Weak Stability and Large Time Behavior for the Cauchy Problem of the Vlasov-Maxwell-Boltzmann Equations
aa r X i v : . [ m a t h . A P ] J a n WEAK STABILITY AND LARGE TIME BEHAVIOR FOR THECAUCHY PROBLEM OF THE VLASOV-MAXWELL-BOLTZMANNEQUATIONS
XIANPENG HU AND DEHUA WANG
Abstract.
The Cauchy problem for the Vlasov-Maxwell-Boltzmann equations (VMB)is considered. First the renormalized solution to the Vlasov equation with the Lorentzforce is discussed and the difficulty on the partial differentiability of the coefficients isovercome. Then the weak stability of the renormalized solutions to the Cauchy problemof VMB is established using the compactness of velocity averages and a renormalizedformulation. Furthermore, the large time behavior of the renormalized solutions to VMBis studied and it is proved that the density of particles tends to a local Maxwellian asthe time goes to infinity. Introduction
Since the work of DiPerna and Lions [10] on the Cauchy problem for the Boltzmannequation twenty years ago, it has been a well-known open problem to extend their the-ory to the Vlasov-Maxwell-Boltzmann equations. Among the difficulties, how to definethe characteristics of the Vlasov-Maxwell-Boltzmann equations is a major obstacle. Inthis paper, we will give the following partial results: the weak stability and large timebehavior of the renormalized solutions to the Vlasov-Maxwell-Boltzmann equations, andexistence of the renormalized solutions to the Vlasov equation with the Lorentz force. Thefundamental model for dynamics of dilute charged particles is described by the Vlasov-Maxwell-Boltzmann equations (VMB) of the following form [5, 7, 16, 18, 23, 27]: ∂f∂t + ξ · ∇ x f + ( E + ξ × B ) · ∇ ξ f = Q ( f, f ) , x ∈ R , ξ ∈ R , t ≥ , (1.1a) ∂E∂t − ∇ × B = − j, div B = 0 , on R x × (0 , ∞ ) , (1.1b) ∂B∂t + ∇ × E = 0 , div E = ρ, on R x × (0 , ∞ ) , (1.1c) ρ = Z R f dξ, j = Z R f ξdξ, on R x × (0 , ∞ ) , (1.1d)where f = f ( t, x, ξ ) is a nonnegative function for the density of particles which at time t and position x move with velocity ξ under the Lorentz force E + ξ × B, Date : September 23, 2017.1991
Mathematics Subject Classification.
Key words and phrases.
The Vlasov-Maxwell-Boltzmann equations, renormalized solutions, weak sta-bility, large time behavior. E is the electric field, B is the magnetic field, the function j is called the current density,and the function ρ is the charge density. The collison operator Q ( f, f ), which acts onlyon the velocity dependence of f (this reflects the physical assumption that collisions arelocalized in space and time), is defined as Q ( f, f ) = Z R dξ ∗ Z S dω b ( ξ − ξ ∗ , ω )( f ′ f ′∗ − f f ∗ ) , with ω ∈ S , the unit sphere in R , where b = b ( z, ω ) denotes the collision kernel which isa given nonnegative function defined on R × S , and f ∗ = f ( t, x, ξ ∗ ) , f ′ = f ( t, x, ξ ′ ) , f ′∗ = f ( t, x, ξ ′∗ ) , with ξ ′ = ξ − ( ξ − ξ ∗ , ω ) ω,ξ ′∗ = ξ ∗ + ( ξ − ξ ∗ , ω ) ω, which yield one convenient parametrization of the set of solutions to the law of elasticcollisions ξ ′ + ξ ′∗ = ξ + ξ ∗ , | ξ ′ | + | ξ ′∗ | = | ξ | + | ξ ∗ | . The interpretation of ξ , ξ ∗ , ξ ′ , ξ ′∗ is the following: ξ, ξ ∗ are the velocities of two collidingmolecules immediately before collision, while ξ ′ , ξ ′∗ are the velocities immediately after thecollision. Those unknown functions f , E , and B are strongly coupled, and the constrainton the divergence of E will be ensured provided that the conservation of charge holds;that is, ∂ρ∂t + div x j = 0 , since 0 = ∂∂t (div x E − ρ ) = div x E t − ρ t = div x ( ∇ x × B − j ) − ρ t = − ρ t − div x j, due to the fact div( ∇ × v ) = 0 for any vector-valued function v . Similarly, the magneticfield B remains divergence free if it is so initially.The VMB equations are integro-differential equations which provide a mathematicalmodel for the statistical evolution of dilute charged particles. The construction of globalsolutions to VMB has been open for a long time until only a few years ago. In Guo [18],a unique global in time classical solution near a global Maxwellian (independent of spaceand time) was constructed. See also Strain [27] for the extension to the Cauchy problem.Notice that, Lions constructed in [23] a very weak solution to VMB, which is usually calleda measure-valued solution, using Young’s measure to deal with the nonlinearity.For the particles without collision (cf. [4, 9, 15, 16, 24, 26]), or when the molecules are sorare that they do not interact with each other, VMB becomes the so-called Vlasov-Maxwell EAK STABILITY FOR THE VLASOV-MAXWELL-BOLTZMANN EQUATIONS 3 system (VM), ∂f∂t + ξ · ∇ x f + ( E + ξ × B ) · ∇ ξ f = 0 , x ∈ R , ξ ∈ R , t ≥ , (1.2a) ∂E∂t − ∇ × B = − j, div B = 0 , on R x × (0 , ∞ ) , (1.2b) ∂B∂t + ∇ × E = 0 , div E = ρ, on R x × (0 , ∞ ) , (1.2c) ρ = Z R f dξ, j = Z R f ξdξ, on R x × (0 , ∞ ) . (1.2d)Note that (1.2a) is a transport equation with a divergence free coefficient, that isdiv x,ξ ( ξ, E + ξ × B ) = 0 . This property ensures that the solution will remain the same integrability as the initialdata. With the help of this observation and velocity averaging lemma, DiPerna and Lionsproved in [9] the global existence in time of weak solutions to VM with large initial data.For the smooth solutions to VM, we refer the readers to Glassey [16] and Schaeffer [26].The main goal of this paper is to show the weak stability and the large time behaviorof the renormalized solutions to VMB. To this end, we will need an existence result of therenormalized solution to the Vlasov equation (1.2a). Notice that the Vlasov equation isa transport equation with only partially W , loc regularity, since usually we can not expectany differentiability on the magnetic field B and the electric field E from the conservationof energy. Inspirited by the result in Bouchut [3] and Le Bris-Lions [20], we will first showthe existence of renormalized solutions to the Vlasov equation. The presence of a non-trivial magnetic field B ( x, t ), a natural consequence of the celebrated Maxwell theory forelectromagnetism, creates severe mathematical difficulty in studying the weak stability ofweak solutions and the construction of global in time solutions for VMB. Our first resulton weak stability is built on our above mentioned new result about renormalized solutionsto the Vlasov equation with the aid of the velocity average lemma (DiPerna-Lions [9] andDiPerna-Lions-Meyer [13]) and some techniques from Lions [22, 23]. Our second result onrenormalized solutions to VMB is their large time behavior, since from the physical pointof view, the density of particles is assumed to converge to an equilibrium represented bya Maxwellian function of the velocity as the time t becomes large. Our results heavilydepend on, apart from the weak compactness property, • the existence of renormalized solutions to the Vlasov equation; • a renormalized formulation, which is crucial to make sure that the quadratic term Q ( f, f ) is meaningful in D ′ (sense of distributions); and • the velocity averaging lemma [9, 13], which is crucial for the convergence of nonnlin-ear term ( E + ξ × B ) · ∇ ξ f .The stability of renormalized solutions under weak convergence yields a consequenceon the propagation of smoothness for those solutions. Indeed, a sequence of renormalizedsolutions { f n } ∞ n =1 to VMB is relatively strongly compact in L ([0 , T ] × R ) if and only ifthe sequence of the corresponding initial data { f n } ∞ n =1 is relatively strongly compact in L ( R ). In other words, under our assumption on the collision kernel and the integrabilityof the electric field and the magnetic field, no oscillations develop unless they are presentfrom the beginning . XIANPENG HU AND DEHUA WANG
In order to prove our results, the standard a priori estimates derived from the conser-vation laws and H theorem are very useful, and in addition we need some assumptions onthe integrability of the electric field E ( t, x ) and the magnetic field B ( t, x ). More precisely,besides the standard estimate of E and B in L ∞ (0 , T ; L ( R )), we need to assume that E is uniformly bounded in L ∞ (0 , T ; L ( R )) and B is uniformly bounded in L ∞ (0 , T ; L s ( R ))for some s >
5. The reasons for these requirements on E and B are twofold: (I)when we define the characteristics for the Vlasov equation, we need a bound on E in L ∞ (0 , T ; L ( R )); (II) the averaging lemma (cf. [23]), combining with the uniform boundof R R f dξ in L ∞ (0 , T ; L ( R )) and the uniform bound of R R ξf dξ in L ∞ (0 , T ; L ( R )), im-plies the compactness of the first two moments of f on L p (0 , T ; L ploc ( R )) for any 1 ≤ p < ,which is enough to ensure the convergence of the nonlinear Lorentz force term in the senseof the distributions provided that E and B are uniformly bounded in L ∞ (0 , T ; L ( R ))and L ∞ (0 , T ; L s ( R )) for some s > E is uniformly bounded in L ∞ (0 , T ; L ( R )). We notice that the hyperbolic propertyof the Maxwell equations also demonstrates some difficulties if we want to improve theintegrability of the electric field and the magnetic field. How to fulfill this strategy is stillan open question and will be the topic of our future research.When the Lorentz force disappears, that is E + ξ × B = 0, VMB becomes the classicalBoltzmann equation. For the Cauchy problem of the classical Boltzmann equation, in[10] DiPerna and Lions proved the global existence of renormalized solutions with augularcut-off collision kernel and arbitrary initial data, see also [1, 5, 8, 9, 11, 22, 23] and thereferences cited therein. Later, Hamdache extended this existence result to a boundeddomain in [19]. The method explored for the existence result was the analysis of theweak stability of solutions. The argument strongly relied on some compactness properties(see [22]) which hold for sequences of renormalized solutions. In [23], Lions extendedthe similar weak stability and global existence result to the Vlasov-Poission-Boltzmannequations. For the extension to the Landau equation, see Villani [28]. For the long timebehavior of the Boltzmann equations, see [7, 11, 12, 28].This paper will proceed as follows. We will discuss the renormalized solution to theVlasov equation in Section 2. Section 3 is devoted to stating a priori estimates for VMB,main assumptions and main results on the weak stability of renormalized solutions toVMB. Then, Theorem 3.1 on weak stability and Theorem 3.2 on the propagation ofsmoothness will be proved in Section 4 and Section 5, respectively. In Section 6, westudy the large time behavior and establish mathematically the convergence of f to a lo-cal Maxwellian satisfying the Vlasov-Maxwell equations. Finally, in Section 7, we explainan extension of our results to the relativistic Vlasov-Maxwell-Boltzmann equations. EAK STABILITY FOR THE VLASOV-MAXWELL-BOLTZMANN EQUATIONS 5 Renormalized Solutions to the Vlasov Equation
In this section, we consider the Vlasov equation of the form: ∂ t f + ξ · ∇ x f + ( E + ξ × B ) · ∇ ξ f = 0 , (2.1)with B ( x, t ) ∈ L ∞ (0 , T ; L ( R x )) and E ( x, t ) ∈ L ∞ (0 , T ; L ∩ L ( R x )).If we set y = ( x, ξ ) ∈ R , B = ( ξ, E + ξ × B ) ∈ R , then (2.1) becomes a standard transport equation ∂ t f + B · ∇ f = 0 . (2.2)The question of whether the Vlasov equation has renormalized solutions is not only usefulwhen the normalized solution to VMB system is considered, but also has its own inter-est due to the lower regularity of the coefficients. The renormalized solutions mean that(2.2) still holds if we replace f by β ( f ) with a suitable β . Over past twenty years, thereare many important progress about the renormalized solutions to (2.2). More precisely,DiPerna and Lions showed in [8] the existence of renormalized solutions when the co-efficient B ∈ W , ( R ). In 2004, Ambrosio extended the DiPerna-Lions theory to BV(bounded variations) field in [2] (for related work, see [3]). Also, in 2004 Le Bris and Lionsextended in [20] the DiPerna-Lions theory to the case that the coefficient has only partialregularity.For the VMB or the Vlasov equation, the velocity B is no longer in W , x,ξ ) ,loc . In-spired by [3, 20], we claim that we still can prove the existence of a renormalized solutionto (2.1) under the conditions that E ( x, t ) ∈ L ∞ (0 , T ; L ( R ) ∩ L ( R )), and B ( x, t ) ∈ L ∞ (0 , T ; L ( R )). This is a crucial step for establishing renormalized solutions to theVlasov-Maxwell-Boltzmann equations. Theorem 2.1.
Assume that B ( x, t ) ∈ L ∞ (0 , T ; L ( R x )) and E ( x, t ) ∈ L ∞ (0 , T ; L ∩ L ( R x )) . Let f ∈ L ∩ L ∞ ( R ) and | ξ | f ∈ L ( R ) . Then there exists a solution to (2.1) (and hence to (2.2) ) such that f ( t, x, ξ ) ∈ L ∞ ([0 , T ] , L x,ξ ∩ L ∞ x,ξ ( R )) , and | ξ | f ∈ L ∞ (0 , T ; L ( R )) , satisfying the initial condition f | t =0 = f ( x, ξ ) . Further-more, if f ∈ L ∞ x ( L ξ ( R )) , then f ∈ L ∞ (0 , T ; L ∞ x ( L ξ ( R ))) , and hence the solution isunique. To begin with the proof, notice that B = ( B , B ) satisfiesdiv x B = div ξ B = 0 , with B ( x, ξ ) = ξ ∈ W , ξ,loc ( R ) (it does not depend on x ) , B ( x, ξ ) = E + ξ × B ∈ L x,loc ( R , W , ξ,loc ( R )) . The proof of this theorem is divided into three steps. The uniqueness is a crucial issuewhich is the consequence of the following two lemmas, the first one dealing with regular-ization, and the second one stating the uniqueness. Finally, we will show the existencepart.
