THE VLASOV-POISSON-BOLTZMANN SYSTEM FOR SOFT POTENTIALS

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1 May 9, 01 11:15 WSPC/INSTRUCTION FILE M3AS-DYZ-s Mathematical Models and Methods in Applied Sciences c World Scientific Publishing Company THE VLASOV-POISSON-BOLTZMANN SYSTEM FOR SOFT POTENTIALS RENJUN DUAN Department of Mathematics, The Chinese University of Hong Kong Shatin, Hong Kong rjduan@math.cuhk.edu.hk TONG YANG Department of Mathematics, City University of Hong Kong Kowloon, Hong Kong and School of Mathematics and Statistics, Wuhan University Wuhan 43007, P.R. China matyang@cityu.edu.hk HUIJIANG ZHAO School of Mathematics and Statistics, Wuhan University Wuhan 43007, P.R. China hhjjzhao@hotmail.com Received (Day Month Year) Revised (Day Month Year) Communicated by (xxxxxxxxxx) An important physical model describing the dynamics of dilute weakly ionized plasmas in the collisional kinetic theory is the Vlasov-Poisson-Boltzmann system for which the plasma responds strongly to the self-consistent electrostatic force. This paper is concerned with the electron dynamics of kinetic plasmas in the whole space when the positive charged ion flow provides a spatially uniform background. We establish the global existence and optimal convergence rates of solutions near a global Maxwellian to the Cauchy problem on the Vlasov-Poisson-Boltzmann system for angular cutoff soft potentials with γ < 0. The main idea is to introduce a time dependent weight function in the velocity variable to capture the singularity of the cross-section at zero relative velocity. Keywords: Vlasov-Poisson-Boltzmann system; soft potentials; stability; AMS Subject Classification: 76X05, 76P05, 35Q0, 35B35, 35B40 1. Introduction The Vlasov-Poisson-Boltzmann (called VPB in the sequel for simplicity) system is a physical model describing the motion of dilute weakly ionized plasmas (e.g., electrons and ions in the case of two-species) in the collisional kinetic theory, where 1

2 May 9, 01 11:15 WSPC/INSTRUCTION FILE M3AS-DYZ-s Renjun Duan, Tong Yang and Huijiang Zhao plasmas respond strongly to the self-consistent electrostatic force. In physics, the ion mass is usually much larger than the electron mass so that the electrons move much faster than the ions. Thus, the ions are often described by a fixed background n b (x) and only the electrons move rapidly. In this simplified case, the VPB system takes the form of t f + x f + x φ f = Q(f, f), (1.1) x φ = f d 1, φ(x) 0 as x, (1.) R 3 f(0, x, ) = f 0 (x, ). (1.3) Here the unknown f = f(t, x, ) 0 is the density distribution function of the particles located at x = (x 1, x, x 3 ) R 3 with velocity = ( 1,, 3 ) R 3 at time t 0. The potential function φ = φ(t, x) generating the self-consistent electric field x φ in (1.1) is coupled with f(t, x, ) through the Poisson equation (1.), where n b (x) 1 is chosen to be a unit constant. The bilinear collision operator Q acting only on the velocity variable, see Ref. 4, 14, 35, is defined by ( ) Q(f, g) = γ q 0 (θ) f( )g( ) f( )g() dωd, (1.4) R 3 S where (, ) and (, ), denoting velocities of two particles before and after their collisions respectively, satisfy = [( ) ω]ω, = + [( ) ω]ω, ω S, by the conservation of momentum and energy + = +, + = +. Note that the identity = holds. The function γ q 0 (θ) in (1.4) is the cross-section depending only on and cos θ = ( ) ω/, and it satisfies 3 < γ 1 and it is assumed to satisfy the Grad s angular cutoff assumption 0 q 0 (θ) C cos θ. The exponent γ is determined by the potential of intermolecular forces, which is called the soft potential when 3 < γ < 0, the Maxwell model when γ = 0 and the hard potential when 0 < γ 1 including the hard sphere model γ = 1, q 0 (θ) = cont. cos θ. For the soft potential, the case < γ < 0 is called the moderately soft potential and 3 < γ < very soft potential, see Ref. 35. Notice 3,31,35 that the Coulomb potential coincides with the limit at γ = 3 for which the Boltzmann collision operator should be replaced by the Landau operator under the grazing collision. It is an interesting problem to consider the nonlinear stability and convergence rates of a spatially homogeneous steady state M for the Cauchy problem (1.1), (1.), (1.3) when initial data f 0 is sufficiently close to M in a certain sense, where M = (π) 3/ e /

3 May 9, 01 11:15 WSPC/INSTRUCTION FILE M3AS-DYZ-s The Vlasov-Poisson-Boltzmann System for Soft Potentials 3 is a normalized global Maxwellian. This problem was first solved by Guo for the VPB system on torus in the case of the hard sphere model. Since then, the case of non hard sphere models, i.e. 3 < γ < 1, under the angular cutoff assumption, has remained open. In a recent work 13 for the VPB system over the whole space, we studied the problem for the case 0 γ 1 that includes both the Maxwell model and general hard potentials. However, the method used in Ref. 13 can not be applied to the soft potentials, mainly because the collision frequency ν() (1 + ) γ with γ < 0 is degenerate in the large-velocity domain. In this paper we shall study the problem in the soft potential case by further developing the approach in Ref. 13 with the following extra ingredients to overcome the specific mathematical difficulties in dealing with soft potentials: a new time-velocity-dependent weight in the form of exp{ [q + λ/(1 + t) ϑ ]} to capture the dissipation for controlling the velocity growth in the nonlinear term for non hard sphere models, and a time-frequency/time weighted method to overcome the large-velocity degeneracy in the energy dissipation. The approach and techniques that this paper together with Ref. 13 developed can be applied to some other collisional kinetic models for the non hard sphere interaction potential when an external forcing is present Main results To this end, as in Ref., set the perturbation u = u(t, x, ) by f(t, x, ) M = M 1/ u(t, x, ). Then, the Cauchy problem (1.1)-(1.3) of the VPB system is reformulated as t u + x u + x φ u 1 xφu x φ M 1/ = Lu + Γ(u, u), (1.5) x φ = M 1/ u d, φ(x) 0 as x, (1.6) R 3 u(0, x, ) = u 0 (x, ) = M 1/ (f 0 M), (1.7) where the linearized collision operator L and the quadratic nonlinear term Γ are defined by { ( ) ( )} Lu = M 1 Q M, M 1/ u + Q M 1/ u, M, ) Γ(u, u) = M 1 Q (M 1/ u, M 1/ u, respectively. Here, notice that due to (1.6), φ can be determined in terms of u by φ(t, x) = 1 4π x x M 1/ u(t, x, ) d. (1.8) R 3

