Global Existence for the "One and one-half" dimensional relativistic Vlasov-Maxwell-Fokker-Planck system
aa r X i v : . [ m a t h . A P ] D ec GLOBAL EXISTENCE FOR THE “ONE AND ONE-HALF” DIMENSIONALRELATIVISTIC VLASOV-MAXWELL-FOKKER-PLANCK SYSTEM ∗ NICHOLAS MICHALOWSKI † AND
STEPHEN PANKAVICH ‡ Abstract.
In a recent paper Calogero and Alcantara [1] derived a Lorentz-invariant Fokker-Planck equation,which corresponds to the evolution of a particle distribution associated with relativistic Brownian Motion. Westudy the “one and one-half” dimensional version of this problem with nonlinear electromagnetic interactions - therelativistic Vlasov-Maxwell-Fokker-Planck system - and obtain the first results concerning well-posedness of solutions.Specifically, we prove the global-in-time existence and uniqueness of classical solutions to the Cauchy problem and again in regularity of the distribution function in its momentum argument.
Key words.
Kinetic Theory, Vlasov, Fokker-Planck equation, global existence
Subject classifications. 35L60, 35Q83, 82C22, 82D10
1. Introduction
A plasma is a partially or completely ionized gas. Matter exists in thisstate if the velocities of individual particles in a material achieve magnitudes approaching the speedof light. If a plasma is of sufficiently low density or the time scales of interest are small enough, itis deemed to be “collisionless”, as collisions between particles become extremely infrequent. Manyexamples of collisionless plasmas occur in nature, including the solar wind, the Van Allen radiationsbelts, and galactic nebulae.From a mathematical perspective, the fundamental Lorentz-invariant equations which describethe time evolution of a collisionless plasma are given by the relativistic Vlasov-Maxwell system:(RVM) ∂ t f + ˆ v · ∇ x f + ( E + ˆ v × B ) · ∇ v f = 0 ρ ( t,x ) = Z f ( t,x,v ) dv, j ( t,x ) = Z ˆ vf ( t,x,v ) dv∂ t E = ∇ × B − j, ∇ · E = ρ∂ t B = −∇ × E, ∇ · B = 0 . Here, f represents the distribution of (positively-charged) ions in the plasma, while ρ and j arethe charge and current density, and E and B represent electric and magnetic fields generated bythe charge and current. The independent variables, t ≥ x,v ∈ R represent time, position, andmomentum, respectively, and physical constants, such as the charge and mass of particles, as wellas, the speed of light, have been normalized to one. The structure of the velocity terms ˆ v in (RVM)arise due to relativistic corrections, and this quantity is defined byˆ v = vv , v = p | v | . In order to include collisions of particles with a background medium in the physical formulation,often a diffusive Fokker-Planck term is added to the Vlasov equation in (RVM). With this, the systemis referred to as the relativistic Vlasov-Maxwell-Fokker-Planck equation. Since basic questions ofwell-posedness remain unknown even in lower dimensions, we study a dimensionally-reduced versionof this model for which x ∈ R and v ∈ R , the so-called “one and one-half dimensional” analogue, ∗ This work was supported by the National Science Foundation under the awards DMS-0908413 and DMS-1211667. † Department of Mathematical Sciences, New Mexico State University, Las Cruces, New Mexico 88003,( [email protected] ). ‡ Department of Applied Mathematics and Statistics, Colorado School of Mines, Golden, Colorado 80401( [email protected] ). 1
N. MICHALOWSKI AND S. PANKAVICH given by(RVMFP) ∂ t f + ˆ v ∂ x f + K · ∇ v f = ∇ v · ( D ∇ v f ) D = 1 v (cid:20) v v v v v v (cid:21) K = E + ˆ v B, K = E − ˆ v Bρ ( t,x ) = Z f ( t,x,v ) dv − φ ( x ) , j ( t,x ) = Z ˆ vf ( t,x,v ) dv∂ t E = − ∂ x B − j , ∂ t B = − ∂ x E , ∂ x E = ρ, ∂ t E = j . Here, we assume a single species of particles described by f ( t,x,v ) in the presence of a given, fixedbackground φ ∈ C c ( R ) that is neutralizing in the sense that Z ρ (0 ,y ) dy = 0 . The electric and magnetic fields are given by E ( t,x ) = h E ( t,x ) ,E ( t,x ) i and B ( t,x ), respectively.Finally, the matrix D = v − ( I + v ⊗ v ) ∈ R × is the relativistic diffusion operator and possesses somedesirable properties, as discovered for its three-dimensional variant in [1]. We note, however, thatthe operator ∇ v · ( D ∇ v f ) is not uniformly elliptic and provides less dissipation than the Laplacian∆ v f . Namely, for any u ∈ R , D satisfies(1.1) v − | u | ≤ | u · Du | ≤ v | u | . For initial data we take a nonnegative particle density f with compact x -support and boundedmoments v b ∂ kx f ∈ L ( R ), along with fields E ,B ∈ H ( R ). Additionally, we specify particulardata for E , namely E (0 ,x ) = Z x −∞ (cid:18)Z f ( y,w ) dw − φ ( y ) (cid:19) dy. In fact, this particular choice of data for E is the only one which leads to a solution possessingfinite energy (see Lemma 2.2 and [6]). The inclusion of the neutralizing density φ is also necessaryin order to arrive at finite energy solutions for (RVMFP) with a single species of ion.Over the past twenty-five years significant progress has been made in the analysis of (RVM),specifically, the global existence of weak solutions (which also holds for the non-relativistic system(VM); see [5]) and the determination of conditions which ensure global existence of classical solutions(originally discovered in [8], and later in [2] and [10] using different methods) for the Cauchy problem.Additionally, a wide array of results have been obtained regarding electrostatic simplifications of(RVM) - the Vlasov-Poisson and relativistic Vlasov-Poisson systems, obtained by taking the limit as c → ∞ [17] and B ≡
0, respectively. These models do not include magnetic effects, and the electricfield is given by an elliptic equation rather than a hyperbolic PDE. This simplification has led to agreat deal of progress concerning the electrostatic systems, including theorems regarding the well-posedness of solutions [14, 15, 16, 18]. General references on kinetic equations of plasma dynamics,such as (RVM) and (RVMFP), include [7] and [20].Independent of recent advances, many of the most basic existence and regularity questionsremain unsolved for (RVMFP). For much of the existence theory for collisionless models, one ismainly focused on bounding the velocity support of the distribution function f , assuming that f possess compact momentum support, as this condition has been shown to imply global existence[8]. Hence, one of the main difficulties which arises for (RVMFP) is the introduction of particles .5D RELATIVISTIC VLASOV-MAXWELL-FOKKER-PLANCK EQUATION v -support of the distribution function is not bounded, we are able toovercome this issue by controlling large enough moments of the distribution to guarantee sufficientdecay of f in its momentum argument. This also allows us to control the singularities which arisefrom representing derivatives of the fields. As an additional difference arising from the Fokker-Planck operator, we note that when studying collisionless systems, in which D ≡ L ∞ is typicallythe proper space in which to estimate both the particle distribution and the fields. With the additionof the diffusion operator, though, the natural space in which to estimate f is now L . Thus, to takeadvantage of the gain in regularity that should result from the Fokker-Planck term, we iterate in aweighted L setting, estimating moments v γ ∂ kx,v f in L . Other crucial features which appear includethe cone estimate, conservation of mass, and the symmetry and positivity of the diffusive operator.Though this is the first investigation of the well-posedness of (RVMFP), others have studiedVlasov-Maxwell models incorporating a Fokker-Planck term. Both Yu and Yang [23] and Chae [3]constructed global classical solutions to the non-relativistic Vlasov-Maxwell-Fokker-Planck systemfor initial data sufficiently close to Maxwellian using Kawashima estimates and the well-known en-ergy method. Additionally, Lai [11, 12] arrived at a similar result for a one and one-half dimensional“relativistic” Vlasov-Maxwell-Fokker-Planck system using classical estimates. The unfortunate com-monality amongst these models, however, is that they lack invariance properties. Namely, eachcouples the Lorentz-invariant Maxwell equations to either a Galilean-invariant Vlasov equation withnon-relativistic velocities or a hybrid Vlasov equation that includes relativistic velocity corrections,but utilizes the Laplacian ∆ v as the Fokker-Planck term. This latter term destroys the inherentLorentz-invariance of the relativistic Vlasov-Maxwell system. Thus, we consider a diffusive operatorof the form ∇ v · ( D ∇ v f ) which preserves this property. With this structure in place, we can proveglobal existence of classical solutions under relatively relaxed assumptions: Theorem 1.1.
Assume the initial particle distribution satisfies v a f ∈ L ∞ ( R ) and v b − k/ ∂ kx f ∈ L ( R ) for some a > ,b > , and all k = 0 , , . Additionally, assume f possesses compact supportin x with E ,B ∈ H ( R ) and φ ∈ C c ( R ) . Then, for any T > there exist unique functions f ∈ C ((0 ,T ) × R ; C ( R )) ,E ∈ C ((0 ,T ) × R ; R ) , and B ∈ C ((0 ,T ) × R ) satisfying (RVMFP) on (0 ,T ) and the Cauchy data f (0 ,x,v ) = f ( x,v ) , E (0 ,x ) = E ( x ) , and B (0 ,x ) = B ( x ) . We note that a similar global existence theorem for classical solutions can be proven by adaptingthe methods of Lai [11] and Degond [4], which rely only on L ∞ estimates of the density and itsderivatives. The initial data would need to satisfy v a f ∈ C k ( R ) for some a > k ≥ E ,B ∈ C ( R ), which is more restrictive than our assumptions and requires derivatives in v initially.Since we utilize L estimates instead, we are able to gain derivatives in v for the particle distribution.Of course, the methods we employ are also valid in the case D = I , and hence provide an improvedglobal existence theorem for the systems studied by Lai, Yu-Yang, and Chae, but with less regularityimposed on the initial data. Finally, Theorem 1.1 can be altered slightly to accommodate frictionterms which may arise within the formulation of the model. In this case, the Maxwell equations areunchanged and the Vlasov equation undergoes very minor alterations, taking the form ∂ t f + ˆ v ∂ x f + K · ∇ v f = ∇ v · ( D ∇ v f + vf ) . The new terms are lower order and have no additional effect on the results we present. Lai hasalready displayed this within the context of his methods [12], though the additional friction term in[12] is ˆ vf and not vf . Additionally, we note that the friction term destroys the Lorentz-invariantstructure of the equation.This paper proceeds as follows. In the next section, we will derive a priori estimates in orderto simplify the proof of the existence and uniqueness theorem. In Section 3, we prove the lemmas N. MICHALOWSKI AND S. PANKAVICH of Section 2, and then sketch the proof of global-in-time existence and uniqueness in Section 4.Throughout the paper the value
C > C may depend.Regarding norms, we will abuse notation and allow the reader to differentiate certain norms via con-text. For instance, k f ( t ) k ∞ = ess sup x ∈ R ,v ∈ R | f ( t,x,v ) | , whereas k B ( t ) k ∞ = ess sup x ∈ R | B ( t,x ) | , with analogousstatements for k·k and h· , ·i which denote the L norm and inner product, respectively. Finally, forderivative estimates we will use the notation k v γ ∂ jx ∇ kv f ( t ) k = X | α | = k k v γ ∂ jx ∂ αv f ( t ) k for γ ∈ R , j,k = 0 , , ,... , and a multi-index α = ( α ,α ) where we denote ∂ α v ∂ α v by ∂ αv . A priori estimates
Let
T > ,T ) when necessary.To begin, we will first prove a result that will allow us to estimate the particle density and itsmoments. When studying collisionless kinetic equations, one often wishes to integrate along theVlasov characteristics in order to derive estimates. However, the appearance of the Fokker-Planckterm changes the structure of the operator in (RVMFP), and the values of the distribution functionare not conserved along such curves. Hence, the following lemma (similar to that of [4]) will beutilized to estimate the particle distribution in such situations. Lemma 2.1.
