THE VLASOV-MAXWELL-BOLTZMANN SYSTEM NEAR MAXWELLIANS IN THE WHOLE SPACE WITH VERY SOFT POTENTIALS

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1 THE VLASOV-MAXWELL-BOLTZMANN SYSTEM NEAR MAXWELLIANS IN THE WHOLE SPACE WITH VERY SOFT POTENTIALS RENJUN DUAN, YUANJIE LEI, TONG YANG, AND HUIJIANG ZHAO Abstract. Since the work [4] by Guo [Invent. Math , no. 3, ], it has remaine an open problem to establish the global eistence of perturbative classical solutions aroun a global Mawellian to the Vlasov-Mawell-Boltzmann system with the whole range of soft potentials. This is mainly ue to the comple structure of the system, in particular, the egenerate issipation at large velocity, the velocity-growth of the nonlinear term inuce by the Lorentz force, an the regularity-loss of the electromagnetic fiels. This paper solves this problem in the whole space provie that initial perturbation has sufficient regularity an velocity-integrability. Contents. Introuction. Main results 3.. Eisting approaches 4.. Difficulties for very soft potentials 5.3. Main results an ieas 8 3. Proofs of the main results 3.. Preliminaries 3.. Some a priori estimates 3.3. The proof of Theorem The proof of Theorem. 4. Appeni References 40. Introuction The motion of ilute ionize plasmas consisting of two-species particles e.g., electrons an ions uner the influence of binary collisions an the self-consistent electromagnetic fiel can be moelle by the Vlasov- Mawell-Boltzmann system cf. [3, Chapter 9] as well as [, Chapter 6.6] t F v F e E v m c B v F = QF, F QF, F, t F v F e E v m c B v F = QF, F QF, F.. The electromagnetic fiel [E, B] = [Et,, Bt, ] satisfies the Mawell equations t E c B = 4π v e F e F v, R 3 t B c E = 0,. E = 4π e F e F v, R 3 B = 0. Here =,, 3, v = v, v, v3. The unknown functions F ± = F ± t,, v 0 are the number ensity functions for the ions an electrons with position =,, 3 R 3 an velocity v = v, v, v 3 R 3 at time t 0, respectively, e ± an m ± the magnitues of their charges an masses, an c the spee of light.

2 R.-J. DUAN, Y.-J. LEI, T. YANG, AND H.-J. ZHAO Let F v, Gv be two number ensity functions for two types of particles with masses m ± an iameters σ ±, then QF, Gv is efine as cf. [3] QF, Gv = σ σ ω v u u v γ b F v Gu F vgu ωu 4 R 3 S u v Q gain F, G Q loss F, G. Here ω S an b, the angular part of the collision kernel, satisfies the Gra cutoff assumption cf. [9] 0 bcos θ C cos θ.3 for some positive constant C > 0. The eviation angle π θ satisfies cos θ = ω v u/ v u. Moreover, for m, m m, m, v = v m m m [v u ω]ω, u = u m m m [v u ω]ω, which enote velocities v, u after a collision of particles having velocities v, u before the collision an vice versa. Notice that the above ientities follow from the conservation of momentum m v m u an energy m v m u. The eponent γ 3, ] in the kinetic part of the collision kernel is etermine by the potential of intermolecular force, which is classifie into the soft potential case when 3 < γ < 0, the Mawell molecular case when γ = 0, an the har potential case when 0 < γ which inclues the har sphere moel with γ = an bcos θ = C cos θ for some positive constant C > 0. For the soft potentials, the case γ < 0 is calle the moerately soft potentials while 3 < γ < is calle the very soft potentials, cf. [9] by Villani. The importance an the ifficulty in stuying the very soft potentials can be also foun in that review paper. The main purpose of this work is to construct global classical solutions to the Vlasov-Mawell-Boltzmann system.,. for the whole range of soft potentials, in particular, the very soft case when 3 < γ <, near global Mawellians µ v = n 0 m e πk B T 0 µ v = n 0 m e πk B T ep m v k B T 0 ep m v k B T 0 in the whole space R 3, where k B > 0 is the Boltzmann constant, n 0 > 0 an T 0 > 0 are constant reference number ensity an temperature, respectively, an the reference bulk velocities have been chosen to be zero. We consier the Cauchy problem with prescribe initial ata F ± 0,, v = F 0,±, v, E0, = E 0, B0, = B 0,.4 which satisfy the compatibility conitions E 0 = F 0, F 0, v, R 3 B 0 = 0. We remark that the angular non-cutoff case was consiere in [6] basing on the argument cf. [6] that the energy issipations inclue an etra velocity ifferentiation ue to the angular non-cutoff assumption. As will be eplaine later, the techniques use in [6] cannot be applie to the cutoff very soft case uner consieration in the paper. The basic motivation here is to evelop new strategies to eal with such case. We assume in the paper that all the physical constants are chosen to be one. Uner such assumption, accoringly we normalize the above Mawellians as µ = µ v = µ v = π 3 e v. To stuy the stability problem aroun µ, we efine the perturbation f ± = f ± t,, v by F ± t,, v = µ µ / f ± t,, v.,,

3 THE VLASOV-MAXWELL-BOLTZMANN SYSTEM WITH VERY SOFT POTENTIALS 3 Then, the Cauchy problem.,.,.4 is reformulate as t f ± v f ± ± E v B v f ± E vµ / E vf ± L ± f = Γ ± f, f, t E B = vµ / f f v, R 3.5 t B E = 0, E = µ / f f v, B = 0 R 3 with initial ata f ± 0,, v = f 0,±, v, E0, = E 0, B0, = B 0.6 satisfying the compatibility conitions E 0 = µ / f 0, f 0, v, R 3 B 0 = 0..7 Here, as in [4], for later use, setting f = [f, f ], the first equation of.5 can be also written as t f v f q 0 E v B v f E vµ / q Lf = q 0 E vf Γf, f,.8 where q 0 = iag,, q = [, ], an the linearize collision operator L = [L, L ] an the nonlinear collision operator Γ = [Γ, Γ ] are to be given later on. We are now reay to state the main theorem in this paper. Theorem.. Let 3 < γ < an.3 hol. Assume F 0, v = µ µf 0, v 0. Take / ϱ < 3/ an 0 < q. There eist some integer N > 0 an l0 > 0 such that if v l 0 β e q v β α f 0 f0 L v Ḣ ϱ E 0, B 0 H N Ḣ ϱ α β N is sufficiently small, then the Cauchy problem.5,.6,.7 amits a unique global solution [ft,, v, Et,, Bt, ] satisfying F t,, v = µ µft,, v 0. In the net section, the statement of the above theorem will be given more precisely in Theorem. as well as Theorem. for the time ecay property. Basically the result shows that as long as initial ata is small with enough regularity, one can establish the global eistence of small amplitue classical solutions for the full range of cutoff intermolecular interactions with 3 < γ. Note that the case γ is a trivial consequence of [6]; etails for that case will be briefly iscusse in Section.. Here, the boun in the Sobolev space of negative ine is use for obtaining the time ecay of solutions that is neee to close the a priori estimates. The general technique of aopting the negative Sobolev estimates to treat the time-ecay of the Boltzmann equation an other types of issipative equations was firstly introuce in [8]. For the case of the whole space, compare to L initial ata use in [6], the space Ḣ ϱ is much more convenient to euce the fast enough time-ecay rates in terms of only the pure energy metho an the interpolation inequalities. The proof of Theorem. is base on a subtle time-weighte energy metho. For this, in aition to the eisting analytic techniques use in [6] an [6], we evelop a new approach to eal with the weighte estimates involving both the negative power time-weight an the time-velocity epenent w l β,κ t, v weight. The rest of this paper is organize as follows. In Section, we eplain the ifficulty in stuying the case when 3 < γ <, particularly incluing the very soft potential case, an give a complete statement of the main results. In Section 3, we list some basic lemmas for later use. The proof of the main results will be given in Section 4. For clear presentation, the proofs of several technical lemmas an estimates use in Section 4 will be given in the appeni.. Main results In this section, we will first review the previous approaches for stuying the global eistence of classical solutions to Valsov-Mawell-Boltzmann equations, an then we will point out the ifficulties in stuying the very soft potentials an give the complete statements of the main results. First of all, we recall some basic facts concerning the collision operators an the macro-micro ecomposition. L, Γ in.8 are respectively efine by Lf = [L f, L f], Γf, g = [Γ f, g, Γ f, g]