4 May 9, 01 11:15 WSPC/INSTRUCTION FILE M3AS-DYZ-s 4 Renjun Duan, Tong Yang and Huijiang Zhao By plugging the above formula into the equation (1.5) for the reformulated VPB system, one has a single evolution equation for the perturbation u, see Ref. 1, 9. Before stating our main result, we first introduce the following mixed timevelocity weight function λ w τ,q (t, ) = γτ e [ q+ (1+t) ], ϑ (1.9) where τ R, 0 q 1, 0 < λ 1, 0 < ϑ 1/4, and = (1 + ) 1/. Notice that even though w τ,q (t, ) depends also on parameters λ and ϑ, we skip them for notational simplicity, and in many places we also use q(t) to denote q + λ for (1+t) ϑ brevity. For given u(t, x, ) with the corresponding function φ(t, x) given by (1.8), we then define a temporal energy norm u N,l,q (t) = w β l,q (t, ) β α u(t) + x φ(t) H N, (1.10) α + β N where N 0 is an integer, and l N is a constant. The main result of this paper is stated as follows. Some more notations will be explained at the end of this section. } Theorem 1.1. Let γ < 0, N 8, l 0 > 5 {N,, l 1 + max l0 1 γ and q(t) = q + λ (1+t) ϑ f 0 = M + M 1/ u 0 0 and 0 with 0 q 1, 0 < λ 1, and 0 < ϑ 1/4. Assume that R 3 R 3 M 1/ u 0 dxd = 0. (1.11) ) There are constants ɛ 0 > 0, C 0 > 0 such that if w β l,q (0, ) β α u 0 + (1 + x + γl 0 Z1 u 0 ɛ 0, (1.1) sup t 0 α + β N then the Cauchy problem (1.5), (1.6), (1.7) of the VPB system admits a unique global solution u(t, x, ) satisfying f(t, x, ) = M + M 1/ u(t, x, ) 0 and { u N,l,q (t) + (1 + t) 3 4 u N,l 1,q (t) +(1 + t) 5 4 xφ(t) } H N 1 C 0 ɛ 0. (1.13) Remark 1.1. The condition (1.11) implies that the ionized plasma system with electrons and ions are neutral at initial time. Due to the conservation of mass for (1.1) or (1.5), the neutral condition (1.11) holds for all time t > 0. Notice that (1.11) together with (1 + x )u 0 Z1 < can be used to remove the singularity of the Poisson kernel and to induce the same time-decay rate as in the case of the Boltzmann equation; see also Remark 3.1. Moreover, in (1.1), we put the extra space-velocity weight 1 + x + γl 0 on u 0 in Z 1 -norm. This is necessary in the proof of Lemma 6.1 concerning the desired uniform-in-time a priori estimate.

5 May 9, 01 11:15 WSPC/INSTRUCTION FILE M3AS-DYZ-s The Vlasov-Poisson-Boltzmann System for Soft Potentials 5 Remark 1.. The problem of the global existence for the very soft potential case 3 < γ < is still left open. Despite this, we shall discuss in the next subsections the main difficulty. We conclude this subsection by pointing out some possible extensions of Theorem 1.1. First, the arguments used in this paper can be adopted straightforwardly to deal with either the VPB system with soft potentials on torus with additional conservation laws as in Ref. or the two-species VPB system with soft potentials as in Ref. 1, 37. Next, it could be quite interesting to apply the current approach as well as techniques in Ref. 0 and Ref. 31 to consider the Vlasov-Poisson-Landau system 3, where Q in (1.1) is replaced by the Landau operator { } Q L (f, f) = B L ( )[f( ) f() f() f( )] d, R 3 where B L () is a non-negative matrix given by B L ij() = ( δ ij ) i j γ+, γ 3. Note that the soft potential for the Landau operator corresponds to the case 3 γ <. 1.. Strategy of the proof We first recall several elementary properties of the linearized operator L. First of all, L can be written as L = ν + K, where ν = ν() (1 + ) γ denotes the collision frequency, and K is a velocity integral operator with a real symmetric integral kernel K(, ). The explicit representation of ν and K will be given in the next section. It is known that L is non-positive, the null space of L is given by { N = span M 1/, i M 1/ (1 i 3), M 1/}, and L is coercive in the sense that there is a constant κ 0 > 0 such that, cf. 4,18,8 ulu d κ 0 R 3 ν() {I P}u d R 3 (1.14) holds for u = u(), where I means the identity operator and P denotes the orthogonal projection from L to N. As in Ref. 19, 0, for any given any u(t, x, ), one

6 May 9, 01 11:15 WSPC/INSTRUCTION FILE M3AS-DYZ-s 6 Renjun Duan, Tong Yang and Huijiang Zhao can write Pu = { a(t, x) + b(t, x) + c(t, x) ( 3 )} M 1/, a = M 1/ u d = M 1/ Pu d, R 3 R 3 b i = i M 1/ u d = i M 1/ Pu d, 1 i 3, R 3 R 3 c = 1 ( 3 ) M 1/ u d = 1 ( 3 ) M 1/ Pu d. 6 R 6 3 R 3 so that we have the macro-micro decomposition introduced in Ref. 19 u(t, x, ) = Pu(t, x, ) + {I P}u(t, x, ). (1.15) Here, Pu is called the macroscopic component of u(t, x, ) and {I P}u the microscopic component of u(t, x, ). For later use, one can rewrite P as Pu = P 0 u P 1 u, P 0 u = a(t, x)m 1/, P 1 u = { b(t, x) + c(t, x) ( 3 )} M 1/, where P 0 and P 1 are the projectors corresponding to the hyperbolic and parabolic parts of the macroscopic component, respectively, see Ref. 9. To prove Theorem 1.1, we introduce the equivalent temporal energy functional and the corresponding dissipation rate functional D N,l,q (t) = α + β N + E N,l,q (t) u N,l,q(t), (1.16) ν 1/ w β l,q (t, ) β α {I P}u(t) 1 (1 + t) 1+ϑ α + β N w β l,q (t, ) α β {I P}u(t) + a + α N 1 We also introduce the time-weighted temporal sup-energy x α (a, b, c). (1.17) X N,l,q (t) = sup E N,l,q (s) + sup (1 + s) 3 EN,l 1,q (s) 0 s t 0 s t + sup (1 + s) 5 xφ(s). (1.18) H 0 s t N 1 Here, the exact definitions of E N,l,q (t) is given by (5.16) in the proof of Lemma 5.. The strategy to prove Theorem 1.1 is to obtain the uniform-in-time estimates under the a priori assumption that X N,l,q (t) is small enough over 0 t < T for any given T > 0. Indeed,

7 May 9, 01 11:15 WSPC/INSTRUCTION FILE M3AS-DYZ-s The Vlasov-Poisson-Boltzmann System for Soft Potentials 7 one can deduce that there is a temporal energy functional E N,l,q (t) satisfying (1.16) such that d dt E N,l,q(t) + κd N,l,q (t) 0, (1.19) for 0 t < T, where D N,l,q (t) is given by (1.17), and moreover, by combining the time-decay property of the linearized system, one can prove that X N,l,q (t) C { ɛ N,l,q + X N,l,q (t) } (1.0) for 0 t < T, where ɛ N,l,q depends only on initial data u 0. Therefore, by the local existence of solutions as well as the continuity argument, X N,l,q (t) is bounded uniformly in all time t 0 as long as ɛ N,l,q is sufficiently small, which then implies Theorem 1.1. We remark that this kind of strategy used here was first developed in Ref. 11 for the study of the time-decay property of the linearized Boltzmann equation with external forces and was later revisited in Ref. 10 for further generalization of the approach Difficulties and ideas We now outline detailed ideas to carry out the above strategy with emphasize on the specific mathematical difficulties. Before that, it is worth to pointing out that for the VPB system with soft potentials, the only result available so far is Ref. 4 in which global classical solutions near vacuum are constructed. One may expect that the work 3 and its recent improvement 34 by using the spectral analysis and the contraction mapping principle can be adapted to deal with this problem. However, we note that when the self-induced potential force is taken into account, even for the hard-sphere interaction, with the spectral property obtained in Ref. 16, the spectral theory corresponding to Ref. 3 has not been known so far, partially because the Poisson equation produces an additional nonlocal term with singular kernels. Fortunately, the energy method recently developed in 0,19 works well in the presence of the self-induced electric field,38,1 or even electromagnetic field 1,30,7. But due to the appearance of the nonlinear term x φ{i P}u, the direct application of the coercive estimate (1.14) of the linearized collision operator L works only for the hard-sphere interaction with γ = 1. Indeed with a bit weaker (softer) than hard-sphere interaction, such a term is beyond the control by either the usual energy or the energy dissipation rate so that even the local-in-time solutions can not be constructed within this framework. To overcome this difficulty, a new weighed energy method was introduced in Ref. 13 by the authors of this paper to deal with the VPB system with hard potentials. Such a method is based on the use of a mixed time-velocity weight function w hp l (t, ) = l λ e (1+t) ϑ, (1.1)