Let g ∈ L ((0 ,T ) ,L ∞ ( R )) , F ∈ W , ∞ ((0 ,T ) × R ; R ) , and h ∈ L ∞ ( R ) ∩ L ( R ) begiven with D ∈ C ( R ; R × ) positive semi-definite. Assume h ( t,x,v ) is a weak solution of (2.1) ( ∂ t h + ˆ v ∂ x h + F ( t,x,v ) · ∇ v h − ∇ v · ( D ∇ v h ) = g ( t,x,v ) h (0 ,x,v ) = h ( x,v ) . Then, for every t ∈ [0 ,T ] k h ( t ) k ∞ ≤ k h k ∞ + Z t k g ( s ) k ∞ ds. Next, we state a lemma that will allow us to control the fields and moments of the particledistribution.
Lemma 2.2 (
Cone Estimate and Field Bounds ). Assume v a f ∈ L ∞ ( R ) for some a > , f possesses compact support in x , and E ,B ∈ H ( R ) . Then, for any t ∈ [0 ,T ] , x ∈ R , we have (2.2) Z t (cid:18)Z ( v ± v ) f ( s,x ± ( t − s ) ,v ) dv + 12 | E ( s,x ± ( t − s )) | + 12 | E ± B | ( s,x ± ( t − s )) (cid:19) ds ≤ C (1 + t ) , (2.3) Z t | j ( s,x ± ( t − s )) | ds ≤ C (1 + t ) , and (2.4) k E ( t ) k ∞ + k B ( t ) k ∞ ≤ C (1 + t ) . .5D RELATIVISTIC VLASOV-MAXWELL-FOKKER-PLANCK EQUATION Lemma 2.3 (
Estimates on moments ). Let the assumptions of Lemma 2.2 hold. Then, for any γ ∈ [0 ,a ] and t ∈ [0 ,T ](2.5) k v γ f ( t ) k ∞ ≤ C (1 + t ) γ and for any γ ∈ [0 ,a − and t ∈ [0 ,T ](2.6) (cid:13)(cid:13)(cid:13)(cid:13)Z v γ f ( t ) dv (cid:13)(cid:13)(cid:13)(cid:13) ∞ ≤ C (1 + t ) a . With control on moments of the density, we may bound derivatives of the field by adapting awell-known argument [6, 8] that projects these derivatives onto the backward light cone.
Lemma 2.4 (
Estimates on field derivatives ). Let the assumptions of Lemma 2.2 hold, andassume additionally that E ,B ∈ H ( R ) . Then, for any t ∈ [0 ,T ] , we have (2.7) k ∂ t E ( t ) k ∞ + k ∂ x E ( t ) k ∞ + k ∂ t B ( t ) k ∞ + k ∂ x B ( t ) k ∞ ≤ C (1 + t ) a +1) . Thus, we have C estimates on the fields without requiring any regularity of the density. Next, weutilize energy estimates to bound the density and its derivatives in L ( R ). Lemma 2.5.
Assume f ∈ L ( R ) . Then, for every t ∈ [0 ,T ] k f ( t ) k ≤ (cid:13)(cid:13) f (cid:13)(cid:13) . If additionally, v γ f ∈ L ( R ) for some γ > and the hypotheses of Lemma 2.2 hold, then k v γ f ( t ) k ≤ C T for every t ∈ [0 ,T ] . Lemma 2.6.
Assume the hypotheses of Lemma 2.4 hold with v γ +10 f ,v γ ∂ x f ∈ L ( R ) for some γ ≥ . Then for all t ∈ [0 ,T ] we have k v γ ∂ x f ( t ) k ≤ C T . Lemma 2.7.
Assume the hypotheses of Lemma 2.4 hold with v a f ∈ L ∞ ( R ) and v b − k/ ∂ kx f ∈ L ( R ) for some a > , b > and any k = 0 , , . Then, for all t ∈ [0 ,T ] k v γ ∂ xx f ( t ) k + X k =0 (cid:0) k ∂ k E ( t ) k + k ∂ k B ( t ) k (cid:1) ≤ C T for every γ ∈ [0 ,c ] , where c = min (cid:26) a − ,b − (cid:27) and ∂ k is any t or x derivative of order k . Next, we derive dissipative inequalities for lower-order derivatives of the density. Ultimately,these will be used to prove the gain in regularity achieved by Lemma 2.10.
Lemma 2.8 (
Low-order Dissipation ). Assume the hypotheses of Lemma 2.7 hold. Then, for all t ∈ (0 ,T ) , we have the following ddt k v f ( t ) k ≤ C T k v f ( t ) k − k v / ∇ v f ( t ) k N. MICHALOWSKI AND S. PANKAVICH ddt k v / ∇ v f ( t ) k ≤ C T (cid:16) k v / ∇ v f ( t ) k + k v ∂ x f ( t ) k (cid:17) − k v ∇ v f ( t ) k ddt k v / ∂ x f ( t ) k ≤ C T (cid:16) k v / ∂ x f ( t ) k + k v f ( t ) k (cid:17) − (1 − ǫ ) k v ∇ v ∂ x f ( t ) k ddt k v ∇ v ∂ x f ( t ) k ≤ C T (cid:16) k v ∇ v ∂ x f ( t ) k + k ∂ xx f ( t ) k + k v / ∇ v f ( t ) k (cid:17) − (1 − ǫ ) k v / ∇ v ∂ x f ( t ) k . The next lemma contains dissipative inequalities for higher-order derivatives of the density. Inparticular, it will allow us to trade v -derivatives of the density for those which are two orders lesswith the associated penalty of an x -derivative and a v moment. For instance, use of this lemmawill allow us to conclude the estimates k∇ v f ( t ) k . k v / ∇ v ∂ x f ( t ) k . k ∂ xx f ( t ) k ≤ C T along with the previously obtained bound on the second spatial derivative. Lemma 2.9 (
High-order dissipation ). Assume the hypotheses of Lemma 2.7 hold. Then, for all t ∈ (0 ,T ) , we have ddt (cid:13)(cid:13) v γ ∇ kv f ( t ) (cid:13)(cid:13) ≤ C T (cid:18)(cid:13)(cid:13) v γ ∇ kv f ( t ) (cid:13)(cid:13) + k − X j =1 (cid:13)(cid:13)(cid:13) v γ + j − k ∇ jv f ( t ) (cid:13)(cid:13)(cid:13) + (cid:13)(cid:13)(cid:13) v γ +1 / ∂ x ∇ k − v f ( t ) (cid:13)(cid:13)(cid:13) (cid:17) − (1 − ǫ ) (cid:13)(cid:13)(cid:13) v γ − / ∇ k +1 v f ( t ) (cid:13)(cid:13)(cid:13) for every γ ∈ [0 ,b − k/ , k = 2 , , , and ǫ > sufficiently small. Additionally, we have ddt k v / ∇ v ∂ x f ( t ) k ≤ C T (cid:18) k v / ∇ v ∂ x f ( t ) k + k v ∇ v ∂ x f ( t ) k + X j =1 (cid:13)(cid:13)(cid:13) v − j/ ∇ jv f ( t ) (cid:13)(cid:13)(cid:13) + k ∂ xx f ( t ) k (cid:19) − (1 − ǫ ) (cid:13)(cid:13) ∇ v ∂ x f ( t ) (cid:13)(cid:13) for all t ∈ (0 ,T ) and ǫ > sufficiently small. Our final lemma removes the need for regularity of the initial density in v in order to obtainderivative bounds. Hence, solutions achieve a gain in regularity where f and ∂ x f are smooth in v even for initial data which are not. Lemma 2.10.
Assume the hypotheses of Lemma 2.7 hold. Then for all t ∈ (0 ,T ) , X k =0 t k k k ! (cid:13)(cid:13)(cid:13) v (4 − k ) / ∇ kv f ( t ) (cid:13)(cid:13)(cid:13) + X k =0 t k k k ! (cid:13)(cid:13)(cid:13) v (3 − k ) / ∇ kv ∂ x f ( t ) (cid:13)(cid:13)(cid:13) ≤ C T . The gain in regularity achieved from the momentum argument is generally expected from thediffusive term. Additionally, it is possible that the solution gains regularity in its spatial argument aswell, but this feature of the system remains unknown. Precedent exists for this possibility, however,as analogous work of Herau [9] and Villani [22] has determined that this does, in fact, occur for thelinear, non-relativistic Fokker-Planck equation, as long as the given potential is sufficiently smooth. .5D RELATIVISTIC VLASOV-MAXWELL-FOKKER-PLANCK EQUATION
3. Proofs of Lemmas and Estimates
We first prove Lemma 2.1, and this will require an additional result regarding the positivity ofsolutions to the linear Fokker-Planck equations arising from positive initial data.
Proof . [Lemma 2.1] Much of our argument is adapted from ideas of Lions [13], Tartar [19], andDegond [4]. Thus, we sketch the proof of the lemma using results from these papers while correctingfor the differences in the systems, including changes in dimension and the appearance of a diffusionoperator with variable coefficients. Consider the linear equation (2.1) and define L h := ∂ t h + ˆ v ∂ x h + F · ∇ v h − ∇ · ( D ∇ v h ) . We first comment that solutions h ∈ L ∞ ((0 ,T ); L ( R )) of the equation L h = g exist for any T > g ∈ L ∞ ((0 ,T ); L ∞ ( R )), and this follows directly from either a variational argument [4], the useof Green’s functions [21], or by properties of the heat equation on a Riemannian manifold [1]. Withthis, we prove a positivity result: Lemma 3.1.