4 4 R.-J. DUAN, Y.-J. LEI, T. YANG, AND H.-J. ZHAO with L ± f = µ / Q Γ ± f, g =µ / Q µ, µ / f ± f µ / f ±, µ / g ± Q Q µ / f ±, µ, µ / f ±, µ / g. For the linearize collision operator L, it is well known cf. [4] that it is non-negative an the null space N of L is spanne by N = span [, 0]µ /, [0, ]µ /, [v i, v i ]µ / i 3, [ v, v ]µ /. Moreover, uner Gra s angular cutoff assumption.3, it is easy to see that L can be ecompose as Lf = f Kf. with the collision frequency v an the nonlocal integral operator K = [K, K ] being efine by ω v u v = Q loss, µ = v u γ b µuωu v γ,. R 3 S v u an K ± f v = µ Q gain µ f±, µ Q µ, µ f± f ω v u = u v γ b µ u R 3 S v u µ u f ± v µ v f ± f u µ vf± f u ωu,.3 respectively. Define P as the orthogonal projection from L R 3 v L R 3 v to N. Then for any given function ft,, v L R 3 v L R 3 v, one has 3 Pf = a t, [, 0]µ / a t, [0, ]µ / b i t, [, ]v i µ / ct, [, ] v 3µ / i= with a ± = µ / f ± v, b i = v i µ / f f v, c = v 3µ / f f v. R 3 R 3 R 3 Therefore, we have the following macro-micro ecomposition with respect to the given global Mawellian µv, cf. [5], ft,, v = Pft,, v I Pft,, v, where I enotes the ientity operator, an Pf an I Pf are calle the macroscopic an the microscopic component of ft,, v, respectively. Uner the Gra s angular cutoff assumption.3, by [4, Lemma ], L is locally coercive in the sense that f, Lf σ 0 I Pf σ 0 I Pf L R 3 v, v v γ.4 hols for some positive constant σ 0 > 0. Here, enotes the inner prouct in L R 3 v L R 3 v... Eisting approaches. For the problem on the construction of solutions to the Cauchy problem.5,.6,.7, the local eistence an uniqueness of solution [f t,, v, f t,, v, Et,, Bt, ] in certain weighte Sobolev space to be specifie later can be obtaine by combining the arguments use in [4] an [6]. To eten the local solution [f t,, v, f t,, v, Et,, Bt, ] to be global in time, one nees to euce certain a priori estimates in some function spaces. In general, the main ifficulties in this step lies in: How to control the possible velocity-growth inuce by the nonlinearity of the system.8? How to control the convection term v f in the weighte energy estimates? The nonlinear energy metho evelope in [, 5, 3, 4] for the Boltzmann equation provies an effective approach in the perturbative framework; see also the recent progress []. The main iea in those work is to ecompose the solution into the macroscopic part an the microscopic part an then rewrite the original equation as the combination of an equation satisfie by the microscopic part which contains the macroscopic part as source term an a system satisfie by the macroscopic part with the microscopic part as source term. In the perturbative framework, the issipative mechanism on the microscopic part is the coercive estimate

5 THE VLASOV-MAXWELL-BOLTZMANN SYSTEM WITH VERY SOFT POTENTIALS 5.4 of the linearize Boltzmann collision operator or its weighte variants, while for the macroscopic part, the corresponing mechanism comes from the issipation of the compressible Navier-Stokes type system. The corresponing approach to treat the case of non-cutoff cross sections was evelope in [] an [0]. However, as pointe out in [6] an [6], when one applies the energy metho to some comple systems such as the Vlasov-Mawell-Boltzmann system.,., in aition to the ifficulty cause by the nonlinear collision operator mentione above, aitional ifficulties are encountere: How to control the corresponing nonlinear terms inuce by the Lorentz force, such as the terms E v B v f an E vf, that can lea to velocity growth at the rate of the first orer v? How to cope with the regularity loss of the electromagnetic fiel [Et,, Bt, ]? For the har sphere moel, the coercive estimate.4 of L is sufficient to control the nonlinear terms relate to the Lorentz force provie that the electromagnetic fiel [Et,, Bt, ] is suitably small an thus satisfactory global well-poseness theory for the Vlasov-Mawell-Boltzmann system.,. for the har sphere moel has been establishe, cf. [7, 4, 7, 0, 5] an the references therein. But for the corresponing problem involving cutoff non-har sphere intermolecular interactions with γ <, the story is quite ifferent. One can not use the coercive estimate.4 of L to absorb the nonlinear terms relate to the Lorentz force which yiel the velocity growth at the rate of the first orer v. Thus it is interesting an important to fin out how to construct global classical solutions near Mawellians to the Vlasov-Mawell-Boltzmann system.,. for cutoff non-har sphere cases. Certainly, the same applies to the Vlasov-Poisson-Lanau system an the Vlasov-Mawell-Lanau system containing the Coulomb potential, cf. [6, 8, 30] an [4, 6], respectively; an the Vlasov-Poisson-Boltzmann system for non-har sphere interactions cf. [5, 8, 3]. Particularly, a breakthrough was mae in Guo s work [6] on the two-species Vlasov-Poisson-Lanau system in a perioic bo, that leas to the subsequent works for the Vlasov-Poisson-Lanau system in the whole space mentione above. The main ieas can be outline as follows: An eponential weight of electric potential e φ is use to cancel the growth of the velocity in the nonlinear term φ vf ±. A velocity weight w l α β v = v γl α β, v = v, l α β is use to compensate the weak issipation of the linearize Lanau kernel L for the case of 3 γ < ; The ecay of the electric fiel φt, is use to close the energy estimate. However, since the Lorentz force E v B is not of the potential form, the argument in [6] can not be irectly aopte to stuy the Vlasov-Mawell-Boltzmann system.,.. For this, a time-velocity weighte energy metho is introuce in [8] by using the following weight w l, β t, v function: w l β w l β t, v = v γl β e q v t ϑ, 0 < q, β l, 0 < ϑ 4..5 Here it is worth pointing out that, unlike the weight function w l α β v, the algebraic factor w a l β v = v γl β in.5 varies only with the orer of the v erivatives to represent the fact that the issipative effect of the cutoff linearize Boltzmann collision operator L is weaker than that of the linearize Lanau collision operator L... Difficulties for very soft potentials. To illustrate the main ieas use in [6, 8] for γ, an the problem to be stuie in this paper, we first introuce the following general weight function w l β,κ w l β,κ t, v = v κl β e q v t ϑ, κ 0, 0 < q,.6 where the precise range of the parameter ϑ will be specifie later. It is easy to see that w l β, γ t, v w l β t, v. Since for cutoff non-har sphere intermolecular interactions, the macroscopic part can be controlle as for the case of har sphere moel, the main ifficulty for the case of non-har sphere moel is to control the microscopic component I Pft,, v suitably. The iea for that purpose is to use the following two types of issipative mechanisms:

6 6 R.-J. DUAN, Y.-J. LEI, T. YANG, AND H.-J. ZHAO The first one is the issipative term D L α,l β,κ wl β,κ α β I Pf w l β,κ α β I Pf L R 3 v R3 from the coercive estimate of the linearize collision operator L; The secon type is the etra issipative term D W α,l β,κ t ϑ wl β,κ α β I Pf v L R 3 v R3 inuce by the weight function w l β,κ t, v. The most ifficult terms to be stuie are: The term I α,l β,κ lt αei β e i I Pf, wl β,κ α β I Pf relate to the linear transport term v f; The terms containing the electromagnetic fiel [Et,, Bt, ], i.e. I α,l β,κ E α E v α α β I Pf, wl β,κ α β I Pf,.8 an I B α,l β,κ α α v α B v α α β.7 I Pf, wl β,κ α β I Pf..9 Here, enotes the stanar L R 3 v R 3 L R 3 v R 3 inner prouct in R 3 v R 3. To euce the esire estimates on the above terms, the main ingreients use in [6, 8] can be summarize as follows: A time-velocity weighte energy metho is introuce basing on the weight function w l β t, v = w l β, γ t, v. An avantage of this weight function is that the term I α,l β, γ lt relate to the linear transport term v f can be controlle suitably. In fact, then I lt α,l β, γ w l β = w l β w l β ei v γ, ε wl β β α I Pf Cε wl β ei αei β e i I Pf that, by a suitable linear combination with respect to α, can be further controlle by the issipation α β N D L α,l β, γ inuce by the linearize Boltzmann collision operator L. On the other han, this approach leas to an aitional ifficulty on estimating the nonlinear term E v B v f ± that requires a restriction on the range of the parameter γ. In fact, to control the term I B α,l β, γ, by w l β t, v = w l β t, v w l β t, v v γ, we can have I B α,l β, γ 0<α α R 3 R3 v v γ α B wl β β α I Pf ± wl β v α α β I Pf ± v, which can be controlle by the issipation α β N D W α,l β, γ inuce by the eponential factor of the weight function w l β, γ t, v only when an α Bt, ecays sufficiently fast. γ, i.e. γ,,.0

7 THE VLASOV-MAXWELL-BOLTZMANN SYSTEM WITH VERY SOFT POTENTIALS 7 Thus, up to now, the eisting approaches for the construction of global classical solutions to the Vlasov- Mawell-Boltzmann system.,. near Mawellians is limite to the case when γ. An the purpose of this paper is to introuce a new approach for the whole range soft potential, that is, to inclue the case when 3 < γ <. To continue, we first introuce some notations use throughout the paper. C an O enote some positive constants generally large an κ, δ an λ are use to enote some positive constants generally small, where C, O, κ, δ, an λ may take ifferent values in ifferent places; A B means that there is a generic constant C > 0 such that A CB. A B means A B an B A; The multi-inices α = [α, α, α 3 ] an β = [β, β, β 3 ] will be use to recor spatial an velocity erivatives, respectively. An β α = α α α3 3 v β v β v β3 3. Similarly, the notation α will be use when β = 0 an likewise for β. The length of α is enote by α = α α α 3. α α means that no component of α is greater than the corresponing component of α, an α < α means that α α an α < α. An it is convenient to write k = α with α = k;, is use to enote the L v L v inner prouct in R 3 v, with the L norm L. For notational simplicity,, enotes the L L inner prouct either in R 3 R 3 v or in R 3 with the L L norm ; χ Ω is the stanar inicator function of the set Ω; ft,, L p L q v = R 3 R 3 v ft,, v q v p q p, an others like ft,, L p H q v can be efine similarly; B C R 3 enotes the ball of raius C centere at the origin, an L B C L B C stans for the space L L over B C an likewise for other spaces. Recall that v v γ, we set f f vv an for each l R, L R 3 l R3 v L l R3 v enotes the weighte function space with norm f L fv v l v, v = v. l R 3 v Hl kr3 v Hl kr3 v with the norm f H k l etc. can be efine similarly; For s R, Λ s g t,, v = ξ s ĝt, ξ, ve πi ξ ξ = ξ s F[g]t, ξ, ve πi ξ ξ R 3 R 3 with ĝt, ξ, v F[g]t, ξ, v being the Fourier transform of gt,, v with respect to. The homogeneous Sobolev space Ḣs Ḣs is the Banach space consisting of all g satisfying g Ḣs <, where gt Ḣs Λ s g t,, v L,v = ξ s ĝt, ξ, v L ξ,v. For an integer N 0 an l R, the parameter ϑ is suitably chosen so that 0 < ϑ min γ ϱγ4ϱ, ϱγ 3γ 4ϱ 6 8γ 4, when ϱ [, 3 an N 0 5, 0 < ϑ min, when ϱ, 3 an N 0 = γ γ ϱγ4ϱ 4 4γ, ϱγ γ ϱ 4γ. Note that those strictly positive upper bouns for the choice of ϑ above are ue to erivation of estimates 3.9 an 3.35 to be use in the later proof. Define the energy functional E N,l,κ t an the corresponing energy issipation rate functional D N,l,κ t of a given function ft,, v with respect to the weight function w l β,κ t, v efine by.6 as follows: an respectively. Here E N,l,κ t E N,l,κ t Λ ϱ f, E, B, D N,l,κ t D N,l,κ t Λ ϱ a, b, c, E, B Λ ϱ a a, E Λ ϱ I Pf, E N,l,κ t α β N wl β,κ α β f E, B H N,

8 8 R.-J. DUAN, Y.-J. LEI, T. YANG, AND H.-J. ZHAO an D N,l,κ t α N E H N α a ±, b, c B H N α β N wl β,κ α β I Pf a a t ϑ α β N w l β,κ α β I Pf v. Moreover, we also nee to efine E N t, the energy functional without weight, EN k 0 t, the high orer energy functional without weight, an EN k 0,l,κ t, the high orer energy functional with respect to the weight function w l β,κ t, v, as follows: N E N t k f, E, B, an E k N 0,l,κt E k N 0 t α β N 0, α k k=0 N 0 α =k α f, E, B, wl β,κ α β f N 0 α =k α E, B, respectively. The corresponing energy issipation rate functionals D N t, DN k 0 t, an DN k 0,l,κ t are given by D N t E, a a α Pf, E, B α Pf α I Pf, D k N 0 t k E, a a an D k N 0,l,κt k E, a a respectively. α N k α N 0 k α N 0 0 α Pf t ϑ α Pf, E, B α Pf, E, B α β N 0, α k 0 α Pf α β N 0, α k α N wl β,κ α β I Pf v, k α N 0 α I Pf, w l β,κ α β I Pf.3. Main results an ieas. With the above preparation, the precise statement concerning the global in time solvability of the Cauchy problem.5,.6,.7 can be state as follows. Theorem.. Suppose that i F 0, v = µ µf 0, v 0, ϱ < 3, 3 < γ <. Let N0 5, N = N 0, when ϱ [, ], N 0 4, N = N 0, when ϱ, 3 ;. ii The parameter ϑ is chosen to satisfy. an we take σ N,0 = ɛ0, σ n,0 = 0 with n N, σ n,j σ n,j = γ γ ϑ when 0 j n an n N; iii There eists a positive constants l which epens only on γ an N 0 such that a l γ γσ N 0,N 0 ϱ, l γ γσ N,N 3ϱ, an l 3 γ γσ N,N ϱ, b l N, l ma l γ, l3 γ γl, l 0 l 5, l 0 l γ γl 0 l with l = 3 l γ. If we assume further that Y 0 = w l 0 β, β α f 0 w l β, β α f 0 E 0, B 0 H N Ḣ f ϱ 0 Ḣ ϱ α β N 0 N 0 α β N is sufficiently small, then the Cauchy problem.5,.6,.7 amits a unique global solution [ft,, v, Et,, Bt, ] satisfying F t,, v = µ µft,, v 0.

