial }_{y_j}}\left[f\left(t-\left|x-y\right|,y,p\right)\right]\text{d}y\text{d}p$

Now let us integrate the last term using integration by parts in y. Hence, by applying lemma 1.5 in chapter one (integration by parts), this expression reduces to;

$-{\displaystyle {\int }_{\left|y-x\right|=t}}{\displaystyle \int }\frac{w_j{a}_j}{\gamma }f\left(0,y,p\right)\text{d}{S}_y\text{d}p+{\displaystyle {\int }_{\left|y-x\right|\le t}}{\displaystyle \int }f\frac{\partial }{{\partial }_{y_j}}\left(\frac{a_j}{\gamma }\right)\text{d}y\text{d}p$

where

$\text{d}{S}_y={\text{d}}_{y_1}{\text{d}}_{y_2},\text{if}j=3,{\text{d}}_{y_2}{\text{d}}_{y_3}\text{if}j=1\text{or}{\text{d}}_{y_1}{\text{d}}_{y_3}\text{if}j=2$

But

$-{\displaystyle {\int }_{\left|y-x\right|=t}}{\displaystyle \int }\frac{w_j{a}_j}{\gamma }f\left(\mathrm{0,}y\mathrm{,}p\right)\text{d}{S}_y\text{d}p$

is part of ${\left(E^i\right)}_0\left(t\mathrm{,}x\right)$ , hence the above integral reduces to:

${\displaystyle {\int }_{\left|y-x\right|\le t}}{\displaystyle \int }f\frac{\partial }{{\partial }_{y_j}}\left(\frac{a_j}{\gamma }\right)\text{d}y\text{d}p$ (2.8)

But

$\frac{\partial }{{\partial }_{y_j}}\left(\frac{a_j}{\gamma }\right)=\frac{\left(w_i+v_i\right)\left({\left|v\right|}^2-1\right)}{{\gamma }^2{\left(1+v\cdot w\right)}^2}$ (2.9)

see the computation of this expression at the appendix part of [7] . Hence, 2.8 becomes:

${\displaystyle {\int }_{\left|y-x\right|\le t}}{\displaystyle \int }\frac{\left(w_i+v_i\right)\left({\left|v\right|}^2-1\right)}{{\gamma }^2{\left(1+v\cdot w\right)}^2}f\left(t-\left|x-y\right|,y,p\right)\text{d}p\frac{\text{d}y}{{\left|y-x\right|}^2}$ (2.10)

Therefore, substituting 2.10 and $E_S^i\left(t\mathrm{,}x\right)$ term in to 2.7,

$4\text{π}{E}^i\left(t,x\right)={\left(E^i\right)}_0\left(t,x\right)+E_T^i\left(t,x\right)+E_S^i\left(t,x\right)$

Similarly, by using the inhomogeneous wave equation for the field B, we have

$\left({\partial }_t^2-\Delta \right)B^1={\displaystyle \int }\left({\partial }_2{v}_{3}f-{\partial }_3{v}_{2}f\right)\text{d}p$

and following the same step, we have

$4\text{π}{B}^i\left(t\mathrm{,}x\right)={\left(B^i\right)}_0\left(t\mathrm{,}x\right)+B_T^i\left(t\mathrm{,}x\right)+B_S^i\left(t\mathrm{,}x\right)$

This proves theorem 2.2, Proof of uniqueness of Theorem 2.1.

To do this, let $\left(f^{\left(1\right)}\mathrm{,}{E}^{\left(1\right)}\mathrm{,}{B}^{\left(1\right)}\right)$ and $\left(f^{\left(2\right)}\mathrm{,}{E}^{\left(2\right)}\mathrm{,}{B}^{\left(2\right)}\right)$ be two different Classical solutions of 2.1 with the same Cauchy data given. Define

$\begin{array}{l}f=f^{\left(1\right)}-f^{\left(2\right)},E=E^{\left(1\right)}-E^{\left(2\right)},B=B^{\left(1\right)}-B^{\left(2\right)},K=K^{\left(1\right)}-K^{\left(2\right)},\\ \text{where}{K}^{\left(i\right)}=E^{\left(i\right)}+v\times B^{(\; i\; )}\end{array}$

From the Vlasov Equation,

${\partial }_{t}f+v\cdot {\nabla }_{x}f+K\cdot {\nabla }_{p}f=0$

we have

$\begin{array}{c}\left({\partial }_t+v\cdot {\nabla }_x\right)f=Sf=-{\nabla }_p\cdot Kf={\nabla }_p\cdot \left[-K^{\left(1\right)}{f}^{\left(1\right)}+K^{\left(2\right)}{f}^{\left(2\right)}\right]\\ ={\nabla }_p\cdot \left[-K{f}^{\left(1\right)}-K^{\left(2\right)}f\right]\end{array}$