XIANPENG HU AND DEHUA WANG
Now we denote the mollifier κ ε as κ ε = 1 ε n κ (cid:16) · ε (cid:17) , κ ∈ D ( R ) , Z R κ = 1 , κ ≥ , where D ( R ) = C ∞ ( R ). Then, we have the following two lemmas. Lemma 2.1.
Let f = f ( t, x, ξ ) ∈ L ∞ ([0 , T ] , L x,ξ ) ∩ L ∞ ( x,ξ ) ( R )) be a solution of (2.2) , and κ ε and κ µ be two regularizations with two different scalings, respectively, in the variable x and ξ . Then, for any ε > , there exists a number µ ( ε ) with < µ ( ε ) ≤ ε such that f ε,µ ( ε ) = ( f ∗ κ ε ) ∗ κ µ ( ε ) is a smooth (in ( x, ξ ) ) solution of ∂f ε,µ ( ε ) ∂t + B · ∇ f ε,µ ( ε ) = A ε , with lim ε → A ε = 0 , in L ∞ (cid:0) [0 , T ] , L x,ξ ) ,loc ∩ L ∞ ( x,ξ ) ,loc ( R ) (cid:1) . Lemma 2.2.
Let f = f ( t, x, ξ ) ∈ L ∞ (cid:0) [0 , T ] , L x,ξ ) ∩ L ∞ ( x,ξ ) ( R ) (cid:1) be a nonnegative solutionof (2.2) with zero initial data f = 0 . If, in addition, | ξ | f ∈ L ∞ ([0 , T ] , L ( R )) and f ∈ L x ( L ξ ) , then f = 0 for all time t > . We now prove these two lemmas, and then finally complete the proof of Theorem 2.1.2.1.
Proof of Lemma 2.1.
We will use the mollifier to regularize the function f in ξ and x , while we assume that f is differentiable with respect to t (the results below holdalso for the general case from a standard mollification in t with the help of Lebesgue’sdominated theorem.) All the functional spaces used here are local , which is clearly enoughfor such a regularization result.We first regularize in the ξ variable by convoluting (2.2) with κ µ to get ∂f ∗ κ µ ∂t + ( ξ · ∇ x f ) ∗ κ µ + (( E + ξ × B ) · ∇ ξ f ) ∗ κ µ = 0 . (2.3)Denoting by[( E + ξ × B ) · ∇ ξ , κ µ ]( f ) = ( E + ξ × B ) · ∇ ξ ( f ∗ κ µ ) − κ µ ∗ (( E + ξ × B ) · ∇ ξ f ) . Then, (2.3) can be rewritten as ∂f ∗ κ µ ∂t + ( ξ · ∇ x f ) ∗ κ µ + ( E + ξ × B ) · ∇ ξ ( f ∗ κ µ ) = [( E + ξ × B ) · ∇ ξ , κ µ ]( f ) . (2.4)It is a standard fact (see [8]) that I µ := [( E + ξ × B ) · ∇ ξ , κ µ ]( f ) → L x,ξ ) ,loc (2.5)as µ →
0. Indeed, this is clear for smooth coefficients and f , while the general case followsas in [8] by dense property through the estimate k [( E + ξ × B ) · ∇ ξ , κ µ ]( f ) k L ξ,loc ≤ C k E + ξ × B k W , ξ,loc k f k L ∞ ξ , which then implies the following standard estimate by integrating in x , k [( E + ξ × B ) · ∇ ξ , κ µ ]( f ) k L x,ξ ) ,loc ≤ C k E + ξ × B k L x,loc ( W , ξ,loc ) k f k L ∞ x,ξ . EAK STABILITY FOR THE VLASOV-MAXWELL-BOLTZMANN EQUATIONS 7
Next, we regularize in the x variable by convoluting (2.4) with κ ε for f µ = f ∗ κ µ toobtain, ∂f µ ∗ κ ε ∂t + ( ξ · ∇ x f ) ∗ κ µ ∗ κ ε + ( E + ξ × B ) · ∇ ξ ( f µ ∗ κ ε )= [( E + ξ × B ) · ∇ ξ , κ ε ]( f µ ) + I µ ∗ κ ε . (2.6)We now successively deal with each terms on the right-hand side of (2.6). First, it iseasy to observe that for fixed µ , we have I µ ∗ κ ε → I µ , as ε → , in L x,ξ ) ,loc , which together with (2.5) implies thatlim µ → lim ε → I µ ∗ κ ε = 0 , in L x,ξ ) ,loc . (2.7)Second, for the first term on the right-hand side of (2.6), we have[( E + ξ × B ) · ∇ ξ , κ ε ]( f µ ) = ( E + ξ × B ) · ∇ ξ ( κ ε ∗ f µ ) − κ ε ∗ (cid:0) ( E + ξ × B ) · ∇ ξ f µ (cid:1) = ( E + ξ × B ) · (cid:0) ( ∇ ξ f µ ) ∗ κ ε (cid:1) − κ ε ∗ (cid:0) ( E + ξ × B ) · ∇ ξ f µ (cid:1) = (cid:2) ( E + ξ × B ) , κ ε (cid:3) ( ∇ ξ f µ ) . The latter bracket can be controlled as follows: (cid:13)(cid:13)(cid:13)(cid:2) ( E + ξ × B ) , κ ε (cid:3) ( ∇ ξ f µ ) (cid:13)(cid:13)(cid:13) L x,ξ ) ,loc ≤ C k E + ξ × B k L x,ξ ) ,loc k∇ ξ f µ k L ∞ ( x,ξ ) . Hence, for fixed µ , we have lim ε → [( E + ξ × B ) · ∇ ξ , κ µ ]( f µ ) = 0 , in L x,ξ ) ,loc . This implies, lim µ → lim ε → [( E + ξ × B ) · ∇ ξ , κ µ ]( f µ ) = 0 , (2.8)in L x,ξ ) ,loc . By a standard diagonization procedure, for any ε >
0, we can find µ ( ε ) with0 < µ ( ε ) ≤ ε → ε → I µ ( ε )1 ∗ κ ε = 0 , in L x,ξ ) ,loc . and lim ε → [( E + ξ × B ) · ∇ ξ , κ µ ( ε ) ]( f µ ( ε ) ) = 0 , in L x,ξ ) ,loc . To complete the proof of this lemma, it remains to show the following convergence forthe above chosen µ ( ε ): I µ ( ε ) ,ε = ( ξ · ∇ x f ) ∗ κ µ ( ε ) ∗ κ ε − ξ · ∇ x ( f µ ( ε ) ∗ κ ε ) XIANPENG HU AND DEHUA WANG in L x,ξ ) ,loc . Indeed, we can control I µ ( ε ) ,ε as (cid:12)(cid:12)(cid:12) I µ ( ε ) ,ε (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)Z R (cid:2) ( ξ − η ) · ∇ x f ( x − ζ, ξ − η ) − ξ · ∇ x f ( x − ζ, ξ − η ) (cid:3) κ µ ( ε ) κ ε dζdη (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)Z R (cid:2) η · ∇ x f ( x − ζ, ξ − η ) (cid:3) κ µ ( ε ) κ ε dζdη (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)Z R η · ∇ ζ κ ε ( ζ ) κ µ ( ε ) ( η ) f ( x − ζ, ξ − η ) dζdη (cid:12)(cid:12)(cid:12)(cid:12) ≤ µ ( ε ) ε Z R | ε ∇ ζ κ ε | κ µ ( ε ) | f ( x − ζ, ξ − η ) | dζdη. Thus, we deduce that, for any compact subset K ⊂ R , by Fubini’s theorem, k I µ ( ε ) ,ε k L ( K ) = Z K (cid:12)(cid:12)(cid:12)(cid:12)Z R η · ∇ ζ κ ε ( ζ ) κ µ ( ε ) ( η ) f ( x − ζ, ξ − η ) dζdη (cid:12)(cid:12)(cid:12)(cid:12) dxdξ ≤ C µ ( ε ) ε (cid:16) Z R | ε ∇ ζ κ ε | dζ (cid:17) sup | ζ |≤ ε, | η |≤ µ ( ε ) k f ( x − ζ, ξ − η ) k L ( K ) ≤ C µ ( ε ) ε sup | ζ |≤ ε, | η |≤ µ ( ε ) k f ( x − ζ, ξ − η ) k L ( K ) . (2.9)Since f ∈ L x,ξ ) , one has, according to the continuity of translation in L ( K ),sup | ζ |≤ ε, | η |≤ µ ( ε ) k f ( x − ζ, ξ − η ) − f ( x, ξ ) k L ( K ) → , as ε → , and hence, sup | ζ |≤ ε, | η |≤ µ ( ε ) k f ( x − ζ, ξ − η ) k L ( K ) is uniforly bounded for all ε ≤ . Thus, if we let ε → ≤ µ ( ε ) ≤ ε , we deduce from (2.9) that I µ ( ε ) ,ε → , in L x,ξ ) ,loc as ε → . (2.10)Therefore, the lemma follows from (2.7), (2.8), and (2.10), and we complete the proofof this lemma. (cid:3) Next, we turn to the proof of Lemma 2.2.2.2.
Proof of Lemma 2.2.
Let f be a nonnegative solution as claimed in Theorem 2.1.We introduce two cut-off functions, respectively, with respect to each variable x and ξ .For m, n ∈ N , we denote them by ψ m ( x ) = ψ (cid:16) xm (cid:17) , and φ n ( ξ ) = φ (cid:18) ξn (cid:19) , where ψ ∈ D ( R ), ψ ≥ ψ = 1 for | x | ≤ ψ = 0 for | x | ≥
2; and the analogousproperties are required on φ with respect to the variable ξ . We first multiply (2.1) by φ n and integrate over ξ space to obtain ∂∂t Z R f φ n dξ + div x (cid:18)Z R ξf φ n dξ (cid:19) + Z R ( E + ξ × B ) · ∇ ξ f φ n dξ = 0 . (2.11) EAK STABILITY FOR THE VLASOV-MAXWELL-BOLTZMANN EQUATIONS 9
For the last term in (2.11), we deduce, due to div ξ ( E + ξ × B ) = 0, Z R ( E + ξ × B ) · ∇ ξ f φ n dξ = − Z R f ( E + ξ × B ) · ∇ ξ φ n dξ = − Z R f | ξ | n E + ξ × B | ξ | · ∇ ξ φ (cid:18) ξn (cid:19) dξ. Now we multiply (2.11) by ψ m and integrate over x space to deduce ddt Z R f ψ m φ n dxdξ + Z R ψ m div x (cid:18)Z R ξf φ n dξ (cid:19) dx = Z R ψ m Z R f | ξ | n E + ξ × B | ξ | · ∇ ξ φ (cid:18) ξn (cid:19) dξdx. (2.12)Hence, using the integration by parts for the second term in (2.12), we have ddt Z R n f ψ m φ n dxdξ − Z R ∇ x ψ m · (cid:18)Z R ξf φ n dξ (cid:19) dx = Z R ψ m Z R f | ξ | n E + ξ × B | ξ | · ∇ ξ φ (cid:18) ξn (cid:19) dξdx. (2.13)Next, we proceed to control the two integral terms in (2.13). Indeed, for the secondterm in (2.13), we have (cid:12)(cid:12)(cid:12)(cid:12)Z R ∇ x ψ m · (cid:18)Z R ξf φ n dξ (cid:19) dx (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)Z R m ∇ x ψ (cid:16) xm (cid:17) · (cid:18)Z R ξf φ n dξ (cid:19) dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ C m k ξf χ { m ≤| x |≤ m, | ξ |≤ n } k L x,ξ ) → , (2.14)as m → ∞ and n → ∞ . Here we used ξf ∈ L ∞ ([0 , T ] , L ( R )), because Z R Z R | ξ | f dξdx ≤ R Z R Z R ∩{| ξ |≤ R } f dξdx + Z R Z R ∩{| ξ | >R } | ξ | f dξdx ≤ R Z R Z R ∩{| ξ |≤ R } f dξdx + 1 R Z R Z R ∩{| ξ | >R } | ξ | f dξdx ≤ R k f k L ( R ) + 1 R k| ξ | f k L ( R ) ≤ k f k L ( R ) k| ξ | f k L ( R ) , by optimizing the value of R .On the other hand, for m fixed, we claim that the term on the right-hand side of (2.13)goes to zero as n goes to infinity by Lebesgue’s dominated convergence theorem. Indeed, as ∇ φ is L ∞ and supported in the annular { ≤ | ξ | ≤ } , we have for almost all x ∈ R , (cid:12)(cid:12)(cid:12)(cid:12) ψ m Z R f | ξ | n E + ξ × B | ξ | · ∇ ξ φ (cid:18) ξn (cid:19) dξ (cid:12)(cid:12)(cid:12)(cid:12) ≤ ψ m Z R f | ξ | n | E + ξ × B | | ξ | (cid:12)(cid:12)(cid:12)(cid:12) ∇ ξ φ (cid:18) ξn (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) dξ ≤ k∇ φ k L ∞ ψ m k f χ { n ≤| ξ |≤ n } k L ξ ( | E | + | B | ) → , as n → ∞ , since for almost all x ∈ R , f ( x, · ) ∈ L ( R ξ ). In addition, by the Cauchy-Schwarz inequality, we have, (cid:12)(cid:12)(cid:12)(cid:12) ψ m Z R f | ξ | n E + ξ × B | ξ | · ∇ ξ φ (cid:18) ξn (cid:19) dξ (cid:12)(cid:12)(cid:12)(cid:12) ≤ ψ m Z R f | ξ | n | E + ξ × B | | ξ | (cid:12)(cid:12)(cid:12)(cid:12) ∇ ξ φ (cid:18) ξn (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) dξ ≤ k∇ φ k L ∞ k f k L ξ ( | E | + | B | ) ≤ k∇ φ k L ∞ ( k f k L ξ + | E | + | B | ) . and the right-hand side is in L x , since f ∈ L x ( L ξ ) and E, B ∈ L x . Thus, Lebesgue’stheorem applies and we get the convergence of the term on the right-hand side of (2.13)to zero as n goes to infinity, and m being kept fixed.Collecting the behaviors of those two terms, we obtain with (2.13), as n , and next m ,go to infinity, ddt Z R f dxdξ = 0 . As f = 0, this yields f = 0 for all t since f ≥ (cid:3) Having proved Lemma 2.1 and Lemma 2.2, we are now ready to complete the proof ofTheorem 2.1 as follows.2.3.
