arXiv:1411.5470v1 [math.AP] 20 Nov 2014
Vlasov-Poisson-Boltzmann equations
Hai-Liang Li
1, Tong Yang
2, Mingying Zhong
31Department of Mathematics, Capital Normal University, and BCMIIS, Beijing, P.R.China E-mail: [email protected]
2Department of Mathematics, City University of Hong Kong, Hong Kong E-mail: [email protected]
3Department of Mathematics and Information Sciences, Guangxi University, P.R.China.
E-mail: [email protected]
Abstract
In the present paper, we consider the initial value problem for the bipolar Vlasov-Poisson-Boltzmann (bVPB) system and its corresponding modified Vlasov-Poisson-Boltzmann (mVPB). We give the spectrum analysis on the linearized bVPB and mVPB systems around their equilibrium state and show the optimal convergence rate of global solutions. It was showed that the electric field decays exponentially and the distribution function tends to the absolute Maxwellian at the optimal convergence rate (1 +t)−3/4for the bVPB system, yet both the electric field and the distribution function converge to equilibrium state at the optimal rate (1 +t)−3/4for the mVPB system.
Key words. Bipolar Vlasov-Poisson-Boltzmann system, Modified Vlasov-Poisson-Boltzmann, spectrum analysis, optimal time decay rates.
2010 Mathematics Subject Classification. 76P05, 82C40, 82D05.
Contents
1 Introduction 2
2 Main results 4
2.1 bVPB system . . . 4
2.2 mVPB system . . . 7
3 Analysis of spectra and semigroup for linear systems 10 3.1 Spectrum and resolvent of linear bVPB system . . . 10
3.2 Exponential decay of semigroup for linear bVPB . . . 14
3.3 Analysis of spectrum and semigroup for linear mVPB system . . . 18
3.4 Optimal time-decay rates for linear mVPB . . . 21
4 The nonlinear problem for bVPB system 24 4.1 Energy estimates . . . 24
4.2 Convergence rates . . . 30
5 The nonlinear problem for mVPB system 33 5.1 Energy estimates . . . 33 5.2 Convergence rates . . . 36
1 Introduction
The bipolar Vlasov-Poisson-Boltzmann (bVPB) system of two species can be used to model the time evolution of dilute charged particles (e.g., electrons and ions) in the absence of an external magnetic field [12]. In general, the bVPB system for two species of particles in the whole space take the form
∂tF++v· ∇xF++∇xΦ· ∇vF+=Q(F+, F+) +Q(F+, F−), (1.1)
∂tF−+v· ∇xF−− ∇xΦ· ∇vF− =Q(F−, F−) +Q(F−, F+), (1.2)
∆xΦ = Z
R3
(F+−F−)dv, (1.3)
F+(x, v,0) =F+,0(x, v), F−(x, v,0) =F−,0(x, v), (1.4) where F+ =F+(x, v, t) andF− =F−(x, v, t) are number density functions of ions and electrons, and Φ(x, t) denotes the electric potential, respectively. The collision integral Q(F, G) describes the interaction between particles due to binary collisions by
Q(F, G) = Z
R3
Z
S2
|(v−v∗)·ω|(F(v′)G(v′∗)−F(v)G(v∗))dv∗dω, (1.5) where
v′=v−[(v−v∗)·ω]ω, v∗′ =v∗+ [(v−v∗)·ω]ω, ω∈S2.
Assume that the electron density is very rarefied and reaches a local equilibrium state with small electron mass compared with the ions, and that the collisionQ(F+, F−) between the ions and electrons can be neglected, the equation (1.2) can be reduced to
v· ∇xF−− ∇xΦ· ∇vF−= 0 (1.6)
This together with the simple local Maxwellian distribution of electron leads to F− =ρ−(x)M(v) = 1
(2π)32e−Φe−|v|
2 2
with the normalized MaxwellianM(v) given by
M =M(v) = 1
(2π)3/2e−|v|
2 2 .
For rigorous reduction of this type at the hydrodynamical scale, the readers can refer to [2]. Under the above reduction, we can obtain from bVPB system (1.1)–(1.3) the following modified Vlasov-Poisson-Boltzmann (mVPB) system:
Ft+v· ∇xF+∇xΦ· ∇vF=Q(F, F), (1.7)
∆xΦ = Z
R3
F dv−e−Φ, (1.8)
F(x, v,0) =F0(x, v), (1.9)
where F =F(x, v, t) is the distribution function of ions with (x, v, t) ∈ R3×R3×R+, Φ(x, t) is the electric potential, andQ(F, G) is the binary collision operator defined as (1.5).
In the case that the effect of electron is totally neglected, the bVPB system can be simplified to the standard unipolar Vlasov-Poisson-Boltzmann (VPB) model similar to (1.7)–(1.9) with the terme−Φreplaced by a given function, a positive constant for instance.