8 May 9, 01 11:15 WSPC/INSTRUCTION FILE M3AS-DYZ-s 8 Renjun Duan, Tong Yang and Huijiang Zhao where l R, λ > 0 and ϑ > 0 are suitably chosen constants. One of the most important ingredients is to combine the time-decay of solutions with the usual weighted energy inequalities in order to obtain the uniform-in-time estimates. The main purpose of this paper is to generalize the above argument so that it can be adapted to deal with the soft potential case. Compared with the hard potential case, the main difference lies in the fact that at high velocity the dissipation is much weaker than the instant energy. Our main ideas to overcome this are explained as follows. Firstly, similar to the hard potential case 13, we introduce an exponential weight factor [ wτ,q(t, e ) = exp { q + ]} λ (1 + t) ϑ in the weight function w τ,q (t, ) given by (1.9). Notice that q(t) := q + λ/(1 + t) ϑ satisfies 0 < q(t) 1 by choices of q, λ and ϑ as stated in Theorem 1.1, and hence all the estimates on ν and K obtained in Ref. 31 can still be applied with respect to the weight function w τ,q (t, ) considered here. A key observation to use the exponential factor is based on the identity w e τ,q(t, ) t g(t, x, ) = t { w e τ,q (t, )g(t, x, ) } + λϑ (1 + t) 1+ϑ we τ,q(t, )g(t, x, ). Thus one can gain from the second term on the right hand side of the above identity an additional second-order velocity moment with a compensation that the magnitude of the additional dissipation term decays in time with a rate (1+t) (1+ϑ). With this, the nonlinear term x φ{i P}u can be controlled as long as the electric field x φ has the time-decay rate not slower than (1 + t) (1+ϑ). Notice that in the case of the whole space, xφ H N 1 as a part of the high-order energy functional decays at most as (1 + t) 5 4 and thus it is natural to require 0 < ϑ 1/4. Moreover, as used for hard potentials in Ref. 13, the first-order velocity moment in the exponential factor of (1.1) is actually enough to deal with the largevelocity growth in x φ{i P}u. The reason why we have chosen for soft potentials is that one has to control another nonlinear term x φ {I P}u. We shall clarify this point in more details later. The second idea is to control the velocity derivative of u. It is well-known that u may grow in time. To overcome such a difficulty, we apply the algebraic velocity weight factor, introduced in 0,18, w v β l () = γ( β l), l β, in the weight w τ,q (t, ) given in (1.9) with τ = β l. This algebraic factor implies two properties of the velocity weight: one is that w β l v () 1 holds true due to l β, and the other one is that the higher the order of the -derivatives, the more negative velocity weights. The restriction l β, more precisely { l 1 max N, l 0 1 } β, γ

9 May 9, 01 11:15 WSPC/INSTRUCTION FILE M3AS-DYZ-s The Vlasov-Poisson-Boltzmann System for Soft Potentials 9 results from the fact that one has to use the positive-power algebraic velocity weight w β l v () so as to obtain the closed estimate (1.0) on X N,l,q(t), because not only the nonlinear term x φu contains the first-order velocity growth but also initial data is supposed to have an extra positive-power velocity weight γl 0 in the time-decay estimate (3.5) for the evolution operator of the linearized VPB system. As pointed out in Ref. 18 for the study of the Boltzmann equation with soft potentials, the dependence of the weight w β l v () on β makes it possible to control the linear term x u in terms of the purely x-derivative dissipation for the weighted energy estimate on the mixed x- derivatives. However, the algebraic velocity factor w β l v () produces an additional difficulty on the nonlinear term xφ u in the presence of the self-consistent electric field for the VPB system with soft potentials. To obtain the velocity weighted derivative estimate on such a nonlinear term, one should put an extra negative-power function γ in front of {I P}u in the nonlinear term x φ u so that the velocity growth γ comes up to have a balance. Then, as long as γ < 0, it is fortunate that the dissipation functional D N,l,q (t) given by (1.17) containing the second-order moment can be used to control the term x φ u. At this point, it is not known how to use the argument of this paper to deal with the very soft potential case 3 < γ <. However, inspired by Ref. 3, we remark that one possible way is to have a stronger dissipation property of the linearized collision operator L, particularly the possible velocity diffusion effect. In fact, for the Boltzmann equation without angular cutoff assumption, from the coercive estimate on the linearized Boltzmann collision operator L obtained in 1,17, it seems hopeful to deal with the case 3 < γ < for such a physical situation, which will be reported in the future work. Another ingredient of our analysis is the decay of solutions to the VPB system for soft potentials. Recall that the large-time behavior of global solutions has been studied extensively in recent years by using different approaches. One approach which usually leads to slower time-decay than in the linearized level is used in Ref. 38 on the basis of the improved energy estimates together with functional inequalities. The method of thirteen moments and compensation functions proposed by Kawashima in Ref. 7 which gives the optimal time rate without using the spectral theory; see Ref. 15 and Ref. 37 for two applications. Recently, concerning with the optimal time rate, a time-frequency analysis method has been developed in 9,8,7. Precisely, in the same spirit of Ref. 36, some time-frequency functionals or interactive energy functionals are constructed in 9,8,7 to capture the dissipation of the degenerate components of the full system. Back to the nonlinear VPB system, the main difficulty of deducing the decay rates of solutions for the soft potential is caused by the lack of a spectral gap for the linearized collision operator L. Unlike the periodic domain 31, we need a careful and delicate estimate on the time decay of solutions to the corresponding

10 May 9, 01 11:15 WSPC/INSTRUCTION FILE M3AS-DYZ-s 10 Renjun Duan, Tong Yang and Huijiang Zhao linearized equation in the case of the whole space R 3. Following the recent work 9,8, our analysis is based on the weighted energy estimates, a time-frequency analysis method, and the construction of some interactive energy functionals, which gives a new and concise proof of the decay of the solution to the linearized equation with soft potentials. Here, we should mention the recent work 9 by Strain. Different from this work, by starting from the inequality t E l (û) + κd l (û) 0, we use a new time-frequency weighted approach together with the time-frequency splitting technique in order to deduce the time-decay estimate on the solution u for the linearized VPB system; see Step 3 in the proof of Theorem 3.1 and the identity (3.18). Finally we also point out a difference between the soft potential case and hard potentials studied in Ref. 13 for obtaining the time-decay of the energy functional E N,l,q (t). The starting point is the Lyapunov inequality (1.19). In the hard potential case, the Gronwall inequality can be directly used when the lowest-order terms (a, b, c) and x φ are added into D N,l,q (t). However, this fails for the soft potentials because D N,l,q (t) given by (1.17) contains the extra factor ν() γ which is degenerate at large-velocity domain when γ < 0. To overcome this difficulty, we have used the time-weighted estimate on (1.19) as well as an iterative technique on the basis of the inequality D N, l,q (t) + (b, c)(t) + x φ(t) κe N, l 1,q(t) for any l; see Step in the proof of Lemma 6.1 for details. Before concluding this introduction, we mention some previous results concerning the study of the VPB system in other respects, cf. 5,5, the Boltzmann equation with soft potentials, cf. 33,,3, and also the exponential rate for the Boltzmann equation with general potentials in the collision kernel but under additional conditions on the regularity of solutions, cf. 6. The rest of this paper is organized as follows. In Section, we will list some estimates on ν and K proved in Ref. 31. In Section 3, we will study the time-decay property of the linearized VPB system without any source. In Section 4, we give the estimates on the nonlinear terms with respect to the weight function w τ,q (t, ). In Section 5, we will close the a priori estimates on the solution to derive the desired inequality (1.19). In the last section, we will prove (1.0) to conclude Theorem Notations Throughout this paper, C denotes some positive (generally large) constant and κ denotes some positive (generally small) constant, where both C and κ may take different values in different places. A B means A 1 κb and A B means κa B 1 κa, both for a generic constant 0 < κ < 1. For an integer m 0, we use Hx, m, Hm x, H m to denote the usual Hilbert spaces H m (R 3 x R 3 ), Hm (R 3 x),