Proof of Theorem 2.1.
Assume for the time being that we have at hand twosolutions f and f to (2.1) satisfying the regularity stated in Theorem 2.1, and sharingthe same initial value. In view of the interpolation between L and L ∞ , and the fact f i ∈ L ∞ ([0 , T ] , L x,ξ ∩ L ∞ x,ξ ( R )) ∩ L ∞ (cid:16) [0 , T ] , L ∞ x (cid:0) R , L ξ ( R ) (cid:1)(cid:17) for i = 1 ,
2, we deduce that f i ∈ L ∞ (cid:16) [0 , T ] , L x (cid:0) R , L ξ ( R ) (cid:1)(cid:17) . By virtue of Lemma 2.1, their difference f = f − f satisfies ∂ f µ ( ε ) ,ε ∂t + B · ∇ ( x,ξ ) f µ ( ε ) ,ε = A ε , (2.15) EAK STABILITY FOR THE VLASOV-MAXWELL-BOLTZMANN EQUATIONS 11 with the same notation as in Lemma 2.1. Since f µ ( ε ) ,ε ∈ C ∞ ( R ), we multiply (2.15) by β ′ ( f µ ( ε ) ,ε ) for some function β ∈ C ( R ) with β ′ bounded, and obtain ∂β ( f µ ( ε ) ,ε ) ∂t + B · ∇ ( x,ξ ) β ( f µ ( ε ) ,ε ) = A ε β ′ ( f µ ( ε ) ,ε ) . By letting ε go to zero, we obtain the equation ∂β ( f ) ∂t + B · ∇ ( x,ξ ) β ( f ) = 0 , in L ∞ ([0 , T ]; L ∩ L ∞ ( x,ξ ) ,loc ) for such functions β . Now, letting β approximate the absolutevalue function, we end up with ∂ | f | ∂t + B · ∇ ( x,ξ ) | f | = 0 . This implies that we have a nonnegative solution | f | to (2.1), which vanishes at initialtime and belongs to the functional space stated in Lemma 2.2. Applying Lemma 2.2, weget | f | = 0, that is, f = f . There remains now to prove the existence part.Existence in the functional space L ∞ (cid:0) [0 , T ]; L x,ξ ) ∩ L ∞ ( x,ξ ) ( R ) (cid:1) is given in a straightforward way by an application of Proposition 2.1 of [8]. For the sake of consistency, letus only mention here that it is a simple matter of regularization of the vector field B appearing in (2.1). That is, one introduces the solution f α to ∂f α ∂t + B α · ∇ f α = 0 , in (0 , ∞ ) × R , where B α = κ α ∗ B ∈ L ([0 , T ]; C ∞ ( R )) converges to B , then shows the desired estimateson f α , and finally passes to the limit.Next, the non-standard part we have to prove here is the fact that such a solutionnecessarily satisfies | ξ | f ∈ L ∞ ([0 , T ] , L ( R ). This is actually a consequence of the specificform of the transport equation and of the regularization process we have already done.Indeed, first, by the method of characteristics, we know if f ≥ a.e in R , then f ( t ) ≥ a.e in R for all t ≥
0. Then, formally we multiply (2.1) by | ξ | to obtain ∂ ( | ξ | f ) ∂t + | ξ | ξ · ∇ x f + | ξ | ( E + ξ × B ) · ∇ ξ f = 0 . Then we integrate the above identity over ξ on R to deduce ddt Z R | ξ | f dxdξ = Z R div ξ ( | ξ | ( E + ξ × B )) f dxdξ, (2.16)since Z R | ξ | ξ · ∇ x f dxdξ = − Z R div x ( | ξ | ξ ) f dxdξ = 0 . For the term on the right-hand side of (2.16), we have Z R div ξ ( | ξ | ( E + ξ × B )) f dxdξ = 2 Z R ( ξ · Ef ) dxdξ, since div ξ ( | ξ | ξ × B ) = 2 ξ · ( ξ × B ) + | ξ | div ξ ( ξ × B ) = 0 . Also, notice that, for a.e x ∈ R , Z R | ξ | f dξ ≤ Z {| ξ |≤ R } Rf dξ + Z {| ξ | >R } | ξ | f dξ ≤ ω R k f k L ∞ ( R ) + R − Z {| ξ | >R } | ξ | f dξ ≤ C (cid:18)Z R | ξ | f dξ (cid:19) , (2.17)where ω is the volume of the unit ball in R , and in the last inequality R is taken to be R = (cid:18)Z R | ξ | f dξ (cid:19) . Hence, we have the following estimate, by the H¨older inequality, (cid:12)(cid:12)(cid:12)(cid:12)Z R div ξ ( | ξ | ( E + ξ × B )) f dxdξ (cid:12)(cid:12)(cid:12)(cid:12) = 2 (cid:12)(cid:12)(cid:12)(cid:12)Z R ( ξ · Ef ) dxdξ (cid:12)(cid:12)(cid:12)(cid:12) ≤ Z R Z R | ξ | f | E | dxdξ ≤ C Z R (cid:18)Z R | ξ | f dξ (cid:19) | E | dx ≤ C (cid:18)Z R | ξ | f dxdξ (cid:19) k E k L ∞ ([0 ,T ] ,L ( R )) . Substituting this back to (2.16), we obtain ddt Z R | ξ | f dxdξ ≤ C (cid:18)Z R | ξ | f dxdξ (cid:19) k E k L ∞ ([0 ,T ] ,L ( R )) . This implies Z R | ξ | f ( t ) dxdξ ≤ Z R | ξ | f dxdξ + CT , for all t ∈ [0 , T ] . Finally, we show that the solution f necessarily belongs to L ∞ ([0 , T ]; L ∞ x ( R , L ξ ( R )))if f ∈ L ∞ x ( R , L ξ ( R )). This is actually also a consequence of the specific form of thetransport equation and of the regularization process as mentioned earlier. Indeed, wemollify E + ξ · B by κ α to obtain, ∂f α ∂t + ξ · ∇ x f α + ( E + ξ × B ) α · ∇ ξ f α = 0 . (2.18)Integrating (2.18) over ξ in R , one has, thanks to the fact that div ξ ( E + ξ × B ) α = 0, ∂∂t Z R f α dξ + ξ · ∇ x Z R f α dξ = 0 . (2.19) EAK STABILITY FOR THE VLASOV-MAXWELL-BOLTZMANN EQUATIONS 13
That is equivalent to saying that R R f α dξ satisfies a conservation form, which yields theconservation over time of (cid:13)(cid:13)(cid:13)(cid:13)Z R f α dξ (cid:13)(cid:13)(cid:13)(cid:13) L ∞ x . Hence, f α ∈ L ∞ (0 , T ; L ∞ x ( R , L ( R ξ ))). By letting α →
0, one obtain f ∈ L ∞ (0 , T ; L ∞ x ( R , L ( R ξ ))) . The proof of Theorem 2.1 of complete. (cid:3)
Remark . The assumption E ∈ L ∞ (0 , T ; L ( R )) is only needed to show the uniformestimate | ξ | f ∈ L ∞ (0 , T ; L ( R )).We now turn to the extension of the previous result to less regular initial data throughthe notion of renormalized solutions in the spirit of [8]. As in [8], we consider the set L of all measurable functions f on R with value in R such that meas {| f | > λ } < ∞ , for all λ >
0. For any β ∈ C , ( R ), bounded and vanishing near zero, we thus have β ( f ) ∈ L ∩ L ∞ ( R ) for any f ∈ L . As in [8], we shall say that a sequence f n is bounded(respectively, converges) in L whenever β ( f n ) is bounded (respectively, converges) in L for any such β . But now we need some additional assumptions on our initial data, andthat is why we consider the subset L of L consisting of functions f satisfying Z {| f ( x,ξ ) | >δ } | ξ | dxdξ ≤ c δ < ∞ , ∀ δ > . This subset is equipped with the topology induced by that of L . For any f ∈ L , wehave | ξ | β ( f ) ∈ L x,ξ ) ( R ). Indeed, for δ small enough such that β vanishes on [0 , δ ], wehave Z R | ξ | | β ( f ( x, ξ )) | dxdξ = Z {| f ( x,ξ ) | >δ } | ξ | | β ( f ( x, ξ )) | dxdξ + Z {| f ( x,ξ ) |≤ δ } | ξ | | β ( f ( x, ξ )) | dxdξ ≤ k β k L ∞ c δ + 0 < ∞ . It follows that if we choose f in L , then β ( f ) is a convenient initial condition for thetransport equation considered in Theorem 2.1. We therefore say that f is a renormalizedsolution of (2.1) complemented by an initial condition f ∈ L whenever β ( f ) is a solutionof (2.1) in the sense of Theorem 2.1 with the initial condition β ( f ).3. Stability of Vlasov-Maxwell-Boltzmann Equations: Main Results
Let us begin by recalling that the general Vlasov-Maxwell-Boltzmann equations (1.1)has the collison operator Q ( f, f ) which can be written as Q ( f, f ) = Q + ( f, f ) − Q − ( f, f ) , where Q + ( f, f ) = Z R dξ ∗ Z S dωb ( ξ − ξ ∗ , ω ) f ′ f ′∗ , and Q − ( f, f ) = Z R dξ ∗ Z S dωb ( ξ − ξ ∗ , ω ) f f ∗ = f L ( f ) , with L ( f ) = A ∗ ξ f, A ( z ) = Z S b ( z, ω ) dω, z ∈ R . The collision kernel b in the collision operator Q is a given function on R × S . Weshall always assume the so-called angular cut-off kernel throughout the rest of the paper,that is, b satisfies b ∈ L ( B R × S ) for all R ∈ (0 , ∞ ) , b ≥ B R = { z ∈ R : | z | < R } , and ( b ( z, w ) depends only on | z | and | ( z, ω ) | , (1 + | z | ) − (cid:16)R z + B R A ( ξ ) dξ (cid:17) → , as | z | → ∞ , for all R ∈ (0 , ∞ ) . A classical example of such angular cut-off collision kernels is given by the so-called hard-spheres model where we have b ( z, ω ) = | ( z, ω ) | . The VMB system (1.1) is complemented with the initial conditions ( f | t =0 = f , on R , with f ≥ ,E | t =0 = E , B | t =0 = B on R x , (3.1)with the usual compatibility conditiondiv B = 0 , and div E = ρ = Z R f dξ, on R x . We state below our main stability results concerning the Cauchy problem of the Vlasov-Maxwell-Boltzmann system (1.1) and (3.1). We assume that f satisfies Z R f (1 + ν + | ξ | + | log f | ) dxdξ + Z R ( | E | + | B | ) dx < ∞ , (3.2)where ν = ν ( x ) is some function in R satisfying ν ≥ , (1 + ν ) is Lipschitz on R , e − ν ∈ L ( R ) . Using the classical identity (see Lemma 2.1 in [4]), Z R Q ( f, f ) ζ ( ξ ) dξ = 14 Z R dxdξ ∗ Z S dωb ( f ′ f ′∗ − f f ∗ ) (cid:0) ζ + ζ ∗ − ζ ′ − ζ ′∗ (cid:1) , (3.3)we deduce the following local conservation laws of mass, momentum and kinetic energy: ∂ρ∂t + div x j = 0 , (3.4) ∂∂t (cid:18)Z R f ξdξ + E × B (cid:19) + div x (cid:18)Z R ξ ⊗ ξf dξ + (cid:18) | E | + | B | Id − E ⊗ E − B ⊗ B (cid:19)(cid:19) = 0 , (3.5) EAK STABILITY FOR THE VLASOV-MAXWELL-BOLTZMANN EQUATIONS 15 ∂∂t (cid:18)Z R f | ξ | dξ (cid:19) + div x (cid:18)Z R ξ | ξ | f dξ (cid:19) − E · Z R ξf dξ = 0 , (3.6)for ( x, t ) ∈ R × (0 , ∞ ). In fact, while (3.4) and (3.6) are easy to verify, we need to paymore attention to (3.5). To verify (3.5), we first multiply (1.1a) by ξ and integrate withrespect to ξ to obtain ∂∂t Z R f ξdξ + div x Z R ξ ⊗ ξf dξ = − ( ρE + j × B ) . (3.7)Note that E div E + ( ∇ × E ) × E = div( E ⊗ E ) − ∇| E | . Thus it yields the following, combined with (1.1b) and (1.1c), ∂∂t ( E × B ) + div x (cid:18) | E | + | B | Id − E ⊗ E − B ⊗ B (cid:19) = − ( ρE + j × B ) . (3.8)Then, adding (3.7) and (3.8) together gives (3.5). Integrating (3.4)-(3.6) in x over R , wededuce the following global conservation of mass, momentum and total energy ddt Z R f dxdξ = 0 , for t ≥ , (3.9) ddt (cid:18)Z R f ξdxdξ + Z R E × Bdx (cid:19) = 0 , (3.10) ddt Z R f | ξ | dxdξ − Z R E · Z R ξf dξdx = 0 , for t ≥ . (3.11)On the other hand, multiplying (1.1b) by E , multiplying (1.1c) by B , integrating them in x over R , and then summing them together, we obtain ddt Z R ( | E | + | B | ) dx = − Z R E · jdx. Substituting the above identity back to (3.11), one obtains ddt (cid:18)Z R f | ξ | dxdξ + Z R ( | E | + | B | ) dx (cid:19) = 0 , for t ≥ . (3.12)Therefore, if we assume that the initial condition f as (3.2), we deduce from (3.9),(3.10) and (3.12) thatsup t ∈ [0 ,T ] Z R f (1 + ν + | ξ | ) dxdξ + Z R ( | E | + | B | ) dx ≤ C ( T ) (3.13)for some nonnegative constant C ( T ) that depends only on T and on the initial data.Indeed, we observe that we have, multiplying (1.1a) by ν ( x ) and then integrating over ξ , ∂∂t (cid:18)Z R f ν ( x ) dξ (cid:19) + div x (cid:18)Z R f ν ( x ) ξdξ (cid:19) = Z R f ξ · ∇ x ν ( x ) dξ ≤ Z R f | ξ | dξ + 12 ρ ( t, x ) |∇ ν | ≤ Z R f | ξ | dξ + C Z R f dξ + C Z R f νdξ, since (1 + ν ) is Lipschitz. In particular, we deduce ddt Z R f ν ( x ) dxdξ ≤ C + 12 Z R f ( | ξ | + ν ( x )) dxdξ. Then (3.13) follows from the above inequality and Gr¨onwall’s inequality.The final formal bound we wish to obtain is deduced from the entropy identity. Multi-plying (1.1a) by log f , using (3.3), we obtain, at least formally, ddt Z R f log f dxdξ + 14 Z R dx Z R dξdξ ∗ Z S B ( f ′ f ′∗ − f f ∗ ) log f ′ f ′∗ f f ∗ = 0 . (3.14)Since the second term is clearly nonnegative, we deduce in particular thatsup t ≥ Z R f log f dxdξ ≤ Z R f log f dxdξ. (3.15)This inequality together with a lemma in [22] impliessup t ∈ [0 ,T ] Z R f | log f | ≤ C ( T ) . Also, if we go back to (3.14), we deduce that, Z T dt Z R dx Z R dξdξ ∗ Z S b ( f ′ f ′∗ − f f ∗ ) log f ′ f ′∗ f f ∗ ≤ C ( T ) . In conclusion, we obtain the following bounds:sup t ∈ [0 ,T ] (cid:18)Z R f (1 + | ξ | + ν ( x ) + | log f | ) dxdξ + Z R ( | E | + | B | ) dx (cid:19) ≤ C ( T ); Z T Z R dx Z R dξdξ ∗ Z S dωb ( f ′ f ′∗ − f f ∗ ) log f ′ f ′∗ f f ∗ ≤ C ( T ) . (3.16)Now we give the definition of Renormalized Solutions to VMB.
