# Large Time Behavior of the Vlasov-Poisson-Boltzmann System

Authors: Li Li; Shuilin Jin; Li Yang
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

The motion of dilute charged particles can be modeled by Vlasov-Poisson-Boltzmann system. We study the large time stability of the VPB system. To be precise, we prove that when time goes to infinity, the solution of VPB system tends to global Maxwellian state in a rateOt−∞, by using a method developed for Boltzmann equation without force in the work of Desvillettes and Villani (2005). The improvement of the present paper is the removal of condition on parameter λ as in the work of  Li (2008).


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## Body

## 1. Introduction

Large time behavior for the Boltzmann equation and related systems is an important topic for both physicists and mathematicians. We consider the Cauchy problem for Vlasov-Poisson-Boltzmann system in a torus𝕋N:
(1)ft+v·∇xf+∇xϕ·∇vf=Q(f,f),on𝕋N,(2)Δϕ=∫ℝN‍fdv-ρ0,on𝕋N,(3)f(0,x,v)=f0(x,v),(4)Q(f,f)=∫ℝN‍∫𝕊N-1‍(f′f*′-ff*)q(v-v*,σ)dσdv*.f=(t,x,v), which represents the distribution of particles, is a function of time t∈ℝ+, particle velocity v∈ℝN, and position x∈𝕋N. The force ∇ϕ in (1) is controlled by Poisson equation (2), which comes intrinsically by the nonequilibrium distribution of particles.The quadratic termQ(f,f) is the collision operator and q(v-v*,σ) is the corresponding cross-section. It is well-known by the conservation of mass that ρ0=∫𝕋N×ℝN‍f0dxdv is a fixed constant which represents the background charge.Without loss of generality, we can assume|𝕋N|=1, ρ0=1. Define ρ, u, T, which are functions of t and x by
(5)ρ=∫ℝN‍fdv,ρu=∫ℝN‍fvdv,ρ|u|2+NρT=∫ℝN‍f|v|2dv.
Physically, they represent the macroscopic quantities: density, bulk velocity, and temperature, respectively. It is well known that the conservation of mass, momentum, and energy holds:
(6)ddt∫𝕋N‍ρdx=0,ddt∫𝕋N‍ρudx=0,ddt∫𝕋N‍(ρ|u|22+NρT2+|∇ϕ|22)dx=0.
Here, the total energy consists of the kinetic energy ρ|u|2/2, the internal heat energy NρT/2, and the electric potential energy |∇ϕ|2/2. By simple translation and dilation, ρ, u, T can be normalized as
(7)∫𝕋N×ℝN‍fdvdx=∫𝕋N‍ρdx=ρ0=1,∫𝕋N×ℝN‍fvdvdx=∫𝕋N‍ρudx=0,∫𝕋N×ℝN‍(f|v|22+|∇ϕ|22)dvdx=∫𝕋N‍(ρ|u|22+NρT2+|∇ϕ|22)dx=N2.If the initial datumf0 satisfies the conservation laws (7), then the stationary solution is a global Maxwellian M, in the form of
(8)M=M[1,0,1]=1(2π)N/2exp{-|v|22},
where the subscript [1,0,1] represents the corresponding macroscopic quantities: density, bulk velocity, and temperature, respectively.Traditional method for studying the asymptotic behavior is using linearization around local or global Maxwellian state. Without external force, Ukai [1] proved an exponential decay rate for the cutoff hard potential in a torus in 1974. In 1980, Caflisch [2] obtained a rate like O(e-tβ) for the cutoff soft potential with γ≥-1 in a torus, where β=2/(2-γ)∈[0,1]. Strain and Guo [3] extend Caflisch’s result in 2008 and get a convergence rate like O(e-tP)(0<P<1) for the very soft potential case (γ<-1). The previous results all make use of the linearization.However, by using some estimates on systems of second-order differential inequalities, Desvillettes and Villani [4] obtain an almost exponential convergence rate like O(t-∞). The result is weaker than using linearization, but the smallness assumption on initial data f0-M is removed and the conclusion holds for noncutoff collision kernels as well.Our work is inspired by the work of Desvillettes and Villani [4]. We extend their result for Boltzmann equation without external force to the Vlasov-Poisson-Boltzmann system.In a previous work [5], the Vlasov-Poisson-Boltzmann system with (2) replaced by
(9)λΔϕ=∫ℝN‍fdv-ρ0
is proved to satisfy the following theorem.Theorem 1.