There have been a lot of works on the existence and behavior of solutions to the Vlasov-Poisson-Boltzmann system. The global existence of renormalized solution for large initial data was proved in [13]. The first global existence result on classical solution in torus when the initial data is near a global Maxwellian was established in [7]. And the global existence of classical solution in R3 was given [18, 19] in the same setting. The case with general stationary background density function ¯ρ(x) was studied in [4], and the perturbation of vacuum was investigated in [8, 5]. Recently, Li-Yang-Zhong [9] analyze the spectrum of the linearized VPB system and obtain the optimal decay rate of solutions to the nonlinear system near Maxwellian.
However, in contrast to the works on Boltzmann equation [6, 15, 16, 17] and VPB system [9], the spectrums of the linearized bVPB system and modified VPB system have not been given despite of its importance. On the other hand, an interesting phenomenon was shown recently in [3] on the time asymptotic behavior of the solutions which shows that the global classical solution of one species VPB system tends to the equilibrium at (1 +t)−14 inL2-norm. This is slower than the rate for the two species VPB system, that is, (1 +t)−34, obtained in [20]. Therefore, it is natural to investigate whether these rates are optimal.
The main purpose of the present paper is to investigate the spectrum and optimal time-convergence rates of global solutions to the linearized the bVPB system (1.1)–(1.4) in section 2.1 and the mVPB (1.7)–(1.9) in section 2.2 respectively. In particular, the main results established in this paper justifies how the electric field and the interplay interaction between ions and electrons influence the asymptotical behaviors of the global solution to the bVPB system (1.1)–(1.4) and the mVPB (1.7)–(1.9).
The rest of this paper will be organized as follows. The main results about the global existences and the optimal time-convergence rates of strong solution to bVPB system (1.1)–(1.4) and mVPB (1.7)–(1.9) are stated in Section 2. In Section 3, we analyze the spectrum of the bVPB system and mVPB system, and then establish the exponential time decay rates of the linearized bVPB and the algebraic time decay rates of the linearized mVPB equations in Sections 3.1–3.3. In Sections 4 and 5, we prove the optimal time decay rates of the global solution to the original nonlinear bVPB system and mVPB system respectively.
Notations: Define the Fourier transform of f =f(x, v) by ˆf(ξ, v) = Ff(ξ, v) = (2π)13/2
R
R3f(x, v)e−ix·ξdx, where and throughout this paper we denote i =√
−1.
Denote the weight functionw(v) by
w(v) = (1 +|v|2)1/2 and the Sobolev spacesHN andHwN as
HN ={f ∈L2(R3x×R3v)| kfkHN <∞ }, HwN ={f ∈L2(R3x×R3v)| kfkHwN <∞ } equipped with the norms
kfkHN = X
|α|+|β|≤N
k∂xα∂vβfkL2(R3
x×R3
v), kfkHwN = X
|α|+|β|≤N
kw∂xα∂vβfkL2(R3
x×R3
v). Forq≥1, we also define
L2,q=L2(R3v, Lq(R3x)), kfkL2,q = Z
R3
Z
R3
|f(x, v)|qdx 2/q
dv 1/2
.
In the following, we denote byk·kL2x,vandk·kL2ξ,vthe norms of the function spacesL2(R3x×R3v) andL2(R3ξ×R3v) respectively, and denote byk · kL2x, k · kL2ξ and k · kL2v the norms of the function spacesL2(R3
x), L2(R3
ξ) and L2(R3v) respectively. For any integer m ≥ 1, we denote by k · kHmx and k · kL2v(Hxm) the norms in the spaces Hm(R3
x) andL2(R3
v, Hm(R3
x)) respectively.
2 Main results
2.1 bVPB system
First of all, we consider the Cauchy problem of the bVPB system (1.1)–(1.4) in the present paper. Define F1=:F++F−, F2=:F+−F−.
Then Cauchy problem of the bVPB system (1.1)–(1.4) can be rewritten as
∂tF1+v· ∇xF1+∇xΦ· ∇vF2=Q(F1, F1), (2.10)
∂tF2+v· ∇xF2+∇xΦ· ∇vF1=Q(F2, F1), (2.11)
∆xΦ = Z
R3
F2dv, (2.12)
F1(x, v,0) =F1,0(x, v) =F+,0+F−,0, F2(x, v,0) =F2,0(x, v) =F+,0−F−,0. (2.13) The bVPB system (2.10)-(2.12) has an equilibrium state (F1∗, F2∗,Φ∗) = (M(v),0,0). Define the perturba- tionsf1(x, v, t) andf2(x, v, t) by
F1=M +√
M f1, F2=√ M f2.