Definition 3.1.
A triple ( f ( t, x, ξ ) , E ( t, x ) , B ( t, x )) with f ≥ T ∈ (0 , ∞ ), we have • f ( t, x, ξ ) ∈ C ([0 , T ]; L ( R )), E, B ∈ C ([0 , T ]; L ( R )) and (3.16) holds; • for any β ∈ C ([0 , ∞ )) satisfying that β (0) = 0 and β ′ ( t )(1 + t ) is bounded in[0 , ∞ ), ∂∂t β ( f ) + ξ · ∇ x β ( f ) + ( E + ξ × B ) · ∇ ξ β ( f ) = β ′ ( f ) Q ( f, f ) (3.17)holds in D ′ (sense of distributions); and • (1.1b) and (1.1c) hold in D ′ .One of the main objectives in the rest of this paper is devoted to the stability ofrenormalized solutions to VMB. More precisely, we consider a sequence of initial data { ( f n , E n , B n ) } ∞ n =1 satisfying (3.2) with f n ≥ a.e. in R and converging to ( f , E , B ).Then, corresponding to those initial conditions, we suppose that there is a sequence of EAK STABILITY FOR THE VLASOV-MAXWELL-BOLTZMANN EQUATIONS 17 renormalized solutions { ( f n , E n , B n ) } ∞ n =1 to VMB satisfying (3.16). Without loss of gen-erality, we may assume that ( f n , E n , B n ) converges weakly to ( f, E, B ). We will provethat ( f, E, B ) is still a renormalized solution to VMB with the initial data ( f , E , B ). Theorem 3.1 (Weak Stability) . Suppose that { ( f n , E n , B n ) } ∞ n =1 is a sequence of renor-malized solutions to VMB (1.1) satisfying (3.16) , with initial data { ( f n , E n , B n ) } ∞ n =1 satis-fying (3.2) , f n ≥ , a.e. in R and converging weakly to ( f , E , B ) in L ( R ) × (cid:0) L ( R ) (cid:1) ;and ( f, E, B ) is a weak- ∗ limit of { ( f n , E n , B n ) } in L ∞ (0 , T ; L ( R )) × (cid:16) L ∞ (0 , T ; (cid:0) L ( R ) (cid:1) (cid:17) .Then the sequence { f n } satisfies: (1) For all ψ ∈ C ( R ) such that | ψ ( ξ ) | | ξ | → as | ξ | → ∞ , R R f n φdξ converges to R R f ψdξ in L p ([0 , T ] , L loc ( R )) for all ≤ p < ∞ . (2) L ( f n ) converges to L ( f ) in L p ([0 , T ]; L ( R x × K )) for all ≤ p < ∞ , T ∈ (0 , ∞ ) , K compact set in R ξ . (3) For all φ ∈ L ∞ ( R ) with compact support, R R Q ± ( f n , f n ) φdξ converges locallyin measure to R R Q ± ( f, f ) φdξ . And Q ± ( f n , f n )(1 + f n ) − are relatively weaklycompact in L ( R x × K × (0 , T )) for all T ∈ (0 , ∞ ) , compact set K in R ξ . (4) Q + ( f n , f n ) converges locally in measure to Q + ( f, f ) .Moreover, if k E n k L ∞ (0 ,T ; L ( R )) is uniformly bounded , and k B n k L ∞ (0 ,T ; L s ( R )) is uniformly bounded for some s > , then the weak limit ( f, E, B ) is a renormalized solution of (1.1) with the initial data ( f , E , B ) .Remark . Due to the convexity of x ln x and the monotonicity of ( x − y ) ln xy for all x, y >
0, we can show, as in [11], Z R f ( t ) ln f ( t ) dxdξ ≤ lim inf n →∞ Z R f n ( t ) ln f n ( t ) dxdξ, and Z t Z R Z R dξdξ ∗ Z S dωb ( f ′ f ′∗ − f f ∗ ) ln f ′ f ′∗ f f ∗ ≤ lim inf n →∞ Z t Z R Z R dξdξ ∗ Z S dωb ( f n ′ f n ′∗ − f n f n ∗ ) ln f n ′ f n ′∗ f n f n ∗ , for all t ≥
0. This entropy estimate is crucial for the long time behavior of renormalizedsolutions.A consequence of the weak stability is the propagation of smoothness of renormalizedsolutions.
Theorem 3.2 (Propagation of Smoothness) . If, in addition to the assumptions in Theo-rem 3.1, f n converges in L ( R ) to f , then f n converges to f in C ([0 , T ]; L ( R )) for all T ∈ [0 , ∞ ) , and ( f, E, B ) is a renormalized solution of (1.1) if ( f n , E n , B n ) is a sequenceof renormalized solutions. Remark . The assumption that E ( x, t ) is uniformly bounded in L ∞ (0 , T ; L ( R )) iscrucial for Theorems 3.1 and 3.2, because of the nonlinear term associated with the Lorentzforce. Notice that usually from Maxwell’s equations, we can only obtain the a priori estimates on E and B in L ∞ (0 , T ; L ( R )).4. Proof of Theorem 3.1: Weak Stability
This section is devoted to the proof of Theorem 3.1. We divide the proof into two steps.In the first step we show why the first four statements of the theorem hold. Then weconcentrate in the second step on the proof of the fact that the weak limit is indeed arenormalized solution of Vlasov-Maxwell-Boltzmann equations. We remark that the firststep is essentially an adaptation of the results and methods of [10, 22, 23], while the secondone requires a new result of renormalized solutions for the Vlasov-Maxwell equations.4.1.
Step One.
In this subsection, we are aiming at proving the first statement of The-orem 3.1 following the spirit of [10]. Then the second and the third statements can beshown exactly as in [10]. Finally, once the first three statements hold, the fourth statementwill immediately follows from the argument in [22]. Therefore, for the sake of conciseness,we only give the detailed proof of the first statement of Theorem 3.1.In order to prove the first statement, we first recall that for all compact sets K ⊂ R ξ and T ∈ (0 , ∞ ), we have Z R × K (1 + f n ) − Q − ( f n , f n ) dxdξ ≤ Z R × K L ( f n ) dxdξ = Z R dx Z R f n ( x, ξ ∗ , t ) Z K A ( ξ − ξ ∗ ) dξdξ ∗ ≤ C Z R dx Z R f n ( x, ξ ∗ , t )(1 + | ξ ∗ | ) dξ ∗ < ∞ , (4.1)due to the assumption on the collision kernel b , hence,(1 + f n ) − Q − ( f n , f n ) is bounded in L ∞ (0 , T ; L ( R × K )) . Also, we observe that we have (see [10]), Q + ( f n , f n ) ≤ Q − ( f n , f n ) + 1ln 2 Z R dξ ∗ Z S bdω ( f n ′ − f n ∗ − f n f n ∗ ) ln f n ′ f n ′ ∗ f n f n ∗ , which, combining with (3.16) and (4.1), implies(1 + f n ) − Q + ( f n , f n ) is bounded in L (0 , T ; L ( R × K )) (4.2)for all compact sets K in R ξ and T ∈ (0 , ∞ ).Next, we observe that since f n is a renormalized solution of VBM (1.1), we have, for β = β δ = t δt , (cid:18) ∂∂t + ξ · ∇ x (cid:19) β δ ( f n ) = β ′ δ ( f n ) Q ( f n , f n ) − div ξ (( E n + ξ × B n ) β δ ( f n )) (4.3)in D ′ . In order to apply the velocity averaging results in [9, 13], we remark that (4.1)and (4.2) imply that β ′ δ ( f n ) Q ( f n , f n ) is bounded in L (0 , T ; L ( R x × K )) for all compact EAK STABILITY FOR THE VLASOV-MAXWELL-BOLTZMANN EQUATIONS 19 subsets K of R ξ . And also we observe that β δ ( f n ) is bounded in L ∞ ((0 , T ) × R )), andhence, div ξ (( E n + ξ × B n ) β δ ( f n )) is bounded in L ((0 , T ) × R ; H − ξ ( R )). Denoting T δ ( f n ) = β ′ δ ( f n ) Q ( f n , f n ) , and decomposing β δ ( f n ) into u n = β δ ( f n ) χ { ( t,x,ξ ): |T δ ( f n ) |≤ M } ∈ L ((0 , T ) × R )) ,g n , and h n by (cid:18) ∂∂t + ξ · ∇ x (cid:19) u n = T δ ( f n ) χ { ( t,x,ξ ): |T δ ( f n ) |≤ M } − div ξ (cid:0) ( E n + ξ × B n ) β δ ( f n ) χ { ( t,x,ξ ): |T δ ( f n ) |≤ M } (cid:1) , (4.4) (cid:18) ∂∂t + ξ · ∇ x (cid:19) g n = − div ξ (( E n + ξ × B n ) β δ ( f n ) χ { ( t,x,ξ ): |T δ ( f n ) | >M } ) , (4.5) (cid:18) ∂∂t + ξ · ∇ x (cid:19) h n = T δ ( f n ) χ { ( t,x,ξ ): |T δ ( f n ) | >M } , (4.6)for M >
1, where h n | t =0 = g n | t =0 = 0 , u n | t =0 = β δ ( f n ) , and χ is the characteristic function of sets. Because {T δ ( f n ) } ∞ n =1 is weakly compact in L ((0 , T ) × R ) due to the facts that β ′ ( t ) = δt ) ≤ δt and f n Q ( f n , f n ) is weaklycompact in L ((0 , T ) × R ), and because, from (4.6), h n ( t, x + tξ, ξ ) = Z t T δ ( f n ) χ { ( t,x,ξ ): |T δ ( f n ) |≥ M } ( τ, x + ξτ, ξ ) dτ, it follows that, uniformly with respect to n , Z T Z R Z R | h n ( t, x, ξ ) | dξdxdt → , as M → ∞ . (4.7)Similarly, from the compactness of T δ ( f n ), we deduce that S n := ( E n + ξ × B n ) β δ ( f n ) χ { ( t,x,ξ ): |T δ ( f n ) | >M } → L loc ((0 , T ) × R ) as M → ∞ . From (4.5), we have g n ( t, x + tξ, ξ ) = Z t − div ξ (( E n + ξ × B n ) β δ ( f n ) χ { ( t,x,ξ ): |T δ ( f n ) | >M } )( τ, x + ξτ, ξ ) dτ. Thus, for any ψ ∈ D ξ ( R ), we deduce from the above identity that Z R g n ( t, x + tξ, ξ ) ψ ( ξ ) dξ = Z t Z R (( E n + ξ × B n ) β δ ( f n ) χ { ( t,x,ξ ): |T δ ( f n ) | >M } )( τ, x + ξτ, ξ ) · ∇ ξ ψdξdτ. Therefore, from the weak compactness of S n , the above identity with (4.8) implies Z R g n ( t, x, ξ ) ψdξ → , as M → ∞ , (4.9)in L loc ((0 , T ) × R ). On the other hand, since { u n } ∞ n =1 and (cid:8) T δ ( f n ) χ { ( t,x,ξ ): |T δ ( f n ) |≤ M } (cid:9) ∞ n =1 are boundedsequences in L ((0 , T ) × R ), and div ξ (( E n + ξ × B n ) β δ ( f n )) is bounded in L ((0 , T ) × R ; H − ξ ( R )), by the velocity averaging lemma (Theorem 3 in [9]), we deduce that Z R u n ψ ( ξ ) dξ is bounded in H ((0 , T ) × R ) , for all ψ ∈ D ( R ). Thus, (cid:8)R R u n ψ ( ξ ) dξ (cid:9) ∞ n =1 is compact in L ((0 , T ) × R ) and is locallycompact in L ((0 , T ) × R ), which, combining with (4.7) and (4.9), implies that Z R ξ β δ ( f n ) ψdξ is relatively compact in L p ( × (0 , T ) , L loc ( R )) (4.10)for all 1 ≤ p < ∞ , ψ ∈ D ( R ).The first statement of the theorem for ψ ∈ D ( R ) then follows from (4.10) and (3.16),since it suffices to observe that we have for all R > ≤ f n − β δ ( f n ) ≤ δRf n + f n χ { f n >R } ≤ δRf n + f n ln f n ln R , (4.11)and then take the limit as R → ∞ and δ →
0. Next, for a general ψ ∈ C ( R ) such that ψ ( ξ )(1 + | ξ | ) − → | ξ | → ∞ , we introduce η M = η (cid:16) · M (cid:17) , for M >
1, where η ∈ D ( R ), 0 ≤ η ≤ η = 1 on B . Then the first statement holds for ψη M , and the first statement will be valid for such a ψ provided thatsup n Z T dt Z K × R f n | ψ | (1 − η M ) dxdξ → M → ∞ , (4.12)for compact subsets K ∈ R x . Indeed, (4.12) follows from (3.16) since Z T Z K × R f n | ψ | (1 − η M ) dxdξ ≤ C sup | ξ |≥ M | ψ ( ξ ) | | ξ | Z T dt Z K × R f n (1 + | ξ | ) χ {| ξ |≥ M } ≤ C sup | ξ |≥ M | φ ( ξ ) | | ξ | for some C > n .4.2. Step Two.