Letq(v-v*,σ) satisfy
(10)q≥KBmin(|v-v*|γ-,|v-v*|-β-),
and let the collision operator satisfy
(11)∥Q(g,h)∥L2(ℝvN)≤CB∥g∥Hs0k0(ℝvN)∥h∥Hs0k0(ℝvN),
for some k0, s0≥0, where KB and CB are positive constants. Let (f)t≥0 be a smooth solution of the problem (1), (9), and (3), such that, for all k, s>0,
(12)supt≥0∥f∥Hsk(𝕋N×ℝN)≤Ck,s<+∞,
and for all t>0, x∈𝕋N, and v∈ℝN,
(13)f(x,v)≥K0e-A0|v|q0(A0,K0>0;q0≥2).
Then ∃λ0, such that, for all λ>λ0, the solution f converges to M in an almost exponential rate; that is, for any small positive constant ϵ>0,
(14)∥f-M∥=O(1)t-1/700ϵ,
where O(1) depends on KB, γ-, β-, CB, k0, s0, Ck,s, K0, A0, q0, and ϵ.The present paper extends the result of [5] by removing the condition on λ and considers system (1)–(3). To be precise, the main result of this paper is as follows.Theorem 2.
Under condition (10)–(13), the solution f of problem (1)–(3) converges to M in an almost exponential rate; that is, for any small positive constant ϵ>0,
(15)∥f-M∥=O(1)t-1/700ϵ,
where O(1) depends on the constants in (10)–(13) and ϵ.Now, we state some results on the existence of solutions of VPB system. The global existence of solutions is proved in [6] in a torus and [7–9] in the whole space with small perturbed initial data. The existence result in [7] also holds for a more general case, like the Vlasov-Maxwell-Boltzmann system.The following is devoted to the proof of Theorem2. Section 2 gives some lemmas which will be used later. Proof of the main result is given in Section 3.


## 2. Preliminaries

First, denote some local Maxwellian states in forms ofρ, u, T. Define M[ρ,u,T], M[ρ,u,〈T〉], M[ρ,0,〈T〉], M[ρ,0,1] as follows:
(16)M[ρ,u,T](v)=ρ(2πT)N/2exp{-|v-u|22T},M[ρ,u,〈T〉](v)=ρ(2π〈T〉)N/2exp{-|v-u|22〈T〉},M[ρ,0,〈T〉](v)=ρ(2π〈T〉)N/2exp{-|v|22〈T〉},M[ρ,0,1](v)=ρ(2π)N/2exp{-|v|22},
where 〈T〉=∫‍ρTdx stands for the mean temperature.As we will show in Section3, the gradient of temperature prevents f from being close to M[ρ,u,T] for too long; the symmetric gradient of velocity prevents f from being close to M[ρ,u,〈T〉] for long, that is, the local Maxwellians with constant temperature; and finally, the gradient of ρ and ϕ prevents f from being close to M[ρ,0,〈T〉] and M[ρ,0,1] for long. In order to estimate the distance between two distributions, we need to define H functional and relative information (or relative entropy) between two distributions, which is the main measure of the distance between f and the local Maxwellians.Definition 3.
Supposef and g are two distributions on 𝕋N×ℝN, s.t.:
(17)∫𝕋N×ℝN‍f=∫𝕋N×ℝN‍g.
Define the H functional (negative of the entropy) and the Kullback relative information by
(18)H(f)=∫𝕋N×ℝN‍flogf,H(f∣g)=∫𝕋N×ℝN‍flogfg.Proposition 4.
The well-known Csiszár-Kullback inequality asserts(19)H(fg)≥14∥f-g∥L1(x,v)2,
if f and g are two distributions on 𝕋N×ℝN. Moreover, if f is the solution of (1), (2) and satisfies (7), then(20)H(f∣M)=H(f)-H(M)-12∫𝕋N‍|∇ϕ|2.Proof.