Then the bVPB system (2.10)–(2.12) forf1(x, v, t) andf2(x, v, t) is reformulated into
∂tf1+v· ∇xf1−Lf1=1
2(v· ∇xΦ)f2− ∇xΦ· ∇vf2+ Γ(f1, f1), (2.14)
∂tf2+v· ∇xf2−v√
M · ∇xΦ−L1f2= 1
2(v· ∇xΦ)f1− ∇xΦ· ∇vf1+ Γ(f2, f1), (2.15)
∆xΦ = Z
R3
f2
√M dv, (2.16)
f1(x, v,0) =f1,0(x, v) = (F1,0−M)M−12, f2(x, v,0) =f2,0(x, v) =F2,0M−12, (2.17) where the operatorsLf,L1f and Γ(f, f) are defined by
Lf= 1
√M[Q(M,√
M f) +Q(√
M f, M)], (2.18)
L1f = 1
√MQ(√
M f, M), (2.19)
Γ(f, g) = 1
√MQ(√ M f,√
M g). (2.20)
The linearized collision operatorsLandL1can be written as [1, 21]
(Lf)(v) = (Kf)(v)−ν(v)f(v), (L1f)(v) = (K1f)(v)−ν(v)f(v), ν(v) =
Z
R3
Z
S2|(v−v∗)·ω|M∗dωdv∗, (Kf)(v) =
Z
R3
Z
S2
|(v−v∗)·ω|(p
M∗′f′+√
M′f∗′−√
M f∗)p
M∗dωdv∗
= Z
R3
k(v, v∗)f(v∗)dv∗, (K1f)(v) =
Z
R3
Z
S2
|(v−v∗)·ω|p M∗′p
M∗f′dωdv∗= Z
R3
k1(v, v∗)f(v∗)dv∗, whereν(v) is called the collision frequency,K and K1 are self-adjoint compact operators onL2(R3
v) with real symmetric integral kernelsk(v, v∗) andk1(v, v∗). The nullspace of the operatorL, denoted byN0, is a subspace spanned by the orthogonal basis{χj, j= 0,1,· · ·,4} with
χ0=√
M , χj =vj
√M (j= 1,2,3), χ4= (|v|2−3)√
√ M
6 , (2.21)
and the nullspace of the operatorL1, denoted byN1, is a subspace spanned by√ M.
We denoteL2(R3) be a Hilbert space of complex-value functionsf(v) onR3with the inner product and the norm
(f, g) = Z
R3
f(v)g(v)dv, kfk= Z
R3
|f(v)|2dv 1/2
. Let P0, Pdbe the projection operators fromL2(R3v) to the subspaceN0, N1with
P0f =
4
X
i=0
(f, χi)χi, P1=I−P0, (2.22)
Pdf = (f,√ M)√
M , Pr=I−Pd. (2.23)
From the Boltzmann’s H-theorem, the linearized collision operatorsLandL1are non-positive and moreover, LandL1 are locally coercive in the sense that there is a constantµ >0 such that
(Lf, f)≤ −µkP1fk2, f ∈D(L), (2.24)
(L1f, f)≤ −µkPrfk2, f ∈D(L1), (2.25) whereD(L) andD(L1) are the domains ofLandL1 given by
D(L) =D(L1) =
f ∈L2(R3)|ν(v)f ∈L2(R3) . In addition, for the hard sphere model,ν satisfies
ν0(1 +|v|)≤ν(v)≤ν1(1 +|v|). (2.26) From the system (2.14)–(2.17) for (f1, f2), we have the following decoupled linearized system forf1andf2:
∂tf1=Ef1, f1(x, v,0) =f1,0(x, v), (2.27)
∂tf2=Bf2, f2(x, v,0) =f2,0(x, v), (2.28) where
Ef1=Lf1−(v· ∇x)f1, (2.29)
Bf2=L1f2−(v· ∇x)f2−v√
M · ∇x(−∆x)−1 Z
R3
f2
√M dv. (2.30)
The equation (2.27) is the linearized Boltzmann equation, its spectrum analysis and the optimal decay rate of the solution has already been made for instance in [15, 22]. Therefore, we only need to investigate the spectrum analysis and the decay rate of the solution to the linearized Vlasov-Poisson-Boltzmann type equation (2.28).
Indeed, take Fourier transform to (2.27)–(2.28) inxto get
∂tfˆ1= ˆE(ξ) ˆf1, (2.31)
∂tfˆ2= ˆB(ξ) ˆf2, (2.32)
where the operators ˆE(ξ), ˆB(ξ) are defined forξ6= 0 by
E(ξ) =ˆ L1−i(v·ξ), B(ξ) =ˆ L1−i(v·ξ)−i(v·ξ)
|ξ|2 Pd. Then, we have
Theorem 2.1. Letσ( ˆB(ξ))denotes the spectrum of operatorB(ξ)ˆ to the linear equation (2.32). There exist a constanta1>0 such that it holds for all ξ6= 0 that
σ( ˆB(ξ))⊂ {λ∈C|Reλ <−a1}. (2.33)
Let σ( ˆE(ξ))denotes the spectrum of operatorE(ξ)ˆ to the linear equation (2.31). Then, for any r0>0 there existsα=α(r0)>0 so that it holds for|ξ| ≥r0 that
σ( ˆE(ξ))⊂ {λ∈C|Reλ≤ −α}. (2.34) There exists a constantr0>0 such that the spectrumσ( ˆE(ξ))for ξ=sω with |s| ≤r0 andω ∈S2 consists of five points{µj(s), j=−1,0,1,2,3}on the domain Reλ >−µ/2, which areC∞ functions of sfor |s| ≤r0 and satisfy the following asymptotical expansion for|s| ≤r0
µ±1(s) =±i r5
3s−b±1s2+o(s2), µ1(s) =µ−1(s), µ0(s) =−b0s2+o(s2),
µ2(s) =µ3(s) =−b2s2+o(s2),
(2.35)
with constantsbj >0,−1≤j≤2.