We now aim at proving that ( f, E, B ) is a renormalized solution of VBM.First of all, we claim that it is enough to show that
Lemma 4.1. If f ∈ L ∞ (0 , T ; L ( R )) , the equation (3.17) holds if and only if ∂∂t ln(1 + f ) + div x ( ξ ln(1 + f )) + ( E + ξ × B ) · ∇ ξ ln(1 + f ) = 11 + f Q ( f, f ) , (4.13) in D ′ . EAK STABILITY FOR THE VLASOV-MAXWELL-BOLTZMANN EQUATIONS 21
Proof.
On one hand, if f is a renormalized solution to VMB, then (4.13) automaticallyholds since β ( f ) = ln(1 + f ) ∈ C ([0 , ∞ )) with β (0) = 0 and β ′ ( f )(1 + f ) = 1.On the other hand, if (4.13) holds, we claim that f is a renormalized solution to VMB.Indeed, denoting σ ( s ) = β ( e s − β ( t ) ∈ C ([0 , ∞ )) with β (0) = 0 and β ′ ( t )(1 + t ) ≤ C . Then, we have ∂ t σ ( f ) = σ ′ ( f ) ∂ t f ; ∇ x σ ( f ) = σ ′ ( f ) ∇ x f ; ∇ ξ σ ( f ) = σ ′ ( f ) ∇ ξ f ;Multiplying (4.1) by σ ′ (ln(1 + f )), we obtain, ∂∂t σ (ln(1 + f )) + div x ( ξσ (ln(1 + f ))) + ( E + ξ × B ) · ∇ ξ σ (ln(1 + f ))= σ ′ (ln(1 + f )) 11 + f Q ( f, f ) , (4.14)in the sense of distributions. Note that, by the definition σ , we have σ (ln(1 + f )) = β ( f ) , and σ ′ (ln(1 + f )) = β ′ ( f )(1 + f ) . Hence, substituting the above two identities in (4.14), we get ∂∂t β ( f ) + div x ( ξβ ( f )) + ( E + ξ × B ) · ∇ ξ β ( f ) = β ′ ( f ) Q ( f, f ) , in the sense of distributions. The proof of this lemma is complete. (cid:3) The rest of this subsection is devoted to the proof of (4.13). Recall that we deduce,from a priori estimate (3.16) and weak passages to the limit,sup t ∈ [0 ,T ] (cid:18)Z R f (1 + | ξ | + ν ( x ) + | log f | ) dxdξ + Z R ( | E | + | B | ) dx (cid:19) + Z T Z R dx Z R dξdξ ∗ Z S dωb ( f ′ f ′∗ − f f ∗ ) log f ′ f ′∗ f f ∗ < ∞ , (4.15)for all T ∈ (0 , ∞ ). Now the strategy to prove (4.13) is the following: we first consider β δ ( f n ) = f n (1 + δf n ) − for δ ∈ (0 ,
1] and weakly pass to the limit as n goes to ∞ in the equation satisfied by β δ ( f n ); then for the equation satisfied by the limit of β δ ( f n ) as n → ∞ , we use β torenormalize it and let δ go to 0 to recover (4.13). To begin with, without loss of generality,in view of (3.16), we can assume f n → f weakly ∗ in L ∞ (0 , T ; L ( R )); B n → B weakly ∗ in L ∞ (0 , T ; L ( R )); E n → E weakly ∗ in L ∞ (0 , T ; L ( R ) ∩ L ( R )) . Furthermore, without loss of generality, extracting subsequence if necessary, we may as-sume that for all δ > β δ ( f n ) → β δ weakly in L p ( R × (0 , T )); (4.16) h nδ = (1 + δf n ) − → h δ weakly-* in L ∞ ( R × (0 , ∞ )); (4.17) g nδ = f n (1 + δf n ) − → g δ weakly in L p ( R × (0 , T )) , (4.18)for all T ∈ (0 , ∞ ), 1 ≤ p ≤ ∞ . Furthermore, because of the third statement and theequi-integrability, we may assume that(1 + δf n ) − Q ± ( f n , f n ) → Q ± δ weakly in L ( R x × K × (0 , T )) , (4.19)for all compact sets K ⊂ R ξ and T ∈ (0 , ∞ ).Notice that, since f n is a renormalized solution of VMB, (4.3) holds with β ( f n ) replacedby β δ ( f n ) for all δ > n goes to ∞ .To this end, we deduce from the first statement of Theorem 3.1 that ρ n and j n convergein L p (0 , T ; L ( R x )) to ρ and j , respectively for all 1 ≤ p < ∞ and T ∈ (0 , ∞ ). We thenpass to the limit in (4.3) and we obtain ∂∂t β δ + div x ( ξβ δ ) + ( E + ξ × B ) · ∇ β δ = Q + δ − Q − δ , x ∈ R , ξ ∈ R , t ≥ , (4.20a) ∂E∂t − ∇ × B = − j, div B = 0 , on R x × (0 , ∞ ) , (4.20b) ∂B∂t + ∇ × E = 0 , div E = ρ, on R x × (0 , ∞ ) , (4.20c) ρ = Z R f dξ, j = Z R f ξdξ, on R x × (0 , ∞ ) , (4.20d)in D ′ . Here, for the convergence of the nonlinear term ( E n + ξ × B n ) · ∇ β δ ( f n ), we needto show, for all φ ∈ D ((0 , ∞ ) × R ), Z t Z R φ ( E n + ξ × B n ) · ∇ ξ β δ ( f n ) dξdxds = − Z t Z R ∇ ξ φ · ( E n + ξ × B n ) β δ ( f n ) dξdxds, (4.21)since div ξ ( E n + ξ × B n ) = 0 . If we take φ = φ ( t, x )Φ( ξ ) (which is enough by dense property) for φ ∈ D ((0 , ∞ ) × R )and Φ ∈ D ( R ), we can rewrite the term on the right-hand side of (4.21) as − Z t Z R φ ( t, x )( E n + ξ × B n ) · (cid:18)Z R ψ ( ξ ) β δ ( f n ) dξ (cid:19) dxds, by letting ψ = ∇ ξ Φ . In fact, on one hand, by (4.10) or the velocity averaging lemmain [13], R R ψ ( ξ ) β δ ( f n ) dξ and R R ξψ ( ξ ) β δ ( f n ) dξ strongly converge to R R ψ ( ξ ) β δ dξ and R R ξψ ( ξ ) β δ dξ in L p (0 , T ; L loc ( R )) respectively. On the other hand, since ψ ∈ C ( R ) and β δ ( t ) ≤ t , we have, using (2.17), (cid:26)Z R ξψβ δ ( f n ) dξ (cid:27) ∞ n =1 is uniformly bounded in L ∞ (0 , T ; L ( R )) , and (cid:26)Z R ψβ δ ( f n ) dξ (cid:27) ∞ n =1 is uniformly bounded in L ∞ (0 , T ; L ( R )) . EAK STABILITY FOR THE VLASOV-MAXWELL-BOLTZMANN EQUATIONS 23
The latter is true, because, for all
R > Z R | ψ | β δ ( f n ) dξ = Z {| ξ |≤ R } | ψ | β δ ( f n ) dξ + 1 R Z {| ξ | >R } | ξ | | ψ | β δ ( f n ) dξ ≤ k ψ k L ∞ (cid:18) R | B (0 , |k β δ ( f n ) k L ∞ + 1 R k| ξ | β δ ( f n ) k L ξ ( R )) (cid:19) ≤ k ψ k L ∞ (cid:18) R | B (0 , | δ + 1 R k| ξ | f n k L ξ ( R )) (cid:19) , (4.22)where | B (0 , | denotes the Lebesgue measure of the unit ball B (0 ,
1) in R , and by taking R = δ k| ξ | f n k L ξ ( R )) | B (0 , | ! , (4.22) becomes Z R | ψ | β δ ( f n ) dξ ≤ k ψ k L ∞ | B (0 , | k| ξ | f n k L ξ ( R )) δ . Therefore, (cid:26)Z R | ψ | β δ ( f n ) dξ (cid:27) ∞ n =1 is uniformly bounded in L ∞ (0 , T ; L x ( R )) , since {| ξ | f n } ∞ n =1 is uniformly bounded in L ∞ (0 , T ; L ( R )). Thus Z R ξψβ δ ( f n ) dξ → Z R ξψβ δ dξ in L p (0 , T ; L sloc ( R )) for all 1 ≤ s < , and Z R ψβ δ ( f n ) dξ → Z R ψβ δ dξ in L p (0 , T ; L rloc ( R )) for all 1 ≤ r < , for all 1 ≤ p < ∞ .The weak convergence of E n in L ((0 , T ) × R ), combined with the strong convergenceof R R ψ ( ξ ) β δ ( f n ) dξ , implies Z t Z R ∇ ξ φ · E n β δ ( f n ) dξdxds → Z t Z R ∇ ξ φ · Eβ δ dξdxds. The similar argument goes to the second part of the nonlinear term Z t Z R φ ( t, x ) B n × (cid:18)Z R ψ ( ξ ) ξβ δ ( f n ) dξ (cid:19) dxds, due to the weak convergence of B n in L q ((0 , T ) × R ) for q >
5. That is, Z t Z R ∇ ξ φ · ξ × B n β δ ( f n ) dξdxds → Z t Z R ∇ ξ φ · ξ × Bβ δ dξ. Next, since β δ ( f n ) ∈ L ( R ) ∩ L ∞ ( R ), we know that β δ ∈ L ∞ ( R ) ∩ L ( R ). Also, since | ξ | f n ∈ L ∞ (0 , T ; L ( R )), we know that β δ ( f n ) | ξ | ∈ L ∞ (0 , T ; L ( R )) and { β δ ( f n ) } ∞ n =14 XIANPENG HU AND DEHUA WANG is weakly compact in L ∞ (0 , T ; L ( R )). Hence | ξ | β δ ∈ L ∞ (0 , T ; L ( R )). Thus, for any σ >
0, we have Z { β δ >σ } | ξ | dxdξ < σ Z { β δ >σ } | ξ | β δ dxdξ ≤ σ Z R | ξ | β δ dxdξ < ∞ . Therefore, Theorem 2.1 implies that β δ is a renormalized solution of (4.20).As δ →
0, we claim that
Lemma 4.2. β δ → f, in C ([0 , T ]; L ( R )) , as δ → .Proof. We start with proving the continuity of β δ with respect to t ≥ L p ( R ) for all 1 ≤ p < ∞ . To this end, we remark that if we regularize by convolution β δ into β εδ as in Lemma 2.1, we obtain ∂∂t β εδ + ξ · ∇ x β εδ + ( E + ξ × B ) · ∇ ξ β εδ = Q + δ − Q − δ + r ε (4.23)where r ε → L (0 , T ; L loc ( R ) ∩ L ∞ ( R )) as ε goes to 0 for all T ∈ (0 , ∞ ). Hence, itis easy to see from (4.23) that, β εδ ∈ C ([0 , ∞ ); L p ( R )) for 1 ≤ p < ∞ . Note that β δ is arenormalized solution to the VM (4.20a). Subtracting (4.23) from (4.20), multiplying theresult by | β δ − β εδ | p − ( β δ − β εδ ), and then integrating over R , we obtain ddt Z R | β δ − β εδ | p dxdξ → L (0 , T ) , as ε → ≤ p < ∞ , T ∈ (0 , ∞ ). It follows that β δ ∈ C ([0 , T ]; L ( R )).Next, we show that f ∈ C ([0 , ∞ ); L ( R )). Indeed, because of (3.16), we have for all T ∈ (0 , ∞ ), as in (4.11)sup t ∈ [0 ,T ] sup n ≥ k f n − β δ ( f n ) k L ( R ) → δ → . (4.25)Hence, by the lower semi-continuity of the weak convergence, we obtainsup t ∈ [0 ,T ] k f − β δ k L ( R ) ≤ sup t ∈ [0 ,T ] lim inf n →∞ k f n − β δ ( f n ) k L ( R ) ≤ sup t ∈ [0 ,T ] sup n ≥ k f n − β δ ( f n ) k L ( R ) → δ → , and this implies β δ converges in C ([0 , T ]; L ( R )) to f . (cid:3) Now we can state the equation (4.20a) more precisely. To this end, we observe that − t δt , δt ) are convex on [0 , ∞ ), therefore we have β δ ≤ β δ ( f ) , h δ ≥ (1 + δf ) − a.e on R × (0 , ∞ ) . (4.26)In addition t (1+ δt ) = β δ ( t )(1 − δβ δ ( t )), because the function x (1 − δx ) is a concave function,hence g δ ≤ β δ (1 − δβ δ ) a.e on R × (0 , ∞ ) . (4.27)Furthermore, because of the second statement of Theorem 3.1, we deduce that Q − δ = g δ L ( f ) a.e on R × (0 , ∞ ) . (4.28) EAK STABILITY FOR THE VLASOV-MAXWELL-BOLTZMANN EQUATIONS 25