9 THE VLASOV-MAXWELL-BOLTZMANN SYSTEM WITH VERY SOFT POTENTIALS 9 Remark.. Several remarks concerning Theorem. are given. As mentione before, although only the case of 3 < γ < is stuie in this paper, the case of γ is much simpler an similar result hols. Thus, the current work provies a satisfactory wellposeness theory for the Cauchy problem of the two-species Vlasov-Mawell-Boltzmann system.5,.6,.7 in the perturbative framework for the whole range of the cutoff intermolecular interactions. Since in the proof of Lemma 4.3, N is assume to satisfy N > 5 3 N 0 5 3, while in the proof of Lemma 3.5, N is further require to satisfy N N 0 ϱ. Putting these assumptions together, we can take N = N 0 for ϱ [, ] an N = N 0 for ϱ, 3. The minimal regularity ine, i.e., the lower boun on the parameter N, we impose on the initial ata is N = 9, N 0 = 5 for ϱ [, ] an N = 8, N 0 = 4 for ϱ, 3. The precise value of the parameter l will be specifie in the proof of Lemma 4.3. Note that Theorem. is an immeiate consequence of Theorem.. The net result is concerne with the temporal ecay estimates on the global solution [ft,, v, Et,, Bt, ] obtaine in Theorem.. Theorem.. Uner the assumptions of Theorem., we have Taking k = 0,,,, N 0, it follows that EN k 0 t Y0 t ϱk..3 Let 0 i k N 0 3 be an integer. Take l 0,k N 0 with l 0,k l 0,k 3 for k N 0 3. Further take l 0 an l respectively as l 0 = l 0,0 = l 0, ma χ k l 0,k 3k 3, l 5 an l = k l γ in Theorem.. Then it follows that E k N 0,l 0,k i, γt Y0 t k ϱi, i = 0,,, k [ϱ]..4 Here an in the sequel [ϱ] enotes the greatest integer less than ϱ. 3 When N 0 α N, α f Y0 N α N 0 ϱ N N t 0..5 Remark.. In Theorem., we notice that the highest ine k of E k N 0 t is N 0 while the highest ine of E k N 0,l, γ t is N 0 3. The reason is that the highest orer α E appearing in 3.9 oes not belong to the corresponing issipation rate D k N 0,l, γ t. Now we present the main ieas in the proof. To overcome the ifficulties pointe out before for the case when 3 < γ <, the main observation is that two sets of time-velocity weighte energy estimates shoul be performe simultaneously as eplaine in the following. i. First of all, when estimating I α,l β,κ B efine by.9 for κ = γ, there are some error terms with higher weight when 3 < γ <, cf..0 that can not be controlle. However, as long as the solution [ft,, v, Et,, Bt, ] constructe up to t = T > 0 satisfies the a priori assumption Xt = sup 0 s t sup 0 s t EN s E N0,l 0l, γs E N,l, γs N 0 n N N 0 n N α =n n N 0 α β =n, β =j, j n n N 0 α =n α β =n, β =j, j n wl, α f s σn,j wl j, α β I Pf s ɛ 0 s σn,j wl 0 j, α β I Pf w l 0, α f w l 0,I Pf M, wl, α f.6 where M > 0 is sufficiently small, then one can obtain t E N 0,l 0l, γt D N0,l 0l, γt E, B θ D H N 0 N0,l0, t ε α E, 0

10 0 R.-J. DUAN, Y.-J. LEI, T. YANG, AND H.-J. ZHAO an t E Nt D N t E L t E N,l, γt D N,l, γt E, B θ 3 H N 0 E, B H N 0 θ D N,l,t E N te N 0,l 0, γt DN,l, t E N te N 0,l 0, γt, α E µ δ α f, where D N0,l0, t, D N,l,t, an D N,l,t are efine by 3.6 an 3.8 respectively. Notice that θ i i =,, 3 can be chosen sufficiently small as long as lj j = 0, is taken sufficiently large. Thus, one euce some uniform-in-time estimates base on the above three ifferential inequalities provie that i. The electromagnetic fiel [Et,, Bt, ] has certain temporal ecay estimate an EN t 0,l 0, γ L R ; i. There are some upper boun estimates on D N0,l0, t, D N,l,t, an D N,l,t. For eample, even if we can not euce uniform-in-time bouns on D N0,l0, t, D N,l,t, an D N,l, t, it suffices to show that the possible time increasing upper bouns on D N0,l0, t, D N,l,t, an D N,l,t are inepenent of the choices of the parameters lj j = 0, but epen only on N an N 0. To achieve i, first of all, uner the assumption of.6 with M > 0 sufficiently small, we can euce that t E N k 0 t DN k 0 t 0, k = 0,,, N 0 an t E N k 0,l, γt DN k 0,l, γt α E, k = 0, 0 hol for any 0 t T. From these two ifferential inequalities, by using the interpolation technique as in [8, 30], we can euce a temporal ecay rate of EN k 0 t, from which one can further obtain the temporal ecay rates of EN k t with E 0,l 0, γ N t 0,l 0, γ L R. ii. To euce the estimates state in i, we nee the secon set of time-velocity weighte energy estimates with the weight function w l β, t, v for some l that is sufficiently large. In this case, since w l β, t, v = w l β,t, v w l β, t, v v, w l β, t, v = w l β,t, v w l β, t, v v, we can euce that for all 3 < γ <, the terms.8 an.9 can be controlle by the etra issipative term.7 provie that the electromagnetic fiel [Et,, Bt, ] has certain temporal ecay estimates. On the other han, the term.7 relate to the linear transport term v f can only be boune as I α,l β, lt η wl β, β α I Pf C η w l β ei, αei β e i I Pf v γ. Hence, it leas to how to control w l β ei, αei β e i I Pf v γ..7 For.7, observe that Since γ < γ < hols for all 3 < γ <, it oes not lea to the increase of the weight if we neglect the fact t ϑ in the etra issipative term D α,l β, W given by.7; The orer of the erivative with respect to increases by one in.7 so that the corresponing temporal ecay rate in L norm increases, cf. [6, 7]. Therefore, motivate in [9] for eucing the temporal ecay estimates on solutions to some nonlinear equations of regularity-loss type, we set ifferent time increase rate σ n,j for wl j, β α I Pf, α β =n, β =j

11 THE VLASOV-MAXWELL-BOLTZMANN SYSTEM WITH VERY SOFT POTENTIALS where Thus, one can euce that α β =n, β =j, j n α β =n, β =j, j N 0 σ n,j σ n,j = γ ϑ. γ t σn,j wl j, αei β e i I Pf v γ t σn,j ϑ wl j, αei β e i I Pf v t σn,j wl j, αei β e i I Pf Once the above argument is substantiate, we can then close the a priori assumption.6 an the global solvability result follows. An this will be given in etail in the following sections.. 3. Proofs of the main results The proofs of Theorem. an Theorem. will be given in this section. To illustrate the main ieas of the proof clearly an to make the presentation easy to follow, we will just state some key estimates first an then use them to prove our main results. The complete proofs of these key estimates will be given in the net section. To simplify the presentation, we ivie this section into a few parts. 3.. Preliminaries. In this subsection, for later use we collect several basic estimates on the linearize Boltzmann collision operator L an the nonlinear term Γ for cutoff potentials, whose one-species version can be foun in [8, 7]. The first lemma concerns the coercivity estimate.4 on the linearize collision operators L together with its weighte version with respect to the weight w l,κ t, v given by.6. Lemma 3.. Let 3 < γ < 0, one has Lf, f I Pf. 3. Moreover, let β > 0, for η > 0 small enough an any l R, κ 0, 0 < q, ϑ R, there eists C η > 0 such that w l,κ β Lf, β f w l,κ β f η w l,κ β I Pf C η χ v Cηf 3. hols. β < β Proof. For the estimate 3., the case for the har sphere moel has been prove in [4], while for general cutoff soft potentials, recall that L can be ecompose as in. with the collision frequency v an the nonlocal integral operator K being efine by. an.3 respectively, one can euce by using the argument employe in Lemma of [3] for one-species linearize Boltzmann collision operator with cutoff that the operator K can be ecompose into a small part K s an a compact part K c, therefore 3. follows by repeating the argument use in Lemma 3 of [3]. As to 3., it can be prove by a straightforwar moification of the argument use in Lemma of [7], we thus omit the etails for brevity. The secon lemma is concerne with the corresponing weighte estimates on the nonlinear term Γ. For this purpose, similar to that of [7], we can get that β α Γ ± g, g C β0ββ β Cα αα Γ 0 ± α β g, α β g 3.3 C β0ββ β Cα αα v u γ bcos θ β0 [µu ] α β g ± v α β g ± u R 3 S α β g ± v α β g u α β g ± v α β g ± u α β g ± v α β g u ωu, where g i t,, v = [g i t,, v, g i t,, v] i =, an the summations are taken for all β 0 β β = β, α α = α. From which one can euce that