11 May 9, 01 11:15 WSPC/INSTRUCTION FILE M3AS-DYZ-s The Vlasov-Poisson-Boltzmann System for Soft Potentials 11 H m (R 3 ), respectively, and L, L x, L are used for the case when m = 0. When without any confusion, we use H m to denote Hx m and use L to denote L x or L x,. We denote, by the inner product over L x,. For q 1, we also define the mixed velocity-space Lebesgue space Z q = L (Lq x) = L (R 3 ; Lq (R 3 x)) with the norm u Zq = ( R 3 ( ) ) /q 1/ u(x, ) q dx d, u = u(x, ) Z q. R 3 For multi-indices α = (α 1, α, α 3 ) and β = (β 1, β, β 3 ), we denote α β = α x β, that is, β α = α1 x 1 x α x α3 3 β1 1 β β3 3. The length of α is α = α 1 + α + α 3 and the length of β is β = β 1 + β + β 3.. Preliminaries Recall that L = ν + K is defined by ν() = γ q 0 (θ)m( ) dωd (1 + ) γ, R 3 S and Ku() = γ q 0 (θ)m 1/ ( )M 1/ ( )u( ) dωd R 3 S + γ q 0 (θ)m 1/ ( )M 1/ ( )u( ) dωd R 3 S γ q 0 (θ)m 1/ ( )M 1/ ()u( ) dωd R 3 S = K(, )u( )d. R 3 We list in the following lemma velocity weighted estimates on the collision frequency ν() and the integral operator K with respect to the velocity weight function w τ, q () = γτ e q, τ R, 0 q 1. Lemma.1 (Ref. 31). Let 3 < γ < 0, τ R, and 0 q 1. If β > 0, then for any η > 0, there is C η > 0 such that wτ, q() β (νu) β u d ν()wτ, q() β u d R 3 R 3 η ν()wτ, q() β1 u d C η R 3 β 1 β R 3 R 3 χ Cη γτ u d.

12 May 9, 01 11:15 WSPC/INSTRUCTION FILE M3AS-DYZ-s 1 Renjun Duan, Tong Yang and Huijiang Zhao If β 0, then for any η > 0, there is C η > 0 such that wτ, q() β (Kf) g d R η ( ν()wτ, q() β1 f d 3 R 3 β 1 β +C η (R 3 χ Cη γτ f d ) 1 } ) 1 ( ) 1 ν()wτ, q() g d. (.1) R 3 For later use, let us write down the macroscopic equations of the VPB system up to third-order moments by applying the macro-micro decomposition (1.15) introduced in Ref. 19. For that, as in Ref. 9, define moment functions Θ ij ( ) and Λ i ( ), 1 i, j 3, by Θ ij (v) = R 3 ( i j 1)M 1/ v d, Λ i (v) = 1 10 R 3 ( 5) i M 1/ v d, (.) for any v = v(). Then, one can derive from (1.5)-(1.6) a fluid-type system of equations t a + x b = 0, and with t b + x (a + c) + x Θ({I P}u) x φ = x φa, t c x b x Λ({I P}u) = 1 3 xφ b, x φ = a, t Θ ij ({I P}u) + i b j + j b i 3 δ ij x b 10 3 δ ij x Λ({I P}u) = Θ ij (r + G) 3 δ ij x φ b, t Λ i ({I P}u) + i c = Λ i (r + G) (.3) (.4) r = x {I P}u + Lu, G = Γ(u, u) + 1 xφu x φ u, where r is a linear term related only to the micro component {I P}u and G is a quadratic nonlinear term. Here and hereafter, for simplicity, we used j to denote xj for each j = 1,, 3. The above fluid-type system (.3)-(.4) plays a key role in the analysis of the zero-order energy estimate and the dissipation of the macroscopic component (a, b, c) in the case of the whole space; see 19,1,9,8,7. 3. Time decay for the evolution operator Consider the linearized homogeneous equation t u + x u x φ M 1/ = Lu, (3.1)

13 May 9, 01 11:15 WSPC/INSTRUCTION FILE M3AS-DYZ-s The Vlasov-Poisson-Boltzmann System for Soft Potentials 13 with φ = 1 4π x x M 1/ u(t, x, ) d 0 as x. (3.) R 3 Given u 0 = u 0 (x, ), e tb u 0 is the solution to the Cauchy problem (3.1)-(3.) with u t=0 = u 0. For an integer m, set the index σ m of the time-decay rate by σ m = m, which corresponds to the one for the case of the usual heat kernel in three dimensions. The main result of this section is stated as follows. Theorem 3.1. Set µ = µ() := γ. Let 3 < γ < 0, l 0, α 0, m = α, l 0 > σ m, and assume a 0 dx = 0, R 3 (1 + x ) a 0 dx <, R 3 (3.3) and µ l+l 0 u 0 Z1 + µ l+l 0 α u 0 <. (3.4) Then, the evolution operator e tb satisfies µ l α e tb u 0 + α x x 1 P 0 e tb u 0 ( C(1 + t) σm µ l+l0 u Z1 0 + ) µ l+l0 α u 0 + (1 + x )a 0 L 1 x (3.5) for any t 0. Proof. The proof is divided by three steps. Step 1. The Fourier transform of (3.1)-(3.) gives t û + i kû ik ˆφ M 1/ = Lû, (3.6) with k ˆφ = â. By taking the complex inner product of the above equation with û, integrating it over R 3 and using (1.14), as in 9, one has ( ) t û L + â k + κ ν 1/ {I P}û 0. (3.7) L Furthermore, similar to derive (.3) and (.4) from the nonlinear VPB system, one can also derive from (3.1)-(3.) a fluid-type system of linear equations t a + x b = 0, t b + x (a + c) + x Θ({I P}u) x φ = 0, t c x b x Λ({I P}u) = 0, x φ = a,

14 May 9, 01 11:15 WSPC/INSTRUCTION FILE M3AS-DYZ-s 14 Renjun Duan, Tong Yang and Huijiang Zhao and t Θ ij ({I P}u) + i b j + j b i 3 δ ij x b t Λ i ({I P}u) + i c = Λ i (r) 10 3 δ ij x Λ({I P}u) = Θ ij (r), with r = x {I P}u + Lu, where Θ( ) and Λ( ) are defined in (.). From the above fluid-type system, as in 7, one can deduce { k t R E int } (û(t, k)) + κ (a, b, c) + â 1 + k C ν 1/ {I P}û, (3.8) where E int (û(t, k)) is given by L E int (û(t, k)) = { 1 (ikĉ ) Λ({I P}û) 1 + k + ( ik jˆbm + ik mˆbj ) 3 δ jmik ˆb Θ jm ({I P}û) jm +κ 1 (ikâ ˆb) }. Therefore, for 0 < κ 1, a suitable linear combination of (3.7) and (3.8) gives [ ] t û L + â k + κ R E int (û(t, k)) { ν + κ 1/ {I P}û Here, notice that since L E int (û(t, k)) { C {I P}û L + we have taken κ > 0 small enough such that û L } + k (a, b, c) + â 1 + k 0. (3.9) (a, b, c) }, + â k + κ R E int (û(t, k)) û L + â k. (3.10) Step. Applying {I P} to (3.6), we have t {I P}û + i k{i P}û = L{I P}û + Pi kû i kpû. By further taking the complex inner product of the above equation with µ l (){I

15 May 9, 01 11:15 WSPC/INSTRUCTION FILE M3AS-DYZ-s The Vlasov-Poisson-Boltzmann System for Soft Potentials 15 P}û and integrating it over R 3, one has 1 t µ l {I P}û L + κ µ l 1 {I P}û L ( R K{I P}û µ l (){I P}û ) d R 3 ( + R Pi kû i kpû µ l (){I P}û ) d. (3.11) R 3 It follows from (.1) that for any η > 0, ( K{I P}û µ l (){I P}û ) d R 3 { } η ν 1/ µ l χ Cη {I P}û + C L η γl {I P}û ν 1/ µ l {I P}û, (3.1) which further by the Cauchy-Schwarz inequality implies ( K{I P}û µ l (){I P}û ) d R 3 η ν 1/ µ l {I P}û ν + C 1/ η {I P}û. L Notice that over k 1, ( R Pi kû i kpû µ l (){I P}û ) d χ k 1 R 3 ( ν C 1/ {I P}û L + k 1 + k (a, b, c) ). Plugging the above two inequalities into (3.11) and fixing a small enough constant η > 0 give t µ l {I P}û L χ k 1 + κ µ l 1 {I P}û L ( ν C 1/ {I P}û L χ k 1 + k 1 + k (a, b, c) ). (3.13) To obtain the velocity-weighted estimate for the pointwise time-frequency variables over k 1, we directly take the complex inner product of (3.6) with µ l ()û and integrate it over R 3 to get that 1 t µ l û + ν 1/ µ l {I P}û L L ( = R ν(){i P}û µ l ()Pû ) ( d + R K{I P}û µ l ()û ) d R 3 R ( 3 + R ik ˆφ ) M 1/ µ l ()û d. (3.14) R 3