Let
T > be given. Assume h ∈ L ( R ) and g ∈ L ∞ ((0 ,T ); L ( R )) are given with h ∈ L ∞ ((0 ,T ); L ( R )) satisfying L h = g ≥ and h (0 ,x,v ) = h ( x,v ) ≥ . Then, h ( t,x,v ) ≥ for all t ≥ ,x ∈ R ,v ∈ R .Proof . [Lemma 3.1] Let λ > k∇ v · F k ∞ be given. Define u ( t,x,v ) = e − λt h ( t,x,v ) and f ( t,x,v ) = e − λt g ( t,x,v ). These functions then satisfy(3.1) ( L u + λu = fu (0 ,x,v ) = h ( x,v )Let u − ( t,x,v ) = max {− ( u ( t,x,v )) , } . In what follows, we will use the notation h· , ·i to denote the L inner product in ( t,x,v ) and k·k to denote the corresponding induced norm. It follows immediatelyfrom [4, 19] that(3.2) (cid:28) ∂u∂t + ˆ v ∂ x u, u − (cid:29) = 12 (cid:18)ZZ | u − (0 ,x,v ) | dxdv − ZZ | u − ( t,x,v ) | dxdv (cid:19) . Using this, we find h f,u − i = hL u + λu,u − i = (cid:28) ∂u∂t + ˆ v ∂ x u,u − (cid:29) + h F · ∇ v u,u − i − h∇ v · ( D ∇ v · u ) ,u − i + λ h u,u − i For the last term we split the integral into two portions, namely h u,u − i = Z T ZZ u ( t,x,v ) u − ( t,x,v ) dv dxdt = Z A u ( t,x,v ) u − ( t,x,v ) dv dxdt + Z A c u ( t,x,v ) u − ( t,x,v ) dv dxdt where A = { ( t,x,v ) : u ( t,x,v ) ≥ } . On the set A , we have u − ( t,x,v ) = 0, and the correspondingintegrals vanish. On A c we have u − ( t,x,v ) = − u ( t,x,v ) and hence Z T ZZ u ( t,x,v ) u − ( t,x,v ) dv dxdt = − Z A c | u − ( t,x,v ) | dv dxdt = − Z T ZZ | u − ( t,x,v ) | dv dxdt. N. MICHALOWSKI AND S. PANKAVICH
Hence, we find λ h u,u − i = − λ k u k After a similar analysis for the other terms above, we find h F · ∇ v u,u − i = −h F · ∇ v u − ,u − i . For the diffusion term, we proceed similarly and integrate by parts to find −h∇ v · ( D ∇ v · u ) ,u − i = h D ∇ v u, ∇ v u − i = −h D ∇ v u − , ∇ v u − i≤ , since D is positive semi-definite. Therefore, using these identities with (3.2) we have the inequality h f,u − i ≤ (cid:18)ZZ | u − (0 ,x,v ) | dxdv − ZZ | u − ( t,x,v ) | dxdv (cid:19) − h F · ∇ v u − ,u − i − λ k u − k By assumption, h ( x,v ) ≥ u − (0 ,x,v ) = 0. The first term above is then nonpositive and h f,u − i ≤ −h F · ∇ v u − ,u − i − λ k u − k . Lastly, we integrate by parts to find −h F · ∇ v u − ,u − i = − Z T ZZ F ( t,x,v ) · ∇ v (cid:18) | u − | (cid:19) ,dv dxdt = 12 Z T ZZ ∇ v · F ( t,x,v ) | u − | dv dxdt ≤ k∇ v · F k ∞ k u − k We finally have h f,u − i ≤ (cid:18) k∇ v · F k ∞ − λ (cid:19) k u − k ≤ . However, by hypothesis f ( t,x,v ) ≥ u − ( t,x,v ) ≥
0, so h f,u − i ≥
0. Therefore, itmust be the case that k u − k = 0, from which it follows that u − = 0 and hence h ( t,x,v ) ≥ g ∈ L ((0 ,T ); L ∞ ( R )) ,F ∈ W , ∞ ((0 ,T ) × R ; R ), and h ∈ L ( R ) ∩ L ∞ ( R ) be given. Assume h ∈ L ∞ ((0 ,T ); L ( R )) satisfies L h = g ( t,x,v ) in the weak sense and h (0 ,x,v ) = h ( x,v ) . Define w ( t,x,v ) := k h k ∞ + Z t k g ( s ) k ∞ ds − h ( t,x,v ) . Then, we have w (0 ,x,v ) = k h k ∞ − h ( x,v ) ≥ L w = k g ( t ) k ∞ − L h = k g ( t ) k ∞ − g ( t,x,v ) ≥ . .5D RELATIVISTIC VLASOV-MAXWELL-FOKKER-PLANCK EQUATION w ( t,x,v ) ≥
0, by which it follows that h ( t,x,v ) ≤ k h k ∞ + Z t k g ( s ) k ∞ ds for all t,x,v . Finally, taking the supremum in ( x,v ), the conclusion follows. Proof . [Lemma 2.2] To prove the cone estimate, we begin by using conservation of mass. Inte-grating the Vlasov equation over all ( x,v ) we find ddt ZZ f ( t,x,v ) dv dx = 0 . Thus, using the decay of f we find for every t ∈ [0 ,T ](3.3) ZZ f ( t,x,v ) dv dx = ZZ f ( x,v ) dv dx < ∞ . To derive the necessary energy identities, we first rewrite the Fokker-Planck term in the Vlasovequation as ∇ v · ( D ∇ v f ) = v − (cid:18) ∂ v ( v ∂ v f ) + ∂ v ( v ∂ v f ) + 2 v v ∂ v v f + ∆ v f (cid:19) . Then, multiplying the Vlasov equation by v and integrating in v , the Fokker-Planck term becomes Z v ∇ v · ( D ∇ v f ) dv = Z (cid:0) ∂ v ( v ∂ v f ) + ∂ v ( v ∂ v f ) + 2 v v ∂ v v f + ∆ v f (cid:1) = Z v v ∂ v v f dv dv = 2 Z f dv after two integrations by parts. Hence, using the divergence structure of the Vlasov equation, wearrive at the local energy identity(3.4) ∂ t e + ∂ x m = 2 Z f ( t,x,v ) dv where e ( t,x ) = Z v f ( t,x,v ) dv + 12 (cid:0) | E ( t,x ) | + | B ( t,x ) | (cid:1) and m ( t,x ) = Z v f ( t,x,v ) dv + E ( t,x ) B ( t,x ) . Since f has compact support in x with suitable decay in v , we find v f ∈ L ( R ). We can thenintegrate (3.4) over all space to deduce the global energy identity ddt Z e ( t,x ) dx = 2 ZZ f ( x,v ) dx dv N. MICHALOWSKI AND S. PANKAVICH whence we find Z e ( t,x ) dx ≤ C (1 + t )for all t ∈ [0 ,T ) and E ,E ,B ∈ L ∞ ([0 ,T ]; L ( R )).To derive local estimates, we fix ( t,x ), integrate (3.4) along the backwards cone in space-time { ( s,y ) ∈ (0 ,t ) × R : | y − x | ≤ t − s } , and use Green’s Theorem to find Z t (cid:20) ( e + m )( s,x + t − s ) + ( e − m )( s,x − t + s ) (cid:21) ds = Z x + tx − t e (0 ,y ) dy + 2 Z t Z x + t − sx − t + s Z f ( s,y,v ) dvdyds. Using the positivity of the mass and energy, the assumptions on the data, and conservation of mass,the right side satisfies Z x + tx − t e (0 ,y ) dy + 2 Z t Z x + t − sx − t + s Z f ( s,y,v ) dvdyds ≤ Z e (0 ,y ) dy + 2 Z t ZZ f ( s,y,v ) dvdy ds = Z e (0 ,y ) dy + 2 Z t (cid:18)ZZ f ( y,v ) dv dy (cid:19) ds ≤ C (1 + t )and this yields the first result.The other conclusions of the lemma then follow from the first. More specifically, we find(3.5) v ± v = v − v v ∓ v = 1 + v v ∓ v ≥ | v | v ∓ v ≥ | v | v = | ˆ v | and by (2.2) Z t | j ( s,x ± ( t − s )) | ds ≤ Z t Z | ˆ v | f ( s,x ± ( t − s ) ,v ) dv ds ≤ Z t Z ( v ± v ) f ( s,x ± ( t − s ) ,v ) dv ds ≤ C (1 + t ) . Next, we represent the fields in terms of the source j in the associated transport equations.Either adding or subtracting the equations for E and B in (RVMFP) yields ∂ t ( E ± B ) ± ∂ x ( E ± B ) = − j . Thus, we can write the sum or difference of the fields in terms of initial data and an integral of j along one side of the backwards cone, namely(3.6) ( E ± B )( t,x ) = ( E ± B )(0 ,x ∓ t ) − Z t j ( s,x ∓ ( t − s )) ds. Then, in view of the previous conclusion of the lemma and the assumption on the initial fields, wefind k ( E ± B )( t ) k ∞ ≤ C (1 + t ) .5D RELATIVISTIC VLASOV-MAXWELL-FOKKER-PLANCK EQUATION E ( t,x ) = 12 ( E + B )( t,x ) + 12 ( E − B )( t,x ) , and similarly for B , it follows that k E ( t ) k ∞ and k B ( t ) k ∞ are controlled by this same quantity.Finally, control of E follows from conservation of mass and the assumption on the backgrounddensity. Integrating the equation for E and using the assumption on E (0 ,x ) yields E ( t,x ) = Z x −∞ ρ ( t,y ) dy and we find for x ∈ R | E ( t,x ) | ≤ ZZ f ( t,y,v ) dv dy + k φ k ≤ C. The second conclusion of the theorem then follows by adding the field estimates.