Defineφ(h)=hlog(h); then since ∫‍f=∫‍g=1, we have
(21)H(f∣g)=∫‍flogfg=∫‍flogf-flogg-f+g=∫‍flogf-glogg-(logg+1)(f-g)=∫‍φ(f)-φ(g)-φ′(g)(f-g)=12∫‍φ′′(h)|f-g|2=12∫‍1h|f-g|2,
where h stands for a positive function between f and g. The last equality is obtained by using second-order Taylor expansion. By Hölder’s inequality, we have
(22)∫‍|f-g|h-1/2h1/2≤(∫‍1h|f-g|2)1/2(∫‍h)1/2,∀h>0.
Since h lies between f and g, notice that distributions f, g are nonnegative; thus h≤f+g. We have
(23)∫‍1h|f-g|2≥(∫‍|f-g|)2∫‍(f+g)=12(∫‍|f-g|)2,
and (19) is obtained. Equation (20) follows directly from (7).We now state the quantitative version ofH-theorem. See [10] for the proof.Theorem 5 (QuantitativeH-Theorem).
If(f)t≥0 is a smooth solution of the VPB equation (1), (2), then the H functional H(f) is nonincreasing as a function of t, and the decreasing rate
(24)ddtH(f)=-∫𝕋N‍D(f(x,·))dx,
where
(25)D(f)=14∫ℝN×ℝN×𝕊N-1‍(f′f*′-ff*)logf′f*′ff*Bdσdvdv*
is a positive definite functional.
Moreover, if the collision kernelq satisfies (10), and f complies with (12), then
(26)D(f)≥Kϵ(∫ℝN‍flogfM[ρ,u,T])1+ϵ,-ddtH(f)≥KHH(f∣M[ρ,u,T])1+ϵ.
The only set that can make D vanish is the local Maxwellian state.We state some notations here for the fluency of description. LetA and B be matrices; let the operation A:B=∑ij‍AijBij. For a vector-valued function u, the divergence is
(27)∇x·u=∑i‍∂ui∂xi,
the elements of gradient matrix ∇xu satisfy
(28)(∇xu)ij=∂uj∂xi,
the symmetric part of ∇u is
(29)∇xsymu=∇xu+(∇xu)T2,
and the traceless part of ∇xsymu is symbolized by {∇xu}:
(30){∇xu}=∇xsymu-∇x·uNIN.We expect to estimate decay rate of the distance betweenf and M, and the distance is measured by Kullback relative information. By using conservation laws, a direct computation will show that the relative information between f and M can be decomposed into a purely hydrodynamic part and a purely kinetic part:
(31)H(f∣M)=ℋ(ρ,u,T)+H(f∣M[ρ,u,T]),ℋ(ρ,u,T)=∫TN‍ρlogρ+N2∫TN‍ρ(T-logT-1)+∫TN‍ρ|u|22=:ℋ(ρ∣1)+ℋ(T∣1)+ℋ(u∣0),
where
(32)ℋ(u∣0)=∫‍ρ|u|22,ℋ(ρ∣1)=∫TN‍ρlogρ=∫TN‍ρlogρ-ρ+1
are nonnegative since ρlogρ-ρ+1 is convex with the minimum zero at ρ=1.Moreover, denoteΨ(X)=(N/2)(X-lnX-1); we can further decompose ℋ(T∣1) into
(33)ℋ(T∣1)=ℋ(T∣〈T〉)+ℋ(〈T〉∣1),
where
(34)ℋ(T∣〈T〉)=∫‍ρΨ(T)-Ψ(〈T〉),ℋ(〈T〉∣1)=Ψ(〈T〉).
It is easy to check that each of the previous terms is nonnegative by using Jensen’s inequality and convexity of functions Ψ(X).It is easy to verify the following.Lemma 6.
Use the previously mentioned notations; then one has the following additivity roles:(35)H(f∣M[ρ,u,T])+ℋ(T∣〈T〉)=H(f∣M[ρ,u,〈T〉]),H(f∣M[ρ,u,〈T〉])+1〈T〉ℋ(u∣0)=H(f∣M[ρ,0,〈T〉]),H(f∣M[ρ,0,〈T〉])+ℋ(〈T〉∣1)+(1-1〈T〉)ℋ(u∣0)=H(f∣M[ρ,0,1]),H(f∣M[ρ,0,1])+ℋ(ρ∣1)=H(f∣M).
