# Local Gevrey Regularity for Linearized Homogeneous Boltzmann Equation

Authors: Shi-you Lin
Journal: Journal of Function Spaces and Applications
 (2012)

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## Abstract

The local Gevrey regularity of the solutions of the linearized spatially homogeneous Boltzmann equation has been shown in the non-Maxwellian case with mild singularity.


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## Body

## 1. Introduction

This paper focuses on the Gevrey class smoothing property of solutions of the following linear Cauchy problems of the spatially homogeneous Boltzmann equation:(1.1)∂f∂t=Lf=Q(μ,f)+Q(f,μ),v∈ℝ3,t>0,μ(v)=(2π)-3/2e-|v|2/2,f|t=0=f0,
where the initial datum f0≢0 satisfies the natural boundedness on mass, energy, and entropy:
(1.2)f0≥0,∫ℝ3‍f0(v){1+|v|2+log(1+f0(v))}dv<+∞.Q(g,f) is the Boltzmann quadratic operator which has the following form:
(1.3)Q(g,f)=∫ℝ3‍∫𝕊2‍B(v-v*,σ){g(v*′)f(v′)-g(v*)f(v)}dσdv*,
where σ∈𝕊2 (unit sphere of ℝ3); the post- and precollisional velocities are given as follows:
(1.4)v′=v+v*2+|v+v*|2σ,v*′=v+v*2-|v+v*|2σ.
The Boltzmann collision cross-section B(|z|,σ) is a nonnegative function which depends only on |z| and the scalar product 〈z/|z|,σ〉. To capture its main properties, we usually assume
(1.5)B(|v-v*|,σ)=Φ(|v-v*|)b(cosθ),cosθ=〈v-v*|v-v*|,σ〉,θ∈[0,π2].μ is called the normalized Maxwellian distribution in (1.1). Notice that Q(μ,μ)≡0.Recall that the inverse power law potential1/ρs, where s>1, and ρ denotes the distance between two particles, has the form (1.5) with the corresponding kinetic factors:
(1.6)Φ(|v-v*|)≈|v-v*|1-4/s,b(cosθ)≈Kθ2+ν,θ→0,
for a constant K>0 and 0<ν=2/s<2. The cases 1<s<4,s=4, and s>4 correspond to so-called soft, Maxwellian, and hard potentials, respectively.We will concentrate on the modified hard potentials as follows:(1.7)Φ(|v-v*|)=(1+|v-v*|2)γ/2,0<γ<1,b(cosθ)≈Kθ2+ν,θ→0,0<ν<2,
where the singularity is called the mild singularity when 0<ν<1 and the strong singularity when 1≤ν<2. In this paper, we consider only the case of the mild singularity. Before making the discussion, we start by introducing the norms of the weighted function spaces:
(1.8)∥f∥Lrp=∥〈|v|〉rf(v)∥Lp,∥f∥Hrs=∥〈|D|〉s〈|v|〉rf(v)∥L2,
where 〈|v|〉=(1+|v|2)1/2 and 〈|D|〉 is the corresponding pseudodifferential operator. And then, we list the definition of the weak solution in the Cauchy problem (1.1); compare [1].Definition 1.1.
For an initial datumf0(v)∈L21(ℝ3), f(t,v) is called a weak solution of the Cauchy problem (1.1) if it satisfies
(1.9)f(t,v)∈C(ℝ+;𝒟′(ℝ3))∩L2([0,T];L21(ℝ3))∩L∞([0,T];L1(ℝ3)),f(0,v)=f0,∫ℝ3‍f(t,v)φ(t,v)dv-∫ℝ3‍f(0,v)φ(0,v)dv-∫0t‍dτ∫ℝ3‍f(τ,v)∂τφ(τ,v)dv=∫0t‍dτ∫ℝ3‍L(f)(τ,v)φ(τ,v)dv,
for any test function φ∈L∞([0,T];W2,∞(ℝ3)).For the definition of the Gevrey class functions, compare [1–5].Definition 1.2.
Suppose thatW is a bounded open set on ℝ3, for s≥1, u∈Gs(W) which is the Gevrey class function space with index s, if u∈C∞(W) and for any compact subset U⊂W, there exists a constant C=C(U)>0 such that for any k∈ℕ,
(1.10)∥Dku∥L2(U)≤Ck+1(k!)s,
or equivalently,
(1.11)∥〈|D|〉ku∥L2(U)≤Ck+1(k!)s,
where
(1.12)∥Dku∥L2(U)2=∑|β|=k‍∥Dβu∥L2(U)2,〈|D|〉=(1+|Dv|2)1/2.
Particularly, u∈Gs(ℝ3), that is, ||Dku||L2(ℝ3)≤Ck+1(k!)s, is equivalent to the fact that there exists ϵ0>0 such that eϵ0〈|D|〉1/su∈L2(ℝ3).Notice thatG1(ℝ3) is the usual analytic function space. When 0<s<1, we call Gs(ℝ3) the ultra-analytic function space, cpmpare [4, 5].There have been some results about the Gevrey regularity of the solutions for the Boltzmann equation; compare [1, 4, 6–8]. Among them, unique local solutions having the same Gevrey regularity as the initial data are first constructed in [8]. This implies the propagation of the Gevrey regularity. In 2009, Desvillettes et al. improved this result for the nonlinear spatially homogeneous Boltzmann equation, they showed in [6] that, for the Maxwellian molecules model, the Gevrey regularity can propagate globally in time. Other results for the nonlinear case can be found in [4], where the Gevrey regularity of the radially symmetric weak solutions has been proved. Meanwhile, this issue is also considered in [7] for the Maxwellian decay solutions. For the linear case, the best result so far is obtained by the work of Morimoto et al. in [1]; they proved the propagation of Gevrey regularity of the solutions, without any extra assumption for the initial data. We mention that the crucial tools in [1, 6] are the following pseudodifferential operator:
(1.13)Gδ(t,Dv)=1δ+e-t〈|Dv|〉ν/2,0<ν<2.In the Maxwellian case, this pseudodifferential operator can be used successfully, but it seems unsuitable for the non-Maxwellian model. The difficulty comes from the commutator of the kinetic factorΦ and the pseudodifferential operator (1.13) which lacks of the effective estimations. In this paper, we apply a new method which is based on the mathematical induction to overcome it. Compared with [7], we consider only the local space; however, we discuss this issue by using the much weaker preconditions (actually, we do not need any smooth assumption for the initial data). Concerning the same issue for the other related equations, such as the Landau equation and the Kac equation, compare [2–5].Now we can state our main result.Theorem 1.3.
SupposeΦ, b have the forms in (1.7), 0<ν<1. Let W be a bounded open set of ℝ3, and f(t,v) be the weak solution of the Cauchy problem (1.1) satisfying
(1.14)supt∈(0,T]∥f(t,·)∥L2(W)<+∞.
Then for any t∈(0,T], there exists a number s=s(t)>3 satisfying f(t,·)∈Gs(W). More precisely, for any fixed 0<t0≤T and compact subset U⊂W, there exists a constant C=C(U)>0 and a number s>3 such that for any k∈ℕ,
(1.15)supt∈[t0,T]∥Dkf(t,·)∥L2(U)≤Ck+1(k!)s.From Theorem1.3, we have the following remark.Remark 1.4.
Suppose thatΦ, b have the forms in (1.7), 0<ν<1. If the weak solution f(t,v) satisfies that
(1.16)supt∈(0,T]∥f(t,·)∥L2(ℝ3)<+∞,
then for any t∈(0,T], any bounded open set U⊂ℝ3, there exists a constant s=s(t) satisfying f(t,·)∈Gs(U).


