Weak solutions of the relativistic Vlasov–Maxwell system with external currents

DOI: 10.1002/mma.7070

R E S E A R C H A R T I C L E

Weak solutions of the relativistic Vlasov–Maxwell system with external currents

Jörg Weber

Department of Mathematics, University of Bayreuth, Bayreuth, Germany

Correspondence

Jörg Weber, Department of Mathematics, University of Bayreuth, 95440 Bayreuth, Germany.

Email: Joerg.Weber@uni-bayreuth.de Communicated by: R. Picard

The time evolution of a collisionless plasma is modeled by the relativistic Vlasov–Maxwell system which couples the Vlasov equation (the transport equation) with the Maxwell equations of electrodynamics. We consider the case that the plasma consists of N particle species, the particles are located in a bounded containerΩ⊂R³, and are subject to boundary conditions on𝜕Ω. Fur- thermore, there are external currents, typically in the exterior of the container, that may serve as a control of the plasma if adjusted suitably. We do not impose perfect conductor boundary conditions for the electromagnetic fields but con- sider the fields as functions on whole spaceR³and model objects, that are placed in space, via given matrix-valued functions𝜀(the permittivity) and𝜇(the perme- ability). A weak solution concept is introduced and existence of global-in-time solutions is proved, as well as the redundancy of the divergence part of the Maxwell equations in this weak solution concept.

K E Y WO R D S

nonlinear partial differential equations, relativistic Vlasov–Maxwell system M S C C L A S S I F I C AT I O N

35Q61; 35Q83; 82D10

1 I N T RO D U CT I O N

The time evolution of a collisionless plasma is modeled by the relativistic Vlasov–Maxwell system. Collisions among the plasma particles can be neglected if the plasma is sufficiently rarefied or hot. The particles only interact through electromagnetic fields created collectively. We consider the following setting: there areNspecies of particles, all of which are located in a containerΩ⊂R³, which is a bounded domain, for example, a fusion reactor. Thus, boundary conditions on𝜕Ωhave to be imposed. In the exterior ofΩ, there are external currents, for example, in electric coils, that may serve as a control of the plasma if adjusted suitably. In order to model materials that are placed somewhere in space, for example, the reactor wall, electric coils, and (almost perfect) superconductors, we consider the permittivity𝜀and permeability𝜇, which are functions of the space coordinate, take values in the set of symmetric, positive definite matrices of dimension three, and do not depend on time, as given. With this assumption, we can model linear, possibly anisotropic materials that stay fixed in time. We should mention that in reality,𝜀and𝜇will on the one hand additionally depend on the particle density [Correction added on 29 December 2020 after initial online publication. Mathematical intervals throughout the paper have been corrected in this version.]

This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.

© 2020 The Authors. Mathematical Methods in the Applied Sciences published by John Wiley & Sons Ltd

Math Meth Appl Sci. 2020;1–32. wileyonlinelibrary.com/journal/mma 1

insideΩand on the other hand additionally locally on the electromagnetic fields, typically via their frequencies (maybe even nonlocally because of hysteresis). However, this would cause further nonlinearities which we avoid in this work.

The unknowns are on the one hand the particle densities𝑓^𝛼 = 𝑓^𝛼(t,x,v),𝛼 = 1,…,N, which are functions of time t≥0, the space coordinatex∈ Ω, and the momentum coordinatev∈R³. Roughly speaking,𝑓^𝛼(t,x,v)indicates how many particles of the𝛼th species are at timetat positionxwith momentumv. On the other hand, there are the electromagnetic fieldsE=E(t,x),H =H(t,x), which depend on timetand space coordinatex∈R^{3. TheD- and}B-fields are computed fromEandHby the linear constitutive equationsD= 𝜀EandB= 𝜇H. We will only viewEandHas unknowns in the following.

The Vlasov–Maxwell system on a time interval with given final time 0<T_•≤∞, equipped with boundary conditions on𝜕Ωand initial conditions fort=0, is then given by the following set of equations; we explain the appearing notation afterwards:

𝜕t𝑓^𝛼+̂v_𝛼·𝜕x𝑓^𝛼+e_𝛼(

E+̂v_𝛼×H)

·𝜕v𝑓^𝛼 =0 on IT•× Ω ×R³, (VM.1)

𝑓−^𝛼 =𝛼𝑓₊^𝛼+g^𝛼on𝛾_T⁻•, (VM.2)

𝑓^𝛼(0) = ̊𝑓^𝛼 on Ω ×R³, (VM.3)

𝜀𝜕tE−curl_xH= −4𝜋𝑗 on I_T•×R³, (VM.4)

𝜇𝜕tH+curlxE=0 on IT•×R³, (VM.5)

(E,H) (0) =(E, ̊̊ H)

on R³, (VM.6)

where (VM.1) to (VM.3) have to hold for all𝛼 = 1,…,N andI_T• denotes the given time interval. Here and in the following,IT ∶= [0,T]for 0≤T<∞andI_∞∶= [0,∞[. Additionally, the divergence equations

div_x(𝜀E) =4𝜋𝜌 onI_T•×R³, (1a)

divx(𝜇H) =0 onIT•×R^{3 (1b)}

have to hold. In (VM.3) and (VM.6),𝑓^𝛼(0)and(E,H) (0)denote the evaluation off^𝛼 and(E,H)at timet =0, that is, to say the functions𝑓^𝛼(0,·,·)and(E,H) (0,·). We will use this notation often, also similarly for other functions.