Moreover, one has
(36)H(f∣M[ρ,u,T])≥KI∥f-M[ρ,u,T]∥L22(1+ϵ),H(f∣M[ρ,u,〈T〉])≥KI∥f-M[ρ,u,〈T〉]∥L22(1+ϵ),H(f∣M[ρ,0,〈T〉])≥KI∥f-M[ρ,0,〈T〉]∥L22(1+ϵ),H(f∣M[ρ,0,1])≥KI∥f-M[ρ,0,1]∥L22(1+ϵ).
Here nonnegative termsℋ(ρ∣1), ℋ(u∣0), ℋ(T∣〈T〉), ℋ(〈T〉∣1) are parts of the relative entropy, KI>0.Proof.
Additivity rules can be verified by direct computation. By using Csiszár-Kullback inequality and the interpolation fromL2 into L1, we can get (36). See [4] or [5] for more details.Now we assert the key lemma of the paper, which asserts the instability of hydrodynamic descriptions forf.Lemma 7.
The following four second-order differential inequalities hold:(37)d2dt2∥f-M[ρ,u,T]∥L2(x,v)2≥K1[∫TN‍|∇T(x)|2dx+∫TN‍|{∇u(x)}|2dx]-C1δ11-ϵ(∥f-M[ρ,u,T]∥L22)1-ϵ-δ1H(f∣M),(38)d2dt2∥f-M[ρ,u,〈T〉]∥L2(x,v)2≥K2∫TN‍|∇symu|2dx-C2δ21-ϵ(∥f-M[ρ,u,〈T〉]∥L22)1-ϵ-δ2H(f∣M),(39)d2dt2∥f-M[ρ,0,〈T〉]∥L2(x,v)2≥K3[∫TN‍|∇ϕ|2dx+∫TN‍|∇ρ|2dx]-C3δ31-ϵ(∥f-M[ρ,0,〈T〉]∥L22)1-ϵ-δ3H(f∣M),(40)d2dt2∥f-M[ρ,0,1]∥L2(x,v)2≥K4[∫TN‍|∇ϕ|2dx+∫TN‍|∇ρ|2dx]-C4δ41-ϵ(∥f-M[ρ,0,1]∥L22)1-ϵ-δ4H(f∣M).
Here δ1, δ2, δ3, δ4 are small enough constants, and all constants are positive.Roughly speaking, the previous inequalities show thatf cannot stay near local Maxwellian states. The gradient of T prevents f from staying close to M[ρ,u,T] for long; the symmetric gradient of u prevents f from staying close to M[ρ,u,〈T〉] for long; finally, the gradient of ρ prevents f from staying close to M[ρ,0,〈T〉] and M[ρ,0,1]. It left M=M[1,0,1] as the only stable state.To prove Lemma7, the following lemma is needed, whose proof can be found in [4].Lemma 8.
Leth be a smooth function of x, v. Then, for all multi-indexes α, β, and for all η<1,
(41)∫‍(vα∂x,vβh)2dvdx≤∥h∥H|α|/η|β|2η∥h∥H|β|/η2η(1-η)∥h∥L22(1-η)2.Proof of Lemma7.
Most of the proof is similar to that in [4, 5]; the only difference is in estimating terms with ϕ. We will only prove (39) as an example of how to estimate terms with ϕ.
We have(42)d2dt2∥f-M[ρ,0,〈T〉]∥L22=2∫‍(∂∂t(f-M[ρ,0,〈T〉]))2dvdx+2∫‍(f-M[ρ,0,〈T〉])∂2∂t2(f-g)dvdx=A+B.
At the moment whenf=M[ρ,0,〈T〉], B vanishes, so we only need to estimate A:
(43)∂∂t(f-M[ρ,0,〈T〉])=-v·∇xf-∇xϕ·∇vf+Q(f,f)-∂∂tM[ρ,0,〈T〉]=-(∂∂t+v·∇x+∇xϕ·∇v)M[ρ,0,〈T〉].
From (1) we have
(44)ρt+∇x·(ρu)=0,(ρu)t+∇x·(ρu⊗u+ρTIN+D)-ρ∇xϕ=0,(ρ|u|2+NρT)t+∇x·(ρ|u|2u+(N+2)ρuT+2Du+2R)-2ρu·∇xϕ=0.