12 R.-J. DUAN, Y.-J. LEI, T. YANG, AND H.-J. ZHAO Lemma 3.. Assume κ 0, l 0. Let 3 < γ < 0, N 4, g i = g i t,, v = [g i t,, v, g i t,, v] i =,, 3, β 0 β β = β an α α = α, we have the following results: i. When α β N, we have wl,κ Γ0 ± α β g, α β g, β αg 3 m v µ δ α wl,κ β g α wl,κ β g α L wl,κ β g β αg 3 m L 3.4 or wl,κ Γ0 ± α β g, α β g, β αg 3 m v µ δ α wl,κ β g α wl,κ β g α L wl,κ β g β αg 3. m L 3.5 ii. Set ςv = v γ v, l 0, it hols that ς l Γg, g L v β ς l Γg, g L v β ς l β β g ς l g L L ς l g, L ς l β β g Proof. Although the efinition of Γ 0 ±g, g in 3.3 is a little ifferent from Γ 0 g, g of [7], one can still euce 3.4 an 3.5 by employing the similar argument use to yiel the estimates state in Lemma 3 of [7], we thus omit its proof for simplicity. As for 3.6, it can also be prove by repeating the argument use in Lemma.4 of [3]. This completes the proof of Lemma 3.. In what follows, we will collect some analytic tools which will be use in this paper. The first one is on the Sobolev interpolation among the spatial regularity. Lemma 3.3. cf. [, 8] Let p < an k, l, m R, then we have k f L p l f θ m f θ L L. Here 0 θ an l satisfy Moreover, we have that where 0 θ an l satisfy k 3 = l 3 p k 3 = l θ 3 m θ. 3 k f L l f θ m f θ L L, θ m θ, l k, m k. 3 The secon one is concerne with the L p L q estimate on the operator Λ ϱ. Lemma 3.4. Let 0 < ϱ < 3, < p < q <, q ϱ 3 = p, then Λ ϱ f L q f L p. 3.. Some a priori estimates. In this subsection, we will euce some a priori estimates on the solutions [ft,, v, Et,, Bt, ] to the Cauchy problem.5 an.6 uner some aitional assumptions impose on [ft,, v, Et,, Bt, ]. For this purpose, we suppose that the Cauchy problem.5 an.6 amits a unique local solution [ft,, v, Et,, Bt, ] efine on the time interval 0 t T for some 0 < T <. We now turn to euce certain a priori estimates on [ft,, v, Et,, Bt, ]. The first result is concerne with the temporal ecay estimates on the energy functional E k N 0 t for k = 0,,,, N 0 : Lemma 3.5. Let N 0 an N satisfy., n 3 N 0 5 3, an take k = 0,,,, N 0, then one has sup 0 τ T L. 3.6 t E k N 0 t D k N 0 t 0, 0 t T 3.7 provie that there eists a positive constant l whose precise range will be specifie in the proof of Lemma 4.3 such that H ma E N0nτ, sup E N,N, γ τ, sup E N0,N 0 l, γτ is sufficiently small. γ 0 τ T 0 τ T

13 THE VLASOV-MAXWELL-BOLTZMANN SYSTEM WITH VERY SOFT POTENTIALS 3 Furthermore, as a consequence of 3.7, we can get that EN k 0 t ma hols for 0 t T. sup E N0,N 0 kϱ, γτ, sup E N0kϱτ Proof. First notice that uner the smallness assumption H, one can euce that t kϱ 3.8 t E N k 0 t DN k 0 t 0, which is an immeiate consequence of Lemma 4.3 an Lemma 4.4 whose proofs are complicate an thus are postpone to the net section. Now we turn to compare the ifference between EN k 0 t an DN k 0 t. To this en, for the macroscopic component Pft,, v an the electromagnetic fiel [Et,, Bt, ] one has by Lemma 3.3 that k Pf, B k Pf, B kϱ kϱ Λ ϱ Pf, B kϱ an N 0 E, B N 0 E, B kϱ kϱ N 0kϱ E, B kϱ, while for the microscopic component I Pft,, v, we have by employing the Höler inequality that α I Pf α I Pf v γ kϱ kϱ α I Pf v γkϱ kϱ k α N 0 Therefore, we arrive at E k N 0 t D k N 0 t kϱ kϱ which combing with 3.7 yiels that t E N k 0 t ma sup k α N 0 Solving the above inequality irectly gives EN k 0 t ma Here we have use the fact that k α N 0 α I Pf ma sup kϱ kϱ w kϱ, γ α I Pf kϱ E N0,N 0 kϱ, γτ, sup E N0kϱτ kϱ, E N0,N 0 kϱ, γτ, sup E N0kϱτ sup This completes the proof of Lemma 3.5. E N0,N 0 kϱ, γτ, sup E N0kϱτ EN k 0 0 sup E N0,N 0 kϱ, γτ. kϱ E k N0 t t kϱ.. kϱ 0. Base on the above lemma, we can further obtain the temporal time ecay of EN k 0,l, γ t as in the following lemma. Lemma 3.6. Let l N 0, n 3 N an suppose that H ma sup E N0nτ, sup E N0,l τ l γ with l being given in Lemma 3.5, then the following estimates t E N k 0,l, γt DN k 0,l, γt α E χ k 0 α k, α β =N 0 α E, B L is sufficiently small w l β, γ α α βe i I Pf v 3γ 3.9

14 4 R.-J. DUAN, Y.-J. LEI, T. YANG, AND H.-J. ZHAO hol for any 0 t T an k = 0,,, N 0 3. Therefore, letting l 0,k N 0 with l 0 = l 0,0 = l 0, an l 0,k l 0,k 3 for k N 0 3, one has E k N 0,l 0,k i, γt ma sup E N0,N 0 kϱ, γτ, sup E N0kϱτ t k ϱi, i = 0,,, k [ϱ]. 3.0 Proof. We omit the proof of 3.9 as it is similar to the one of 3.7. Here, we point out that the main ifference for proving 3.7 an 3.9: The term α E appears when we eal with the term α E vµ, w l, γ α f ; 0 0 To euce the esire estimates on α E v α α β I Pf, wl β, γ I Pf an α k, α =k, α β =N 0 α k, α =k, α β =N 0 one has to encounter the term α E, B L α k, α =k, α β =N 0 v α B v α α β I Pf, w l β, γ I Pf, v 3γ wl β, γ α α βe i I Pf. With 3.9 in han, we now turn to prove 3.0. For the case k = 0,, the last term on the right han sie of 3.9 isappears, we have by replacing the parameter l in 3.9 by l 0 i i = 0,,, k [ϱ] an then by multiplying the resulting inequality by t kϱ iɛ that t kϱ iɛ E kn0,l0 t i, γt t kϱ iɛ D k N 0,l 0 i, γt t kϱ iɛ α E t kϱ i ɛ E k 3. N 0,l 0 i, γt. 0 Here ɛ is taken as a sufficiently small positive constant. By replacing the parameter l in 3.9 by l 0 k[ϱ], it hols that t E k t N 0,l 0 k[ϱ], γ Dk t α E. 3. N 0,l 0 k[ϱ], γ 0 By using the relation between the energy functional EN k 0,l 0, γ t an its corresponing issipation functional DN k 0,l 0, γ t, we euce by a proper linear combination of 3. an 3. that k[ϱ] C t i t kϱ iɛ E k N 0,l 0 i, γt C k[ϱ]e k t N 0,l 0 k[ϱ], γ i=0 k[ϱ] i=0 t kϱ iɛ D k N 0,l 0 i, γt Dk N 0,l 0 k[ϱ], γ t t kϱɛ α E t kϱ ɛ k Pf, B N 0 B. 0 On the other han, Lemma 3.5 tells us that t kϱɛ α E t kϱ ɛ k Pf, E, B N0 B 0 ma sup E N0,N 0 kϱ, γτ, sup E N0kϱτ t ɛ