Note that throughout this work, we use modified Gaussian units such that the speed of light (in vacuum) is normalized to unity and all rest massesm_𝛼of a particle of the respective species are at least 1. In (VM.1),e_𝛼 is the charge of the𝛼th particle species and̂v_𝛼the velocity, which is computed from the momentumvvia

̂v_𝛼 = v

√

m²_𝛼+|v|² ,

according to special relativity. Clearly,||̂v_𝛼|| < 1, that is, the velocities are bounded by the speed of light. Moreover, we assume that𝜀=𝜇=Id onΩ, Id denoting the 3×3-identity matrix. Thus, the speed of light is constant inΩandB= H onΩ.

Equation (VM.2) describes the boundary condition on𝜕Ω. Typically, one imposes specular boundary conditions. Thus, it is natural to consider the following decompositions:

̃𝛾±

∶={

(x,v) ∈𝜕Ω ×R³|v·n(x)≷0}

, ̃𝛾⁰∶={

(x,v) ∈𝜕Ω ×R³|v·n(x) =0} , 𝛾± ∶= [0,∞[×̃𝛾±, 𝛾⁰∶= [0,∞[×̃𝛾⁰, 𝛾±

T ∶=IT×̃𝛾±, 𝛾_T⁰ ∶=IT×̃𝛾⁰,

wheren(x)is the outer unit normal of𝜕Ωatx∈𝜕Ωand 0<T≤∞. In (VM.2),𝑓±^𝛼 are the restrictions off^𝛼to𝛾±

T•. The operator_𝛼maps functions on𝛾_T⁺•to functions on𝛾_T⁻•. In Section 3, we deal with the case that

_𝛼h=a^𝛼(Kh), (2)

where

(Kh) (t,x,v) =h(t,x,v−2(v·n(x)))

describes reflection on the boundary anda^𝛼, satisfying 0≤a^𝛼≤1, describes how many of the particles hitting the boundary at timetatxwith momentumvare reflected (and not absorbed);g^𝛼≥0 is the source term according to how many particles are added from outside. We will deal with purely reflecting (a^𝛼 =1 andg^𝛼 =0) and partially absorbing (a^𝛼 ≤a^𝛼₀for some a^𝛼₀ <1) boundary conditions and also with a “hybrid” of these two (there is no sucha^𝛼₀, andg^𝛼=0).

In (VM.4) and (1a),jand𝜌are the current and charge density. Typically, they are the sum of the internal current and charge densities

𝑗^int∶=

∑N 𝛼=1

e_𝛼

∫R³

̂v_𝛼𝑓^𝛼dv, 𝜌^int∶=

∑N 𝛼=1

e_𝛼

∫R³ 𝑓^𝛼dv

and some external current densityu, which is supported in some open setΓ⊂R³, and charge density𝜌^uresulting fromu.

We will always extendj^int,𝜌^int(u) by zero outsideΩ(Γ). Usually, the divergence Equations (1) are known to be redundant if all functions are smooth enough, local conservation of charge is satisfied, that is,

𝜕t𝜌+divx𝑗=0,

and (1) holds initially, which we then view as a constraint on the initial data. Therefore, in the first sections, we ignore (1) and discuss in Section 4 in what sense (1) is satisfied in the context of a weak solution concept.

The paper is organized as follows: In Section 2.3, we state our main two theorems. The first regards the existence of weak solutions to (VM). In Section 3, we prove this theorem. To this end, we state some basic results about linear Vlasov and Maxwell equations (Section 3.1), approximate the given functions in a proper way (Section 3.2), consider a cut-off system (Section 3.3), and finally remove the cut-off (Section 3.4). The second main result regards the redundancy of the divergence equations in our weak solution concept. We prove this theorem in Section 4 and give some comments on the physical interpretation of the obtained equations.

In Section 3, we proceed similarly to Guo,¹who proved existence of weak solutions in the case that𝜀=𝜇=Id,u=0, and the electromagnetic fields are subject to perfect conductor boundary conditions on𝜕Ω, that is,E×n=0. However, there is no need of artificially inserting the factore^−tas is done throughout that paper. The more important motivation of our paper is the following: the papers concerning plasma in a domain we are aware of deal with perfect conductor boundary conditions for the electromagnetic fields. Such a setup can model no interaction between this domain and the exterior.