Here, D and R are matrix-valued and vector-valued functions, respectively, defined by
(45)Dij(x)=∫RN‍f(x,v)=∫RN‍×[(v-u)i(v-u)j-|v-u|2Nδij]dv,R(x)=∫RN‍f(x,v)|v-u|22(v-u)dv.
Then, we obtain
(46)(∂t+u·∇)ρ+ρ∇·u=0,(∂t+u·∇)u+∇T+T∇ρρ+∇·Dρ-∇xϕ=0,(∂t+u·∇)T+2TN∇·u+2ρN(∇u:D+∇·R)=0.
Also, we get
(47)∂t〈T〉=[∫‍(∂tρ)T+∫‍ρ(∂tT)]=[-∫‍∇·(ρu)T-∫‍ρu·∇T-2N∫‍ρT∇·u-2N∫‍∇u:D-2N∫‍∇·R]=[-2N∫‍ρT∇·u-2N∫‍∇u:D].
Then the equations ofM[ρ,0,〈T〉] can be stated as follows:
(48)(∂t+v·∇x+∇xϕ·∇v)M[ρ,0,〈T〉]=M[ρ,0,〈T〉]×{[∂tρρ-N2∂t〈T〉〈T〉]×+v·[∇ρρ-∇xϕ〈T〉]+|v|2[∂t〈T〉2〈T〉2]}.
From (46) and (47), we have
(49)∂t(f-M[ρ,0,〈T〉])=-M[ρ,0,〈T〉]×{[∂tρρ-N2∂t〈T〉〈T〉]×+v·[∇ρρ-∇xϕ〈T〉]+|v|2[∂t〈T〉2〈T〉2]}.
Note thatM[ρ,0,〈T〉], |v|2M[ρ,0,〈T〉], |v|2M[ρ,0,〈T〉] are linearly independent in weighted L2((1/M[ρ,0,〈T〉])dv) space. Therefore,
(50)d2dt2∥f-M[ρ,0,〈T〉]∥L22|f=M[ρ,0,〈T〉]≥κ∫TN‍|∇ρρ-∇xϕ〈T〉|2dx=κ[∫‍|∇ρρ|2+∫‍|∇xϕ〈T〉|2-2∫‍∇ρ·∇ϕρ〈T〉],
where
(51)-2∫‍∇ρ·∇ϕρ〈T〉=-2〈T〉∫‍∇lnρ·∇ϕ=2〈T〉∫‍lnρΔϕ=2〈T〉∫‍lnρ(ρ-1).
It is easy to verify the convexity and nonnegativity of lnρ(ρ-1). Therefore,
(52)d2dt2∥f-M[ρ,0,〈T〉]∥L22|f=M[ρ,0,〈T〉]≥κ[∫‍|∇ρρ|2+∫‍|∇xϕ〈T〉|2]≥K3[∫TN‍|∇ρ|2dx+∫TN‍|∇xϕ|2dx].
Whenf does not coincide with M[ρ,0,〈T〉], we need to estimate two terms A and B of (42) separately. The detailed calculation can be found in [4, 5]. Also, we just emphasize the estimates for terms with ϕ here.
Notice that, when estimatingB, we need to control ∥∂2f/∂t2∥L2 by ∥f-M∥L21-α. Substitute the Vlasov-Poisson-Boltzmann equation (1) into ∂2f/∂t2; we get terms of ϕ.
(a)L2 norm estimate of ∇xϕt·∇vf.
It is obvious that(53)∥∇xϕt·∇vf∥L2≤∥∇xϕt·∇v(f-M)∥L2+∥∇xϕt·∇vM∥L2≤C(∫‍|∇v(f-M)|2dxdv)1/2+C(∫‍|∇xϕt|2dx)1/2.
The first term is bounded by C∥f-M∥L21-α by interpolation lemma. As for the second term, since
(54)∫‍|∇xϕt|2dx=-∫‍Δϕt·ϕtdx=-∫‍ρtϕtdx=∫‍∇x·(ρu)ϕtdx=-∫‍ρu·(∇xϕt)dx≤(∫‍ρ2u2dx)1/2(∫‍|∇xϕt|2dx)1/2,
we have
(55)∫‍|∇xϕt|2dx≤C∫‍ρ2u2dx≤Cℋ(u∣0)≤CH(f∣M)≤C∥f-M∥L22.