With above spectrum analysis, we can obtain the global existence and the time-asymptotical behavior of unique solution to the Cauchy problem for the linear bVPB system (2.31)–(2.32) as follows.
Theorem 2.2. Assume that f1,0 ∈L2v(HxN)∩L2,q for N ≥1 andq ∈[1,2]. Then there is a globally unique solutionf1(x, v, t) =etEf1,0(x, v)to the linearized Boltzmann equation (2.27), which satisfies for anyα, α′ ∈N3 with|α| ≤N,α′ ≤αandm=|α−α′| that
k(∂xαetEf1,0, χj)kL2x ≤C(1 +t)−32(1q−12)−m2(k∂xαf0kL2x,v+k∂xα′f0kL2,q), j= 0,1,2,3,4, (2.36) kP1(∂xαetEf1,0)kL2x,v ≤C(1 +t)−32(1q−12)−m+12 (k∂αxf0kL2x,v+k∂xα′f0kL2,q). (2.37) In addition, assume thatf1,0∈L2(R3v;HN(R3x)∩L1(R3x))forN ≥1and there exist positive constantsd0, d1>0 and a small constantr0 >0 so that the Fourier transform fˆ1,0(ξ, v) of the initial data f1,0(x, v) satisfies that inf|ξ|≤r0|( ˆf1,0,√
M)| ≥d0,inf|ξ|≤r0|( ˆf1,0, χ4)| ≥d1sup|ξ|≤r0|( ˆf1,0,√
M)|andsup|ξ|≤r0|( ˆf1,0, v√
M)|= 0. Then global solutionf(x, v, t) =etEf1,0(x, v)satisfies for two positive constantsC2≥C1 that
C1(1 +t)−34−k2 ≤ k∇kx(etEf1,0, χj)kL2x ≤C2(1 +t)−34−k2, j= 0,1,2,3,4, (2.38) C1(1 +t)−54−k2 ≤ k∇kxP1(etEf1,0)kL2x,v≤C2(1 +t)−54−k2, (2.39) for t >0 sufficiently large andk≥0.
Furthermore, if f2,0 ∈ L2v(HxN)∩L2,1 for N ≥1, then the global solution f2(x, v, t) =etBf2,0(x, v) to the linear Vlasov-Poisson-Boltzmann type equation (2.28) exists globally in time and satisfies fort >0
k∂αxf2(t)kL2x,v+k∂xα∇xΦ(t)kL2x≤Ce−12a1t(k∂xαf2,0kL2x,v+kf2,0kL2,1) (2.40) for 0≤ |α| ≤N, where∇xΦ(t) =∇x∆−x1(etBf2,0,√
M).
With the help of optimal time decay rates on the linearized bVPB (2.27)–(2.28) given by Theorem 2.2, we can obtain the optimal decay rates of the global solution to original bVPB system (1.1)–(1.3) as follows.
Theorem 2.3. Assume that f±,0 = (F±,0− 12M)M−12 ∈ HwN ∩L2,1 for N ≥ 4 and kf±,0kHwN∩L2,1 ≤ δ0
for a constant δ0 > 0 small enough. Then there exists a globally unique solution (F±,Φ) with F±(x, v, t) =
1 2M +√
M f±(x, v, t) to the bVPB system (1.1)–(1.3), which satisfies
k∂xk(f+, f−)(t)kL2x,v ≤Cδ0(1 +t)−34−k2, (2.41) k∂xk∇xΦ(t)kL2x≤Cδ0e−dt, (2.42)
and in particular
k∂xk(f+(t), χj)kL2x+k∂xk(f−(t), χj)kL2x ≤Cδ0(1 +t)−34−k2, (2.43) k∂xk(P1f+,P1f−)(t)kL2x,v≤Cδ0(1 +t)−54−k2, (2.44) k(P1f+,P1f−)(t)kHwN+k∇x(P0f+,P0f−)(t)kL2v(HxN−1)≤Cδ0(1 +t)−54, (2.45) for j= 0,1,2,3,4, k= 0,1 and a constant d >0.
Moreover, the recombination (f1, f2)with f1 =:f++f−, f2 =:f+−f− is the global solution to the system (2.14)–(2.17) satisfies
k∂xk(f1(t), χj)kL2x≤Cδ0(1 +t)−34−k2, j= 0,1,2,3,4, (2.46) k∂xkP1f1(t)kL2x,v≤Cδ0(1 +t)−54−k2, (2.47) k∂xkf2(t)kL2x,v+k∂xk∇xΦ(t)kL2x ≤Cδ0e−dt, (2.48) k(P1f1, Prf2)(t)kHwN +k∇x(P0f1, Pdf2)(t)kL2v(HxN−1)≤Cδ0(1 +t)−54, (2.49) for k= 0,1.