And, using the fourth statement of Theorem 3.1, we could also deduce that Q + δ = h δ Q + ( f, f ) a.e on R × (0 , ∞ ) . (4.29)We finally use the fact that β δ is a renormalized solution of (4.20) to write ∂∂t β ( β δ ) + div x ( ξβ ( β δ )) + ( E + ξ × B ) · ∇ ξ β ( β δ )= (1 + β δ ) − Q + δ − (1 + β δ ) − Q − δ . (4.30)And we wish to recover (4.13) by letting δ go to 0. Recall that we already showed inLemma 4.2 that β δ converges to f in C ([0 , T ]; L ( R )) for all T ∈ (0 , ∞ ). Therefore, inorder to complete the proof of Theorem 3.1, it only remains to show Lemma 4.3. Q ± δ (1 + β δ ) − are weakly relatively compact in L ( R x × K × (0 , T )) (4.31) for all compact sets K ⊂ R ξ and T ∈ (0 , ∞ ) , and (1 + β δ ) − Q − δ → (1 + f ) − Q − ( f, f ) , a.e (4.32a)(1 + β δ ) − Q + δ → (1 + f ) − Q + ( f, f ) , a.e (4.32b) as δ goes to .Proof. We will follow the lines of the argument in [23] and begin with Q − δ . Without lossof generality, we may assume that β δ converges a.e. to f as δ goes to 0. Then, (4.32a)follows since (1 + β δ ) − Q − δ = (1 + β δ ) − g δ L ( f ) → (1 + f ) − f L ( f )a.e. as δ → g δ converges a.e. to f .This is easy since we have for all R > ≤ f n − f n (1 + δf n ) − ≤ Rδf n + f n χ { f n >R } , hence g δ converges to f in C ([0 , T ]; L ( R )) for all T ∈ (0 , ∞ ) by the uniform integrabilityof f n and the lower semi-continuity of the weak convergence. We now prove (4.31) for Q − δ by first observing that (4.28) yields0 ≤ (1 + β δ ) − Q − δ = (1 + β δ ) − g δ L ( f ) ≤ (1 − δβ δ ) β δ β δ L ( f ) ≤ L ( f ) , a.e.And we conclude the proof of (4.31) for Q − δ by the equi-integrability, since L ( f ) ∈ L ∞ (0 , T ; L ( R x × K )) for all compact sets K ⊂ R x and T ∈ (0 , ∞ ).Next, we turn to the proof of (4.31) for Q + δ and (4.32b). We begin with (4.31). Werecall the following classical inequality for all M > Q + ( f n , f n ) ≤ M Q − ( f n , f n ) + 1ln M ˜ e n (4.33)where ˜ e n = Z R dξ ∗ Z S b dω ( f n ′ f n ∗ ′ − f n f n ∗ ) ln f n ′ f n ∗ ′ f n f n ∗ is positive and bounded in L ( R × (0 , T )) for all T ∈ (0 , ∞ ). Without loss of generality,we may assume that ˜ e n converges weakly in the sense of measures to some bounded nonnegative measure ˜ e on R × [0 , ∞ ) and we denote by ˜ e its regular part with respectto the usual Lebesgue measure, that is, ˜ e = D ˜ eDyDt , ( y, t ) ∈ R × (0 , T ). Dividing (4.33)by (1 + δf n ) and letting n go to ∞ , we obtain Q + δ ≤ M Q − δ + 1ln M ˜ e, hence Q + δ ≤ M Q − δ + 1ln M ˜ e a.e. on R × (0 , ∞ ) . Then (4.31) for Q + δ follows since we already show it for Q − δ and the integrability of ˜ e .We finally prove (4.32) for Q + δ . We first remark that we have for all R > Q + ( f n , f n ) ≥ (1 + δf n ) − Q + ( f n , f n ) ≥ (1 + δR ) − Q + ( f n , f n ) χ { f n 0, we find by letting n go to ∞ and using the third statement of Theorem 3.1, Z R Q + ( f, f ) ψdξ ≥ Z R Q + δ ψdξ a.e. on R x × (0 , ∞ ) . Indeed, the integrated left-hand side converges locally in measure while the right-handside converges weakly in L and this is enough to pass to the limit in the inequality a.e.on R x × (0 , ∞ ). Therefore, we have for all δ ∈ (0 , Q + ( f, f ) ≥ Q + δ . (4.35)Next, we use the other part of the inequality (4.34) and we write for τ ∈ (0 , δR ) − (1 + τ L ( f n )) − Q + ( f n , f n ) ≤ (1 + δf n ) − Q + ( f n , f n ) + (1 + τ L ( f n )) − χ { f n >R } Q + ( f n , f n ) ≤ (1 + δf n ) − Q + ( f n , f n ) + 1ln M e n + Mτ f n χ { f n >R } . (4.36)We then observe that Q + ( f n , f n )(1 + τ L ( f n )) − is relatively weakly compact in L ( R × (0 , T )) for all T ∈ (0 , ∞ ) since it is bounded above by M e n + M τ f n for all M > 1. Hence,we may assume without loss of generality that it converges weakly in L ( R × (0 , T )) forall T ∈ (0 , ∞ ). We claim that its weak limit is given by (1 + τ L ( f )) − Q + ( f, f ). Indeed,if ψ ∈ L ∞ ( R ξ ) with compact support, we have Z R (1 + τ L ( f n )) − Q + ( f n , f n ) ψdξ = Z R Q + ( f n , f n ) ψ nτ dξ, where ψ nτ is uniformly bounded in L ∞ ( R ξ ), and has a uniform compact support and ψ nτ → ψ τ = (1 + τ L ( f )) − ψ in L p ((0 , T ) × R ) for all 1 ≤ p < ∞ . This is enough to enableus to deduce Z R Q + ( f n , f n ) ψ nτ dξ → Z R Q + ( f, f ) ψ τ dξ locally in measure on R x × [0 , ∞ ), which yields the claim. EAK STABILITY FOR THE VLASOV-MAXWELL-BOLTZMANN EQUATIONS 27 We then pass to the limit in (4.36) and deduce as above(1 + δR ) − (1 + τ L ( f )) − Q + ( f, f ) ≤ Q + δ + 1ln M ˜ e + Mτ f R , a.e. , (4.37)where f R is the weak limit of f n χ { f n >R } in L ( R ). Since we have Z R f R dxdξ = lim n →∞ Z R f n χ { f n >R } dxdξ ≤ C ln R , we deduce from (4.37), by letting first δ go to 0, then M go to ∞ , then R go to ∞ , andfinally τ go to 0, that Q + ( f, f ) ≤ lim δ → Q + δ a.e. , which, combining with (4.35), implies that Q + ( f, f ) = lim δ → Q + δ a.e.The proof is complete. (cid:3) Putting together the conclusion of Step One, Lemmas 4.1-4.3, we finish the proof ofTheorem 3.1.5. Proof of Theorem 3.2: Propagation of Smoothness In this section, we prove Theorem 3.2. First, without loss of generality, in view of (3.16),we can assume f n → f weakly ∗ in L ∞ (0 , T ; L ( R )); B n → B weakly ∗ in L ∞ (0 , T ; L ( R ) ∩ L ( R )); E n → E weakly ∗ in L ∞ (0 , T ; L ( R ) ∩ L ( R )) . Applying Theorem 3.1, we know that f ∈ C ([0 , ∞ ); L ( R )) is a renormalized solution ofVBM. In particular, we know that we have, setting γ δ ( f ) = δ ln(1 + δf ), ∂∂t γ δ ( f ) + div x ( ξγ δ ( f )) + ( E + ξ × B ) · ∇ ξ γ δ ( f )= γ ′ δ ( f ) Q + ( f, f ) − f γ ′ δ ( f ) L ( f ) , (5.1)in D ′ . It is easy to deduce that, γ δ ( f ) ∈ C ([0 , ∞ ); L p ( R )) for all 1 ≤ p < ∞ since γ δ ( f ) ∈ C ([0 , ∞ ); L ( R )) ∩ L ∞ (0 , ∞ ; L ( R )) , hence γ δ ( f ) | t =0 = γ δ ( f ) a.e. on R . The strategy of the proof of Theorem 3.2 goes as follows. First of all, we introduce,without loss of generality, the weak limit of γ δ ( f n ) in L p ( R × (0 , T )) for all T ∈ (0 , ∞ )and 1 ≤ p < ∞ , and we denote it by γ δ (note the difference from the notation γ δ ( f n )throughout this section). The first step is to show that γ δ is a supersolution of (5.1).In the second step, we deduce that γ δ = γ δ ( f ) and that f n converges to f a.e. or in L ( R × (0 , T )) for all T ∈ (0 , ∞ ). Finally in the third step, we show that f n convergesto f in C ([0 , T ]; L ( R )), thus proving Theorem 3.2. Applying Theorem 3.1 and a similar argument in Section 4, we can show that γ δ satisfies: γ δ ∈ L ∞ (0 , T ; L p ( R )) for all T ∈ (0 , ∞ ) and 1 ≤ p < ∞ ,0 ≤ γ δ ≤ γ δ ( f ) a.e. on R × (0 , ∞ ) , (5.2)and ∂γ δ ∂t + div x ( ξγ δ ) + ( E + ξ × B ) · ∇ ξ γ δ = Q + δ − Q − δ , (5.3)in D ′ , where Q + δ , Q − δ are respectively the weak limits in L ( R × K × (0 , T )) for all compactsets K ⊂ R ξ of (1+ δf n ) − Q + ( f n , f n ), (1+ δf n ) − Q − ( f n , f n ). For the weak limit function γ δ , we claim Lemma 5.1. γ δ ∈ C ([0 , ∞ ); L p ( R )) for all ≤ p < ∞ .Proof. In fact, we claim that the weak limit γ δ is a renormalized solution of (5.3) and then γ δ ∈ C ([0 , ∞ ); L p ( R ))for all 1 ≤ p < ∞ . For this purpose, we introduce γ εδ ( f n ) = γ δ ( β ε ( f n ))for ε ∈ (0 , 1] and denote its weak limit by γ εδ . Then, the proof in Section 4 applies andshows that the weak limit γ εδ ∈ C ([0 , ∞ ); L p ( R )) is a renormalized solution of ∂∂t γ εδ + div x ( ξγ εδ ) + ( E + ξ × B ) · ∇ ξ γ εδ = γ ′ δ ( β ε ( f )) β ′ ε ( f ) Q + δ,ε − γ ′ δ ( β ε ( f )) β ′ ε ( f ) Q − δ,ε , (5.4)where the notation g means the weak limit of the sequence { g n } ∞ n =1 in L loc . Next, weclaim 0 ≤ γ δ ( f n ) − γ εδ ( f n ) ≤ f n − β ε ( f n ) → L ( R ) (5.5)uniformly in n ≥ t ∈ [0 , T ]. Indeed, since the sequence { f n } ∞ n =1 is equi-integrable, forany η > 0, there exists two positive numbers D and R such thatsup n ∈ N Z ([0 ,T ] × B R × B R ) c f n dtdxdξ ≤ η, and sup n ∈ N Z { f n ≥ D } f n dtdxdξ ≤ η. Hence, in particular, sup n ∈ N Z { f n ≥ D }∩ [0 ,T ] × B R × B R f n dtdxdξ ≤ η. EAK STABILITY FOR THE VLASOV-MAXWELL-BOLTZMANN EQUATIONS 29 Therefore, we have sup n ∈ N Z [0 ,T ] × R × R ( f n − β ε ( f n )) dtdxdξ ≤ sup n ∈ N Z ([0 ,T ] × B R × B R ) c f n dtdxdξ + sup n ∈ N Z { f n ≥ D }∩ [0 ,T ] × B R × B R f n dtdxdξ + sup n ∈ N Z { f n ≤ D }∩ [0 ,T ] × B R × B R ( f n − β ε ( f n )) dtdxdξ ≤ η + D R T ε, (5.6)since f n − β ε ( f n ) ≤ D ε if f n ≤ D . Thus, letting first ε go to 0 and then η go to 0 in(5.6), we deduce (5.5).Similarly, we have 0 ≤ − β ′ ε ( f n ) → L ( R )uniformly in n ≥ t ∈ [0 , T ];0 ≤ ( γ ′ δ ( β ε ( f n )) − γ ′ δ ( f n )) β ′ ε ( f n ) Q − ( f n , f n ) ≤ εf n δf n Q − ( f n , f n ) → L (0 , T ; L ( R x × K ))uniformly in n ≥ 1, for all compact sets K ⊂ R ξ ; and0 ≤ ( γ ′ δ ( β ε ( f n ) − γ ′ δ ( f n )) β ′ ε ( f n )) Q + ( f n , f n ) ≤ εf n δf n Q + ( f n , f n ) → L (0 , T ; L ( R x × K ))uniformly in n ≥ 1, for all compact sets K ⊂ R ξ . Here, we used( γ ′ δ ( β ε ( f n )) − γ ′ δ ( f n )) β ′ ε ( f n ) ≤ εf n δf n . Thus, letting ε go to 0 in (5.4), we deduce that γ δ is a renormalized solution to (5.3).Hence, from (5.3), we deduce that ∂γ δ ∂t ∈ L (0 , T ; W − n, ( R ))for n > γ δ ( t ) is a strictly concave function,0 ≤ γ δ ≤ γ δ ( f ) ≤ f ∈ L ∞ (0 , T ; L ( R )) . Hence, by the Aubin-Lions lemma in [21], we know that γ δ ∈ C ([0 , T ]; W − s, ( R )) . But actually, we know γ δ ∈ L ∞ ([0 , T ] , L p ( R ))for all 1 ≤ p < ∞ . Thus, by the interpolation, we know that γ δ ∈ C ([0 , T ]; L p ( R )) for all 1 ≤ p < ∞ . (cid:3) Step One: γ δ is a supersolution of (5.1) . Without loss of generality, we mayassume that we have γ ′ δ ( f n ) = 11 + δf n → ζ δ weakly* in L ∞ ( R × (0 , ∞ )) , and f n γ ′ δ ( f n ) = f n δf n → θ δ weakly* in L ∞ ( R × (0 , ∞ )) . Furthermore, since γ ′ δ ( f ), − tγ ′ δ ( f ) are convex on [0 , ∞ ), we deduce the following inequal-ities: ζ δ ≥ 