Existence and Uniqueness of Global Smooth Solutions for Vlasov Maxwell Equations

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 2 0 f C ∈ , 3 0 0 , E B C ∈ . A small data global existence result is presented as well.


Introduction and Preliminaries
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 where N is the total number of charges per unit volume, and 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: ( ) T e N > we have that e N e N i.e.T Ne a a γ < < < 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 frequency ν ω < -that is the frequency of variation of E, B. In this case collision term.f t ∂ > ∂

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 α and e α .According to statistical physics the set of the particles of this species is denoted by a distribution function ( ) which is the probability density to find a particle at a time 0 t > , at a position x with momentum p.Here in the vlasov Maxwell system the motion of the particles is governed by Vlasov's equation; where v α is the relativistic speed of a particle α , c is the speed of light and E and B are electric and magnetic fields respectively and p is momentum.Here where m α is the mass of the particle α .
From this we can observe that v c α < (hence relativistic system).
The electric field ( ) , E t x and the magnetic field ( ) , B t x satisfies the fol- lowing Maxwell equations. ; ; 0 where ρ and j are the densities of charge and current respectively, and hence they can be computed by: 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 1  C by taking initial data 0 0 , E B in .Here our work has many similarities with their work, the only difference is taking 0 0 , E B from 3  C and 0 f is from 2 0 C .In another work Robert T. Glassey and Walter A. Strauss [2], also showed uniqueness and existence of 1 C solution by taking sufficiently small 2 C 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 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 1  C , and finally we try to show that the sequences are Cauchy sequences in 1 C .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.

Preliminaries
solves the initial value problem: Kirchhoff's states that an explicit formula for u in terms of g and h in three dimensions is: .; 0 in , .; 0, .; ., on Duhamel's principle asserts that this is a solution of where ( ) 0 , u x t is the solution of the homogeneous equation 0

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  is continuous and bounded above on each closed interval [ ] 0,T and satisfies for increasing function ( ) a t and positive (integrable) function ( ) differentiating both sides with respect to t and applying 1.4, we have: Integrating both sides and using the increasing property of the functions gives then using 1.9, and the above bound, we have Proof: On account of the property ( ) ( ) . Hence the fundamental theorem of differential and integral calculus yields: ( ) hence the result.

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 The term E v B + × can be represented by K .That is which we call Lorentz force.
Theorem 2.1.[6] Let Then there exists a unique 1 C 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: Hence the solution of this system is: , , , ,  , , ,  X s t For the next two sections, the reader can consult the material [10] for more details.

Representation of Electric and Magnetic Fields
x in a similar way.Proof: Here, i T is the tangential derivative along the surface of a backward cha- racteristic cone.Now let us replace the usual operators t ∂ and i ∂ by i T and S .From For relativistic Vlasov Maxwell system, the fields satisfy the inhomogeneous wave equation: Hence, substituting 2.5 in to 2.4, we have ( ) By applying Equation (1.8) in chapter one, we have ( where ( ) ( ) x is the solution of the homogeneous wave equation.From this we can easily see that the second term is i S E .Let ( ) Hence, we can re-write the last integral as: ( ) 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; ( ) where x , hence the above integral reduces to: see the computation of this expression at the appendix part of [7].Hence, 2.8 Therefore, substituting 2.10 and ( ) Similarly, by using the inhomogeneous wave equation for the field B, we have and following the same step, we have This proves theorem 2.2 Proof of uniqueness of Theorem 2.1.
To do this, let , , f E B be two different Classical solutions of 2.1 with the same Cauchy data given.Define where From the Vlasov Equation, where , E B and ( ) the characteristics of this equation are the solutions of: , , when f is written as a line integral over such a characteristics curve, we have is bounded , we can write it as: Now adding 2.13 and 2.14, we have Applying Gronwall lemma, we have 0 E B f = = = .This implies the solution is unique.This proves the uniqueness of theorem 2.1.

Representations of Derivatives of Electric and Magnetic Fields
Theorem 2.3.Assume that ( ) exists as in the hypothesis of theorem 2.1.Then the derivatives of the fields can be represented as: , , f Sf S f without explicit arguments are evaluated at ( ) Now using the fact that j T is a perfect j y derivative, integrating the last integral using integration by parts in y, is equal to: Here the last expression is part of ( ) , a w v .Here, the most singular term is the j T term, which appears in the first expression, it is: Simplifying this we get: since the first term depends only on initial data, hence part of ( ) . Now the second term simplifies as: (see an elementary computation of this in the appendix part of [1]).Hence, this expression is the value of ( ) Now first compute the third term, we have: Now the integrand in the first term simplifies as: Now the second integral becomes: ( ) Adding the Equations (2.15), (2.16) and (2.17), we get ( ) We can have the same result for the magnetic field see [1], in this case the singular term is TT B .This completes the proof of 2.3.