Theorem 2.4. Let the assumptions of Theorem 2.3 hold. Assume further that there exist positive constants d0, d1>0and a small constantr0>0so thatfˆ±,0= ( ˆF±,0−12M)M−12 satisfies thatinf|ξ|≤r0|( ˆf1,0,√
M)| ≥d0, sup|ξ|≤r0|( ˆf1,0, χj)|= 0 (j= 1,2,3)andinf|ξ|≤r0|( ˆf1,0, χ4)| ≥d1sup|ξ|≤r0|( ˆf1,0,√
M)|with fˆ1,0= ˆf+,0+ ˆf−,0. Then, the global solution(F±,Φ)withF±(x, v, t) =12M+√
M f±(x, v, t)to the bVPB system(1.1)–(1.3)satisfies C1δ0(1 +t)−34−k2 ≤ k∇kxf±(t)kL2x,v≤C2δ0(1 +t)−34−k2, (2.50) k∇kx∇xΦ(t)kL2x ≤Cδ0e−dt, (2.51) and in particular
C1δ0(1 +t)−34−k2 ≤ k∇kx(f±(t), χj)kL2x ≤C2δ0(1 +t)−34−k2, (2.52) C1δ0(1 +t)−54−k2 ≤ k∇kxP1f±(t)kL2x,v ≤C2δ0(1 +t)−54−k2, (2.53) for t >0 large with two positive constantsC2> C1,j= 0,1,2,3,4,andk= 0,1.
Moreover, the recombinationf1=f++f−, f2=f+−f− to the system (2.14)–(2.17)satisfies
C1δ0(1 +t)−34−k2 ≤ k∇kx(f1(t), χj)kL2x≤C2δ0(1 +t)−34−k2, (2.54) C1δ0(1 +t)−54−k2 ≤ k∇kxP1f1(t)kL2x,v ≤C2δ0(1 +t)−54−k2, (2.55) C1δ0(1 +t)−34 ≤ kf1(t)kHwN ≤C2δ0(1 +t)−34, (2.56) k∇kxf2(t)kL2x,v+k∂xk∇xΦ(t)kL2x ≤Cδ0e−dt, (2.57) for t >0 large with two constantsC2> C1,j = 0,1,2,3,4,andk= 0,1.
2.2 mVPB system
Next, we deal with the global existence and uniqueness of solution to the Cauchy problem for the mVPB system (1.7)-(1.9) and the optimal time-convergence rate of the global solutions. The mVPB system (1.7)-(1.8) has an equilibrium state (F∗,Φ∗) = (M,0) with M =M(v) being the normalized global Maxwellian defined above. Define the perturbationf(x, v, t) ofF nearM by
f = (F−M)M−12,
then the modified Vlasov-Poisson-Boltzmann system (1.7)-(1.9) forf(x, v, t) reads
∂tf +v· ∇xf −v√
M· ∇xΦ−Lf= 1
2(v· ∇xΦ)f− ∇xΦ· ∇vf+ Γ(f, f), (2.58) (I−∆x)Φ =−
Z
R3
f√
M dv+ (e−Φ+ Φ−1), (2.59)
f(x, v,0) =f0(x, v) =: (F0−M)M−1/2, (2.60) where the operatorsLf and Γ(f, f) are defined by (2.18) and (2.20) respectively.
From the modified VPB system (2.58)–(2.60), we have the following the linearized mVPB equation
∂tf =Bmf, t >0, (2.61)
f(x, v,0) =f0(x, v), (x, v)∈R3x×R3v, (2.62) where the linear operatorBmis defined by
Bmf =Lf−v· ∇xf −v√
M· ∇x(I−∆x)−1 Z
R3
f√ M dv
. Take the Fourier transform to (2.61) with respect toxto get
∂tfˆ= ˆBm(ξ) ˆf , (2.63)
where
Bˆm(ξ) =L−i(v·ξ)− i(v·ξ) 1 +|ξ|2Pd.
Then, we have the spectrum analysis of the operator ˆBm(ξ) and the time-decay rates of the global solution to the linearized mVPB system (2.61)–(2.62) and establish its optimal time-decay rates as follows.
Theorem 2.5. Let σ( ˆBm(ξ)) denotes the spectrum of operator Bˆm(ξ) to the linear equation (2.63) for all ξ∈R3. Then, for anyr0>0there exists α=α(r0)>0so that it holds for |ξ| ≥r0 that
σ( ˆBm(ξ))⊂ {λ∈C|Reλ≤ −α}.
There exists a constant r0 >0 so that the spectrum λ∈σ(Bm(ξ))⊂C for ξ =sω with |s| ≤ r0 and ω ∈ S2 consists of five points {λj(s), j =−1,0,1,2,3} on the domainReλ >−µ/2, which areC∞ functions of s for
|s| ≤r0and satisfy the following asymptotical expansion for |s| ≤r0
λ±1(s) =±i2 r2
3s−a±1s2+o(s2), λ1(s) =λ−1(s), λ0(s) =−a0s2+o(s2),
λ2(s) =λ3(s) =−a2s2+o(s2),
(2.64)
with constantsaj>0,−1≤j ≤2, defined in Lemma 3.16.
With above spectrum analysis, we can obtain the global existence and the time-asymptotical behavior of unique solution to the Cauchy problem for the linear mVPB system (2.61)–(2.62) as follows.