16 May 9, 01 11:15 WSPC/INSTRUCTION FILE M3AS-DYZ-s 16 Renjun Duan, Tong Yang and Huijiang Zhao Here, Cauchy-Schwarz s inequality implies ( R ν(){i P}û µ l ()Pû ) d C R 3 and R R 3 ( ν 1/ {I P}û + (a, b, c) ), L ( ik ˆφ ) M 1/ µ l ()û d ( = R ik ˆφ ) M 1/ µ l ()[Pû + {I P}û] d R 3 { ν C 1/ {I P}û + } (a, b, c) + â L k, where k ˆφ = â / k due to the Poisson equation k ˆφ = â was used. From (.1), similar to the proof of (3.1), one has ( R K{I P}û µ l ()û ) d R 3 ( = R K{I P}û µ l ()[Pû + {I P}û] ) d R 3 ( ν η ν 1/ µ l {I P}û + C 1/ η {I P}û + (a, b, c) ). L L By plugging the above two estimates into (3.14), fixing a small constant η > 0 and using 1 k 1+ k χ k 1, it follows that t µlû χ L k 1 + κ µ l 1 {I P}û χ L k 1 ( ν C 1/ {I P}û L + k 1 + k (a, b, c) ). (3.15) Now, for properly chosen constants 0 < κ 3, κ 4 1, a suitable linear combination of (3.9), (3.13) and (3.15) yields that whenever l 0, t E l (û) + κd l (û) 0, (3.16) holds true for any t 0, k R 3, where E l (û) and D l (û) are given by E l (û) = û L + â k + κ R E int (û(t, k)) +κ 3 µ l {I P}û L χ k 1 + κ 4 µ l û L χ k 1, D l (û) = µ l 1 {I P}û L + k 1 + k (a, b, c) + â. Due to (3.10) and the fact that Pû decays exponentially in, it is straightforward to check that for l 0, E l (û) µlû L + â k.

17 May 9, 01 11:15 WSPC/INSTRUCTION FILE M3AS-DYZ-s The Vlasov-Poisson-Boltzmann System for Soft Potentials 17 Step 3. To the end, set ρ(k) = k 1 + k. Let 0 < ɛ 1 and J > 0 be chosen later. Multiplying (3.16) by [1 + ɛρ(k)t] J, d dt {[1 +ɛρ(k)t]j E l (û)} + κ[1 +ɛρ(k)t] J D l (û) J[1+ɛρ(k)t] J 1 ɛρ(k)e l (û). (3.17) To estimate the right-hand term, we bound E l (û) in the way that That is, E l (û) µlû L = µlû L + â k ( µ l {I P}û L µ l {I P}û L (χ k 1 + χ k >1 ) + â k E l (û) E I l (û) + E II (û), + µ l Pû )χ L k 1 + µlû χ L k 1 + â k χ k 1 + µlû χ L k 1 + (a, b, c) + â k. El I (û) = µ l {I P}û χ L k 1 + µ l û χ L k 1, E II (û) = (a, b, c) + â k. Therefore, for the right-hand term of (3.17), one has J[1 + ɛρ(k)t] J 1 ɛρ(k)e l (û) J[1 + ɛρ(k)t] J 1 ɛρ(k)e I l (û) + J[1 + ɛρ(k)t] J 1 ɛρ(k)e II (û) := R 1 + R. We estimate R 1 and R as follows. First, for R, Thus, one can let R = J[1 + ɛρ(k)t] J 1 ɛρ(k)e II (û) = J[1 + ɛρ(k)t] J 1 ɛρ(k)( (a, b, c) + â k ) { k = ɛj[1 + ɛρ(k)t] J 1 1 (a, b, c) k 1 + k â { } ɛj[1 + ɛρ(k)t] J k â (a, b, c) k k ɛj[1 + ɛρ(k)t] J D l (û). } ɛj < κ 4

18 May 9, 01 11:15 WSPC/INSTRUCTION FILE M3AS-DYZ-s 18 Renjun Duan, Tong Yang and Huijiang Zhao where κ > 0 is given in (3.16), such that For R 1, we use the splitting so that with R κ 4 [1 + ɛρ(k)t]j D l (û). 1 = χ µ () [1+ɛρ(k)t] + χ µ ()>[1+ɛρ(k)t], (3.18) El I (û) = El I (ûχ µ () [1+ɛρ(k)t]) + El I (ûχ µ ()>[1+ɛρ(k)t]) := El I< (û) + El I> (û) El I< (û) = + El I> (û) = + µ () [1+ɛρ(k)t] µ () [1+ɛρ(k)t] µ () [1+ɛρ(k)t] µ () [1+ɛρ(k)t] Thus, with this splitting, R 1 is bounded by where we have used For R 11, notice that R 1 = J[1 + ɛρ(k)t] J 1 ɛρ(k)e I l (û) µ l () {I P}û dχ k 1 µ l () û dχ k 1, µ l () {I P}û dχ k 1 µ l () û dχ k 1. J[1 + ɛρ(k)t] J 1 ɛρ(k){el I< (û) + El I> (û)} ɛj[1 + ɛρ(k)t] J ρ(k)el 1 I< (û) +J[1 + ɛρ(k)t] J 1 ɛρ(k)el I> (û) := R 11 + R 1, E I< l (û) [1 + ɛρ(k)t]e I< l 1 (û). ρ(k)el 1 I< (û) ρ(k)( µ l 1 {I P}û χ L k 1 + µ l 1 û χ L k 1 ) µ l 1 {I P}û + ρ(k) (a, b, c) L C 1 D l (û) for a generic constant C 1 1, and hence One can make ɛj further small such that and hence R 11 C 1 ɛj[1 + ɛρ(k)t] J D l (û). C 1 ɛj κ 4, R 11 κ 4 [1 + ɛρ(k)t]j D l (û).

19 May 9, 01 11:15 WSPC/INSTRUCTION FILE M3AS-DYZ-s The Vlasov-Poisson-Boltzmann System for Soft Potentials 19 To estimate R 1, let p > 1 be chosen later and compute R 1 = J[1 + ɛρ(k)t] p ɛρ(k) [1 + ɛρ(k)t] J+p 1 El I> (û) J[1 + ɛρ(k)t] p ɛρ(k) El+J+p 1 I> (û). Noticing that the estimate E I> l+j+p 1 (û) EI l+j+p 1(û) E l+j+p 1 (û) E l+j+p 1 (û 0 ) holds true by (3.16) due to l + J + p 1 0, one has R 1 J[1 + ɛρ(k)t] p ɛρ(k)e l+j+p 1 (û 0 ). Therefore, collecting all estimates above, it follows from (3.17) that d dt {[1 + ɛρ(k)t]j E l (û)} + κ [1 + ɛρ(k)t]j D l (û) Integrating this inequality, using t 0 [1 + ɛρ(k)s] p ɛρ(k)ds J[1 + ɛρ(k)t] p ɛρ(k)e l+j+p 1 (û 0 ). for p > 1, and noticing J + p 1 > 0, one has 0 (1 + η) p dη = C p < [1 + ɛρ(k)t] J E l (û) E l (û 0 ) + JC p E l+j+p 1 (û 0 ) (1 + JC p )E l+j+p 1 (û 0 ), that is, whenever l 0, E l (û) [1 + ɛρ(k)t] J E l+j+p 1 (û 0 ), for any t 0, k R 3, where the parameters p, ɛ and J with p > 1, 0 < ɛ 1, J > 0, C 1 ɛj κ 4. are still to be chosen. Now, for any given l 0 > σ m, we fix J > σ m, p > 1 such that l 0 = J + p 1 to have E l (û) [1 + ɛρ(k)t] J E l+l0 (û 0 ). Since J > σ m, in the completely same way as in 9 and 13 by considering the frequency integration over R 3 k = { k 1} { k 1} with a little modification of the proof in 6,7, one can derive the desired time-decay property (3.5) under conditions (3.3) and (3.4); the details are omitted for brevity. This completes the proof of Theorem 3.1. Remark 3.1. The rate index σ m in (3.5) is optimal in the sense that it is the same as in the case of the Boltzmann equation. In fact, from 3,34, one has α e t( x+l) u 0 C(1 + t) σ α ( u 0 Z1 + α u ),