However, considering fusion reactors, there are external currents in the exterior, for example, in field coils. These external currents induce electromagnetic fields and thus influence the behavior of the internal plasma. Even more important, the main aim of fusion plasma research is to adjust these external currents “suitably.” Thus, we impose Maxwell's equations globally in space and model objects like the reactor wall, electric coils, and almost perfect superconductors via𝜀and𝜇. In order to make use of an energy consideration, we note that for classical solutions of (VM) one can easily derive the energy balance

d dt

⎛⎜

⎜⎝

∑N 𝛼=1∫

Ω

∫R³

√

m²_𝛼+|v|²𝑓^𝛼dvdx+ 1 8𝜋∫

R³

(𝜀E·E+𝜇H·H)dx

⎞⎟

⎟⎠

≤C−∫ R³

E·u dx,

whereCis some expression in theg^𝛼; ifa^𝛼 = 1 for all𝛼, equality holds above. In order to apply a quadratic Gronwall argument and to conclude that the left bracket is bounded for each time, the map

(E,H)→⎛

⎜⎜

⎝∫ R³

(𝜀E·E+𝜇H·H)dx

⎞⎟

⎟⎠

1 2

should be a norm onL²(

R^3;R⁶⁾which is equivalent to the standardL²-norm. Thus, assumptions about uniform positive definiteness of𝜀and𝜇will be made.

Especially, the second main result, regarding the redundancy of the divergence part of Maxwell's equations, in our setting is much harder to prove than a similar result in the setting that was considered in Guo.¹ The main difficulty is that (1) has to hold on whole spaceR³ in the sense of distributions. Thus, we have to extend the weak formulation of (VM) to a larger class of test functions and somehow have to “cross over”𝜕Ω.

Vlasov–Maxwell systems have been studied extensively. In case of no reactor wall, that is, the Vlasov equation is imposed globally in space (as well as Maxwell's equations), global well-posedness of the Cauchy problem is a famous open problem.

Global existence and uniqueness of classical solutions has been proved in lower dimensional settings; see Glassey and

Schaeffer.^2-5In the full three-dimensional setting, a continuation criterion was given by Glassey and Strauss.⁶Further- more, global existence of weak solutions was proved by Di Perna and Lions.⁷ Their momentum-averaging lemma is fundamental for proving existence of weak solutions in any setting (with or without boundary, with or without perfect conductor boundary conditions, etc.), since it handles the nonlinearity in the Vlasov equation. However, uniqueness of these weak solutions is not known. The regularity of such weak solutions in free space was studied by Bouchut et al⁸and by Besse and Bechouche.⁹ However, in case of the presence of boundary conditions for the plasma particles, one can- not expectC¹-solutions in general; this was observed by Guo¹⁰ even in a one-dimensional setting. For a more detailed overview, we refer to Rein¹¹and to the book of Glassey,¹²which also deals with other PDE systems in kinetic theory.

2 P R E L I M I NA R I E S 2.1 Some notation

Throughout this work,C^k-spaces (k ∈ N∪ {∞}) on the closure of some open set Uare defined to be the space of C^k-functionshonUsuch that all derivatives ofhof order less or equalkcan be continuously extended toU. Moreover, the indexbinC^k_bindicates that all derivatives of order less or equalkof such functions shall be bounded, and the indexc inC^k_cindicates that such functions shall be compactly supported. As usual,C^k,s(k∈N0, 0<s≤1) denotes Hölder spaces.

It will be convenient to introduce the surface measure

d𝛾_𝛼 =||̂v_𝛼·n(x)||dvdSxdt on[0,∞[×𝜕Ω ×R^3.

Furthermore, we denote by𝜒Mthe characteristic function of some setMand by𝜒Tthe characteristic function of[0,T].

For 1≤p<∞, we define

L^p_𝛼kin(A,da) ∶=

⎧⎪

⎨⎪

⎩

u∈L^p(A,da)|∫

A

v⁰_𝛼|u|^pda<∞

⎫⎪

⎬⎪

⎭ ,

equipped with the corresponding weighted norm. Here,A⊂R^3×R^3orA⊂R^×R^3×R³is some Borel set equipped with a measurea, and the weightv⁰_𝛼 is given by

v⁰_𝛼∶=

√

m²_𝛼+|v|². Bym_𝛼≥1, we havev⁰_𝛼 ≥1. Moreover, we write

L^p_lt(A,da) ∶={

u∶A→R|𝜒Tu∈L^p(A,da)for all T >0}

for 1≤p≤∞. Ifais the Lebesgue measure, we writeL^p_𝛼kin(A)andL^p_lt(A), respectively. A combinationL^p_𝛼kin,lt(A,da)is defined accordingly. Furthermore, we abbreviate

Glt(I;X) ∶= {u∶I→X|u∈G([0,T] ;X)for all T∈I},

where 0∈I ⊂[0,∞[is some interval,Gis someC^korL^p, andXis a normed, separable vector space. Also, the somewhat sloppy notation

L^∞(I;L^∞(A)) ∶=L^∞(I×A) and

G(I;X∩Y) ∶=G(I;X) ∩G(I;Y) (and likewise with index “lt”, respectively) occur.