15 THE VLASOV-MAXWELL-BOLTZMANN SYSTEM WITH VERY SOFT POTENTIALS 5 Plugging 3.4 into 3.3 an taking the time integration, one can get that k[ϱ] t kϱ iɛ E k N 0,l 0 i, γt E k N 0,l 0 k[ϱ] i=0 t, γ t k[ϱ] τ kϱ iɛ D k 0 N 0,l 0 i, γτ Dk τ N 0,l 0 k[ϱ], γ τ i=0 ma sup E N0,N 0 kϱ, γτ, sup E N0kϱτ t ɛ, an the estimate 3.0 with the case k = 0, follows by multiplying the above inequality by t ɛ where we take l 0 = l 0, = l 0,. As to the case of k N 0 3, noticing that γ 3,, let l 0,k N 0 an l 0,k l 0,k 3, l 0 = l 0,0 = l 0,, 3.0 with the case k N 0 3 follows by using inuction in k. Thus the proof of Lemma 3.6 is complete. The above two lemmas are for the temporal time ecay estimates on EN k 0 t an EN k 0,l, γ t respectively which are base on the following two assumptions: Set n > 3 N an N 0 n N. It is easy to see that if N 0 an N are suitably chosen such that. hols, one can be able to fin such an ine n; The assumptions H an H hol, that is, both ma sup E N τ, sup E N,N, γ τ, sup E N0,N 0 l, γτ γ an are assume to be small. ma sup E N0,l 0 k l γ l = k, γτ, sup E N τ l γ, 3.5 the above computation tells us that to guarantee the valiity of the assumptions impose in Lemma 3.5 an Lemma 3.6, we nee to control E N0,l 0l, γt, E N t, an E N,N, γ t suitably. To this en, we only outline the main ieas to yiel these estimates an since the proofs are quite complicate, we leave the etails to the net section. In fact, as pointe out in the introuction, if we perform the weighte energy estimate with respect to the weight function w l β, γ, it is easy to see that the corresponing term I α,l β, γ lt efine by.7 relate to the linear transport term v f can be controlle suitably. In fact, ue to w l β, γ t, v = w l β, γt, v w l β ei, γt, v v γ, the above term can be controlle by I α,l β, γ lt wl β ei, γ αei β e i I Pf ε wl β, γ β α I Pf. On the other han, since w l β, γ t, v = w l β, γt, v w l βei, γt, v v γ, one can euce that for γ <, the terms.8 an.9 containing the electromagnetic fiel [Et,, Bt, ] can not be controlle by the etra issipation term inuce by the weight w l β, γ. t ϑ wl β, γ t, v α β I Pf v

16 6 R.-J. DUAN, Y.-J. LEI, T. YANG, AND H.-J. ZHAO To overcome such a ifficulty, our main trick is to use the interpolation metho for v to boun these terms by E, B θ D H N 0 N0,l0, t with D N0,l0, t n N 0 α β =n, β =j, j n wl 0 j, α β I Pf n N α =n an some other similar terms. In fact, for E N0,l, γt, we can euce that wl 0, α f wl 0,I Pf Lemma 3.7. Let N 0 3, l N 0, l > γ, θ = γ an l l 0 l γ γ γl, then one has t E N 0,l, γt D N0,l, γt E provie that γ γ L α β N 0 wl β, γ α β I Pf v E, B θ H N 0 H 3 E N0,lt is sufficiently small. D N0,l 0, t ε α E 3.7 Note that ε > 0 is an arbitrary small constant, an for brevity of presentation, here an in the sequel the epenence of coefficient constants on ε similarly as on the right of 3.7 is skippe, since the orer of those terms are strictly higher than that of the quaratic term. Similar to the efinition of D N0,l0, t given in Lemma 3.7, for m = N or N, D m,l,t is efine by D m,l,t N 0 n m α β =n, β =j, j n w l j, α β I Pf N 0 n m α =n w l, α f. 3.8 Here we emphasize that for the functional D N,l,t or D N,l,t, the ifferentiation orer in an v starts from N 0, i.e. α β N 0. We have the following two lemmas for E N t an E N,l, γ t respectively: Lemma 3.8. Assume N 0 3, N 0 N N 0, l > γ, θ = γ l γ, l 0 3 γ, an l l γ, we can euce that t E Nt D N t provie that E L E, B H N 0 θ H 4 E N t is sufficiently small. DN,l, t E N te N 0,l 0, γt 3.9 Lemma 3.9. Take N 0 3, N 0 N N 0, l 3 > γ, θ 3 = γ l 3 γ N, l 0 l 5, an l l 3 γ γl, one has t E N,l, γt D N,l, γt γ γ E wl β, γ β α I Pf v L α β N E, B θ 3 H N 0 where we have use the assumption that D N,l,t E N te N 0,l 0, γt H 5 E N,l t is sufficiently small. α E µ δ α f, 3.0 Lemmas together with the fact E N 0,l 0, γ t L R which is a irect consequence of the estimates 3.0 imply that to euce the esire estimates on E N0,l, γt, E N t, an E N,l, γt, one nees to boun D N0,l 0, t an D N,l,t suitably. To this en, we have to perform the weighte energy estimates by replacing the weight w l β, γ by w l β, an in such a case, as eplaine in the introuction, the terms I α,l β, E an I α,l β, B corresponing to.8 an.9 can be controlle by the corresponing etra issipation rate

17 THE VLASOV-MAXWELL-BOLTZMANN SYSTEM WITH VERY SOFT POTENTIALS 7 D α,l β, W given by.7 inuce by the eponential factor of the weight w l β,t, v provie that the electromagnetic fiel [Et,, Bt, ] enjoys certain temporal ecay estimates. However, compare with the weighte energy estimate with respect to the weight w l, γ, the linear term I α,l β, lt efine by.7 leas to a new ifficult term wl β ei, αei β e i I Pf v γ, which can not be controlle irectly by combining the issipative effects D α,l β, L inuce by the linearize collision operator L. Motivate by the argument evelope in [9] to euce the temporal ecay estimates on solutions to some nonlinear equations of regularity-loss type, we want to esign ifferent time increase rate σ n,j for wl j, β α I Pf, α β =n, β =j where σ n,j σ n,j = γ γ ϑ. For result in this irection, we have the following two lemmas whose proof will be given in the net section. The first one is concerne with the case of N 0 n N. N 0 n N α β =n, β =j, j n Lemma 3.0. Assume N 0 4, σ n,j σ n,j = γ γ ϑ, l N, an l 0 l 5, one can get that t σn,j wl t j, β α I Pf t σn,0 wl, α f N 0 n N α β =n, α =n β =j, j n t σn,j w l j, β α I Pf t σn,0 w l, α f N 0 n N α β =n, β =j, j n N 0 n N α =n α N α =n t ϑ σn,j wl j, α β I Pf v 3. t ϑ σn,0 wl, α f v α f E N td N t I Pf α E t σ N,0 N 0 n N, 0 j n t σn,j E n tri,jt η N 0 n N, j n where Etri,j n t is efine by Etri,jt n γ γ E wl L j, β α I Pf v α β =n, β =j α β =n, β =j α β =n, β =j α β =n, β =j α β =n, β =j α α jm N 0, α N 0,m α α jm N 0, N 0 α N 0,m α α jm N 0, α N 0,m α α jm N 0, N 0 α N 0,m ma E N0,l 0, γt, E m,m, γ t α β =n, β =j, β <j N E E N te N 0,l 0, γt t σn,j w l β, α β I Pf, t ϑ α B L w l j m, m v α α β I Pf v t ϑ α B wl j m, m v α α β I Pf v t ϑ α E L w l j m, m v α α β I Pf t ϑ α E wl j m, m v α α β α β n, β β =j w l β, α β I Pf I Pf L v L L v L. 3.