20 May 9, 01 11:15 WSPC/INSTRUCTION FILE M3AS-DYZ-s 0 Renjun Duan, Tong Yang and Huijiang Zhao for any t 0, where L the linearized Boltzmann operator in the case 0 γ 1. Here, different from the hard potential case, the extra velocity weight γl 0 in (3.5) on initial data u 0 for soft potentials 3 < γ < 0 is needed. Moreover, as we mentioned before, the condition (3.3) is used to remove the singularity of the Poisson kernel and recover the optimal rate index σ m, otherwise, only the rate (1+t) α can be obtained, see 9 and references therein. 4. Estimate on nonlinear terms This section concerns some estimates on each nonlinear term in G = Γ(u, u) + 1 x φu x φ u which will be needed in the next section. For this purpose, to simplify the notations, in the proof of all subsequent lemmas, w τ is used to denote w τ,q (t, ). That is w τ w τ,q (t, ) = γτ e q(t), q(t) = q + λ (1 + t) ϑ, where 0 < q(t) 1 is fixed and τ = β l 0 depending on the order of velocity derivatives may take different values in different places. Moreover, we also use in the proof u 1 = Pu, u = {I P}u. Now we turn to state the estimates on the nonlinear terms. To make the presentation clear, we divide it into several subsections. The first subsection is concerned with the estimates on Γ(u, u) Estimate on Γ(u, u) We always use the decomposition Γ(u, u) = Γ(Pu, Pu) + Γ(Pu, {I P}u) + Γ({I P}u, Pu) Recall + Γ({I P}u, {I P}u). (4.1) Γ(f, g) = Γ + (f, g) Γ (f, g) [ ] = γ M 1/ ( ) q 0 (θ)f( )g( ) dω d R 3 S [ ] g() γ M 1/ ( ) q 0 (θ) dω f( ) d. R 3 S Our first result in this subsection is concerned with the estimate on Γ(u, u) both without and with the weight.

21 May 9, 01 11:15 WSPC/INSTRUCTION FILE M3AS-DYZ-s The Vlasov-Poisson-Boltzmann System for Soft Potentials 1 Lemma 4.1. Let N 4, l 0, 0 < q(t) 1. It holds that Γ(u, u), u C (a, b, c) H 1 x (a, b, c) ν 1/ {I P}u + C x(a, b, c) H 1 + α β {I P}u ν 1/ {I P}u (4.) and Γ(u, u), w l,q (t, ){I P}u α + β 4 C (a, b, c) H 1 x (a, b, c) ν 1/ {I P}u + C {E N,l,q (t)} 1/ ν 1/ w l,q (t, ){I P}u. (4.3) Proof. We will prove only (4.3), since (4.) can be obtained in the similar way by noticing Γ(u, u), u = Γ(u, u), {I P}u. To this end, set I 1 = Γ(u, u), w l,q(t, ){I P}u, and denote I 1,11, I 1,1, I 1,1, I 1, to be the terms corresponding to the decomposition (4.1). Now we turn to estimate these terms term by term. Estimate on I 1,11 : Since Γ(u 1, u 1 ) decays exponentially in, I 1,11 C (a, b, c) ν 1/ L u dx C (a, b, c) R 3 H 1 x (a, b, c) ν 1/ u, where Hölder s inequality with 1 = 1/3 + 1/6 + 1/ and Sobolev s inequalities have been used. (a, b, c) L 3 C (a, b, c) H 1, (a, b, c) L 6 C x (a, b, c) Estimate on I 1,1 : For the loss term, I 1,1 C R 3 R 3 u () (a, b, c) ν() w l() u () dxd C sup (a, b, c) ν 1/ w l ()u C x (a, b, c) H 1 ν 1/ w l ()u, x R 3 where we have used in the first inequality γ M q / ( ) d C γ (4.4) R 3 for q > 0 and 3 < γ < 0. For the gain term, using Hölder s inequality, I 1,1 + is bounded by { } 1/ γ q 0 (θ)m 1/ ( )w l() u 1 ( )u ( ) dxdd dω { γ q 0 (θ)m 1/ ( )w l() u () dxdd dω} 1/. (4.5)

22 May 9, 01 11:15 WSPC/INSTRUCTION FILE M3AS-DYZ-s Renjun Duan, Tong Yang and Huijiang Zhao From (4.4), the second term in (4.5) is bounded by ν 1/ w l ()u. By noticing min{, + } and w l () = γl e q(t) C γl γl e q(t) e q(t) Cw l ( )w l ( ) (4.6) for l 0, the first term in (4.5) is bounded by { } 1/ C γ q 0 (θ)m 1 ( )w l()m q ( ) (a, b, c) u ( ) x,,,ω { C { = C { = C } 1/ γ q 0 (θ)w l( )M q ( ) (a, b, c) u ( ) x,,,ω } 1/ γ q 0 (θ)w l( )M q ( ) (a, b, c) u ( ) x,,,ω γ q 0 (θ)w l()m q x,,,ω ( ) (a, b, c) u () } 1/, where we have used Pu( ) C (a, b, c) M q / ( ) with 0 < q < 1 in the first equation, M 1/ ( ) C, the inequality (4.6) and w l ( )M q / ( ) C in the second equation, the identity = in the third equation and the change of variable (, ) (, ) with the unit Jacobian in the last equation. Thus, from (4.4), the first term in (4.5) is bounded by C sup x (a, b, c) ν 1/ w l ()u. Therefore, I 1,1 + C x(a, b, c) H 1 ν 1/ w l ()u. Estimate on I 1,1 : This can be done in the completely same way as for I 1,1, so that I 1,1 C x (a, b, c) H 1 ν 1/ w l ()u. Estimate on I 1, : By 31, I 1, C {E N,l,q (t)} 1/ ν 1/ w l {I P}u, (4.7) and for brevity we skip the proof of this term. Now, the desired inequality (4.3) follows by collecting all the above estimates. This completes the proof of Lemma 4.1. For the weighted estimates on β α Γ(u, u), we have Lemma 4.. Let N 8, 1 α + β N, l β and 0 < q(t) 1. For given u = u(t, x, ), define u αβ as u αβ = α u if β = 0 and u αβ = β α {I P}u if β 1.