Proof . [Lemma 2.3] We begin by noting that v is an eigenvector of D since(3.7) Dv = v − [ I + v × v ] = v − [ v + ( v · v ) v ] = v − (1 + | v | ) v = v v. Now, let γ ≥ v γ , we find(3.8) ∂ t ( v γ f ) + ∂ x (ˆ v v γ f ) + ∇ v · [ Kv γ f ] − ∇ v ( v γ ) · Kf = v γ ∇ v · [ D ∇ v f ] . We first compute the right side of this equation. Using (1.1) and (3.7), we find v γ ∇ v · [ D ∇ v f ] = ∇ v · [ v γ D ∇ v f ] − ∇ v ( v γ ) · D ∇ v f = ∇ v · [ D ∇ v ( v γ f )] − ∇ v · [ D ∇ v ( v γ ) f ] − ∇ v ( v γ ) · D ∇ v f = ∇ v · [ D ∇ v ( v γ f )] − γ ∇ v · [ v γ − f Dv ] − γv γ − v · D ∇ v f = ∇ v · [ D ∇ v ( v γ f )] − γ ∇ v · [ v γ − f v ] − γv γ − Dv · ∇ v f = ∇ v · [ D ∇ v ( v γ f )] − γ [( γ − v γ − | v | + v γ − v · ∇ v f + 2 v γ − f ] − γv γ − v · ∇ v f = ∇ v · [ D ∇ v ( v γ f )] − γ ( γ − v γ − | v | − γv γ − v · ∇ v f − γv γ − f. The next to last term here can be rewritten as − γv γ − v · ∇ v f = − γv − v · ∇ v ( v γ f ) + 2 γ v γ − v · vf = − γ ˆ v · ∇ v ( v γ f ) + 2 γ v γ − (cid:18) | v | | v | (cid:19) f. Combining this with (3.8) yields(3.9) ∂ t ( v γ f ) + ∂ x (ˆ v v γ f ) + ∇ v · [ Kv γ f ] − ∇ v ( v γ ) · Kf = ∇ v · [ D ∇ v ( v γ f )] − γ ( γ − v γ − | v | − γ ˆ v · ∇ v ( v γ f ) + 2 γ v γ − (cid:18) | v | | v | (cid:19) f − γv γ − f. Thus, if we rearrange terms and use the operator V h := ∂ t h + ˆ v ∂ x h + ( K + 2 γ ˆ v ) · ∇ v h − ∇ · ( D ∇ v h ) , we have(3.10) V ( v γ f ) = g ( t,x,v )2 N. MICHALOWSKI AND S. PANKAVICH where g ( t,x,v ) = ∇ v ( v γ ) · Kf − γ ( γ − v γ − | v | + 2 γ v γ − (cid:18) | v | | v | (cid:19) f − γv γ − f. Estimating g , we find | g ( t,x,v ) | ≤ γv γ − | ˆ v · K | f + γ ( γ − v γ − | ˆ v | + 2 γ v γ − f + 2 γv γ − f ≤ γv γ − ( k E ( t ) k ∞ + k B ( t ) k ∞ ) f + Cv γ − f ≤ C (1 + t ) k v γ − f ( t ) k ∞ Since the coefficients of V satisfy the hypotheses of Lemma 2.1, we use this result with h = v γ f , L = V , and g defined as above. This yields k v γ f ( t ) k ∞ ≤ k v γ f k ∞ + C Z t (1 + s ) k v γ − f ( s ) k ∞ ds. Of course, the same lemma can be invoked with h = f and g = 0 using the Vlasov equation in orderto find k f ( t ) k ∞ ≤ k f k ∞ for all t ∈ [0 ,T ]. With this bound on the particle distribution, which represents the γ = 0 caseabove, we use induction to bound k v γ f ( t ) k ∞ for any γ ≥ k v γ f k ∞ is finite, and the firstconclusion follows.The second conclusion is a straightforward application of the first. Namely, for any γ ∈ [0 ,a − Z v γ f ( t,x,v ) dv ≤ k v a f ( t ) k ∞ (cid:18)Z v γ − a dv (cid:19) ≤ C (1 + t ) a since γ − a < − Proof . [Lemma 2.4] We begin by noting that E can be handled separately from the other fieldterms, since by Lemma 2.3 ∂ x E = Z f ( t,x,v ) dv + φ ( x ) ≤ (cid:13)(cid:13)(cid:13)(cid:13)Z f ( t ) dv (cid:13)(cid:13)(cid:13)(cid:13) ∞ + k φ k ∞ ≤ C (1 + t ) a . The same bound holds using this argument for ∂ t E = j since | ˆ v | ≤ E and B as in the proof of Lemma 2.2. We willconsider only x -derivatives and the term ( E + B )( t,x ), but note that the same computations belowcan be done for ( E − B )( t,x ) and time derivatives. Using (3.6) and differentiating in x , we find ∂ x ( E + B )( t,x ) = ( E + B ) ′ (0 ,x − t ) − Z t Z ˆ v ∂ x f ( s,x − ( t − s ) ,v ) dvds. At this point, we wish to project ∂ x onto the directions of “good” derivatives included in the fieldrepresentation. This idea was used by Glassey and Schaeffer [6] for the collisionless problem andoriginally developed for the three-dimensional relativistic Vlasov-Maxwell system by Glassey andStrauss [8]. We introduce the operators ( T + = ∂ t + ∂ x S = ∂ t + ˆ v ∂ x .5D RELATIVISTIC VLASOV-MAXWELL-FOKKER-PLANCK EQUATION x -derivatives on the density as ∂ x = 11 − ˆ v ( T + − S ) . Contrastingly, the operator T − = ∂ t − ∂ x would be needed for an estimate of E − B . Using theVlasov equation, we can write Sf = ∂ t f + ˆ v ∂ x f = −∇ v ( Kf ) + ∇ v · ( D ∇ v f )so that integrating by parts yields Z t Z ˆ v ∂ x f ( s,x − t + s,v ) dvds = Z t Z ˆ v − ˆ v ( T + f − Sf )( s,x − t + s,v ) dvds = Z t Z ˆ v − ˆ v (cid:20) dds ( f ( s,x − t + s,v )) + ∇ v · ( Kf )( s,x − t + s,v ) −∇ v · ( D ∇ v f )( s,x − t + s,v ) (cid:21) dvds = Z ˆ v − ˆ v [ f ( t,x,v ) − f ( x − t,v )] dv + Z t Z ˆ v − ˆ v ∇ v · ( Kf )( s,x − t + s,v ) dvds − Z t Z ˆ v − ˆ v ∇ v · ( D ∇ v f )( s,x − t + s,v ) (cid:21) dvds =: I + I + III
The first term is easily estimated since moments of the density are bounded. We use (3.5) and a > I = Z v v − v [ f ( t,x,v ) − f ( x − t,v )] dv ≤ Z | v | ( v + v )1 + v f ( t,x,v ) dv ≤ k v a f ( t ) k ∞ Z v − a dv ≤ C (1 + t ) a . To estimate II , we first integrate by parts to find Z t Z ˆ v − ˆ v ∇ v · ( Kf )( s,x − t + s,v ) dvds = − Z t Z ∇ v (cid:18) ˆ v − ˆ v (cid:19) · ( Kf )( s,x − t + s,v ) dvds + lim | v |→∞ Z t ˆ v − ˆ v ( Kf )( s,x − t + s,v ) · v ⊥ | v | ds where v ⊥ = h v , − v i . The boundary term vanishes on (0 ,T ) because K and v a f are bounded in L ∞ , and thus moments can be used to introduce sufficient decay in v . For the remaining term, wecompute the gradient ∇ v (cid:18) ˆ v − ˆ v (cid:19) = (cid:28) ˆ v v − v , v − v − ˆ v v ( v − v ) (cid:29) N. MICHALOWSKI AND S. PANKAVICH
The first term is bounded since (3.5) implies | ˆ v | ≤ v − v . Similarly, one can show the second termis bounded by 3 v using (3.5). Hence, using a >
3, we have II ≤ Z t k K ( s ) k ∞ k v a f ( s ) k ∞ (cid:18)Z v − a dv (cid:19) ds ≤ C (1 + t ) a +1) Finally, we use the symmetry of D and integrate by parts twice in III to find Z t Z ˆ v − ˆ v ∇ v · ( D ∇ v f ) (cid:12)(cid:12)(cid:12)(cid:12) ( s,x − t + s,v ) dvds = Z t Z ∇ v · (cid:20) D ∇ v (cid:18) ˆ v − ˆ v (cid:19)(cid:21) f ( s,x − t + s,v ) dvds + lim | v |→∞ Z t ˆ v − ˆ v ∇ v f ( s,x − t + s,v ) · Dv ⊥ | v | ds − lim | v |→∞ Z t ∇ v (cid:18) ˆ v − ˆ v (cid:19) · Dv ⊥ | v | f ( s,x − t + s ) ,v ) ds For the boundary terms, we use the property Dv ⊥ = v − v ⊥ so that an extra order of decay appears,and these terms vanish on (0 ,T ). To estimate the remaining term, a long computation yields thebound (cid:12)(cid:12)(cid:12)(cid:12) ∇ v · (cid:20) D ∇ v (cid:18) ˆ v − ˆ v (cid:19)(cid:21)(cid:12)(cid:12)(cid:12)(cid:12) ≤ . Thus, we find for a > III ≤ Z t Z f ( s,x − t + s,v ) dvds ≤ (cid:18)Z t k v a f ( s ) k ∞ ds (cid:19)(cid:18)Z v − a dv (cid:19) ≤ C (1 + t ) a +1 . Combining the estimates and using the regularity of the initial fields, each term is controlled by C (1 + t ) a +1) . Thus, the bound on k ∂ x ( E + B )( t ) k ∞ follows, as does the conclusion of the lemma. Proof . [Lemma 2.5] We proceed by using energy estimates. We calculate:12 ddt k f ( t ) k = h− ˆ v ∂ x f − K · ∇ v f + ∇ v · D ∇ v f,f i = −h ˆ v ∂ x f,f i − h K · ∇ v f,f i + h∇ v · D ∇ v f,f i . Notice that the first two terms are pure derivatives in x and v , respectively. Thus, h ˆ v ∂ x f,f i = 12 ZZ ∂ x (cid:0) ˆ v f (cid:1) dv dx = 0and h K · ∇ v f,f i = 12 ZZ ∇ v · (cid:0) Kf (cid:1) dv dx = 0 . Finally h∇ v · D ∇ v f,f i = − (cid:13)(cid:13)(cid:13) D / ∇ v f ( t ) (cid:13)(cid:13)(cid:13) . Hence ddt k f ( t ) k ≤ v γ and proceed in the same manner12 ddt k v γ f ( t ) k = −h v γ ˆ v ∂ x f,v γ f i − h v γ K · ∇ v f,v γ f i + h v γ ∇ v · D ∇ v f,v γ f i .5D RELATIVISTIC VLASOV-MAXWELL-FOKKER-PLANCK EQUATION −h v γ K · ∇ v f,v γ f i = ZZ v γ K · ∇ v (cid:18) f (cid:19) dv dx = − γ ZZ v γ − (ˆ v · K ) 12 f dv dx Hence, this yields |h v γ K · ∇ v f,v γ f i| ≤ C k K ( t ) k ∞ (cid:13)(cid:13)(cid:13) v γ − f ( t ) (cid:13)(cid:13)(cid:13) . For the last term we integrate by parts and use the symmetry of D , h v γ ∇ v · ( D ∇ v f ) ,v γ f i = − γ D v γ − ˆ v · D ∇ v f,f E − (cid:13)(cid:13)(cid:13) v γ D / ∇ v f ( t ) (cid:13)(cid:13)(cid:13) = − γ D v γ − v · ∇ v f,f E − (cid:13)(cid:13)(cid:13) v γ D / ∇ v f ( t ) (cid:13)(cid:13)(cid:13) = − γ ZZ v γ − v · ∇ v (cid:18) f (cid:19) dv dx − (cid:13)(cid:13)(cid:13) v γ D / ∇ v f ( t ) (cid:13)(cid:13)(cid:13) We may drop the latter term. After integrating by parts again in v we can bound the former termby C (cid:13)(cid:13)(cid:13) v γ − f ( t ) (cid:13)(cid:13)(cid:13) . Putting the estimates together and using the field bound of Lemma 2.2, we find12 ddt k v γ f ( t ) k ≤ C (1 + t ) (cid:13)(cid:13)(cid:13) v γ − f ( t ) (cid:13)(cid:13)(cid:13) . Using the first conclusion of the lemma for the γ = 1 / k v γ f ( t ) k ≤ C (1 + t ) γ (cid:13)(cid:13) v γ f (cid:13)(cid:13) ≤ C T . for every γ ≥ Proof . [Lemma 2.6] To begin, we estimate derivatives of the density in x , and first define somenotation. Since density derivatives will depend upon field derivatives, we let F ( t ) = k E ( t ) k ∞ + k B ( t ) k ∞ + k ∂ x E ( t ) k ∞ + k ∂ x B ( t ) k ∞ and note that kFk ∞ ≤ C T by Lemmas 2.2 and 2.4. We differentiate the Vlasov equation in x ,multiply by v γ ∂ x f and integrate to yield12 ddt k v γ ∂ x f ( t ) k = − ZZ ∂ x (cid:18)
12 ˆ v v γ | ∂ x f | (cid:19) dv dx − ZZ v γ ∂ x f ∇ v · ( ∂ x Kf + K∂ x f ) dx dv + ZZ v γ ∂ x f ∇ v · ( D ∇ v ∂ x f ) dv dx = ZZ (cid:20) v γ − (2 γ ˆ v∂ x f + v ∇ v ∂ x f ) · ( ∂ x Kf ) + 12 v γ K · ∇ v ( | ∂ x f | ) (cid:21) dx dv − ZZ v γ − (2 γ ˆ v∂ x f + v ∇ v ∂ x f ) · ( D ∇ v ∂ x f ) dv dx =: I + II N. MICHALOWSKI AND S. PANKAVICH
Here, we have integrated by parts in v and used the divergence-free structure of K , as well as, thefact that the transport term above is a pure x -derivative along with the compact x -support of theparticle distribution. Using Cauchy’s inequality with ǫ we find for any ǫ > I ≤ C ZZ h k ∂ x K ( t ) k ∞ (cid:16) v γ − | ∂ x f | f + v γ |∇ v ∂ x f | f (cid:17) + k K ( t ) k ∞ v γ − | ∂ x f | i dxdv ≤ C F ( t ) (cid:18)(cid:13)(cid:13)(cid:13) v γ − ∂ x f ( t ) (cid:13)(cid:13)(cid:13) + (cid:13)(cid:13)(cid:13) v γ − f ( t ) (cid:13)(cid:13)(cid:13) + ǫ (cid:13)(cid:13)(cid:13) v γ − ∇ v ∂ x f ( t ) (cid:13)(cid:13)(cid:13) + 1 ǫ (cid:13)(cid:13)(cid:13) v γ + f ( t ) (cid:13)(cid:13)(cid:13) (cid:19) ≤ C T (cid:18)(cid:13)(cid:13)(cid:13) v γ − ∂ x f ( t ) (cid:13)(cid:13)(cid:13) + (cid:13)(cid:13)(cid:13) v γ + f ( t ) (cid:13)(cid:13)(cid:13) + ǫ (cid:13)(cid:13)(cid:13) v γ − ∇ v ∂ x f ( t ) (cid:13)(cid:13)(cid:13) (cid:19) Then, the symmetry of D along with D ˆ v = v implies II = − ZZ v γ − (2 γ ˆ v∂ x f + v ∇ v ∂ x f ) · ( D ∇ v ∂ x f ) dv dx = − γ ZZ v γ − v · ∇ v ( | ∂ x f | ) dv dx − k v γ D / ∇ v ∂ x f k ≤ C (cid:13)(cid:13)(cid:13) v γ − ∂ x f ( t ) (cid:13)(cid:13)(cid:13) − (cid:13)(cid:13)(cid:13) v γ − ∇ v ∂ x f ( t ) (cid:13)(cid:13)(cid:13) . Combining I and II , we use Lemma 2.5 to find for ǫ sufficiently small ddt k v γ ∂ x f ( t ) k ≤ C T (cid:18)(cid:13)(cid:13)(cid:13) v γ − ∂ x f ( t ) (cid:13)(cid:13)(cid:13) + (cid:13)(cid:13)(cid:13) v γ + f ( t ) (cid:13)(cid:13)(cid:13) (cid:19) ≤ C T (cid:18) (cid:13)(cid:13)(cid:13) v γ − ∂ x f ( t ) (cid:13)(cid:13)(cid:13) (cid:19) . If we compute this for γ = 0 and use the bound on F , the result is just ddt k ∂ x f ( t ) k ≤ C T k f ( t ) k which, by Lemma 2.5, leads to k ∂ x f ( t ) k ≤ C T (cid:16)(cid:13)(cid:13) ∂ x f (cid:13)(cid:13) + (cid:13)(cid:13) f (cid:13)(cid:13) (cid:17) ≤ C T for every t ∈ [0 ,T ]. Then, by induction, for every γ ≥ v γ ∂ x f ∈ L ( R ) we have(3.11) k v γ ∂ x f ( t ) k ≤ C T (cid:16) (cid:13)(cid:13) v γ ∂ x f (cid:13)(cid:13) (cid:17) ≤ C T for all t ∈ [0 ,T ]. Proof . [Lemma 2.7] To begin, we estimate second derivatives of the density. These involvesecond derivatives of the fields, which must be estimated in L rather than L ∞ . As before, denote F ( t ) = k E ( t ) k ∞ + k B ( t ) k ∞ + k ∂ x E ( t ) k ∞ + k ∂ x B ( t ) k ∞ and now let G ( t ) = k ∂ xx E ( t ) k + k ∂ xx B ( t ) k . We differentiate the Vlasov equation twice in x , multiply by v γ ∂ xx f and integrate to yield .5D RELATIVISTIC VLASOV-MAXWELL-FOKKER-PLANCK EQUATION ddt k v γ ∂ xx f ( t ) k = − ZZ ∂ x (cid:18)
12 ˆ v v γ | ∂ xx f | (cid:19) dv dx − ZZ v γ ∂ xx f ∇ v · ( ∂ xx Kf + 2 ∂ x K∂ x f + K∂ xx f ) dx dv + ZZ v γ ∂ xx f ∇ v · ( D ∇ v ∂ xx f ) dv dx = ZZ (cid:20) v γ − (2 γ ˆ v∂ xx f + v ∇ v ∂ xx f ) · ( ∂ xx Kf + ∂ x K∂ x f ) + 12 v γ K · ∇ v ( | ∂ xx f | ) (cid:21) dxdv − ZZ v γ − (2 γ ˆ v∂ xx f + v ∇ v ∂ xx f ) · ( D ∇ v ∂ xx f ) dv dx = I + II As before, we have integrated by parts in v and used the compact x -support of the particle distribu-tion. With this, bounds for I follow as in Lemma 2.6 with the exception of terms involving ∂ xx K .More specifically, we use Lemmas 2.2, 2.5, 2.6, Cauchy-Schwarz, and Cauchy’s inequality to find I ≤ C G ( t ) / "Z (cid:18)Z ( v γ − | ∂ xx f | + v γ |∇ v ∂ xx f | ) f dv (cid:19) dx / + C F ( t ) (cid:18) k v γ − ∂ xx f ( t ) k + k ∂ x f ( t ) k + ǫ k v γ − ∇ v ∂ xx f ( t ) k + 1 ǫ k v γ + ∂ x f ( t ) k (cid:19) ≤ C G ( t ) / (cid:20)Z (cid:18)Z v γ − | ∂ xx f | dv (cid:19) · (cid:18)Z v γ − f dv (cid:19) dx + Z (cid:18)Z v γ − |∇ v ∂ xx f | dv (cid:19) · (cid:18)Z v γ +10 f dv (cid:19) dx (cid:21) / + C T (cid:18) (cid:13)(cid:13)(cid:13) v γ − ∂ xx f ( t ) (cid:13)(cid:13)(cid:13) + ǫ (cid:13)(cid:13)(cid:13) v γ − ∇ v ∂ xx f ( t ) (cid:13)(cid:13)(cid:13) (cid:19) ≤ C T (cid:18) (cid:18) ǫ (cid:19) G ( t ) + (cid:13)(cid:13) f (cid:13)(cid:13) ∞ (cid:13)(cid:13)(cid:13)(cid:13)Z v γ +10 f ( t ) (cid:13)(cid:13)(cid:13)(cid:13) ∞ (cid:20)(cid:13)(cid:13)(cid:13) v γ − ∂ xx f ( t ) (cid:13)(cid:13)(cid:13) + ǫ (cid:13)(cid:13)(cid:13) v γ − ∇ v ∂ xx f ( t ) (cid:13)(cid:13)(cid:13) (cid:21)(cid:19) ≤ C T (cid:18) G ( t ) + (cid:13)(cid:13)(cid:13) v γ − ∂ xx f ( t ) (cid:13)(cid:13)(cid:13) + ǫ (cid:13)(cid:13)(cid:13) v γ − ∇ v ∂ xx f ( t ) (cid:13)(cid:13)(cid:13) (cid:19) for 2 γ < a −