18 8 R.-J. DUAN, Y.-J. LEI, T. YANG, AND H.-J. ZHAO Similar to Lemma 3.0, we can also get for the case of n N 0 that Lemma 3.. Uner the assumptions of Lemma 3.0, for l0 N 0, we have t σn,j w l t 0 j, β α I Pf t σn,0 w l 0, α f n N 0 α β =n, β =j, j n α =n t σ0,0 wl 0,I Pf t σn,j wl 0 j, β α I Pf n N 0 α β =n, β =j, j n t σn,0 w l 0, α f t σ0,0 w l 0,I Pf α =n t ϑ σn,j wl 0 j, β α I Pf v n N 0 α β =n, β =j, j n t ϑ σn,0 wl 0, α f v wl t ϑ σ0,0 0,I Pf v α =n α N 0 0 n N 0, 0 j n f α I Pf α E N 0 E E N0 td N0 t t σn,j F n tri,jt η n N 0, j n, α β =n, β =j, β <j where Ftri,j n t is efine by Ftri,jt n γ γ E wl L 0 j, β α I Pf v α β =n, β =j α β =n, β =j α β =n, β =j α β =n, β =j α β =n, β =j α minn j,n 0, m N 0 α N 0, m α minn j,n 0, m N 0 α N 0, m E N0,0t wl 0,f L v H α β =n, β =j α β n t σn,j wl 0 β, α β I Pf, t ϑ α B L w l 0 m j, m v α α β I Pf v t ϑ α B wl 0 m j, m v α α β I Pf v L v L t ϑ α E L w l 0 m j, m v α α β I Pf t ϑ α E wl 0 m j, m v α α I Pf L v L α β n, α, β j w l 0, α β f L v L3 w l 0 β, α β f w l 0, α α β β f L L The proof of Theorem.. We now prove Theorem. in this subsection. For this purpose, suppose that the Cauchy problem.5 an.6 amits a unique local solution [ft,, v, Et,, Bt, ] efine on the time interval 0 t T for some 0 < T < an ft,, v satisfies the a priori assumption.6, where the parameters m, N 0, N, l 0, l, an l, l 0, l, σ n,j are given in Theorem. an M is a sufficiently small positive constant. Then to use the continuity argument to eten such a solution step by step to a global one, one only nee to euce certain uniform-in-time energy type estimates on ft,, v such that the a priori assumption.6 can be close, which is the main result of the following lemma.

19 THE VLASOV-MAXWELL-BOLTZMANN SYSTEM WITH VERY SOFT POTENTIALS 9 Lemma 3.. Assume that The assumptions of Lemma 3.0 hol; ϑ is chosen to satisfy., N 0 an N satisfy.; σ N,0 = ɛ0, σ n,0 = 0 for n N ; l γ γσ N 0,N 0 ϱ, l γ γσ N,N 3ϱ an l 3 γ γσ N,N l N, l ma l γ, l3 γ γl ϱ ;, l 0 l 5, l 0 l γ γl 0 l with l = 3 l γ being given in Lemma 3.5; The a priori assumption.6 hols for some sufficiently small M > 0. Then it hols that E N t E N0,l 0l, γt E N,l, γt t σn,j wl j, β α I Pf N 0 n N α β =n, β =j, j n t ɛ0/ w l, α f Y 0 n N 0 α =n for all 0 t T. n N 0 wl 0, α f wl 0,I Pf α β =n, β =j, j n N 0 n N α =n wl, α f t σn,j w l 0 j, α β I Pf with l 3.5 Proof. Before proving 3.5, we first point out that if the assumptions state in Lemma 3. hol, especially the a priori assumption.6 is satisfie an the parameters such as ϑ, ϱ, N 0, N, σ n,j, l, l, l 3, l, l, l 0, an l satisfy the conitions liste in Lemma 3., then all the conitions liste in Lemma 3.5, Lemma 3.6, Lemma 3.7, Lemma 3.8, Lemma 3.9, Lemma 3.0, an Lemma 3. are satisfie, an base on the results obtaine in these lemmas, we can euce that: i. If we take an notice that we can euce that ma σ n,j = σ N,N, N 0 n N,0 j n σ n,0 = σ n,j σ n,j = ɛ0, n = N, 0, n N γ ϑ, γ ma σ n,j = σ N,N, N 0 n N,0 j n ma σ n,j = σ N0,N 0 ; 0 n N 0,0 j n ii. If we choose l γ γσ N,N 3ϱ an l l γ, then we can euce that θ = γ 3ϱ l γ 4σ N,N. Consequently, we have from Lemma 3.5 that E L E, B H N 0 E L E, B H N 0 θ DN,l, t θ t σ N,N t σ N,N DN,l, t Xt θ t 3 4 ϱ θ t σ N,N t σ N,N DN,l,t Xt θ t σ N,N DN,l,t; iii. If we take l 3 γ γσ N,N ϱ an l l 3 γ γl, then θ 3 = γ ϱ l 3 γ σ N,N from Lemma 3.5 that E, B θ 3 D H N 0 N,l,t Xt θ 3 t ϱ θ t σ N,N t σ N,N DN,l,t Xt θ 3 t σ N,N DN,l,t; 3.6 an we have 3.7

20 0 R.-J. DUAN, Y.-J. LEI, T. YANG, AND H.-J. ZHAO iv. For l γ γσ N 0,N 0 ϱ an l0 l γ γl 0 l, it is easy to see that θ = γ ϱ l γ σ N0,N 0 consequently we have from Lemma 3.5 that E, B θ H N 0 v. Since N 0 4, by 3.8, we take 0 < ϑ γ ϱγ4ϱ 4 4γ γ γ E L α β N or N 0 E γ γ E γ γ sup an D N0,l 0, t Xt θ t σ N 0,N 0 DN0,l 0, t; 3.8 such that wl β, γ α β I Pf v α β N or N 0 E N0,N 0 kϱ, γτ, sup E N0kϱτ α β N or N 0 Xt γ γ wl β, γ α β I Pf v γ γ wl β, γ α β I Pf v t ϑ α β N or N 0 t 3 4 ϱ γ γ wl β, γ α β I Pf v. 3.9 With the above preparations in han, we now turn to prove 3.5. To this en, we first multiply 3.9 by t ɛ0 an get by employing 3.6 that t ɛ 0 E N t ɛ 0 t ɛ0 E N t t ɛ0 D N t t t ɛ0 E L E, B H N 0 θ DN,l,t t ɛ0 E N te N 0,l 0, γt t ɛ0 Xt θ t σ N,N DN,l,t t ɛ0 E N te N 0,l 0, γt It is worth pointing out that the term ɛ 0 t ɛ0 E N t on the left han sie of the above inequality can be use to control the term t σ N,0 α E on the right han of 3.3. Seconly, plugging 3.6 into 3.9 gives t E N t D N t Xt θ t σ N,N DN,l,t E N ten 0,l 0, γt. 3.3 Thirly, by combing 3.7, 3.9 with 3.0, one has t E N,l, γt D N,l, γt Xt θ 3 t σ N,N DN,l,t E N te N 0,l 0, γt α E µ δ α f. 3.3 Thus if l is suitably chosen such that l ma l γ, l 3 γ γl, then the estimates 3.3 an 3.3 hol an from these we can euce that If we choose l N, then once we euce the estimate on E N,l, γt, the estimate on E N,N, γ t follows immeiately; A sufficient conition to control the term E N ten 0,l 0, γ t which appears on the right han sie of 3.3, 3.3, an 3. is to show that EN t 0,l 0, γ L R 3. In fact Lemma 3.6 provies us with such a nice estimate provie that sup E N0,l 0l, γτ is sufficiently small. Now we turn to estimate E N0,l 0l, γt an for this purpose, we first notice from 3.5 that since k =, l is now taken as l = 3 l γ, then for l 0 l 5, we have by replacing l in the estimate 3.7 with l 0 l an the estimate 3.8 that t E N 0,l 0l, γt D N0,l 0l, γt Xt θ t σ N 0,N 0 DN0,l 0, t 0 ε α E, 3.33