23 May 9, 01 11:15 WSPC/INSTRUCTION FILE M3AS-DYZ-s The Vlasov-Poisson-Boltzmann System for Soft Potentials 3 Then, it holds that β α Γ(u, u), w β l,q (t, )u αβ C (a, b, c) H N x (a, b, c) H N 1 ν 1/ u αβ + C {EN,l,q (t)} 1/ D N,l,q (t). (4.8) Proof. Take α, β with 1 α + β N. Write β α Γ(f, g) = ) C β β 0β 1β Cα α 1,α Γ 0 ( α1 β 1 f, α β g, where the summation is taken over β 0 + β 1 + β = β and α 1 + α = α, and Γ 0 is given by ) ( ) ( ) Γ 0 ( α1 β 1 f, α β g = Γ + 0 α1 β 1 f, α β g Γ 0 α1 β 1 f, α β g [ ] = γ q 0 (θ) β0 M 1/ ( ) α1 β 1 f( ) α β g( ) ddω R 3 S [ ] α β g() γ q 0 (θ) β0 M 1/ ( ) α1 β 1 f( ) d dω. R 3 S In what follows we set I = ( ) Γ 0 α1 β 1 u, α β u, w β l,q (t, )u αβ and denote the terms corresponding to the decomposition (4.1) as I,11, I,1, I,1 and I,. These terms will be estimated term by term as follows. ) Estimate on I,11 : Notice that Γ 0 ( α1 β 1 u 1, α β u 1 decays exponentially in since 3 < γ < 0. Then, I,11 C α1 (a, b, c) α (a, b, c) ν 1/ L u αβ dx R 3 C (a, b, c) H N x (a, b, c) H N 1 ν 1/ u αβ. Estimate on I,1 : For the loss term, since } I,1 γ q 0 (θ)m q ( ) {M q ( ) α1 (a, b, c) x,,,ω α β u () w β l () u αβ() C dx α1 (a, b, c) d γ w β l () α β u ()u αβ (), we have for α 1 N/ that I,1 C sup α1 (a, b, c) ν 1/ w β l () α β u x ν 1/ w β l ()u αβ C {EN,l,q (t)} 1/ D N,l,q (t),

24 May 9, 01 11:15 WSPC/INSTRUCTION FILE M3AS-DYZ-s 4 Renjun Duan, Tong Yang and Huijiang Zhao while for the case α 1 N/, we can get that I,1 C dx α1 (a, b, c) ν 1/ w β l () α L ν β u 1/ L w β l ()u αβ C sup ν 1/ w β l () α L β u α1 (a, b, c) ν 1/ w β l ()u αβ x C {E N,l,q (t)} 1/ D N,l,q (t). For the gain term I,1 +, } I,1 + γ q 0 (θ)m q ( ) {M q ( ) α1 (a, b, c) x,,,ω α β u ( ) w β l () u αβ(). We now turn to deduce an estimate on the right-hand side of the above inequality by considering the two cases α 1 N/ and α 1 > N/, respectively. In the case of α 1 N/, we have from Hölder s inequality that I +,1 dx α1 (a, b, c) { γ q 0 (θ)m q ( )M q ( )w β l () α β u ( ) } 1/,,ω { } 1/ γ q 0 (θ)m q ( )M q ( )w β l () u αβ(). (4.9),,ω As for estimating the first factor in (4.5), we have γ q 0 (θ)m q ( )M q ( )w β l () α β u ( ),,ω C γ q 0 (θ)m q ( )M q ( )w β l ( )w β l ( ) α β u ( ),,ω C γ q 0 (θ)m q 4 ( )w β l ( ) α β u ( ),,ω = C γ q 0 (θ)m q 4 ( )w β l () α β u (),,,ω which is bounded by ν 1/ w β l () α β u. Moreover,,,ω γ q 0 (θ)m q ( )M q ( )w β l () u αβ() L C γ M q ( )w β l () u αβ(), C ν 1/ w β l ()u αβ. L

25 May 9, 01 11:15 WSPC/INSTRUCTION FILE M3AS-DYZ-s The Vlasov-Poisson-Boltzmann System for Soft Potentials 5 Therefore, plugging the above two estimates into (4.9), in the case when α 1 N/, we have I,1 + C sup α1 (a, b, c) ν 1/ w β l () α β u x ν 1/ w β l ()u αβ C {EN,l,q (t)} 1/ D N,l,q (t). The discussion for the case α 1 N/ is divided into the following three subcases: In D 1 = { 1 }, we have I,11 + C dx α1 (a, b, c) γ q 0 (θ)m q q ()M ( ) α β u ( )u αβ ().,,ω Here, by Hölder s inequality, we can deduce that γ q 0 (θ)m q q ()M ( ) α β u ( )u αβ (),,ω { γ q 0 (θ)m q q ()M ( ) α β u ( ) } 1,,ω { } 1 γ q 0 (θ)m q q ()M ( ) u αβ (),,ω C ν 1/ α L ν β u 1/ L u αβ. Therefore, we have in this subcase that I,1 + C sup ν 1/ α L β u α1 (a, b, c) ν 1/ u αβ x C {E N,l,q (t)} 1/ D N,l,q (t). (4.10) Secondly, we consider the region D = { 1, 1}. In this case, we can get that I,1 + C dx α1 (a, b, c) γ q 0 (θ)m q ( )M q ( ) α β u ( )u αβ ().,,ω Similarly as in (4.9), γ q 0 (θ)m q ( )M q ( ) α β u ( )u αβ (),,ω C ν 1/ α L ν β u 1/ L u αβ.

26 May 9, 01 11:15 WSPC/INSTRUCTION FILE M3AS-DYZ-s 6 Renjun Duan, Tong Yang and Huijiang Zhao Hence, as in (4.10), the above two estimates imply I +,1 C {E N,l,q(t)} 1/ D N,l,q (t). At last, we consider the region D 3 = { 1, 1}. In this domain, 1 and hence γ C γ C(1 + + ) γ. Thus, I +,13 C dx α1 (a, b, c),,ω ( ) γ q 0 (θ) M q ( )M q ( )w β l () α β u ( )u αβ (). (4.11) From Hölder s inequality, (1 + + ) γ q0 (θ)m q ( )M q ( )w β l () α β u ( )u αβ (),,ω { (1 + + ) γ q0 (θ)m q ( )M q ( )w β l () α β u ( ) } 1,,ω { } 1 (1 + + ) γ q0 (θ)m q ( )M q ( )w β l () u αβ(),,ω C ν 1/ w β l () α L β u ν 1/ L w β l ()u αβ, where the computation similar to the estimate on the first factor in (4.5) by using + = + instead of = has been performed. Therefore, I,13 + C dx α1 (a, b, c) ν 1/ w β l () α L β u C sup x ν 1/ L w β l ()u αβ ν 1/ w β l () α L β u α1 (a, b, c) ν 1/ w β l ()u αβ,,ω C {E N,l,q (t)} 1/ D N,l,q (t). Estimate on I,1 : For the loss term, due to I,1 dx α (a, b, c) γ q 0 (θ)m q q ()M ( ) α1 β 1 u ( )u αβ (), we have by Hölder s inequality that I,1 C sup α (a, b, c) ν 1/ ν α1 β 1 u 1/ u αβ x

27 May 9, 01 11:15 WSPC/INSTRUCTION FILE M3AS-DYZ-s The Vlasov-Poisson-Boltzmann System for Soft Potentials 7 for α N/, and I,1 C sup x for α N/. Hence, in both cases, For the gain term, by noticing that I +,1 C ν 1/ M q /4 α1 L β 1 u α (a, b, c) ν 1/ u αβ I,1 C {E N,l,q(t)} 1/ D N,l,q (t). dx α (a, b, c) γ q 0 (θ),,ω M q ( )M q ( )w β l () α 1 β 1 u ( )u αβ (), then, in the same way as for I +,1, when α N/, we have I +,1 C sup x α (a, b, c) ν 1/ w β l () α1 β 1 u ν 1/ w β l ()u αβ C {EN,l,q (t)} 1/ D N,l,q (t). While when α N/, sup ν 1 α 1 L β 1 u α (a, b, c) ν 1/ u αβ over D1 D, x I,1 + C sup ν 1/ w β l () α1 L β 1 u x α (a, b, c) ν 1/ w β l ()u αβ over D3. Therefore, for each α, I +,1 C {E N,l,q(t)} 1/ D N,l,q (t). Estimate on I, : As for (4.7), by 31, I, C {E N,l,q (t)} 1/ D N,l,q (t), and we also skip the proof of this term for brevity. Now, (4.8) follows by collecting all the above estimates. This completes the proof of Lemma Estimate on x φu This subsection concerns the estimate on x φu. To this end, we first have the following result on the case of zeroth order derivative.