Hence, ∥∇xϕt·∇vf∥L2≤C∥f-M∥L21-α.
(b)L2 norm estimate of (v⊗∇vM):(∇x2ϕ).
Note thatM is a Gaussian distribution, so that M2 times any polynomials of v is integrable:
(56)∥(v⊗∇vM):(∇x2ϕ)∥L22=∥(v⊗v):(∇x2ϕ)M∥L22=∑i,j,k,l‍∫‍vivjvkvlM2∂xixjϕ∂xkxlϕdvdx≤C∑i,j‍∫‍∂xixiϕ∂xjxjϕdx=C∫‍(Δϕ)2dx=C∫‍|ρ-1|2dx≤Cℋ(ρ∣1)≤CH(f∣M)≤C∥f-M∥L22.
(c)L2 norm estimate of (∇xϕ⊗∇xϕ):∇v2M.
Similarly as in the previous argument,M2 times any polynomials of v is integrable. Also, ϕ and ∂ϕ are bounded by Schauder estimate because it is constrained by a Poisson equation.
Note thatΔϕ=ρ-1; we have
(57)∫TN‍|∇ϕ|2dx=-∫‍Δϕ(ϕ-ϕ-)dx=-∫‍(ρ-1)(ϕ-ϕ-)dx≤(∫‍(ρ-1)2)1/2(∫‍(ϕ-ϕ-)2)1/2≤KP1/2(∫‍(ρ-1)2)1/2(∫‍|∇ϕ|2)1/2.
Here, KP is the constant appearing in the Poincaré inequality, which is only relevant to the domain TN. Thus,
(58)∫‍|∇ϕ|2≤KP∫‍(ρ-1)2≤KP2∫‍|∇ρ|2.
Therefore,(59)∥(∇xϕ⊗∇xϕ):∇v2M∥L22=∥(∇xϕ⊗∇xϕ):(v⊗v-I)M∥L22=∑i,j,k,l‍∫‍∂iϕ∂jϕ∂kϕ∂lϕ(vivj-δij)=∑i,j,k,l‍∫‍×(vkvl-δkl)M2dvdx≤C∑i,j‍∫‍(∂iϕ∂jϕ)2dx≤C∑i‍∫‍(∂iϕ)2dx=C∫‍|∇ϕ|2dx≤C∫‍|ρ-1|2dx≤Cℋ(ρ∣1)≤CH(f∣M)≤C∥f-M∥L22.
(d)L2 norm estimate of Qsym(f,∇xϕ·∇vM).
From the momentum and energy conservation of particle collisions, it is easy to verify that(60)Qsym(M,viM)=0.
Thus,
(61)Qsym(f,∇xϕ·∇vM)=Qsym(f-M,∇xϕ·vM).
Then, using our continuity assumption (11) on Q(g,h) and the interpolation Lemma 8, we can estimate L2 norm of Qsym(f,∇xϕ·∇vM) by ∥f-M∥L21-α. Therefore, we have
(62)∀0<α<η<1,∥∂2f∂t2∥L2≤Cα∥f-M∥L21-α≤Cη∥f-M∥L11-η.
The rest of the proof is similar to that in [5]. Now we complete the proof of the lemma.Notice that there is the symmetric gradient ofu in (38); the next lemma can provide a method to control this term.Lemma 9.
One has the Korn-type inequality:(63)∫TN‍|∇symu|2dx≥KK∫Ω‍|∇u|2dx
and the following Poincaré-type inequalities:
(64)∫TN‍|∇T|2dx≥KTℋ(T∣〈T〉),∫TN‍|∇u|2dx≥Kuℋ(u∣0),∫TN‍|∇ρ|2dx≥Kρℋ(ρ∣1).
Here all constants are positive.Lemma 10.