Since𝜀is already used for the permittivity, the letter𝜄, and not𝜀, will always denote a small positive number.

For a matrixA∈R^n×n(n∈N) and a positive number𝜎 >0, we writeA≥𝜎(A≤𝜎) ifAx·x≥𝜎|x|²(Ax·x≤𝜎|x|²) for all x∈Rⁿ. For a measurableA∶Rⁿ→R^n×nand𝜎 >0, we writeA≥𝜎(A≤𝜎) ifA(x)≥𝜎(A(x)≤𝜎) for almost allx∈R^n.

Finally, for a normed spaceX, somex∈X, and r>0,Br(x) denotes the open ball inXwith centerxand radiusr.

Furthermore, we abbreviateBr∶=Br(0).

2.2 Weak formulation

The space of test functions for (VM.1) to (VM.3) will beΨ_T• where ΨT∶=

{𝜓 ∈C^∞ (

IT× Ω ×R³⁾|supp𝜓 ⊂[ 0,T

[

×Ω ×R^3compact,dist(

supp𝜓, 𝛾_T⁰)

>0, dist(

supp𝜓,{0} ×𝜕Ω ×R³⁾>0}

for 0<T≤∞. On the other hand,ΘT• will be the space of test functions for (VM.4) to (VM.6), where ΘT ∶={

𝜗∈C^∞(

I_T×R³;R³⁾|supp𝜗 ⊂[0,T[ ×R^3compact} for 0<T≤∞.

We start with the definition of what we call solutions to (VM).

Definition 1. Let 0<T•≤∞,u∈ L¹_loc(

R^3;R³⁾. We call a tuple((

𝑓^𝛼, 𝑓₊^𝛼)

𝛼,E,H, 𝑗)

a weak solution of (VM) on the time intervalI_T• with external currentuif (for all𝛼):

(i) 𝑓^𝛼 ∈L¹_loc (

IT•× Ω ×R^3),𝑓₊^𝛼 ∈L¹_loc

(𝛾_T⁺•,d𝛾𝛼

)

,E,H, 𝑗∈L¹_loc(

IT•×R^3;R^3). (ii) For all𝜓 ∈ ΨT• it holds that

0= −

T•

∫

0

∫

Ω

∫R³

(𝜕t𝜓+̂v_𝛼·𝜕x𝜓+e_𝛼(

E+̂v_𝛼×H)

·𝜕v𝜓)

𝑓^𝛼dvdxdt

+∫

𝛾T⁺•

𝑓+^𝛼𝜓d𝛾𝛼−

∫𝛾_T⁻•

(𝛼𝑓+^𝛼+g^𝛼) 𝜓d𝛾𝛼−

∫

Ω

∫R³

̊𝑓^𝛼𝜓(0)dvdx

(3)

(in particular, especially the integral of(

E+̂v_𝛼×H)

𝑓^𝛼·𝜕v𝜓is supposed to exist).

(iii) For all𝜗∈ ΘT• , it holds that

0=

T•

∫

0

∫R³

(𝜀E·𝜕t𝜗−H·curlx𝜗−4𝜋𝑗·𝜗)dxdt+∫ R³

𝜀 ̊E·𝜗(0)dx, (4a)

0=

T•

∫

0 ∫ R³

(𝜇H·𝜕t𝜗+E·curlx𝜗)dxdt+

∫R³

𝜇 ̊H·𝜗(0)dx. (4b)

(iv) The currentjis the sum of the internal and the external currents, that is,

𝑗=𝑗^int+u∶=

∑N 𝛼=1

e_𝛼

∫R³

̂v_𝛼𝑓^𝛼dv+u.

We easily derive this weak formulation after multiplying the respective equations of (VM) with the respective test function and integrating by parts, assuming all functions are smooth enough.

2.3 Statement of main results

We have two main results: the first is about existence of weak solutions in the case of partially absorbing boundary condi- tions for particle species 1,…,N^′and purely reflecting or “hybrid” boundary conditions for particle speciesN^′+1,…,N.

We assume that the following conditions hold:

Condition 1.