Theorem 2.6. Assume thatf0∈L2(R3v;HN(R3x)∩Lq(R3x))for N ≥1 andq∈[1,2]. Then there is a globally unique solution f(x, v, t) =etBmf0(x, v) to the linearized mVPB system (2.61)–(2.62), which satisfies for any α, α′∈N3 with|α| ≤N,α′ ≤αand k=|α−α′|that
P4
j=0k∂xα(etBmf0, χj)kL2x ≤C(1 +t)−32(1q−12)−k2(k∂xαf0kL2x,v+k∂xα′f0kL2,q), (2.65) k(I−∆x)−1(∂αxetBmf0,√
M)kHx1 ≤C(1 +t)−32(1q−12)−k2(k∂xαf0kL2x,v+k∂xα′f0kL2,q), (2.66)
kP1(∂xαetBmf0)kL2x,v ≤C(1 +t)−32(1q−12)−k+12 (k∂xαf0kL2x,v+k∂xα′f0kL2,q). (2.67) In addition, assume thatf0∈L2(R3
v;HN(R3
x)∩L1(R3
x))for N≥1 and there exist positive constantsd0, d1>0 and a small constant r0 > 0 so that the Fourier transform fˆ0(ξ, v) of the initial data f0(x, v) satisfies that inf|ξ|≤r0|( ˆf0, χ0)| ≥ d0, inf|ξ|≤r0|( ˆf0, χ4)| ≥ d1sup|ξ|≤r0|( ˆf0, χ0)| and sup|ξ|≤r0|( ˆf0, v√
M)| = 0. Then global solutionf(x, v, t) =etBmf0(x, v) satisfies for two positive constantsC2≥C1 that
C1(1 +t)−34−k2 ≤ k∇kx(etBmf0, χj)kL2x≤C2(1 +t)−34−k2, (2.68) C1(1 +t)−34−k2 ≤ k∇kx(I−∆x)−1(etBmf0,√
M)kH1x≤C2(1 +t)−34−k2, (2.69) C1(1 +t)−54−k2 ≤ k∇kxP1(etBmf0)kL2x,v ≤C2(1 +t)−54−k2, (2.70) for t >0 sufficiently large,j= 0,1,2,3,4,andk≥0.
Then, we state the results on the global existence and the optimal time-asymptotical behavior of unique solution to the Cauchy problem for the mVPB system (2.58)–(2.60) blow.
Theorem 2.7. Assume that f0 ∈ HwN ∩L2,1 for N ≥ 4 and kf0kHwN∩L2,1 ≤ δ0 for a constant δ0 >0 small enough. Then, there exists a globally unique strong solution f =f(x, v, t) to the mVPB system (2.58)-(2.60) satisfying
P4
j=0k∂xk(f(t), χj)kL2x+k∂xkΦ(t)kHx1 ≤Cδ0(1 +t)−34−k2, (2.71) k∂xkP1f(t)kL2x,v ≤Cδ0(1 +t)−54−k2, (2.72) kP1f(t)kHwN +k∇xP0f(t)kL2v(HxN−1)≤Cδ0(1 +t)−54, (2.73) for k= 0,1 andt >0.
We shall prove that the above convergence rates are indeed optimal in the following sense.
Theorem 2.8. Let the assumptions of Theorem 2.7 hold. Assume further that there exist positive constants d0, d1>0 and a small constant r0 >0 so that the Fourier transform fˆ0(ξ, v) satisfiesinf|ξ|≤r0|( ˆf0, χ0)| ≥d0, sup|ξ|≤r0|( ˆf0, χj)|= 0 (j = 1,2,3)andinf|ξ|≤r0|( ˆf0, χ4)| ≥d1sup|ξ|≤r0|( ˆf0, χ0)|. Then, the global solutionf to the mVPB system (2.58)-(2.60)satisfies
C1δ0(1 +t)−34−k2 ≤ k∇kx(f(t), χj)kL2x≤C2δ0(1 +t)−34−k2, (2.74) C1δ0(1 +t)−34−k2 ≤ k∇kx∇xΦ(t)kL2x≤C2δ0(1 +t)−34−k2, (2.75) C1δ0(1 +t)−54−k2 ≤ k∇kxP1f(t)kL2x,v≤C2δ0(1 +t)−54−k2, (2.76) C1δ0(1 +t)−34 ≤ kf(t)kHwN ≤C2δ0(1 +t)−34, (2.77) for t >0 sufficiently large, two positive constantsC2≥C1,j= 0,1,2,3,4,andk= 0,1.
Remark 2.9(Example). The initial dataf0=f1(x, v)defined below satisfies the assumptions of Theorem 2.8 f1(x, v) =d0e
r2 0 2 e−x
2
2 χ0+d1d0e
r2 0 2 e−x
2 2 χ4. for a small positive constantd0.
Remark 2.10. The conditions on initial data in above theorems can be applied to the VPB system and the Boltzmann equation to obtain corresponding the optimal time decay rate (refer to [22]). Although the global solution to the modified Vlasov-Poisson-Boltzmann (mVPB) equation system takes the same optimal time decay rate (1 +t)−3/4 as the Boltzmann equation, yet it observed that the hyperbolic waves of the mVPB system propagates at a faster speed due to the influence of electric field.