3. We estimate II exactly as before to find II ≤ C (cid:13)(cid:13)(cid:13) v γ − ∂ xx f ( t ) (cid:13)(cid:13)(cid:13) − (cid:13)(cid:13)(cid:13) v γ − ∇ v ∂ xx f ( t ) (cid:13)(cid:13)(cid:13) . Hence, combining I and II , we find for ǫ small enough, ddt k v γ ∂ xx f ( t ) k ≤ C T (cid:18) G ( t ) + (cid:13)(cid:13)(cid:13) v γ − ∂ xx f ( t ) (cid:13)(cid:13)(cid:13) (cid:19) ≤ C T (cid:16) G ( t ) + k v γ ∂ xx f ( t ) k (cid:17) . By Gronwall’s Lemma we have(3.12) k v γ ∂ xx f ( t ) k ≤ C T (1 + G ( t ))for all t ∈ [0 ,T ] and γ < min (cid:26) a − ,b − (cid:27) .Before turning to field derivatives, we will need a way to relate the current density and itsderivatives to that of the particle distribution. So, for k = 0 , , k ∂ kx j ( t ) k ≤ Z (cid:18)Z | ∂ kx f | dv (cid:19) dx ≤ (cid:18)ZZ v γ | ∂ kx f | dvdx (cid:19)(cid:18)Z v − γ dv (cid:19) ≤ C k v γ ∂ kx f ( t ) k . N. MICHALOWSKI AND S. PANKAVICH for γ >
1. Additionally, we will need to bound ∂ t j in L , which can be done using (3.11). Using theVlasov equation and integrating by parts in v , we see ∂ t j = − Z ˆ v ˆ v ∂ x f + Z ∇ v (ˆ v ) · Kf dv + Z ∇ v · [ D ∇ v (ˆ v )] f dv ≤ Z | ∂ x f | dv + Z (1 + k K ( t ) k ∞ ) f dv Thus, it follows by Lemmas 2.2 and 2.5 that(3.14) k ∂ t j ( t ) k ≤ C T (cid:0) k v γ ∂ x f ( t ) k (cid:1) ≤ C T . for every t ∈ [0 ,T ] where γ > ∂ x E = ρ , we find for all t ∈ [0 ,T ] k ∂ x E ( t ) k ≤ C Z (cid:18)Z f ( t,x,v ) dv (cid:19) dx + k φ k ≤ C (cid:18) k v γ f ( t ) k (cid:18)Z v − γ dv (cid:19)(cid:19) ≤ C T by Lemma 2.5 where γ > ∂ xx E and use φ ∈ C c so that by Lemma 2.6 with γ > k ∂ xx E ( t ) k ≤ C (cid:18) k φ ′ k + k v γ ∂ x f ( t ) k (cid:18)Z v − γ dv (cid:19)(cid:19) ≤ C T . Using the transport equations of (RVMFP) for E and B , it follows that these quantities andtheir derivatives satisfy wave equations with derivatives of j as source terms, namely ✷ B = ∂ x j , ✷ E = − ∂ t j . Using standard L estimates for the wave equation, we multiply the first equation by ∂ t B andintegrate in x . After integrating by parts and using Cauchy’s inequality, this yields ddt (cid:0) k ∂ t B ( t ) k + k ∂ x B ( t ) k (cid:1) ≤ k ∂ x j ( t ) k + k ∂ t B ( t ) k . Using Lemma 2.6 with (3.13), this becomes ddt (cid:0) k ∂ t B ( t ) k + k ∂ x B ( t ) k (cid:1) ≤ C T (cid:0) k ∂ t B ( t ) k (cid:1) which, by Gronwall’s inequality, yields k ∂ t B ( t ) k + k ∂ x B ( t ) k ≤ C T . Since ∂ x E = − ∂ t B and ∂ x E = − ∂ x B − j , the same bounds hold for derivatives of E .We may now proceed in a similar fashion for second derivatives of the field. From the fieldequations, we see ✷ ( ∂ x B ) = ∂ xx j .5D RELATIVISTIC VLASOV-MAXWELL-FOKKER-PLANCK EQUATION ddt (cid:0) k ∂ tx B ( t ) k + k ∂ xx B ( t ) k (cid:1) ≤ k ∂ xx j ( t ) k + k ∂ tx B ( t ) k . Since ∂ tx B = − ∂ xx E , this is equivalent to ddt (cid:0) k ∂ xx E ( t ) k + k ∂ xx B ( t ) k (cid:1) ≤ k ∂ xx j ( t ) k + k ∂ xx E ( t ) k . Using (3.12) and (3.13), this implies G ′ ( t ) ≤ C T (1 + G ( t )) , and using Gronwall’s inequality and the assumption on the initial fields, we find k ∂ xx E ( t ) k + k ∂ xx B ( t ) k ≤ C T . With this, (3.12) provides an a priori bound on k v γ ∂ xx f ( t ) k for all t ∈ [0 ,T ]. Since ✷ B = − ∂ x j , we see that ∂ tt B = ∂ xx B − ∂ x j ∈ L ∞ ([0 ,T ]; L ( R )) by (3.11) and (3.13). Then, ∂ tx E = − ∂ tt B ∈ L ∞ ([0 ,T ]; L ( R )) and ∂ tx B = − ∂ xx E ∈ L ∞ ([0 ,T ]; L ( R )), and finally ∂ tt E = − ∂ tx B − ∂ t j ∈ L ∞ ([0 ,T ]; L ( R )) by (3.14). Proof . [Lemma 2.8] Throughout, we will use v ≥ R γ ( v ) to generically denote a function of v such that | R γ ( v ) | ≤ C T v γ , but the specific value of R γ ( v )may change from line to line. We first estimate moments of the density. Computing12 ddt k v f ( t ) k = ZZ v f [ − ˆ v ∂ x f − K · ∇ v f + ∇ v · ( D ∇ v f )] dvdx = I + II + III.
The first term vanishes as it is a pure x -derivative. For II , we integrate by parts and use the fieldbounds of Lemma 2.2 so that II = − ZZ v ∇ v · ( Kf ) dvdx = 4 ZZ v ˆ v · Kf dvdx ≤ C T k v / f ( t ) k . To estimate
III , we integrate by parts, then use the property D ˆ v = v and integrate by parts againin the first term. Also, we use (1.1) in the second term to find III = − ZZ ∇ v ( v f ) · D ∇ v f dvdx = − ZZ (4 v ˆ vf + v ∇ v f ) · D ∇ v f dvdx ≤ C k v / f ( t ) k − k v D / ∇ v f ( t ) k . ≤ C k v / f ( t ) k − k v / ∇ v f ( t ) k . Combining the estimates, the first inequality follows.0
N. MICHALOWSKI AND S. PANKAVICH
Next, we let ∂ v be either first-order derivative and compute12 ddt k v / ∂ v f ( t ) k = ZZ v ∂ v f [ − ˆ v ∂ v ∂ x f − ∂ v ˆ v ∂ x f − ∂ v K · ∇ v f − K · ∇ v ∂ v f + ∇ v · (( ∂ v D ) ∇ v f ) + ∇ v · ( D ∇ v ∂ v f )] dvdx = I + II + III.
The first term in I vanishes as before and thus using Cauchy’s inequality I = − ZZ v R − ( v ) ∂ v f ∂ x f dvdx ≤ C (cid:0) k v ∂ v f ( t ) k + k v ∂ x f ( t ) k (cid:1) . For II , we use the field bounds of Lemma 2.2 and integrate by parts in the second term to find II = − ZZ v ∂ v f [ R − ( v ) ∇ v f + K · ∇ v ∂ v f ] dvdx ≤ C T k v / ∇ v f ( t ) k . Finally, in
III we integrate by parts while using D ˆ v = v and boundedness of derivatives of D to find III = − ZZ (cid:2) v ˆ v∂ v f + v ∇ v ∂ v f (cid:3) [ ∂ v D ∇ v f + D ∇ v ∂ v f ] dvdx = − ZZ (cid:20) v R ( v ) ∂ v f ∇ v f + 12 v R ( v ) ∂ v |∇ v f | + 32 v D ˆ v · ∇ v | ∂ v f | + v ∇ v ∂ v f · D ∇ v ∂ v f (cid:21) dvdx ≤ C k v ∇ v f ( t ) k − k v ∇ v ∂ v f ( t ) k . We collect these estimates, use k ∂ v f ( t ) k ≤ k∇ v f ( t ) k , and then sum over first-order v -derivativesto arrive at an estimate on ddt k v / ∇ v f ( t ) k . With this, the second result follows.The final two results concern x -derivatives of the density, so we first compute12 ddt k v / ∂ x f ( t ) k = ZZ v ∂ x f [ − ˆ v ∂ xx f − ∇ v · ( ∂ x Kf ) − K · ∇ v ∂ x f + ∇ v · ( D ∇ v ∂ x f )] dvdx = I + II + III + IV.
As in the other estimates, I vanishes. For II , we integrate by parts and use the bounds on fieldderivatives provided by Lemma 2.4 and Cauchy’s inequality to find II ≤ C T (cid:18) k v / ∂ x f ( t ) k + (cid:18) ǫ (cid:19) k v f ( t ) k + ǫ k v ∇ v ∂ x f ( t ) k (cid:19) . We note that for ǫ sufficiently small, the last term can be controlled by the final term arising in IV below. Next, we integrate by parts in III to find
III = − ZZ v ∇ v · ( K | ∂ x f | ) dvdx = 3 ZZ v ˆ v · ( K | ∂ x f | ) dvdx ≤ C T k v ∂ x f ( t ) k . In the last term, we again integrate by parts and use D ˆ v = v along with (1.1) to find IV = − ZZ (cid:0) v ˆ v∂ x f + v ∇ v ∂ x f (cid:1) · D ∇ v ∂ x f dvdx ≤ k v ∂ x f ( t ) k − k v ∇ v ∂ x f ( t ) k . .5D RELATIVISTIC VLASOV-MAXWELL-FOKKER-PLANCK EQUATION ∂ v be either first-order derivative and compute12 ddt k v ∂ v ∂ x f ( t ) k = ZZ v ∂ v ∂ x f [ − ˆ v ∂ v ∂ xx f − ∂ v ˆ v ∂ xx f − ∂ v K · ∇ v ∂ x f − ∂ x K · ∇ v ∂ v f − ∂ v ∂ x K · ∇ v f − K · ∇ v ∂ v ∂ x f + ∇ v · (( ∂ v D ) ∇ v ∂ x f ) + ∇ v · ( D ∇ v ∂ v ∂ x f )] dvdx = I + II + III.