21 THE VLASOV-MAXWELL-BOLTZMANN SYSTEM WITH VERY SOFT POTENTIALS where we have use the estimate 3.9. Taking a proper linear combination of 3.3, 3.3, 3.33, 3., 3.3, an 3.30 an by using the smallness of Xt an ε, we can euce by taking the time integration from 0 to t to the resulting ifferential inequality that E N t E N0,l 0l, γt E N,l, γt t σn,j w l j, β α I Pf N 0 n N Y 0. α β =n, β =j, j n t ɛ 0 n N 0 α =n wl, α f n N 0 α β =n, β =j, j n w l 0, α f w l 0,I Pf Here we have use the following estimate t σn,j Etri,jt n N 0 n N, 0 j n N 0 α β N 0 n N 0, 0 j n N 0 n N α =n w l, α f t σn,j wl 0 j, α β I Pf t σn,j F n tri,jt Xt t σ α β, β D α, β l, t 3.34 σ α β, β Xt t D α, β l0, t 0 α β N 0 Xt γ γ t σ α β, β D α, β or,t XtD l N 0 t, α β N provie that the parameters ϑ, ϱ, N, an N 0 satisfy the conitions liste in Lemma 3.. Here to state briefly, α, β we use D l, t to enote t ϑ w l β, α β I Pf v w l β, α β I Pf. Without loss of generality, we only verify the estimate 3.34 for the term t σn,j t ϑ α B L w l 0 j, v α α β I Pf v α β =n, α =, β =j, since the other terms can be estimate in a similar way. In such a case, Lemma 3.5 tells us that α B L Xt t 5 ϱ, N 0 5, Xt t ϱ, N 0 = 4 which implies α = α = α B L if the parameters ϑ an ϱ are suitably chosen such that 0 < ϑ ϱγ 3γ 4ϱ 6 8γ 4, ϱ [, 3, N 0 5, 0 < ϑ ϱγ γ ϱ 4γ, ϱ, 3, N 0 = 4. Now ue to σ n,j σ n,j = we can get from the estimate 3.35 that t σn,j t ϑ α B L α β =n, α =, β =j, l 0 γ ϑ Xt t γ ϑ 3.35 γ ϑ, γ w l 0 j, v α α β I Pf v

22 R.-J. DUAN, Y.-J. LEI, T. YANG, AND H.-J. ZHAO Xt t σn,j ϑ α β =n, β =j, wl 0 j, α β I Pf v, that is eactly what we wante. Finally, Lemma 3.5 implies that N = N0, when ϱ [, ], Thus the proof of Lemma 3. is complete. N = N 0, when ϱ, 3. Now we turn to prove Theorem.. To this en, recall the efinition of the Xt norm. Lemma 3. tells that for the local solution [ft,, v, Et,, Bt, ] to the Cauchy problem.5 an.6 efine on the time interval [0, T ] for some 0 < T, if Xt M, t [0, T ], then there eists a sufficiently small positive constant δ 0 > 0 such that if there eists a positive constant C > 0 such that M δ 0, Xt C Y 0 hols for all 0 t T. Thus if the initial perturbation Y 0 is assume to be sufficiently small such that Y 0 δ 0 C, then the global eistence follows by combining the local solvability result with the continuation argument in the usual way. This completes the proof of Theorem The proof of Theorem.. Base on Theorem. an by taking k = 0,,,, N 0, we can get firstly from Lemma 3.5 that EN k 0 t Y0 t kϱ, that gives.3. As to.4, as long as one takes l 0 an l respectively as l 0 = l 0,0 = l 0, ma χ k l 0,k 3k 3, l 5 an l = k l γ in Theorem., then.4 follows from Lemma 3.6. Finally, to prove.5, we have by the interpolation metho with respect to space erivative for N 0 α N an by using the time ecay of N0 f an the boun of N f that, α f N f α N 0 N N 0 This is.5 an the proof of Theorem. is complete. N0 f N α N N 0 Y0 N α N 0 ϱ N N t Appeni We will complete the proofs of some lemmas an estimates use in the previous section. 4. The proof of the key estimate in Lemma 3.5. First of all, the following lemmas are for proving 3.7. Lemma 4.. Assume 3 < γ <, N 0 an N satisfying. an n 3 N 0 5 3, there eist a positive integer m satisfying N 0 m N an a sufficiently large number l, which both epen only γ an N 0, such that when k N 0, k v B v f, k f ma E m,m, γ t, E N0,N 0 l k B k I Pf k f ε, γt γ k I Pf k f, 4.

23 THE VLASOV-MAXWELL-BOLTZMANN SYSTEM WITH VERY SOFT POTENTIALS 3 when k = N 0, it hols that k v B v f, k f ma E m,m, γ t, E N0,N 0 l, γt γ N0 B N0 f ε N0 f 4., an as for k = N 0, it follows that k v B v f, k f ma E N0nt, E m,m, γ t, E N0,N 0 l, γt γ k=n 0 N0 B N0 I Pf N0 f N0 f 4.3 ε N0 I Pf N0 f. H L Proof. To obtain 4., by using the macro-micro ecomposition, one has k v B v f, k f v j B v k j f, k f = j k j k v j B v k j Pf, k f j k j k v j B v k j I Pf, k Pf I B, I B, v j B v k j I Pf, k I Pf. I B,3 Applying the interpolation metho with respect to space erivative, so we euce from Lemma 3.3 that I B, I B, j B L 3 k j µ δ f k µ δ f j k j k k j Λ k3 B k B j k3 Λ µ δ f j k3 k µ δ f k j k3 k µ δ f E 0,0, γ t k B k f ε k f. As for I 3,3, when j = k, taking L 6 L 3 L type inequality an applying Lemma 3.3, one has I B,3 k B v L 6 I Pf v γ k I Pf v γ L v L 3 E, 5 γ, γ t k B ε k I Pf. L v L While for the case j k, by the similar virtue of the estimates on I B,3 for j = k, one also has I B,3 j B v L k j I Pf k I Pf v j k j k Λ ϱ B k j kϱ k B jϱ3 kϱ m j v k j I Pf m j Λ ϱ I Pf m j j m j kϱ k I Pf m j k jϱ m j kϱ k I Pf v Λ ϱ B k j kϱ k B jϱ3 kϱ m j v k j I Pf m j j k Λ ϱ I Pf m j j m j kϱ k I Pf v γ m j k jϱβ j m j kϱ k I Pf m j k jϱ β j v lj m j kϱ ma k I Pf v γ k I Pf v lj β j E km,m, γt, E t ˆlγ k B k I Pf k,, γ βj ε k I Pf.