One has estimates on damping of hydrodynamic oscillations withCS>0,
(65)|ddtℋ(ρ∣1),ddtℋ(u∣0),ddtℋ(〈T〉∣1),ddtℋ(T∣〈T〉)|≤CSH(f∣M)1-ϵ.See [4] or [5] for the proof of the previous two lemmas. Inequalities of Lemma 9 provide estimates of the right-hand side of second-order differential inequalities in Lemma 7. Lemma 10 provides the decay rate for hydrodynamic oscillations.


## 3. Proof of the Main Result

Use the previous lemmas; we are now ready to prove Theorem2. The main idea is similar to that in [5]; for convenience of the reader, we restate the sketch of the proof and make it more complete by proving Lemma 11.FromH-theorem (Theorem 5), the convergence rate of H(f) to H(M) is determined by entropy production functional D(f). But there are many local Maxwellians, which make our entropy production functional D(f) vanish. Therefore it is impossible to get a uniform lower bound on the entropy production. To overcome this difficulty, it is natural to estimate the average value of entropy production. Suppose that
(66)α0=H(f)-H(M)|t=t0.We wish to find an upper bound on a durationT0 (it is possible since H(f) is monotone nonincreasing), such that
(67)H(f)-H(M)|t=t0+T0=σα0,
where σ∈(0,1) is fixed; say σ=4/5. Therefore, we have
(68)45α0≤H(f)-H(M)≤α0.Lemma 11.
Choose thatϵ>0 is small enough, like ϵ<0.01, if one can show
(69)T0≤C0(ϵ)α0-699ϵ,
where C0 depends on ϵ and the various constants appearing in lemmas of Section 2. Then
(70)H(f)-H(M)=O(t-1/700ϵ).Proof.
Fixϵ>0 sufficiently small. Denote H(f)-H(M) by g(t). It is not hard to prove the continuity of g(t). From the boundedness of initial data f0, we can denote t0:=0, g(t)|t=0=g(0)=:β0. It is sufficient to prove that, for all t>0, t1/700ϵg(t) or equivalently tg(t)700ϵ is uniformly bounded.
Define a sequence{ti}, such that
(71)g(t)|t=ti=σiβ0.
Correspondingly, we can define{Ti}, Ti=ti-ti-1. From the estimate of T0 in (69), we have
(72)Ti≤C0(ϵ)(σi-1β0)-699ϵ.
Therefore,(73)ti=∑k=1i‍Tk≤C0(ϵ)(β0)-699ϵσ-699ϵ(i-1)-1σ-699ϵ-1.
It is obvious that ti→∞, as i→∞.
For anyt>0, we can find an interval such that t∈[ti-1,ti]. Now we are ready to estimate tg(t)700ϵ. From the monotonicity of g(t), we have
(74)tg(t)700ϵ≤tig(ti-1)700ϵ≤C0(ϵ)(β0)-699ϵσ-699ϵ(i-1)-1σ-699ϵ-1×(σi-1β0)700ϵ≤C,
where the constant is independent of i, since σ<1 is fixed and ϵ>0 can be chosen to be sufficiently small.Once condition (69) is proved, the main theorem is a direct consequence of H(f∣M)=O(H(f)-H(M)). Indeed, from (58) and (20), we have(75)∫‍|∇ϕ|2≤KP∫‍(ρ-1)2≤K′ℋ(ρ∣1)≤K′′H(f∣M),H(f∣M)≤H(f)-H(M)=H(f∣M)+∫‍12|∇ϕ|2≤(1+K′′2)H(f∣M).Therefore, it remains to prove condition (69). Detailed proof can be found in the last part of [5] for Vlasov-Poisson-Boltzmann equations; we only describe the idea of the proof for the completion of this paper. Consider
(76)45α0≤H(f)-H(M)≤α0,
on interval I:=[t0,t0+T0]; that is, H(f)-H(M) has variation α0/5. In order to prove (69), it is sufficient to prove that the average value of -(d/dt)H(f) on interval I satisfies
(77)〈-H˙(f)〉I≥Cα0C0(ϵ)α0-699ϵ=Cα01+699ϵ.Now we proceed the proof of Theorem2 step by step.(
1
)
I
G
: Subinterval of
I
Where
H
(
f
∣
M
[
ρ
,
u
,
T
]
)