Because the first term of I vanishes yet again, we use Cauchy’s inequality to find I = − ZZ v R − ( v ) ∂ v ∂ x f ∂ xx f dvdx ≤ C T (cid:0) k v ∂ v ∂ x f ( t ) k + k ∂ xx f ( t ) k (cid:1) . To estimate II , we integrate by parts in the third and fourth terms below and use the bounds onfields and field derivatives (Lemmas 2.2 and 2.4) as well as Cauchy’s inequality so that II = − ZZ v ∂ v ∂ x f [ ∂ v K · ∇ v ∂ x f + ∂ x K · ∇ v ∂ v f + ∂ v ∂ x K · ∇ v f + K · ∇ v ∂ v ∂ x f ] dvdx = − ZZ v R − ( v ) ∂ v ∂ x f [ ∇ v ∂ x f + ∇ v f ] dvdx − ZZ v ∂ v ∂ x f ∇ v · ( ∂ x K∂ v f ) dvdx − ZZ v ∇ v · ( K∂ v ∂ x f ) dvdx ≤ C T (cid:18) k v ∇ v ∂ x f ( t ) k + (cid:18) ǫ (cid:19) k v ∇ v f ( t ) k + ǫ k v ∇ v ∂ v ∂ x f ( t ) k (cid:19) We note that for ǫ sufficiently small, the last term can be controlled by the final term arising in III below. Lastly, we estimate
III exactly as in the proof of the second inequality, but for ∂ x f insteadof f , to find III ≤ C k v ∇ v ∂ x f ( t ) k − k v ∇ v ∂ v ∂ x f ( t ) k . With this, we combine the estimates, sum over all first-order v -derivatives, and proceed as for thesecond inequality, which yields the final estimate. We note that throughout we have rescaled ǫ > C T > Proof . [Lemma 2.9] For each result the proof is made more difficult because of the structure of D and its derivatives, while in the case D = I derivatives commute with the Fokker-Planck operatorand the computations are straightforward. Let k = 2 , , t ∈ (0 ,T ). As in the proof ofthe previous lemma, we will use the notation R γ ( v ) for a generic function satisfying | R γ ( v ) | ≤ C T v γ .Now, fix a multi-index α = ( α ,α ) where we denote ∂ α v ∂ α v by ∂ αv , and consider12 ddt k v γ ∂ αv f ( t ) k = −h v γ ˆ v ∂ x ∂ αv f,v γ ∂ αv f i − h v γ K · ∇ v ∂ αv f,v γ ∂ αv f i + h v γ ∇ v · ( D ∇ v ∂ αv f ) ,v γ ∂ αv f i + X β + α ′ = α | β | > (cid:18) αα ′ β (cid:19)hD R −| β | + γ ( v ) ∂ x ∂ α ′ v f,v γ ∂ αv f E + D BR −| β | + γ ( v ) ∂ v ∂ α ′ v f,v γ ∂ αv f E + D BR −| β | + γ ( v ) ∂ v ∂ α ′ v f,v γ ∂ αv f E + D v γ ∇ v · ( ∂ βv ( D ) ∇ v ∂ α ′ v f ) ,v γ ∂ αv f Ei =: I + II + III + X β + α ′ = α | β | > (cid:18) αα ′ β (cid:19)h IV αβ + IV αβ + IV αβ + IV αβ i N. MICHALOWSKI AND S. PANKAVICH
For I , we integrate by parts in x so that h v γ ˆ v ∂ x ∂ αv f,v γ ∂ αv f i = −h v γ ∂ αv f,v γ ˆ v ∂ x ∂ αv f i and hencethe first term vanishes. For II we integrate by parts in v to find − h v γ K · ∇ v ∂ αv f,v γ ∂ αv f i = D ( ∇ v v γ ) · K∂ αv f,∂ αv f E + h v γ ( ∇ v · K ) ∂ αv f,v γ ∂ αv f i The second term vanishes by the divergence-free structure of K , while the first term is bounded byfield estimates so that II ≤ C ( k B (( t ) k ∞ + k E ( t ) k ∞ ) k v γ ∂ αv f ( t ) k . To estimate
III , we integrate by parts in v to find h v γ ∇ v · ( D ∇ v ∂ αv f ) ,v γ ∂ αv f i = − (cid:13)(cid:13)(cid:13) v γ D / ∇ v ∂ αv f ( t ) (cid:13)(cid:13)(cid:13) − D ∇ v ( v γ ) · D ∇ v ∂ αv f,∂ αv f E . Integrating by parts again in the second of these two terms yields (cid:10) R γ − ( v ) ∂ αv f,v γ ∂ αv f (cid:11) . So we have III ≤ − (cid:13)(cid:13)(cid:13) v γ − / ∇ v ∂ αv f ( t ) (cid:13)(cid:13)(cid:13) + C k v γ ∂ αv f ( t ) k Next, we estimate the terms IV αβ . If | α ′ | = 0 then we may use Cauchy’s inequality and hence IV αβ ≤ (cid:13)(cid:13)(cid:13) v − k + γ ∂ x f ( t ) (cid:13)(cid:13)(cid:13) + k v γ ∂ αv f ( t ) k . Otherwise we may write ∂ α ′ v = ∂ v i ∂ α ′′ v with | α ′′ | = | α | −
2. Then, we integrate by parts in v i and writethis term as D R −| β | + γ ( v ) ∂ x ∂ α ′ v f,v γ ∂ αv f E = − D R −| β | + γ ( v ) ∂ x ∂ α ′′ v f,v γ ∂ αv f E − D R −| β | + γ ( v ) ∂ x ∂ α ′′ v f,v γ ∂ v i ∂ αv f E Applying Cauchy’s inequality with ǫ > IV αβ ≤ Cǫ (cid:13)(cid:13)(cid:13) v γ +1 / ∂ α ′′ v ∂ x f ( t ) (cid:13)(cid:13)(cid:13) + k v γ ∂ αv f ( t ) k + ǫ (cid:13)(cid:13)(cid:13) v γ − / ∂ v i ∂ αv f ( t ) (cid:13)(cid:13)(cid:13) and we can choose ǫ small enough so that the last term here is absorbed by the first term in theestimate of III . Both IV αβ and IV αβ possess the form D BR −| β | + γ ( v ) ∂ v j ∂ α ′ v f,v γ ∂ αv f E . Hence, afterapplying Cauchy’s inequality we find IV αβ + IV αβ ≤ k B ( t ) k ∞ (cid:18)(cid:13)(cid:13)(cid:13) R −| β | + γ ( v ) ∂ v j ∂ α ′ v f ( t ) (cid:13)(cid:13)(cid:13) + k v γ ∂ αv f ( t ) k (cid:19) (3.15) ≤ k B ( t ) k ∞ k v γ ∂ αv f ( t ) k + X ≤| α | 1, we use Cauchy’s inequality to find IV αβ ≤ C (cid:16)(cid:13)(cid:13)(cid:13) v γ −| β | ∂ v i ∂ α ′ v f ( t ) (cid:13)(cid:13)(cid:13) + (cid:13)(cid:13)(cid:13) v γ +1 −| β | ∂ v i ∂ v j ∂ α ′ v f ( t ) (cid:13)(cid:13)(cid:13) + k v γ ∂ αv f ( t ) k (cid:17) ≤ C k v γ ∂ αv f ( t ) k + X ≤| α | 1, suppose ∂ v i ∂ α ′ v = ∂ αv . Then the terms involving ∂ v i ∂ α ′ v can be handled using Cauchy-Schwarz. The terms involving ∂ v i v j ∂ α ′ v f , after integration by parts, are bounded by k v γ ∂ αv f ( t ) k .Collecting the estimates, summing over all α with | α | = k , and writing k v γ ∇ kv f ( t ) k = X | α | = k k v γ ∂ α f ( t ) k we find k v γ ∇ kv f ( t ) k ≤ C k v γ ∇ kv f ( t ) k + k − X j =1 (cid:13)(cid:13)(cid:13) v γ + j − k ∇ jv f ( t ) (cid:13)(cid:13)(cid:13) + (cid:13)(cid:13)(cid:13) v γ +1 / ∂ x ∇ k +2 v f ( t ) (cid:13)(cid:13)(cid:13) − (1 − ǫ ) (cid:13)(cid:13)(cid:13) v γ − / ∇ k +1 v f ( t ) (cid:13)(cid:13)(cid:13) which proves the first result.Next, we turn to the second result. Let ∂ v be any second-order v -derivative. We compute12 ddt k v / ∂ v ∂ x f ( t ) k = ZZ v ∂ v ∂ x f (cid:2) − ∂ v (ˆ v ∂ xx f ) − ∂ v ∂ x ( K · ∇ v f ) + ∂ v ( ∇ v · ( D ∇ v ∂ x f )) (cid:3) dvdx = I + II + III. As usual, one of the terms in I vanishes. So, we integrate by parts in the latter term below and useCauchy’s inequality with ǫ > I = − ZZ v ∂ v ∂ x f (cid:2) R − ( v ) ∂ xx f + 2 R − ( v ) ∂ v ∂ xx f (cid:3) dvdx ≤ C (cid:18) k ∂ v ∂ x f ( t ) k + (cid:18) ǫ (cid:19) k ∂ xx f ( t ) k + ǫ k ∂ v ∂ x f ( t ) k (cid:19) . We note that for ǫ sufficiently small, the last term can be controlled by the final term arising in III below. To estimate II , we integrate by parts in the third and last terms below, use the control offield and field derivative terms guaranteed by Lemmas 2.2 and 2.4, and utilize Cauchy’s inequalityso that II = − ZZ v ∂ v ∂ x f (cid:2) R − ( v ) ∇ v f + 2 R − ( v ) ∇ v ∂ v f + R ( v ) ∇ v ∂ v f + R − ( v ) ∇ v ∂ x f + 2 R − ( v ) ∇ v ∂ v ∂ x f + K · ∇ v ∂ v ∂ x f (cid:3) dvdx ≤ C T (cid:18) k v / ∇ v ∂ x f ( t ) k + k∇ v f ( t ) k + k∇ v ∂ x f ( t ) k + (cid:18) ǫ (cid:19) k v ∇ v ∂ v f ( t ) k + ǫ k∇ v ∂ v ∂ x f ( t ) k (cid:19) . Again, for ǫ sufficiently small, the last term can be controlled by the final term arising in III below.We integrate by parts, then use aforementioned properties of D and Cauchy’s inequality with ǫ > III to find III = − ZZ (cid:2) ˆ v∂ v ∂ x f + v ∇ v ∂ v ∂ x f (cid:3) [ ∂ v D ∇ v ∂ x f + 2 ∂ v D ∇ v ∂ v ∂ x f + D ∇ v ∂ v ∂ x f ] dvdx = − ZZ (cid:2) ˆ v∂ v ∂ x f + v ∇ v ∂ v ∂ x f (cid:3) [ R − ( v ) ∇ v ∂ x f + R ( v ) ∇ v ∂ v ∂ x f + D ∇ v ∂ v ∂ x f ] dvdx ≤ C (cid:18) k ∂ v ∂ x f ( t ) k + (cid:18) ǫ (cid:19) k∇ v ∂ x f ( t ) k + k∇ v ∂ v ∂ x f ( t ) k (cid:19) − (1 − ǫ ) k∇ v ∂ v ∂ x f ( t ) k . N. MICHALOWSKI AND S. PANKAVICH Finally, we collect these estimates, so that I + II + III ≤ C T (cid:16) k ∂ v ∂ x f ( t ) k + k ∂ xx f ( t ) k + k v / ∇ v ∂ x f ( t ) k + k∇ v f ( t ) k + k∇ v ∂ v f ( t ) k + k∇ v ∂ x f ( t ) k + k ∂ v ∂ x f ( t ) k (cid:1) − (1 − C T ǫ ) k∇ v ∂ v ∂ x f ( t ) k . Then, we use k ∂ v ∂ x f ( t ) k ≤ k∇ v ∂ x f ( t ) k , sum over all v -derivatives to arrive at an estimate on ddt k v / ∇ v ∂ x f ( t ) k , and the claim then follows. As for Lemma 2.8 we have rescaled ǫ > C T > Proof . [Lemma 2.10] We will prove the result in a hierarchical fashion by building pairs ofconsecutive terms and adding higher-order derivatives as we go. To begin the proof, we consider t ∈ (0 ,T ) and define M ( t ) = k v f ( t ) k + 12 t k v / ∇ v f ( t ) k + 18 t k v ∇ v f ( t ) k and differentiate to find M ′ ( t ) = ddt k v f ( t ) k + 12 t ddt k v / ∇ v f ( t ) k + 18 t ddt k v ∇ v f ( t ) k + 12 k v / ∇ v f ( t ) k + 14 t k v ∇ v f ( t ) k Using Lemma 2.8, we find ddt k v f ( t ) k ≤ C