• 0≤ ̊𝑓^𝛼∈(

L¹_𝛼kin∩L^∞) (

Ω ×R^{3)for all}𝛼=1,…,N;

• _𝛼is given by (2) for𝛼=1,…,N;

• 0≤a^𝛼 ∈L^∞ (𝛾_T⁻•

)

,a^𝛼₀∶=||a^𝛼||_L∞( 𝛾T⁻•

) <1, 0≤g^𝛼∈ (

L¹_𝛼kin,lt∩L^∞_lt ) (𝛾_T⁻•

)

for𝛼=1,…,N^′;

• 0≤a^𝛼 ∈L^∞ (𝛾_T⁻•

)

,||a^𝛼||_L∞( 𝛾_T⁻•

) =1,g^𝛼=0 for𝛼=N^′+1,…,N;

• E̊, ̊H∈L²(

R³;R^3);

• 𝜀, 𝜇∈L^∞(

R^3;R^3×3)such that there are𝜎,𝜎^′>0 satisfying𝜎≤𝜀,𝜇≤𝜎^′, and𝜀=𝜇=Id onΩ;

• u∈L¹_lt(

I_T•;L²(

Γ;R^3)).

Then, our first main result is (see Section 3):

Theorem 1. Let T• ∈ ]0,∞],Ω ⊂ R³be a bounded domain such that𝜕Ωis of class C^1,𝜅for some 0< 𝜅≤1, and let Condition 1 hold. Then, there exist functions

• 𝑓^𝛼∈L^∞_lt ( IT•;(

L¹_𝛼kin∩L^∞) (

Ω ×R³⁾⁾, 𝑓₊^𝛼 ∈ (

L¹_𝛼kin,lt∩L^∞_lt

) (𝛾_T⁺•,d𝛾_𝛼)

,𝛼=1,…,N^′, all nonnegative,

• 𝑓^𝛼∈L^∞(

I_T•× Ω ×R^3)∩L∞_lt ^(IT•;L¹_𝛼kin(

Ω ×R³⁾⁾, 𝑓₊^𝛼 ∈L^∞ (𝛾_T⁺•

)

,𝛼=N^′+1,…,N, all nonnegative,

• (E,H) ∈L^∞_lt (

IT•;L²(

R^3;R⁶⁾⁾ such that((

𝑓^𝛼, 𝑓₊^𝛼)

𝛼,E,H, 𝑗)

is a weak solution of (VM) on the time interval IT• with external current u in the sense of Definition 1, where

𝑗=𝑗^int+u=

∑N 𝛼=1

e_𝛼

∫R³

̂v_𝛼𝑓^𝛼dv+u, 𝑗^int∈L^∞_lt (

I_T•; (

L¹∩L⁴³) (

Ω;R³⁾⁾.

Furthermore, we have the following estimates for any 1≤p≤∞and0<T ∈I_T• : Estimates on𝑓^𝛼, 𝑓₊^𝛼:

‖𝑓^𝛼‖L^∞(^[0,T];L^p(^Ω×R³))≤‖‖‖ ̊𝑓^𝛼‖‖‖^Lp(^Ω×R³)+(

1−a^𝛼₀)_p¹−1

‖g^𝛼‖L^p(𝛾⁻T,d𝛾𝛼), (5)

‖‖𝑓₊^𝛼‖‖^Lp(𝛾T⁺,d𝛾𝛼)≤(

1−a^𝛼₀)−¹_p‖‖‖ ̊𝑓^𝛼‖‖‖^Lp(^Ω×R³)+(

1−a^𝛼₀)−1

‖g^𝛼‖_L^p(𝛾T⁻,d𝛾𝛼), (6)

for𝛼=1,…,N^′and

‖𝑓^𝛼‖L^∞(^[0,T];L^p(^Ω×R³))≤‖‖‖ ̊𝑓^𝛼‖‖‖^Lp(^Ω×R³), (7)

‖‖𝑓₊^𝛼‖‖^L∞(𝛾T⁺)≤‖‖‖ ̊𝑓^𝛼‖‖‖^L∞(^Ω×R³), (8) for𝛼=N^′+1,…,N.

Energy-like estimate:

⎛⎜

⎜⎜

⎝

N^′

∑

𝛼=1

(1−a^𝛼₀)

∫

𝛾T⁺

v⁰_𝛼𝑓₊^𝛼d𝛾𝛼+

‖‖‖‖

‖‖‖

∑N 𝛼=1∫

Ω

∫R³

v⁰_𝛼𝑓^𝛼(·)dvdx+ 𝜎

8𝜋‖(E,H) (·)‖²_L2(R^3;R⁶)

‖‖‖‖

‖‖‖L^∞([0,T])

⎞⎟

⎟⎟

⎠

1 2

≤

⎛⎜

⎜⎜

⎝

∑N 𝛼=1∫

Ω

∫R³

v⁰_𝛼 ̊𝑓^𝛼dvdx+

N^′

∑

𝛼=1∫

𝛾T⁻

v⁰_𝛼g^𝛼d𝛾_𝛼+ 𝜎^′ 8𝜋‖‖‖

(E, ̊̊ H)‖

‖‖^2L2(R^3;R⁶)

⎞⎟

⎟⎟

⎠

1 2

+√

2𝜋𝜎⁻¹²‖u‖_L¹(^[0,T];L²(^Γ;R³)).