Using theorem 2.2 above, we get

$4\text{π}E\left(t,x\right)=E_T\left(t,x\right)+E_S\left(t,x\right)$ (2.11)

where

$E_T\left(t,x\right)=-{\displaystyle {\int }_{\left|y-x\right|\le t}}{\displaystyle \int }\frac{\left(w+v\right)\left(1-{\left|v\right|}^2\right)}{{\left(1+v\cdot w\right)}^2}f\left(t-\left|y-x\right|,y,p\right)\text{d}p\frac{\text{d}y}{{\left|y-x\right|}^2}$

$E_S\left(t,x\right)=-{\displaystyle \underset{\left|y-x\right|\le t}{\iint }}{\nabla }_p\left[\frac{w+v}{1+v\cdot w}\right]\cdot \left(-K{f}^{\left(1\right)}-K^{\left(2\right)}f\right)\text{d}p\frac{\text{d}y}{\left|y-x\right|}$

Here, in the $E_S^i$ term using the fact that $Sf$ is a pure p divergent, we have integrated by parts in p. Similarly we can represent the field B as

$4\text{π}B\left(t,x\right)=B_T\left(t,x\right)+B_S\left(t,x\right)$ (2.12)

Since, f has a compact support in p, the expression $1+v\cdot w$ is bounded away from zero. Now since the fields are bounded (from hypothesis), adding Equations (2.11) and (2.12), estimating using the support property, for $t\le T$ and using Equation (1.8), in chapter one, we have

$\begin{array}{c}{\left|E\left(t\right)\right|}_0+{\left|B\left(t\right)\right|}_0\le C_T{\displaystyle {\int }_0^t}({\left|f\left(\xi \right)\right|}_0+\left[{\left|E^{\left(2\right)}\left(\xi \right)\right|}_0+{\left|B^{\left(2\right)}\left(\xi \right)\right|}_0\right]{\left|f\left(\xi \right)\right|}_0\\ +\left({\left|E\left(\xi \right)\right|}_0+{\left|B\left(\xi \right)\right|}_0\right){\left|f^{\left(1\right)}\left(\xi \right)\right|}_0)\text{d}\xi \end{array}$

where, ${\left|\mathrm{.}\right|}_0$ represents the maximum norm.

Now $E^{\left(2\right)},B^{\left(2\right)}$ and $f^{\left(1\right)}$ bounded implies that

${\left|E\left(t\right)\right|}_0+{\left|B\left(t\right)\right|}_0\le C_T{\displaystyle {\int }_0^t}\left({\left|f\left(\xi \right)\right|}_0+{\left|E\left(\xi \right)\right|}_0+{\left|B\left(\xi \right)\right|}_0\right)\text{d}\xi$ (2.13)

Again from

$\left({\partial }_t+v\cdot {\nabla }_x\right)f={\nabla }_p\cdot \left[-K{f}^{\left(1\right)}-K^{\left(2\right)}f\right]$

${\partial }_{t}f+v\cdot {\nabla }_{x}f+K^{\left(1\right)}\cdot {\nabla }_{p}f=-K\cdot {\nabla }_p{f}^{\left(2\right)},\text{then}$

the characteristics of this equation are the solutions of:

$\stackrel{\dot{}}{x}=v,\stackrel{\dot{}}{p}=K^{\left(1\right)},\stackrel{\dot{}}{f}=-K\cdot {\nabla }_p{f}^{(\; 2\; )}$

Thus, from

$\stackrel{\dot{}}{f}=-K\cdot {\nabla }_p{f}^{(\; 2\; )}$

when f is written as a line integral over such a characteristics curve, we have

${\left|f\left(t\right)\right|}_0\le C{\displaystyle {\int }_0^t}{\left|K\cdot {\nabla }_p{f}^{\left(2\right)}\left(\xi \right)\right|}_0\text{d}\xi$

Since, ${\nabla }_p{f}^{\left(2\right)}$ is bounded , we can write it as:

${\left|f\left(t\right)\right|}_0\le C_T{\displaystyle {\int }_0^t}{\left|K\left(\xi \right)\right|}_0\text{d}\xi \le C_T{\displaystyle {\int }_0^t}\left({\left|E\left(\xi \right)\right|}_0+{\left|B\left(\xi \right)\right|}_0\right)\text{d}\xi$ (2.14)

Now adding 2.13 and 2.14, we have

${\left|E\left(t\right)\right|}_0+{\left|B\left(t\right)\right|}_0+{\left|f\left(t\right)\right|}_0\le C_T{\displaystyle {\int }_0^t}\left({\left|E\left(\xi \right)\right|}_0+{\left|B\left(\xi \right)\right|}_0+{\left|f\left(\xi \right)\right|}_0\right)\text{d}\xi$

Applying Gronwall lemma, we have $E=B=f=0$ . This implies the solution is unique. This proves the uniqueness of theorem 2.1.