T k v f ( t ) k − k v / ∇ v f ( t ) k and ddt k v / ∇ v f ( t ) k ≤ C T (cid:16) k v / ∇ v f ( t ) k + k v ∂ x f ( t ) k (cid:17) − k v ∇ v f ( t ) k . Additionally, applying the first result of Lemma 2.9 for γ = 1, k = 2 we find for any ǫ > ddt k v ∇ v f ( t ) k ≤ C T (cid:16) k v ∇ v f ( t ) k + k v / ∇ v f ( t ) k + k v / ∂ x f ( t ) k (cid:17) − (1 − ǫ ) k v / ∇ v f ( t ) k . We combine these results, use the bounds on x -derivatives of the particle distribution (Lemma 2.6),and choose ǫ < M ′ ( t ) ≤ C T (cid:16) M ( t ) + k v / ∂ x f ( t ) k (cid:17) − k v / ∇ v f ( t ) k − t k v ∇ v f ( t ) k − (1 − ǫ )8 t k v / ∇ v f ( t ) k ≤ C T (1 + M ( t ))Thus, by Gronwall’s inequality, we conclude M ( t ) ≤ C T M (0) = C T k v f k Hence, for t ∈ (0 ,T )(3.17) k v / ∇ v f ( t ) k ≤ C T t .5D RELATIVISTIC VLASOV-MAXWELL-FOKKER-PLANCK EQUATION k v ∇ v f ( t ) k ≤ C T t . Next, define M ( t ) = k v / ∂ x f ( t ) k + 12 t k v ∇ v ∂ x f ( t ) k and differentiate to find M ′ ( t ) = ddt k v / ∂ x f ( t ) k + 12 t ddt k v ∇ v ∂ x f ( t ) k + 12 k v ∇ v ∂ x f ( t ) k Using Lemma 2.8, we find ddt k v / ∂ x f ( t ) k ≤ C T (cid:16) k v / ∂ x f ( t ) k + k v f ( t ) k (cid:17) − (1 − ǫ ) k v ∇ v ∂ x f ( t ) k and ddt k v ∇ v ∂ x f ( t ) k ≤ C T (cid:16) k v ∇ v ∂ x f ( t ) k + k ∂ xx f ( t ) k + k v / ∇ v f ( t ) k (cid:17) − (1 − ǫ ) k v / ∇ v ∂ x f ( t ) k . Combining these results while using the L -bounds on second x -derivatives of the density (Lemma2.7) and (3.17), we find M ′ ( t ) ≤ C T (cid:16) M ( t ) + k v f ( t ) k + t k v / ∇ v f ( t ) k + k ∂ xx f ( t ) k (cid:17) − (cid:18) − ǫ (cid:19) k v ∇ v ∂ x f ( t ) k − 12 (1 − ǫ ) t k v / ∇ v ∂ x f ( t ) k ≤ C T (1 + M ( t ))Thus, for ǫ < / 2, we use Gronwall’s inequality to conclude M ( t ) ≤ C T M (0) = C T k v / ∂ x f k Hence, for t ∈ (0 ,T )(3.19) k v ∇ v ∂ x f ( t ) k ≤ C T t . Building onto previous terms, we next define M ( t ) = M ( t ) + 148 t k v / ∇ v f ( t ) k . Hence, using the estimate of M ′ ( t ) we find M ′ ( t ) = M ′ ( t ) + 116 t k v / ∇ v f ( t ) k + 148 t ddt k v / ∇ v f ( t ) k ≤ C T (1 + M ( t )) + (cid:18) − (1 − ǫ )8 (cid:19) t k v / ∇ v f ( t ) k + 148 t ddt k v / ∇ v f ( t ) k ≤ C T (1 + M ( t )) + 148 t ddt k v / ∇ v f ( t ) k N. MICHALOWSKI AND S. PANKAVICH for ǫ < / 2. By the first result of Lemma 2.9 with γ = 1 / k = 3, we find for any ǫ > ddt k v / ∇ v f ( t ) k ≤ C T (cid:16) k v / ∇ v f ( t ) k + k v / ∇ v f ( t ) k + k v ∇ v f ( t ) k + k v ∇ v ∂ x f ( t ) k (cid:17) − (1 − ǫ ) k∇ v f ( t ) k Therefore, using the previous bounds obtained from (3.17), (3.18), and (3.19), we have M ′ ( t ) ≤ C T (1 + M ( t )) − 148 (1 − ǫ ) t k∇ v f ( t ) k . Since ǫ < / k v / ∇ v f ( t ) k ≤ C T t for t ∈ (0 ,T ).Again building onto previous terms, we next define M ( t ) = M ( t ) + 18 t k v / ∇ v ∂ x f ( t ) k so that M ′ ( t ) = M ′ ( t ) + 14 t k v / ∇ v ∂ x f ( t ) k + 18 t ddt k v / ∇ v ∂ x f ( t ) k Using the second result of Lemma 2.9 along with the bound on k ∂ xx f ( t ) k from Lemma 2.7 and theprevious bounds obtained from (3.17), (3.18), and (3.19), we find for ǫ > ddt k v / ∇ v ∂ x f ( t ) k ≤ C T (cid:18) k v / ∇ v ∂ x f ( t ) k + k v ∇ v ∂ x f ( t ) k + X j =1 (cid:13)(cid:13)(cid:13) v − j ∇ jv f ( t ) (cid:13)(cid:13)(cid:13) + k ∂ xx f ( t ) k (cid:19) − (1 − ǫ ) (cid:13)(cid:13) ∇ v ∂ x f ( t ) (cid:13)(cid:13) ≤ C T (cid:16) t + k v / ∇ v ∂ x f ( t ) k (cid:17) . Hence, we incorporate this and use the estimate of M ′ ( t ) to find M ′ ( t ) ≤ C T (1 + M ( t )) − (cid:18) − − ǫ (cid:19) t k v / ∇ v ∂ x f ( t ) k ≤ C T (1 + M ( t ))and upon choosing ǫ < / k v / ∇ v ∂ x f ( t ) k ≤ C T t for t ∈ (0 ,T ). Finally, to obtain bounds on fourth-order v -derivatives of the density, we define M ( t ) = M ( t ) + 12 t k∇ v f ( t ) k .5D RELATIVISTIC VLASOV-MAXWELL-FOKKER-PLANCK EQUATION M ′ ( t ) = M ′ ( t ) + 196 t k∇ v f ( t ) k + 12 t ddt k∇ v f ( t ) k . Using Lemma 2.9 one final time with γ = 0 and k = 4 and utilizing the bounds obtained from (3.17)-(3.21), we find ddt k∇ v f ( t ) k ≤ C T (cid:18) t + k∇ v f ( t ) k (cid:19) . Applying this to M ( t ) and using the estimate of M ′ ( t ), we see M ′ ( t ) ≤ C T (1 + M ( t )) + (cid:18) − (1 − ǫ )48 (cid:19) t k∇ v f ( t ) k . and choosing ǫ < / k∇ v f ( t ) k ≤ C T t for t ∈ (0 ,T ). Lastly, combining the estimates above, the proof of the lemma is complete.We remark that this same argument can be applied to the eight term power series expansion X k =0 t k k k ! (cid:13)(cid:13)(cid:13) v (4 − k ) / ∇ kv f ( t ) (cid:13)(cid:13)(cid:13) + X k =0 t k k k ! (cid:13)(cid:13)(cid:13) v (3 − k ) / ∇ kv ∂ x f ( t ) (cid:13)(cid:13)(cid:13) in order to arrive at an identical result. However, the above argument is perhaps clearer. Also,estimates of higher derivatives can be obtained if one imposes additional spatial regularity on thedensity and field terms, as this requires control of second-order field derivatives in L ∞ . 4. Proof of Theorem 1.1 To conclude the paper, we utilize the previous lemmas to sketchthe proof of Theorem 1.1. Proof . As is typical, the proof utilizes a standard iterative argument. We define a sequence ofsolutions to the corresponding linear equations and show that it must converge to a solution of thenonlinear system (RVMFP). Define the initial iterates in terms of the given initial data f ( t,x,v ) = f ( x,v ) ,E ( t,x ) = E ( x ) B ( t,x ) = B ( x ) . Additionally, for every n ∈ N , given E n ,E n ,B ∈ L ∞ ([0 , ∞ ); H ( R )) we obtain f n ∈ L ∞ ([0 , ∞ ) × R )by solving the linear initial value problems(4.1) ( ∂ t f n + ˆ v ∂ x f n + K n − · ∇ v f n = ∇ v · ( D ∇ v f n ) f n (0 ,x,v ) = f ( x,v ) , where K n = h E n + ˆ v B n ,E n − ˆ v B n i N. MICHALOWSKI AND S. PANKAVICH and the fields satisfy(4.2) ∂ t E n + ∂ x B n = − j , ∂ t B n + ∂ x E n = 0 E n = Z x −∞ (cid:18)Z f n ( t,y,v ) dv − φ ( y ) (cid:19) dyE n (0 ,x ) = E (0 ,x ) B n (0 ,x ) = B (0 ,x )respectively. Let T > f n ,E n ,B n ) be a sequence of weak solutions to the above linearsystem on [0 ,T ]. Using the assumptions on initial data, we apply the estimates of Section 2 andfind E n and B n converge (up to a subsequence) weakly in L ∞ ([0 ,T ]; H ( R )) to functions E and B ,respectively. Then, we proceed by estimating successive differences of iterates (e.g., see [11]). First,we use (3.6) and the linearity of the transport equation to find k K n ( t ) − K n − ( t ) k ∞ ≤ Ct sup s ∈ [0 ,t ] k v a f n ( s ) − v a f n − ( s ) k ∞ . Next, we write the Vlasov equation for the difference of consecutive iterates and use (3.10) andLemmas 2.1, 2.2, and 2.3 to conclude k v a f n +1 ( t ) − v a f n ( t ) k ∞ ≤ C T Z t (cid:0) k K n ( s ) − K n − ( s ) k ∞ + k v a f n +1 ( s ) − v a f n ( s ) k ∞ (cid:1) ds and thus(4.3) k v a f n +1 ( t ) − v a f n ( t ) k ∞ ≤ C T Z t sup τ ∈ [0 ,s ] k v a f n ( τ ) − v a f n − ( τ ) k ∞ ds. It follows from this estimate that f n converges strongly to a function f in L ∞ ([0 ,T ] × R ). Similarestimates can be used to show f ∈ L ∞ ([0 ,T ]; L ( R )) as in [4]. It can then be shown that theselimiting functions satisfy (RVMFP) in the weak sense. Applying the regularizing estimates, we find f ∈ L ∞ ((0 ,T ); H x ( R ; H v ( R ))). By the Sobolev Embedding Theorem, H ( R ) ⊂ C b ( R ) and H ( R ) ⊂ C b ( R ). Thus we find f , E , and B possess a continuous partial derivative in x , and f possessestwo continuous partial derivatives in either v component. Using the Vlasov and transport equations,we see that ∂ t B , ∂ t E , and ∂ t f are all continuous. Hence, we find f ∈ C ((0 ,T ) × R ; C ( R )) and E ,B ∈ C ((0 ,T ) × R ). Finally, from the regularity of f we deduce E ∈ C ((0 ,T ) × R ) as well. Ofcourse, with this additional regularity we conclude that the triple ( f,E ,B ) is, in fact, a classicalsolution of (RVMFP).The uniqueness of solutions follows from another standard argument. We consider the differenceof solutions h ( t,x,v ) = v a ( f ( t,x,v ) − f ( t,x,v ))where f and f are any two solutions of (RVMFP) which share the same initial data, and we derivethe same estimate (4.3) for h , namely k h ( t ) k ∞ ≤ C T Z t sup τ ∈ [0 ,s ] k h ( τ ) k ∞ ds. 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