(9)

Estimate on j^int:

‖‖‖𝑗^int‖‖‖L^∞ (

[0,T];L⁴³(^Ω;R³)

)

≤⎛

⎜⎜

⎝

∑N 𝛼=1

|e_𝛼|⁴ (4𝜋

3 ‖‖‖ ̊𝑓^𝛼‖‖‖L^∞(^Ω×R³)+1+ { _4𝜋

3(^1−a𝛼0)‖g^𝛼‖_L^∞(𝛾_T⁻), 𝛼≤N^′

0, 𝛼 >N^′

)⁴⎞

⎟⎟

⎠

1 4

·

⎛⎜

⎜⎜

⎜⎝

⎛⎜

⎜⎜

⎝

∑N 𝛼=1∫

Ω

∫R³

v⁰_𝛼 ̊𝑓^𝛼dvdx+

N^′

∑

𝛼=1∫

𝛾T⁻

v⁰_𝛼g^𝛼d𝛾_𝛼+ 𝜎^′ 8𝜋‖‖‖

(E, ̊̊ H)‖

‖‖^2L2(R^3;R⁶)

⎞⎟

⎟⎟

⎠

1 2

+√

2𝜋𝜎⁻¹²‖u‖_L¹([0,T];L²(^Γ;R³))

⎞⎟

⎟⎟

⎟⎠

3 2

.

(10)

The second main result answers the question whether the divergence equations (1) are automatically satisfied if we have a weak solution of (VM). To this end, we have to introduce an external charge density𝜌^ucorresponding touand assume that local conservation of the external charge holds:

Condition 2. There are𝜌^u∈L¹_loc(

I_T•× Γ)

and ̊𝜌^u∈L¹_loc(Γ)such that𝜕t𝜌^u+div_xu=0 on]0,T•[×R^3and𝜌^u(0) = ̊𝜌^u onΓ, which is to be understood in the following weak sense:

0=

T•

∫

0

∫R³

(𝜌^u𝜕t𝜓+u·𝜕x𝜓)dxdt+∫ R³

̊𝜌^u𝜓(0)dx

for any𝜓 ∈C^∞(

I_T•×R^{3)with supp}𝜓 ⊂[0,T•[×R³compact. Here,𝜌^uand ̊𝜌^uare extended by zero outsideΓ.

We should point out that this condition very mild: on the one hand, from a physical point of view, there always exists an external charge density, and it is very natural to assume local conservation of external charge. On the other hand, if the charge density is known (or prescribed) initially and divxuis locally integrable, then one can integrate𝜕t𝜌^u= −divxu in time to obtain a suitable external charge density on the whole time interval.

Our second main result is (see Section 4):

Theorem 2. LetΩ ⊂ R³ be a bounded domain with boundary𝜕Ωof class C¹∩W^2,∞. Furthermore, let, for all𝛼 ∈ {1,…,N},𝑓^𝛼 ∈

(

L¹_lt∩L²_𝛼kin,lt∩L^∞_lt) (

IT•× Ω ×R^3),𝑓₊^𝛼 ∈ L^∞_lt (𝛾_T⁺•

)

,(E,H) ∈ L^q_lt( IT•;L²(

R^3;R^{6))for some q}>2,

_𝛼 ∶L^∞_lt (𝛾_T⁺•

)→L^∞_lt (𝛾_T⁻•

)

, g^𝛼 ∈L^∞_lt (𝛾_T⁻•

)

, ̊𝑓^𝛼 ∈(

L¹∩L^∞) (

Ω ×R^3),(E, ̊^̊ H)

∈L²(

R^3;R^6),𝜀, 𝜇 ∈L^∞_loc(

R^3;R^3×3) with𝜀 = 𝜇 = Id onΩ, and u ∈ L¹

loc

(I_T•× Γ;R³⁾such that the tuple((

𝑓^𝛼, 𝑓₊^𝛼)

𝛼,E,H, 𝑗^int+u)

is a weak solution of (VM) on the time interval IT• with external current u in the sense of Definition 1. Furthermore, assume that Condition 2 holds. Moreover, let initially

div_x( 𝜀 ̊E)

=4𝜋(

̊𝜌^int+ ̊𝜌^u)