11 + δf = γ ′ δ ( f ) , θ δ ≤ f δf = f γ ′ δ ( f ) , (5.7a) γ δ ≤ δ ln(1 + δf ) = γ δ ( f ) a.e. in R × (0 , ∞ ) . (5.7b)We claim Lemma 5.2. Q − δ = θ δ L ( f ) a.e. on R × (0 , ∞ ) . (5.8) Proof. In fact, it is enough to verify that (5.8) holds in [0 , T ] × B R × B R , where B R isthe ball with radius R and centered at the origin in R . Due to the second statement ofTheorem 3.1, we know that L ( f n ) converges a.e. to L ( f ) in [0 , T ] × B R × B R . By Egorov’sTheorem ([25]), for any ε > 0, there exists a subset E ⊂ [0 , T ] × B R ⊗ B R with | E | ≤ ε such that L ( f n ) converges uniformly to L ( f ) on E c . Thus, for all φ ∈ L ∞ ( R × (0 , T )), (cid:12)(cid:12)(cid:12)(cid:12)Z T Z B R Z B R φ (cid:18) f n δf n L ( f n ) − θ δ L ( f ) (cid:19) dxdξdt (cid:12)(cid:12)(cid:12)(cid:12) ≤ k φ k L ∞ sup n Z E | L ( f n ) | + | θ δ || L ( f ) | dxdξdt + k φ k L ∞ (cid:12)(cid:12)(cid:12)(cid:12)Z E c (cid:18) f n δf n − θ δ (cid:19) L ( f ) dxdξds (cid:12)(cid:12)(cid:12)(cid:12) + k φ k L ∞ | E c | sup E c | L ( f n ) − L ( f ) | . The first term can be made arbitrarily small uniformly in n , due to the equi-integrabilityof { L ( f n ) } ∞ n =1 . The second term also goes to 0 since L ( f ) ∈ L ((0 , T ) × B R × B R ). Andthe third term goes to 0 as n goes to ∞ since the uniform convergence of L ( f n ) to L ( f )in E c . Thus, (5.8) is verified. (cid:3) Similarly, we have Lemma 5.3. Q + δ = ζ δ Q + ( f, f ) a.e. on R × (0 , ∞ ) . (5.9) Proof. Indeed, let A be an arbitrary compact subset of R × [0 , ∞ ). By the Egorov’stheorem and the fourth statement of Theorem 3.1, for each ε > E with the measure of E not greater than ε (i.e., | E | ≤ ε ), up to a subsequence EAK STABILITY FOR THE VLASOV-MAXWELL-BOLTZMANN EQUATIONS 31 Q + ( f n , f n ) converges uniformly to Q + ( f, f ) on E c and Q + ( f, f ) is integrable on E c .Then, for all φ ∈ L ∞ ( R × (0 , ∞ )) supported in A , we have (cid:12)(cid:12)(cid:12)(cid:12)Z A φ { γ ′ δ ( f n ) Q + ( f n , f n ) − ζ δ Q + ( f, f ) } dxdξdt (cid:12)(cid:12)(cid:12)(cid:12) ≤ k φ k L ∞ Z E (cid:12)(cid:12) γ ′ δ ( f n ) Q + ( f n , f n ) − ζ δ Q + ( f, f ) (cid:12)(cid:12) dxdξdt + (cid:12)(cid:12)(cid:12)(cid:12)Z E c ∩A φ { γ ′ δ ( f n ) − ζ δ } Q + ( f, f ) dxdξdt (cid:12)(cid:12)(cid:12)(cid:12) + k φ k L ∞ | E c ∩ A| sup E c | Q + ( f n , f n ) − Q + ( f, f ) | , where the third term goes to 0 as n goes to ∞ , for each ε > Q + ( f n , f n ) to Q + ( f, f ) on E c . And so does the second term since φχ E c Q + ( f, f ) ∈ L ( R × (0 , ∞ )) . Finally, since γ ′ δ ( f n ) Q + ( f n , f n ) is weakly relatively compact in L ( R x × K × (0 , T )) forall compact sets K ⊂ R ξ , the first term can be made arbitrarily small uniformly in n ifwe let ε go to 0.Notice also that ζ δ Q + ( f, f ) ∈ L ( R x × K × (0 , T )) by following the similar argumentas before, we can show that ζ δ ( Q + ( f, f ) ∧ R ) is the weak limit of δf n ( Q + ( f n , f n ) ∧ R ),where a ∧ b = min { a, b } . Thus, (5.9) follows. (cid:3) Now, we use (5.7)-(5.9) in (5.1) to obtain ∂γ δ ∂t + div x ( ξγ δ ) + ( E + ξ × B ) · ∇ γ δ ≥ γ ′ δ ( f ) Q ( f, f ) (5.10)in D ′ . We conclude this first step by proving that γ δ satisfies the initial condition: γ δ | t =0 = γ δ ( f ) . Indeed, in view of the equation satisfied by γ δ ( f n ), we know that ∂γ δ ( f n ) ∂t ∈ L (0 , T, W − n, ( R ))for n > γ δ ( f n ) ∈ L ∞ (0 , T ; L ( R )) and theAubin-Lions lemma, implies that γ δ ( f n ) → γ δ in C ([0 , T ]; W − s, ( R ))for any s > 1. But, by the assumption, γ δ ( f n ) | t =0 = γ δ ( f n ) converges in L ( R ) and thusin W − s, ( R ) to γ δ ( f ). Thus, we conclude that γ δ satisfies the initial condition.5.2. Step Two: γ δ = γ δ ( f ) and f n converges in L to f . To this end, we consider γ δ ( f ) − γ δ = τ δ ∈ C ([0 , ∞ ); L p ( R ))and observe that τ δ satisfies, in view of (5.1) and (5.10), ∂∂t τ δ + div x ( ξτ δ ) + ( E + ξ × B ) · ∇ ξ τ δ ≤ in D ′ with τ δ ≥ R × (0 , ∞ ) , τ δ | t =0 = 0 a.e. on R . (5.12)Then, for τ δ we have Lemma 5.4. τ δ = 0 . Proof. Formally, we only need to integrate (5.11) over R to get ddt Z R τ δ dxdξ ≤ D ′ (0 , ∞ ) . (5.13)Then (5.13) with (5.12) yield: τ δ = 0 on R × (0 , ∞ ).Our main objective now is to justify (5.13). In order to do so, we introduce the function φ ∈ C ∞ ( R ) with φ ( z ) = ( , if | z | ≤ , if | z | ≥ . Notice that β ε ( τ δ ) = τ δ ετ δ also satisfies (5.11) and (5.12), and β ε ( τ δ ) ∈ C ([0 , ∞ ); L p ( R ))for 1 ≤ p < ∞ and ε > 0, since | β ε ( x ) − β ε ( y ) | ≤ | x − y | for all x, y ≥ 0. Then we multiply (5.11) by φ (cid:0) xn (cid:1) φ (cid:16) ξn (cid:17) , and integrate the resultinginequality over R × (0 , t ) for all t ≥ Z R β ε ( τ δ ) φ (cid:16) xn (cid:17) φ (cid:18) ξn (cid:19) dxdξ ≤ Z t ds Z R dxdξβ ε ( τ δ ) · (cid:16) ξn · ∇ φ (cid:16) xn (cid:17) · φ (cid:18) ξn (cid:19) + 1 n ( E + ξ × B ) · ∇ φ (cid:18) ξn (cid:19) φ (cid:16) xn (cid:17) (cid:17) . (5.14)Recall that sup (cid:26)Z R β ε ( τ δ ) | ξ | dxdξ : t ∈ [0 , T ] , ε ≥ , δ ≥ (cid:27) ≤ sup t ∈ [0 ,T ] (cid:26)Z R f | ξ | dxdξ (cid:27) < ∞ , (5.15)since β ε ( τ δ ) ≤ τ δ ≤ γ δ ( f ) ≤ f for all T ∈ (0 , ∞ ). Hence, for the terms on the right handside of (5.14), we have Z t ds Z R dxdξβ ε ( τ δ ) (cid:12)(cid:12)(cid:12)(cid:12) ξn · ∇ φ (cid:16) xn (cid:17)(cid:12)(cid:12)(cid:12)(cid:12) φ (cid:18) ξn (cid:19) ≤ Z t Z R β ε ( τ δ ) χ { n ≤| x |≤ n } k φ k L ∞ k∇ φ k L ∞ dxdξ ≤ Z t Z R f χ { n ≤| x |≤ n } k φ k L ∞ k∇ φ k L ∞ dxdξ → EAK STABILITY FOR THE VLASOV-MAXWELL-BOLTZMANN EQUATIONS 33 as n → ∞ and Z t ds Z R dxdξβ ε ( τ δ ) 1 n φ (cid:16) xn (cid:17) (cid:12)(cid:12)(cid:12)(cid:12) ( E + ξ × B ) · ∇ φ (cid:18) ξn (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:18)Z t ds Z R dxdξβ ε ( τ δ ) 1 n | E + ξ × B | χ { n ≤| ξ |≤ n } dxdξ (cid:19) k φ k L ∞ k∇ φ k L ∞ . Observing that, because of (5.15), (cid:13)(cid:13)(cid:13)(cid:13)Z R dξβ ε ( τ δ ) 1 n χ { n ≤| ξ |≤ n } (cid:13)(cid:13)(cid:13)(cid:13) L (0 ,t ; L ( R x )) ≤ n (cid:13)(cid:13)(cid:13)(cid:13)Z R dξf | ξ | χ { n ≤| ξ |≤ n } (cid:13)(cid:13)(cid:13)(cid:13) L (0 ,t ; L ( R x )) = 1 n ε n , with ε n → , while of course we have for some C ε > (cid:13)(cid:13)(cid:13)(cid:13)Z R dξβ ε ( τ δ ) 1 n χ { n ≤| ξ |≤ n } (cid:13)(cid:13)(cid:13)(cid:13) L (0 ,t ; L ∞ ( R x )) ≤ C ε n . Therefore, we deduce from the H¨older inequality that we have for all ε > Z R dξβ ε ( τ δ ) 1 n χ { n ≤| ξ |≤ n } → , in L (0 , t ; L p ( R x )) for all 1 ≤ p ≤ as n → ∞ , and hence in particular in L (0 , t ; L ( R )).This implies Z t ds Z R Z R β ε ( τ δ ) 1 n χ { n ≤| ξ |≤ n } dξ | E | dx → , in L ((0 , t ) × R ) (5.16)as n → ∞ , since E ∈ L ∞ (0 , t ; L ( R x )).On the other hand, using (2.17) and the fact β ε ( τ δ ) ≤ f , we obtain Z t ds Z R dxdξβ ε ( τ δ ) 1 n | ξ || B | χ { n ≤| ξ |≤ n } ≤ n Z t ds (cid:18)Z R f | ξ | dξ (cid:19) | B | dx ≤ Cn Z t ds (cid:18)Z R f | ξ | dξ (cid:19) | B | dx ≤ Cn t sup s ∈ (0 ,t ) (cid:26)Z R f | ξ | dξ (cid:27) k B k L ∞ (0 ,t ; L ( R x )) → , (5.17)as n → ∞ . Hence, combining (5.16) and (5.17) together, we get Z t ds Z R dxdξβ ε ( τ δ ) 1 n | E + ξ × B | χ { n ≤| ξ |≤ n } → n → ∞ .Finally, letting first n go to ∞ and then ε go to 0 in (5.14), we deduce, by Fatou’slemma, Z R τ δ ( x, ξ, t ) dxdξ ≤ , for all t ≥ , which, combined with (5.12), implies that τ δ = 0 on R × (0 , ∞ ) almost everywhere. (cid:3) In other words, γ δ ( f n ) weakly converges to γ δ ( f ). Since γ δ is strictly concave on [0 , ∞ ),we deduce from classical functional analysis arguments that f n converges in measure to f on R × (0 , T ) for all T ∈ (0 , ∞ ), see [14]. This convergence implies that f n → f in L p (0 , T ; L ( R )) , (5.18)for all 1 ≤ p < ∞ and T ∈ (0 , ∞ ). Indeed, by the equi-integrability of the sequence { f n } ∞ n =1 and the integrability of f in L ([0 , T ] × R ), we know that for any ε > 0, thereexists R > δ > n ∈ N Z ([0 ,T ] × B R × B R ) c | f n − f | dtdxdξ ≤ ε, (5.19)and sup n ∈ N Z G | f n − f | dtdxdξ ≤ ε, (5.20)for all set G ⊂ [0 , T ] × R with | G | ≤ δ .On the other hand, on the set [0 , T ] × B R × B R , since up to a subsequence, f n convergesto f almost everywhere, by Egorov’s theorem, for the given δ > H ⊂ [0 , T ] × B R × B R with | H | ≤ δ such that f n converges uniformly to f on([0 , T ] × B R × B R ) ∩ H c . Therefore, using (5.20), Z [0 ,T ] × B R × B R | f n − f | dtdxdξ = Z ([0 ,T ] × B R × B R ) ∩ H c | f n − f | dtdxdξ + Z H | f n − f | dtdxdξ ≤ ε + Z ([0 ,T ] × B R × B R ) ∩ H c | f n − f | dtdxdξ. (5.21)Notice that the last term in (5.21) tends to 0 as n → ∞ since the uniform convergence of f n to f in H c . Hence, combining (5.19), (5.20) and (5.21), we conclude that f n → f in L (0 , T ; L ( R )) , which, with the uniform bound of f n in L ∞ (0 , T ; L ( R )), implies (5.18).5.3. Step Three: The convergence in C ([0 , T ]; L ( R )) . It only remains to show that f n converges to f in C ([0 , T ]; L ( R )) using (5.18). Indeed, because of (3.16) and (4.25),it is clearly enough to show that, for each δ > T ∈ (0 , ∞ ), K compact set in R , wehave β δ ( f n ) → β δ ( f ) in C ([0 , T ]; L ( K )) . (5.22)For this purpose, we take φ ∈ C ∞ ( R ) such that φ = 1 on K , φ ≥ 0, and we use (4.3)to deduce that for all t ≥ Z R β δ ( f n ) φdxdξ = Z t Z R dxdξ (cid:16) β δ ( f n )1 + δf n × Q ( f n , f n ) φ + β δ ( f n ) ( ξ · ∇ x φ + ( E n + ξ × B n ) · ∇ ξ φ ) (cid:17) . (5.23) EAK STABILITY FOR THE VLASOV-MAXWELL-BOLTZMANN EQUATIONS 35 Then, due to (5.18), β δ ( f n ) converges to β δ ( f ) in L p ( R × (0 , T )) for all 1 ≤ p < ∞ and T ∈ (0 , ∞ ), and one can check easily that the right-hand side of (5.23) converges uniformlyin t ∈ [0 , T ] to the same expression with f n replaced by f . Since β δ ( f ) is a renormalizedsolution, this expression is also given by R R β δ ( f ) φdxdξ . In other words, we have Z R β δ ( f n ) φdxdξ → Z R β δ ( f ) φdxdξ, (5.24)uniformly in t ∈ [0 , T ], for all T ∈ (0 , ∞ ).In addition, since (4.3) implies ∂β δ ( f n ) ∂t ∈ L (0 , T ; W − n, ( R ))for large enough n > 0, and β δ ( f n ) ∈ L (0 , T ; L ( R )) , by the Aubin-Lions lemma, we know that β δ ( f n ) converges to β δ ( f ) in C ([0 , T ]; W − s, loc ( R ))for any s > 1. Therefore, if we consider L φ = L (supp φ, φdx ), since { β δ ( f n ) } n is boundedin L ∞ (0 , T ; L φ ), we deduce that β δ ( f n ) converges uniformly on [0 , T ] to β δ ( f ) in L φ endowed with the weak topology, which, combined with (5.24) and the fact that β δ ( f ) ∈ C ([0 , ∞ ); L φ ) implies that β δ ( f n ) converges to β δ ( f ) in L φ strongly and uniformly in [0 , T ].Hence, (5.22) follows. 6. Large Time Behavior In this section, we are devoted to the study of the large time behavior of the renormalizedsolution to VMB. Indeed, let f ( t, x, ξ ) be a renormalized solution to VMB with finiteenergy and finite entropy in view of (3.16). Then, for every sequence { t n } ∞ n =1 going toinfinity, there exists a subsequence { t n k } ∞ k =1 and a local time-dependent Maxwellian m such that f n k ( t, x, ξ ) = f ( t + t n k , x, ξ ) converges weakly in L ((0 , T ) × R ) to m for every T > 0. More precisely, we have the following theorem: Theorem 6.1. Let f ( t, x, ξ ) be a renormalized solution to VMB and assume that b > al-most everywhere. Then, for every sequences t n going to infinity, there exists a subsequence t n k and a local time-dependent Maxwellian m ( t, x, ξ ) such that f n k ( t, x, ξ ) = f ( t + t n k , x, ξ ) converges weakly in L ((0 , T ) × R ) to m ( t, x, ξ ) for every T > . Moreover, the Maxwelliansatisfies the Vlasov-Maxwell equations: ∂m∂t + ξ · ∇ x m + ( E + ξ × B ) · ∇ ξ m = 0 , (6.1a) ∂E∂t − ∇ × B = − Z R mξdξ, div B = 0 , (6.1b) ∂B∂t + ∇ × E = 0 , div E = Z R mdξ, (6.1c) in the sense of renormalizations. Remark . When the spatial domain is a periodic box or a bounded domain with thereverse reflexion boundary or the specular reflexion boundary, we can expect, as in [7, 12],that the local Maxwellian m in Theorem 6.1 is actually global; that is, m is independentof t, x . Remark . Our large time behavior result is only sequential; that is, the Maxwelliancould depend on our choice of the sequence { t n } ∞ n =1 . Proof of Theorem 6.1. Notice that since f ( t, x, ξ ) is a renormalized solution to VMB, itautomatically holds:sup t ∈ [0 , ∞ ) (cid:18)Z R f (1 + | ξ | + ν ( x ) + | log f | ) dxdξ + Z R ( | E | + | B | ) dx (cid:19) + Z ∞ Z R dx Z R dξdξ ∗ Z S dωb ( f ′ f ′∗ − f f ∗ ) log f ′ f ′∗ f f ∗ < ∞ . (6.2)Therefore, f n ( t, x, ξ ) = f ( t + t n , x, ξ ) is weakly compact in L ((0 , T ) × R ) for every T > { t n } ∞ n =1 going to ∞ . Similarly, E n ( t, x ) = E ( t + t n , x ), B n ( t, x ) = B ( t + t n , x ) are weakly compact in L ∞ (0 , T ; L ( R )). Then, theweak compactness of f n ( t, x, ξ ) in L ((0 , T ) × R ) implies that there exists a subsequence { t n k } ∞ k =1 and a function m ∈ L ((0 , T ) × R ) such that the function f n k converges weaklyto m in L ((0 , T ) × R ) while the weak compactness of B n ( t, x ) and E n ( t, x ) implies thatwe can choose t n k such that B n k and E n k converge weakly* to B and E respectively in L ∞ (0 , T ; L ( R )). Notice that, applying the velocity average lemma, we know Z R f n k dξ → Z R mdξ in L (0 , T ; L ( R )) , and Z R f n k ξdξ → Z R mξdξ in L (0 , T ; L ( R )) . Hence, according to (1.1b) and (1.1c), the electric field E and the magnetic field B satisfies ∂E∂t − ∇ × B = − Z R mξdξ,∂B∂t + ∇ × E = 0 , with div B = 0 , div E = Z R mdξ, in the sense of distributions.In order to prove that m is a Maxwellian, we denote d k := Z T Z R dx Z R dξdξ ∗ Z S dωb ( f n k ′ f n k ′∗ − f n k f n k ∗ ) log f n k ′ f n k ′∗ f n k f n k ∗ = Z T + t nk t nk Z R dx Z R dξdξ ∗ Z S dωb ( f ′ f ′∗ − f f ∗ ) log f ′ f ′∗ f f ∗ . Then, the estimate (6.2) implies that d k converges to 0 as k goes to ∞ . EAK STABILITY FOR THE VLASOV-MAXWELL-BOLTZMANN EQUATIONS 37 On the other hand, in view of the first statement of Theorem 3.1 or arguing as [11], for allsmooth nonnegative functions ψ, φ with compact support, we have, up to a subsequence, Z R dξdξ ∗ Z S dωbf n k ′ f n k ′∗ φ ( ξ ) ψ ( ξ ∗ ) → Z R dξdξ ∗ Z S dωbm ( t, x, ξ ′ ) m ( t, x, ξ ′∗ ) φ ( ξ ) ψ ( ξ ∗ ) , (6.3)and Z R dξdξ ∗ Z S dωbf n k f n k ∗ φ ( ξ ) ψ ( ξ ∗ ) → Z R dξdξ ∗ Z S dωbm ( t, x, ξ ) m ( t, x, ξ ∗ ) φ ( ξ ) ψ ( ξ ∗ ) , (6.4)for almost all ( t, x ) ∈ [0 , T ] × R .Furthermore, since C ( R ) is separable, we can also assume the convergence in (6.3)and (6.4) holds for all nonnegative function in C ( R ). Since P ( x, y ) = ( x − y ) ln( xy ) is anonnegative convex function for x, y > 0, we have,0 ≤ Z R dξdξ ∗ Z S dω b ( m ′ m ′∗ − mm ∗ ) log m ′ m ′∗ mm ∗ ψ ( ξ ∗ ) φ ( ξ ) ≤ lim inf k →∞ d k = 0 , for almost all ( t, x ) ∈ [0 , T ] × R . Hence, b ( m ′ m ′∗ − mm ∗ ) log m ′ m ′∗ mm ∗ ψ ( ξ ′ ) φ ( ξ ) = 0 , almost all ( t, x ) ∈ [0 , T ] × R . The nonnegativity of the function P ( x, y ) and the strictpositivity of b ensure that m ′ m ′∗ = mm ∗ , for almost all ( t, x, ξ, ξ ∗ , ω ) ∈ (0 , T ) × R × S . According to Lemma 2.2 of [4], ] or Section3.2 of [5], m is a Maxwellian. Thus, Q ( m, m ) = 0 . Also, in view of Theorem 3.1, m is still a renormalized solution to VMB, hence ∂m∂t + ξ · ∇ x m + ( E + ξ × B ) · ∇ ξ m = 0 , in the sense of renormalizations. The proof is complete. (cid:3) Remark on The Relativistic Vlasov-Maxwell-Boltmann Equations An extension of our analysis is possible to the relativistic Vlasov-Maxwell-Boltzmannequations of the form (cf. [6, 17]): ∂f∂t + ˆ ξ · ∇ x f + ( E + ˆ ξ × B ) · ∇ ξ f = Q ( f, f ) , x ∈ R , ξ ∈ R , t ≥ , (7.1a) ∂E∂t − ∇ × B = − j, div B = 0 , on R x × (0 , ∞ ) , (7.1b) ∂B∂t + ∇ × E = 0 , div E = ρ, on R x × (0 , ∞ ) , (7.1c) ρ = Z R f dξ, j = Z R f ˆ ξdξ, on R x × (0 , ∞ ) , (7.1d)with ˆ ξ := ξ p | ξ | , and Q ( f, f ) = Z R dξ ∗ Z S dω b ( ξ − ξ ∗ , ω ) p | ξ | p | ξ ∗ | ( f ′ f ′∗ − f f ∗ ) . The corresponding conservation laws are given by ∂ρ∂t + div x j = 0 ,∂∂t (cid:18)Z R f ξdξ + E × B (cid:19) + div x (cid:18)Z R ξ ⊗ ˆ ξf dξ + (cid:18) | E | + | B | Id − E ⊗ E − B ⊗ B (cid:19)(cid:19) = 0 , and ∂∂t (cid:18)Z R f p | ξ | dξ + | E ( t, x ) | + | B ( t, x ) | (cid:19) + div x (cid:18)Z R ˆ ξ | ξ | f dξ + 2 E ( t, x ) × B ( t, x ) (cid:19) = 0 . Then, we can deduce that for any t ∈ [0 , T ] Z R f ( t, x, ξ )( p | ξ | + p | x | ) dxdξ ≤ C ( T ) , (7.2)which implies f | ξ | ∈ L ∞ (0 , T ; L ( R ). Hence, following the lines in Section 2, the exis-tence of renormalized solution to the relativistic Vlasov equation can be verified. Moreimportantly, we can further release the requirement on the integrability of the electric fieldin L , since we no longer need the estimate on f | ξ | in L ∞ (0 , T ; L ( R )).Note that for the relativistic VMB, the magnetic field has the same integrability in thevariable x as the magnetic field due to equivalence between ξ and p | ξ | when ξ issufficiently large. More precisely, we have EAK STABILITY FOR THE VLASOV-MAXWELL-BOLTZMANN EQUATIONS 39 Proposition . Assume that f ∈ L ∞ ((0 , T ) × R ). Then for any solution satisfying theabove conservation laws, one has k ρ ( t, x ) k L ∞ (0 ,T ; L ( R )) ≤ C, k j ( t, x ) k L ∞ (0 ,T ; L ( R )) ≤ C, where the constant C depends on the energy of the initial data and on k f k L ∞ ((0 ,T ) × R ) . Proof. Indeed, we have ρ ( t, x ) = Z | ξ |≤ R f ( t, x, ξ ) dξ + Z | ξ |≤ R f ( t, x, ξ ) dξ ≤ C R k f k L ∞ + R − Z R f p | ξ | dξ ≤ C (cid:18)Z R f p | ξ | dξ (cid:19) where for the last inequality, we optimize R by taking R = (cid:18)Z R f p | ξ | dξ (cid:19) . The same computation also works for j . (cid:3) For any sequence of f n as in Section 3, by the H Theorem and (7.2), f n is weaklycompact in L ((0 , T ) × ( R ). And then, we can follow the lines in Section 4 and Section 5to show the corresponding weak stability for the relativistic VMB. One difference is that,due to Proposition 7.1, we need to assume the electric field E ( t, x ), and the magnetic field B ( t, x ) are uniformly bound in L ∞ (0 , T ; L α ( R )) for some α > 4. When the weak stabilityand the existence of renormalized solutions to (7.1) are concerned, a different assumptionon the collision kernel need to assume, that is,(1 + | z | ) − Z z + B R A ( ξ ) p | ξ | dξ ! → , as | z | → ∞ , for all R ∈ (0 , ∞ ) . 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