Estimation of the Particle Density
To estimate the particle density take  .The characteristics of the Vlasov equation are solutions of the , , 0, , , , 0, , , f t x p f X t x p P t x p = and since ( ) , that is f is non-negative and bounded.Now, we claim that ( )

( ) ( ) ( )
, supsup A similar definition can be done for the electric field E.
Now by applying the norm properties above, the expression in 2.3 can be reduced to Again by taking , we can have a similar bound, since ( ) Therefore, again by applying the norm properties above we have, (

Bounds on the Electric and Magnetic Fields
We already proved in theorem 2.2 that the fields can be represented as: ( ) By our hypothesis we have We have that ( ) Similarly, for S E , we use ( ) . Then integrating this by parts in p, we get: By the support hypothesis, the v-gradient factor is bounded (say by T C ). Hence, ( ) A similar estimate holds for B, See ( [1]).Hence Adding Equations (2.24) and (2.25), we have:

Bounds on the Gradient of the Field
Theorem 2.4.[1] Let in the form of theorem 2.3 above as:

∫∫
Here the first term(data term) which is ( ) just depends on the derivative of the initial data.For the second term, From this the most singular term is the TT E term.Hence Here ω is integrated over the unit sphere 2 S and p is over 3  .We break the ξ integral into two integrals, over [ ] . Since the support of f is bounded in p, the kernel ( ) Hence, for any , 0 ) Hence, from expressions 2.28 and 2.29, we have: For the Sf term, let us integrate by parts in p: For the 2 S term, we write which is bounded and the y-integrals are over the ball y x t − ≤ and III satisfy the same bound as II.Now again split ( )  IV , integrating by parts in p, and the resulting kernel is bounded for v and p K B ∇ ≤ , hence we have; IV , we recall in Section 2.1, j T is a perfect y derivative, hence we can integrate by part in y.Since ( ) Combining these results, we get: Now adding 2.30, 2.31 and 2.37, we get: To get the same result for B, we repeat the same process, (see [7]), and This proves theorem 2.5.
Here putting 2.21 into this expression, we get: is bounded, and hence ( ) K t is also bounded.
Using this estimates, we will proof the existence of the solutions for theorem 2.1.

Existence of Solutions
From the hypothesis we have smooth initial data ( ) ( ) ( ) which is a linear equation (for a single unknown) of the form and with initial condition 0 f , where c and 0 f are 2 C functions.Since ( ) The characteristics of 2.38 are the solutions of: , , 0 d and d with initial data ( ) ( ) ( ) ( ) Lemma 2.5.[10].Given that C .Now let us proceed by induction on n to show the solutions are 2 C .
From the representation theorem 2.2, where ( ) 0 , E t x is the solution of the homogeneous wave equation with the same Cauchy data, and Substituting this in to ( ) ( ) , n S E t x , we can integrate by parts in p. From the induction hypothesis ( ) This proves lemma 2.5.Now let us claim that the estimates 2.21 and 2.26 holds uniformly in n for To show this we follow the same process as we did for , f E and B , the only difference is replacing with the superscripts ( ) and, the expression analogous to 2.21 is; and the analogue of 2.26 is; with constants C depending on T. Now iterating 2.43, we have This tells us that the fields Now an analogue of the result of theorem 2.5 is: Substituting 2.45 in to 2.46, we conclude that;  as in such a way given in theorem 2.5, and then subtract this expressions.We can write the TT term as written in 2.28-2.30and then estimate it, we have; Similarly TS and ST terms are written as in 2.31 and then estimated as: For the SS term, let us break up in to several pieces as in 2.33.Following the same procedure and using the known bound in 1 C , we conclude that Now by the known bounds, we have Hence by Gronwall's lemma, the sequences ( ) ( ) x s p s converges uniformly on 0 s T ≤ ≤ .Here, on the parameter , , t x p , the convergence is also uniform, After integrating this along the characteristics, we have Subtracting the second from the first and estimating, we get: From 2.55 and the known bound in 1 C , the first term goes to zero uniformly on [ ] 0,T .The second term in the integrand is dominated by    f respectively.Therefore ( ) , , E B f will be the unique solution of the system 2.1 for the simplified case of a single species.
To generalize for n species, we need just a little modification.The operator S now depends on α .

. . t x i e S v α α =∂ + ∇
In this case each f α remains bounded.In the representations of the fields and their derivatives, ρ and j are written as for each α separately.Hence, with these simple modifications, we conclude for several species case.

Uniqueness and Existence with Small Initial Data
In the previous chapter, we have seen that the sufficient condition for the existence of a global 1 C solution for the relativistic Vlasov Maxwell's equations was the existence of a continuous function ( ) with supports in { } x k ≤ which satisfy the constraints, If the data satisfy   To prove this theorem, the key step is to show that the paths of the particles spread out with time.Since the paths of the particles are given by the equations ( ) Thus we need to prove that the electromagnetic field decays as t → ∞ .Hence, to prove this theorem, let us introduce a weighted L ∞ norm for the field, as was introduced by [11].Therefore, we use the weight ( )( )

The Structure of the Proof
The main structure of the proof is as established in the last chapter.To prove uniqueness, we use the same step as we did in chapter two, and for the existence, the following construction was used.For given functions That is given the ( ) iteration, then we define ( ) ( )  Finally we define ( ) ( ) , n n E B as the solutions of the Maxwell's equations, (

Characteristics
Here the characteristics are curves defined by the solutions of the equations 3.9 and 3.10.Because E and B are 1  C , the solutions exist as

4 ) 5 )∫∂ 1 . 1 . 3 .
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: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 In the entire work, we are going to use, the partial derivatives with respect to , while any derivative of order k with respect to t or x or p will be denoted by k and so on with the convention o D f f = .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.
field representation in theorem 2.2, we have

plying 2 S
f , and ( ) , b w v is the term multiplying 2 Sf γ , which we can see it easily.Now let us determine ( ) , d w v and ( ) we must use the fact that the kernel has zero average.

1
, the p-derivative of the difference can be estimated in terms of the x-

(
where e α is the charge of particles of species α .Again n move approximately in straight lines if E and B are small. , we define ( ) ( ) ( ) ( ) , To show the sequences are Cauchy, let us fix two indices m and n.For j = 0, 1, is an upper bound for the 1 C norm, of the field, we thus have "which is the largest momentum up to time t emanating from the support of 0 f α " [2].Hence, ( ) u t is a continuous function of t for * 0 t T ≤ ≤ .