3 Analysis of spectra and semigroup for linear systems
3.1 Spectrum and resolvent of linear bVPB system
We investigate the spectrum analysis and the decay rate of the solution to the linearized Vlasov-Poisson- Boltzmann type equation (2.28). In the followings, we are concerned with the spectral analysis of the operator B(ξ) and optimal time-decay rate of solution to linear VPB type equation (2.32).ˆ
Introduce a weighted Hilbert spaceL2ξ(R3) forξ6= 0 as L2ξ(R3) ={f ∈L2(R3)| kfkξ =
q
(f, f)ξ <∞}, with the inner product defined by
(f, g)ξ = (f, g) + 1
|ξ|2(Pdf, Pdg).
SincePdis a self-adjoint projection operator, it follows that (Pdf, Pdg) = (Pdf, g) = (f, Pdg) and hence (f, g)ξ= (f, g+ 1
|ξ|2Pdg) = (f + 1
|ξ|2Pdf, g). (3.1)
By (3.1), we have for anyf, g∈L2ξ(R3
v)∩D( ˆB(ξ)), ( ˆB(ξ)f, g)ξ = ( ˆB(ξ)f, g+ 1
|ξ|2Pdg) = (f,(L+ i(v·ξ) +i(v·ξ)
|ξ|2 Pd)g) = (f,B(ˆ −ξ)g)ξ. (3.2) We can regard ˆB(ξ) as a linear operator from the spaceL2ξ(R3) to itself because
kfk2≤ kfk2ξ ≤(1 +|ξ|−2)kfk2, ξ6= 0.
Similarly to the proofs of Lemmas 2.6–2.7 in [9], we have the following lemmas.
Lemma 3.1. The operator B(ξ)ˆ generates a strongly continuous contraction semigroup on L2ξ(R3), which satisfies
ketB(ξ)ˆ fkξ ≤ kfkξ, for anyt >0, f ∈L2ξ(R3v). (3.3) Lemma 3.2. For each ξ6= 0, the spectrum of B(ξ)ˆ on the domain Reλ≥ −ν0+δ for any δ >0 consists of isolated eigenvalues{λj(ξ)} withReλj(ξ)<0.
Now denote byT a linear operator onL2(R3
v) orL2ξ(R3
v), and we define the corresponding norms ofT by kTk= sup
kfk=1kT fk, kTkξ= sup
kfkξ=1kT fkξ. Obviously,
(1 +|ξ|−2)−1kTk ≤ kTkξ≤(1 +|ξ|−2)kTk. (3.4) First, we consider the spectrum and resolvent sets of ˆB(ξ) at high frequency. To this end, we define
c(ξ) =−ν(v)−i(v·ξ), (3.5)
and decompose ˆB(ξ) into
λ−Bˆ(ξ) =λ−c(ξ)−K1+i(v·ξ)
|ξ|2 Pd
= (I−K1(λ−c(ξ))−1+i(v·ξ)
|ξ|2 Pd(λ−c(ξ))−1)(λ−c(ξ)). (3.6) Then, we have the estimates on the right hand terms of (3.6) as follows.
Lemma 3.3. There exists a constantC >0 so that it holds:
1. For any δ >0, we have sup
x≥−ν0+δ,y∈RkK1(x+ iy−c(ξ))−1k ≤Cδ−15/13(1 +|ξ|)−2/13, (3.7) 2. For any δ >0, r0>0, there is a constant y0= (2r0)5/3δ−2/3>0 such that if|y| ≥y0, we have
sup
x≥−ν0+δ,|ξ|≤r0
kK1(x+ iy−c(ξ))−1k ≤Cδ−7/5(1 +|y|)−2/5, (3.8)
3. For any δ >0, r0>0, we have sup
x≥−ν0+δ,y∈Rk(v·ξ)|ξ|−2Pd(x+ iy−c(ξ))−1k ≤Cδ−1|ξ|−1, (3.9) sup
x≥−ν0+δ,|ξ|≥r0
k(v·ξ)|ξ|−2Pd(x+ iy−c(ξ))−1k ≤C(r0−1+ 1)(δ−1+ 1)|y|−1. (3.10)
Proof. The proof of (3.9) and (3.10) can be found in Lemma 2.3 in [9]. SinceK1satisfies the same properties asK (see [21]):
Z
R3|k1(v, v∗)|dv∗≤C(1 +|v|)−1, Z
R3|k1(v, v∗)|2dv∗≤C, we can prove (3.7) and (3.8) by a same argument as Lemma 2.2.6 in [17].
By Lemma 3.3 and a similar argument as Lemma 2.4 in [9], we have the spectral gap of the operator ˆB(ξ) for high frequency.
Lemma 3.4. Let λ(ξ)∈σ( ˆB(ξ)) be any eigenvalue of B(ξ)ˆ in the domain Reλ≥ −ν0+δ with δ >0 being a constant. Then, for anyr0>0, there existsα(r0)>0 so thatReλ(ξ)≤ −α(r0)for all|ξ| ≥r0.