Is Large. From quantitative H-Theorem 5, 〈-H˙(f)〉IG can be estimated directly on subinterval of I where H(f∣M[ρ,u,T]) is large. The subinterval can be called IG, which means good interval. Other interval is called IB, bad interval.Notice the entropy additivity rules in Lemma6; we actually have
(78)H(f)-H(M)=H(f∣M)+∫‍12|∇ϕ|2≤(1+K′2)ℋ(ρ∣1)+ℋ(u∣0)+ℋ(T∣〈T〉)+ℋ(〈T〉∣1)+H(f∣M[ρ,u,T]).(
2
)
I
B
G
: Subinterval of
I
B
Where
ℋ
(
T
∣
〈
T
〉
)
Is Large. On interval IB, H(f∣M[ρ,u,T]) is small, while H(f)-H(M) has lower bound (4/5)α0. Then by entropy additivity rules, we must have that ℋ(ρ∣1)+ℋ(u∣0)+ℋ(T∣〈T〉)+ℋ(〈T〉∣1) cannot be small.Denote the subinterval ofIB by IBG where ℋ(T∣〈T〉) is large. Then from the Poincaré-type inequalities of Lemma 9, we have that ∫TN‍|∇T|2dx is large. Therefore, the right hand side of (37) is large. By an argument for second-order differential inequalities (Lemma  12 of Desvillettes and Villani in [4]), we can conclude that either the average value of ∥f-M[ρ,u,T]∥L2 is large (so is H(f|M[ρ,u,T])) or the length of interval is small enough to be absorbed. H-theorem then asserts that average value 〈-H˙(f)〉IBG is large.(
3
)
I
B
B
G
: Subinterval of
I
B
B
Where
ℋ
(
u
∣
0
)
Is Large. On interval IBB, H(f∣M[ρ,u,T]), ℋ(T∣〈T〉) is small, while H(f)-H(M) has lower bound (4/5)α0. Then similarly, ℋ(ρ∣1)+ℋ(u∣0)+ℋ(〈T〉∣1) cannot be small.Denote the subinterval ofIBB by IBBG where ℋ(u∣0) is large. Then from the Poincaré-type and Korn-type inequalities of Lemma 9, we have that ∫‍|∇symu|2dx is large. Therefore, the right-hand side of (38) is large. By an argument for second-order differential inequalities (Lemma  12 of Desvillettes and Villani in [4]), we can conclude thateither the average value of ||f-M[ρ,u,〈T〉]||L2 is large (so is H(f∣M[ρ,u,〈T〉])) or the length of interval is small enough to be absorbed. But the first line of (35) shows that H(f∣M[ρ,u,T]) must be large in average. H-theorem then asserts that average value 〈-H˙(f)〉IBBG is large.(4)IBBBG: subinterval ofIBBBwhereℋ(〈T〉|1)is large. On interval IBBB, H(f∣M[ρ,u,T]), ℋ(T∣〈T〉), ℋ(u∣0) is small, while H(f)-H(M) has lower bound (4/5)α0. Then similarly, ℋ(ρ∣1)+ℋ(〈T〉∣1) cannot be small.Denote the subinterval ofIBBB by IBBBG where ℋ(〈T〉∣1) is large. From the conservation of energy, we have
(79)|〈T〉-1|=2Nℋ(u∣0)+1N∫TN‍|∇ϕ|2dx.
Because of the Lipschitz continuity of ℋ(〈T〉∣1)=Ψ(〈T〉),
(80)ℋ(〈T〉∣1)≤L|〈T〉-1|.
Since ℋ(u∣0) is sufficiently small in IBBBG, therefore, (79) turns to be
(81)ℋ(〈T〉∣1)≤L|〈T〉-1|≤C∫TN‍|∇ϕ|2dx≤Cℋ(ρ∣1).Therefore, the right-hand side of (40) and (39) is large. By a similar argument as in previous subintervals, we can also show that average value 〈-H˙(f)〉IBBBG is large. By a careful calculation to absorb all the bad intervals into good ones, we can prove that average value 〈-H˙(f)〉I is large on interval I. Thus, the whole proof is complete.To conclude the paper, we remove the condition in Theorem1 by making a crucial estimates on terms with ϕ. The main differences with previous works [5] are in proving Lemma 7. We also complete the gap in the last part of [4, 5] by proving Lemma 11.



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*Source: 632903-2013-08-01.xml*