∶=4𝜋⎛

⎜⎜

⎝

∑N 𝛼=1

e_𝛼

∫R³

̊𝑓^𝛼dv+ ̊𝜌^u⎞

⎟⎟

⎠ , divx

(𝜇 ̊H)

=0, onR³be satisfied in the sense of distributions. Then,

(i) It holds that

div_x(𝜇H) =0

on]0,T•[×R³in the sense of distributions. (For this, only Equation (4b) is needed.) (ii) We have

divx(𝜀E) =4𝜋(

𝜌^int+𝜌^u)

on]0,T•[×(

R³∖𝜕Ω)

in the sense of distributions, that is,

0=

T•

∫

0 ∫ R³

(𝜀E·𝜕x𝜑+4𝜋(

𝜌^int+𝜌^u) 𝜑)

dxdt

for all𝜑∈C^∞_c (

]0,T•[×(

R³∖𝜕Ω)) .

(iii) If, additionally to the given assumptions, 𝑓₊^𝛼 ∈ L¹_lt

(𝛾_T⁺•,d𝛾_𝛼)

, g^𝛼 ∈ L¹_lt

(𝛾_T⁻•,d𝛾_𝛼)

, and _𝛼 ∶ (L¹_lt∩L^∞_lt) (

𝛾_T⁺•,d𝛾𝛼

)→(

L¹_lt∩L^∞_lt) ( 𝛾_T⁻•,d𝛾𝛼

)

for all𝛼∈ {1,…,N}, then

divx(𝜀E) =4𝜋(

𝜌^int+𝜌^u+S_𝜕Ω)

(11) on]0,T•[×R³in the sense of distributions, that is,

0=

T•

∫

0

∫R³

(𝜀E·𝜕x𝜑+4𝜋(

𝜌^int+𝜌^u) 𝜑)

dxdt+4𝜋S_𝜕Ω𝜑

for all𝜑∈C^∞_c (

]0,T•[×R³⁾. Here, the distribution S_𝜕Ω, satisfying suppS_𝜕Ω⊂I_T•×𝜕Ω, is given by

S_𝜕Ω𝜑=

T•

∫

0

∫𝜕Ω

𝜑(t,x)

t

∫

0

n(x) ·

⎛⎜

⎜⎜

⎝

∑N 𝛼=1

e_𝛼 ∫ {^v∈R³|n(x)·v>0}

̂v_𝛼𝑓₊^𝛼(s,x,v)dv

+

∑N 𝛼=1

e_𝛼

∫ {^v∈R³|n(x)·v<0}

̂v_𝛼(

𝛼𝑓+^𝛼+g^𝛼)

(s,x,v)dv

⎞⎟

⎟⎟

⎠

dsdS_xdt.

Note that we do not need Condition 1 in Theorem 2; in particular,_𝛼 need not take the form (2).

3 E X I ST E N C E O F W E A K S O LU T I O N S

In this section, we proceed similarly to Guo¹with necessary modifications being made, who considered the problem with 𝜀=𝜇=Id,u=0, and perfect conductor boundary conditions for the electromagnetic fields on𝜕Ω. Citations of this paper always refer to the relativistic version of the respective lemma, theorem, and so on; see Guo.,¹section 5

3.1 Results about linear Vlasov and Maxwell equations

The strategy is to consider an iteration scheme where we decouple Vlasov's equations from Maxwell's equations in each iteration step and hence only have to solve linear problems. Thus, it is natural to consider linear Vlasov and Maxwell equations first. Regarding the Vlasov part, we refer to Beals and Protopopescu.¹³Considering the linear problem (on some [0,T])

Y𝑓 ∶=𝜕t𝑓 +̂v_𝛼·𝜕x𝑓+F·𝜕v𝑓 =0, (12a)

𝑓−=𝑓++g, (12b)

𝑓(0) = ̊𝑓, (12c)

with a Lipschitz continuous, bounded force fieldF, that is divergence free with respect tov, they introduced a space of test functions associated toF. As in Guo,¹lemma 2.1. we can show that our test function spaceΨTbelongs to that test function space for eachFandT, where one needs the assumption that𝜕Ωbe of classC^1,𝜅 and that the support of any

𝜓∈ ΨTbe away from𝛾_T⁰and{0} ×𝜕Ω ×R³. In Beals and Protopopescu,¹³“strong” solutions in a set ofL^p-functions for which a trace on the boundary exists in the sense of the following extended Green's identity were searched for:

T

∫

0

∫

Ω

∫R³

(𝜙Y𝑓+𝑓Y𝜙)dvdxdt=∫

D⁺_T

𝑓⁺𝜙d𝜈⁺−∫

D⁻_T

𝑓⁻𝜙d𝜈⁻,

which is supposed to hold for all test functions𝜙. Here,D±

T are the outgoing/incoming sets associated to the characteristic flow ofYandd𝜈^±are associated measures. In our case, we can splitD⁺_T ≈𝛾_T⁺∪(

{T} × Ω ×R^3),D−_T ^≈𝛾_T⁻∪(

{0} × Ω ×R³⁾ up to negligible sets (cf. Beals and Protopopescu¹³). Then,d𝜈^±=d𝛾_𝛼 on𝛾±

T andd𝜈^±=dvdxon{t=0}and{t=T}, and we decompose𝑓⁺= (𝑓+, 𝑓(T)),𝑓⁻= (𝑓−, 𝑓(0))accordingly.