28 May 9, 01 11:15 WSPC/INSTRUCTION FILE M3AS-DYZ-s 8 Renjun Duan, Tong Yang and Huijiang Zhao Lemma 4.3. Assume that u and φ satisfy the second equation of (.3). It holds that 1 xφu, u 1 d b (a + c) dx dt R 3 + C { (a, b, c) H + x φ H 1 + x φ x b } { x (a, b, c) + ν 1/ {I P}u } m=1 + C xφ H 1 1/ {I P}u. (4.1) Proof. Setting I 3 = 1 xφu, u and noticing u = Pu + {I P}u = u 1 + u, I 3 = I 3,m = xφ, u 1 + x φ, u 1 u + xφ, u. First, for I 3,1, as in 1,7, using the second equation of (.3) to replace x φ yields I 3,1 = x φ b(a + c) dx = 1 d b (a + c) dx R dt 3 R { b R 6 x b x Λ(u ) 1 } 3 xφ b dx 3 + b(a + c) { x (a + c) + x Θ(u ) x φa} dx, R 3 where Θ( ) and Λ( ) are defined in (.). Further by the Hölder, Sobolev and Cauchy- Schwarz inequalities, the above equation implies I 3,1 1 d b (a + c) dx dt R 3 + C { (a, b, c) H + x φ x b } x (a, b, c) + C (a, b, c) H { Λ(u ) + Θ(u ) }. Noticing Λ(u ) + Θ(u ) C ν 1/ u, then I 3,1 is bounded by the right-hand term of (4.1). Next, for I 3, and I 3,3, one has I 3, x φ ν 1/ L u 1 ν 1/ L u R 3 dx C x φ (a, b, c) ν 1/ L u dx R 3 { C x φ H 1 x (a, b, c) } + ν 1/ u

29 May 9, 01 11:15 WSPC/INSTRUCTION FILE M3AS-DYZ-s The Vlasov-Poisson-Boltzmann System for Soft Potentials 9 and I 3,3 C x φ L x 1/ u C x φ H 1 1/ u. Therefore, (4.1) follows from all the above estimates. This completes the proof of Lemma 4.3. For the case of higher order derivatives with respect to x variable, we have Lemma 4.4. Let N 4, 1 α N, and l 0. It holds that ( ) 1 α xφu, w l,q(t, ) α u { } C xφ 1/ H N 1 w l,q(t, ) α {I P}u + α (a, b, c). 1 α N (4.13) Proof. Let I 4 be the left-hand term of (4.13). Corresponding to u = Pu+{I P}u, set I 4 = + 3 I 4,m = m=1 ( ) 1 α xφu 1, w l() α u α ( 1 xφu ), w l() α u 1 + α ( 1 xφu ), w l() α u. (4.14) For I 4,1, one has I 4,1 = α 1 α 1 Cα α 1 x α α1 φ α1 u 1, w α l () α u C α 1 α R 3 x α α1 φ α 1 (a, b, c) α u L dx. Here, when α 1 N/ and α 1 < α, the integral term in the summation above is bounded by sup α1 (a, b, c) x α α1 φ α u, x and when α 1 N/ or α 1 = α, it is bounded by sup x α α1 φ α 1 (a, b, c) α u. x

30 May 9, 01 11:15 WSPC/INSTRUCTION FILE M3AS-DYZ-s 30 Renjun Duan, Tong Yang and Huijiang Zhao Thus, by the Sobolev inequality, I 4,1 is bounded by the right-hand term of (4.13). Similarly, we have for I 4, that I 4, C α 1 α C x α α1 φ α 1 u L α (a, b, c) dx R 3 sup α1 u L x α α1 φ α (a, b, c) x + sup x α α1 φ α 1 u α (a, b, c), x { α 1 N/} {α 1<α} { α 1 N/} {α 1=α} which is also bounded by the right-hand term of (4.13) by Sobolev s inequality. Finally, for I 4,3, since I 4,3 C α 1 α x α1 φ 1/ w l () α α1 L u R 3 1/ w l () α L u dx, it can be further estimated as for I 4,1 and I 4,. Therefore, (4.13) follows from (4.14) by collecting all the above estimates. This completes the proof of Lemma 4.4. For the case of higher order mixed derivatives with respect to both and x variables, we have Lemma 4.5. Let N 4, 1 α + β N with β 1, and l β. It holds that ( ) 1 β α xφ{i P}u, w β l,q (t, ) α β {I P}u C xφ H N 1 1/ w β l,q (t, ) β α {I P}u. (4.15) α + β N Proof. Due to ( ) 1 β α xφu, w β l () α β u = C α α 1α C β β 1β 1 β 1 x α1 φ α β u, w β l () α β u, where the summation is taken over α 1 + α = α and β 1 + β = β with β 1 1, and for simplicity we denote each integration term in the summation as I 5. We now prove (4.15) by considering the following two cases: For the case α 1 N/, we have I 5 C sup x x α1 φ 1/ w β l () α β u 1/ w β l () β α u,

31 May 9, 01 11:15 WSPC/INSTRUCTION FILE M3AS-DYZ-s The Vlasov-Poisson-Boltzmann System for Soft Potentials 31 which is bounded by the right-hand term of (4.15). For the case α 1 N/, it is easy to see that I 5 C sup 1/ w β l () α L β u x α1 φ 1/ w β l () α β u, x which is also bounded by the right-hand term of (4.15). Therefore, (4.5) follows by combining both cases above. This completes the proof of Lemma Estimate on x φ u Now we turn to estimate x φ u. For the case with only derivatives with respect to x variable, we have Lemma 4.6. Assume γ < 0. Let N 4, 1 α N, and l 0. It holds that α ( x φ u), w l,q(t, ) α u C x φ H N 1 { α + β N, β 1 w β l α β {I P}u + x (a, b, c) H N 1 }. (4.16) Proof. Denote the left-hand side of (4.16) by I 6 and write it as I 6 = I 6,1 + Cα α 1α {I 6,1 (α 1 ) + I 6, (α 1 )} α 1+α =α, α 1 1 with I 6,1 = x φ α u, w l() α u, I 6,1 (α 1 ) = x α1 φ α u 1, w l() α u, I 6, (α 1 ) = x α1 φ α u, w l() α u. We estimate term by term as follows. To estimate I 6,1, notice w l() = ( γl) γl 1 e q(t) + γl e q(t) q(t) C 1 γl e q(t) = C w l(), where 1 and the fact that both q(t) = q + λ/(1 + t) ϑ and are bounded by a constant independent of t and have been used. Then, from integration by part, I 6,1 = 1 xφ {w l()}, α u C sup x φ 1/ w l () α u, x

32 May 9, 01 11:15 WSPC/INSTRUCTION FILE M3AS-DYZ-s 3 Renjun Duan, Tong Yang and Huijiang Zhao which is bounded by the right-hand side of (4.16). For I 6,1 (α 1 ), it is straightforward to estimate it by I 6,1 (α 1 ) C x α1 φ α (a, b, c) α u L dx R 3 C x φ H N 1 { x (a, b, c) H N 1 + α u }. To estimate I 6, (α 1 ), notice γ+ 1 due to γ < 0 so that w l() w 1 l () w l (). Thus, I 6, (α 1 ) C x α1 φ w 1 l () α u L w l () α u R 3 L C x φ H N 1 α + β N, β 1 w β l α β u dx + w l () α u }, which is further bounded by the right-hand side of (4.16). By collecting all the above estimates, it then completes the proof of Lemma 4.6. For the case of higher order mixed derivatives with respect to the and x variables, we have Lemma 4.7. Assume γ < 0. Let 1 α + β N with β 1, and l β. It holds that β α ( x φ {I P}u), w β l,q (t, ) α β {I P}u C x φ H N 1 w β l,q (t, ) β α {I P}u. (4.17) α + β N Proof. Similar to the proof of Lemma 4.6, one can rewrite the left-hand side of (4.17) as x φ β α {I P}u, w β l () α β {I P}u + Cα α 1α x α1 φ α β {I P}u, w β l () α β {I P}u. α 1+α =α, α 1 1 Then, similar to arguments used to deal with I 6,1 and I 6, (α 1 ) in Lemma 4.6, (4.17) follows and the details are omitted for brevity. This completes the proof of Lemma 4.7.