2.2. Representations of Derivatives of Electric and Magnetic Fields

Theorem 2.3. Assume that $\beta \left(t\right)$ exists as in the hypothesis of theorem 2.1. Then the derivatives of the fields can be represented as:

$\begin{array}{c}{\partial }_k{E}^i={\left({\partial }_k{E}^i\right)}_0+{\displaystyle \underset{\left|y-x\right|\le t}{\iint }}a\left(w,v\right)f\text{d}p\frac{\text{d}y}{{\left|y-x\right|}^3}+{\displaystyle {\int }_{\left|w\right|=1}}{\displaystyle \int }d\left(w,v\right)f\left(t,x,p\right)\text{d}w\text{d}p\\ +{\displaystyle \underset{\left|y-x\right|\le t}{\iint }}b\left(w,v\right)\left(Sf\right)\text{d}p\frac{\text{d}y}{{\left|y-x\right|}^2}+{\displaystyle \underset{\left|y-x\right|\le t}{\iint }}c\left(w,v\right)S^{2}f\text{d}p\frac{\text{d}y}{\left|y-x\right|}\end{array}$

Note that $f\mathrm{,}Sf\mathrm{,}{S}^{2}f$ without explicit arguments are evaluated at $\left(t-\left|x-y\right|\mathrm{,}y\mathrm{,}p\right)$ and $\left|y-x\right|=\gamma$ . Except at $1+v\cdot w$ , the functions $a\left(w\mathrm{,}v\right)\mathrm{,}b\left(w\mathrm{,}v\right)\mathrm{,}c\left(w\mathrm{,}v\right)\mathrm{,}d\left(w\mathrm{,}v\right)$ are $C^{\infty }$ . Moreover

${\displaystyle \underset{\left|w\right|=1}{\int }}a\left(w\mathrm{,}v\right)\text{d}w=0$

In a similar way, we can represent ${\partial }_k{B}^i$ .

Proof:

Applying $\frac{\partial }{\partial x_k}$ in to the field representation in theorem 2.2, we have

$\begin{array}{l}4\text{π}{\partial }_k{E}^i={\left({\partial }_k{E}^i\right)}_0-{\displaystyle \underset{\left|y-x\right|\le t}{\iint }}\frac{\left(w_i+v_i\right)\left(1-{\left|v\right|}^2\right)}{{\left(1+v\cdot w\right)}^2}{\partial }_{k}f\left(t-\left|y-x\right|,y,p\right)\text{d}p\frac{\text{d}y}{{\left|y-x\right|}^2}\\ -{\displaystyle \underset{\left|y-x\right|\le t}{\iint }}\frac{\left(w_i+v_i\right)}{\left(1+v\cdot w\right)}{\partial }_k\left(Sf\right)\left(t-\left|y-x\right|,y,p\right)\text{d}p\frac{\text{d}y}{\left|y-x\right|}\\ ={\left({\partial }_k{E}^i\right)}_0-{\displaystyle \underset{\left|y-x\right|\le t}{\iint }}\frac{\left(w_i+v_i\right)\left(1-{\left|v\right|}^2\right)}{{\left(1+v\cdot w\right)}^2}\left[\left({\delta }_{jk}-\frac{w_k{v}_j}{1+v\cdot w}\right)T_j+\frac{w_{k}S}{1+v\cdot w}\right]f\text{d}p\frac{\text{d}y}{{\left|y-x\right|}^2}\\ -{\displaystyle \underset{\left|y-x\right|\le t}{\iint }}\frac{\left(w_i+v_i\right)}{\left(1+v\cdot w\right)}\left[\left({\delta }_{jk}-\frac{w_k{v}_j}{1+v\cdot w}\right)T_j+\frac{w_{k}S}{1+v\cdot w}\right]\left(Sf\right)\text{d}p\frac{\text{d}y}{\left|y-x\right|}\end{array}$

Now using the fact that $T_j$ is a perfect $y_j$ derivative, integrating the last integral using integration by parts in y, is equal to:

$\begin{array}{l}{\displaystyle \underset{\left|y-x\right|\le t}{\iint }}{\partial }_j\left[\frac{\left(w_i+v_i\right)}{\left(1+v\cdot w\right)}\left({\delta }_{jk}-\frac{w_k{v}_j}{1+v\cdot w}\right)\frac{1}{\gamma }\right]\left(Sf\right)\left(t-\left|y-x\right|,y,p\right)\text{d}p\text{d}y\\ -{\displaystyle \underset{\left|y-x\right|\le t}{\iint }}\frac{\left(w_i+v_i\right)w_k}{{\left(1+v\cdot w\right)}^2}\frac{1}{\gamma }{S}^{2}f\left(t-\left|y-x\right|,y,p\right)\text{d}p\text{d}y\\ -\frac{1}{t}{\displaystyle {\int }_{\gamma =t}}{\displaystyle \int }\frac{w_k\left(w_i+v_i\right)}{{\left(1+v\cdot w\right)}^2}Sf\left(0,y,p\right)\text{d}p\text{d}{S}_y\end{array}$