Then, we investigate the spectrum and resolvent sets of ˆB(ξ) at low frequency. To this end, we decompose λ−B(ξ) as followsˆ
λ−B(ξ) =ˆ λPd+λPr−Q(ξ) + iPd(v·ξ)Pr+ iPr(v·ξ)(1 + 1
|ξ|2)Pd, (3.11) where
Q(ξ) =L1−iPr(v·ξ)Pr. (3.12)
Lemma 3.5. Let ξ6= 0 andQ(ξ)defined by (3.12). We have 1. If λ6= 0, then
kλ−1Pr(v·ξ)(1 + 1
|ξ|2)Pdkξ ≤C(|ξ|+ 1)|λ|−1. (3.13) 2. If Reλ >−µ, then the operator λPr−Q(ξ)is invertible on N1⊥ and satisfies
k(λPr−Q(ξ))−1k ≤(Reλ+µ)−1, (3.14) kPd(v·ξ)Pr(λPr−Q(ξ))−1Prkξ≤C(1 +|λ|)−1[(Reλ+µ)−1+ 1](1 +|ξ|)2. (3.15) Proof. Since
kλ−1Pr(v·ξ)(1 + 1
|ξ|2)Pdfkξ ≤C|λ|−1(|ξ|+ 1
|ξ|)kPdfk ≤C|λ|−1(|ξ|+ 1)kfkξ, we prove (3.13).
Then, we show that for anyλ∈Cwith Reλ > −µ, the operatorλPr−Q(ξ) =λPr−L1+ iPr(v·ξ)Pr is invertible fromN1⊥ to itself. Indeed, by (2.25), we obtain for anyf ∈N1⊥∩D(L1) that
Re([λPr−L1+ iPr(v·ξ)Pr]f, f) = Reλ(f, f)−(L1f, f)≥(µ+ Reλ)kfk2, (3.16) which implies that the operatorλPr−Q(ξ) is an one-to-one map fromN1⊥ to itself so long as Reλ >−µ. The estimate (3.14) follows directly from (3.16).
By (3.14) andkPd(v·ξ)Prfkξ≤C(|ξ|+ 1)kPrfk, we have
kPd(v·ξ)Pr(λPr−Q(ξ))−1Prfkξ ≤C(|ξ|+ 1)(Reλ+µ)−1kfk. (3.17) Meanwhile, we can decompose the operatorPd(v·ξ)Pr(λPr−Q(ξ))−1Pr as
Pd(v·ξ)Pr(λPr−Q(ξ))−1Pr= 1
λPd(v·ξ)Pr+ 1
λPd(v·ξ)PrQ(ξ)(λPr−Q(ξ))−1Pr. This together with (3.14) and the factkPd(v·ξ)PrQ(ξ)k ≤C(1 +|ξ|)2 give
kPd(v·ξ)Pr(λPr−Q(ξ))−1Prfkξ ≤C|λ|−1[(Reλ+µ)−1+ 1](1 +|ξ|)2kfk. (3.18) The combination of the two cases (3.17) and (3.18) yields (3.15).
Consider the eigenvalue problem
λf = (L1−i(v·ξ))f −i√
M(v·ξ)
|ξ|2 Z
R3
f√
M dv. (3.19)
We shall prove that ˆB(ξ) has a spectral gap when |ξ| is sufficiently small. For convenience, we shall use the parametrizationξ=sω wheres∈R1, ω∈S2.
Letf be the eigenfunction of (3.19), we rewrite f in the form f =f0+f1, wheref0=Pdf =C0
√M and f1= (I−Pd)f =Prf. The eigenvalue problem (3.19) can be decomposed into
λf0=−Pd[i(v·ξ)(f0+f1)], (3.20)
λf1=L1f1−Pr[i(v·ξ)(f0+f1)]−i(v·ξ)
|ξ|2 f0. (3.21)
From Lemma 3.5 and (3.21), we obtain that for any Reλ >−µ
f1= i[L1−λPr−iPr(v·ξ)Pr]−1Pr((v·ξ)f0+(v·ξ)
|ξ|2 f0). (3.22)
Substituting (3.22) into (3.20) and taking inner product the resulted equation with√
M gives λC0= (1 + 1
|ξ|2)(R(λ, ξ)(v·ξ)√
M ,(v·ξ)√
M)C0. (3.23)
whereR(λ, ξ) = [L1−λPr−iPr(v·ξ)Pr]−1.
By changing variable (v·ξ) → sv1 and using the rotational invariance of the operator L1, we have the following transformation.
Lemma 3.6. Let e1= (1,0,0),ξ=sω withs∈R, ω∈S2. Then (R(λ, ξ)(v·ξ)√
M ,(v·ξ)√
M) =s2(R(λ, se1)(v1
√M), v1
√M). (3.24)
With the help of (3.24), we rewrite (3.23) in the form
λC0= (1 +s2)(R(λ, se1)χ1, χ1)C0. (3.25) Denote
D(λ, s) = (1 +s2)(R(λ, se1)χ1, χ1). (3.26)