Proposition 1. Consider a fixed 𝛼 ∈ {1,…,N} and  = aK, where 0 ≤ a ∈ L^∞ (𝛾_T⁻•

)

such that a0 ∶=

‖a‖_L∞( 𝛾⁻T•

) < 1. Let F be Lipschitz continuous, bounded, and divergence free with respect to v, and let ̊𝑓 ∈ (L¹∩L^∞) (

Ω ×R^{3), g} ∈ ( L¹

lt∩L^∞

lt

) (𝛾_T⁻•,d𝛾𝛼

)

both be nonnegative. Then, there is a unique, nonnegative strong solu- tion𝑓 ∈L^∞_lt (

IT•;(

L¹∩L^∞) (

Ω ×R³⁾⁾of (12) on IT• with nonnegative trace𝑓± ∈(

L¹_lt∩L^∞_lt) ( 𝛾±

T•,d𝛾_𝛼)

. In particular, Definition 1(ii) holds for(𝑓, 𝑓+), where the Lorentz force is replaced by F. Moreover, we have

(1−a₀)

1

p‖𝑓+‖L^p(𝛾T⁺,d𝛾𝛼),‖𝑓(T)‖L^p(^Ω×R³)≤‖‖‖ ̊𝑓‖‖‖L^p(^Ω×R³)+ (1−a₀)

1 p−1

‖g‖L^p(𝛾T⁻,d𝛾𝛼) (13)

for any0<T∈IT• and 1≤p≤∞. If additionally ̊𝑓∈L¹_𝛼kin(

Ω ×R^{3)and g∈L1}_𝛼kin,lt⁽𝛾_T⁻•,d𝛾𝛼

) , then

(1−a0) ∫

𝛾_T⁺∩{|v|<R}

v⁰_𝛼𝑓+d𝛾_𝛼+∫

Ω

∫

B_R

v⁰_𝛼𝑓(T)dvdx≤ ∫

Ω

∫R³

v⁰_𝛼 ̊𝑓dvdx+∫

𝛾T⁻

v⁰_𝛼g d𝛾_𝛼+

T

∫

0

∫

Ω

∫

B_R

F·̂v_𝛼𝑓dvdxdt (14)

and

‖‖‖‖

‖‖‖∫

B_R

𝑓(T,·,v)dv‖‖

‖‖‖‖

‖L⁴³(Ω)

≤ (4𝜋

3 ‖‖‖ ̊𝑓‖‖‖^L∞(^Ω×R³)+4𝜋

3 (1−a₀)⁻¹‖g‖_L^∞(𝛾_T⁻) +1 ) ⎛⎜

⎜⎝∫

Ω

∫

B_R

v⁰_𝛼𝑓(T)dvdx

⎞⎟

⎟⎠

3 4

(15)

for any0<T∈IT• and 0<R<∞.

Proof. By Beals and Protopopescu,¹³theorem 1 there is a unique, strong solution of (12) for each 0<T ∈I_T• . Since Tis arbitrary, we get𝑓 ∈L^p_lt(

I_T•× Ω ×R^3)and𝑓± ∈L^p_lt (𝛾±

T•,d𝛾𝛼

)

for all 1≤p<∞. By Beals and Protopopescu,¹³ proposition 1 we have the followingp-norm estimate forT∈IT• :

∫

𝛾T⁺

𝑓₊^pd𝛾𝛼+∫

Ω

∫R³

𝑓(T)^pdvdx≤ ∫

Ω

∫

R³

̊𝑓^pdvdx+∫

𝛾⁻_T

(aK𝑓++g)^pd𝛾𝛼

≤ ∫

Ω

∫

R³

̊𝑓^pdvdx+a₀

∫

𝛾⁺T

𝑓₊^pd𝛾𝛼+ (1−a₀)^1−p∫

𝛾_T⁻

g^pd𝛾𝛼

using the convexity of thepth power. This yields (1−a₀)∫

𝛾T⁺

𝑓₊^pd𝛾𝛼+∫

Ω

∫R³

𝑓(T)^pdvdx≤ ∫

Ω

∫R³

̊𝑓^pdvdx+ (1−a₀)^1−p∫

𝛾_T⁻

g^pd𝛾𝛼,