## 2.  Useful Lammas for the Main Result

In order to gain the main result, we need to prove the following lemmas in this section.Lemma 2.1.
SupposeΦ(v)=〈|v|〉γ=(1+|v|2)γ/2 where γ∈(0,1),v∈ℝn, and n∈ℕ. Then the kth order derivative of Φ satisfies
(2.1)|Φ(k)(v)|≤4kk!Φ(v)〈|v|〉-k.Proof.
Without loss of generality, we only consider the case ofn=1; the other cases are similar. By direct calculation, we have
(2.2)Φ(2m)(v)=∑i=0m‍Ci,2mγ(γ-2)⋯(γ-2i-2m+2)(1+v2)γ/2-i-mv2i,Φ(2m+1)(v)=∑i=0m‍Ai,2m+1γ(γ-2)⋯(γ-2i-2m)(1+v2)γ/2-i-m-1v2i+1.
In addition,
(2.3)Ci,2m+2(i+1)Ci+1,2m=Ai,2m+1,(2i+1)Ai,2m+1+Ai-1,2m+1=Ci,2m+2.
Thus we obtain
(2.4)Ci,2m+2=Ci-1,2m+(4i+1)Ci,2m+(2i+1)(2i+2)Ci+1,2m
and then we will prove the following inequality:
(2.5)|Ci,2m|≤22m|(γ-2i-2m)⋯(γ-4m+2)|.
The inequality is obviously true for m=1. Suppose it is valid for 1≤m≤M, then
(2.6)|Ci,2M+2|=|Ci-1,2M+(4i+1)Ci,2M+(2i+1)(2i+2)Ci+1,2M|≤22M[(4i+1)|γ-2i-2M|+(γ-2i+2-2M)|γ-2i-2M|+(2i+1)(2i+2)|γ-2i-2M|]|(γ-2i-2-2M)⋯(γ-4M+2)|≤22(M+1)|(γ-2i-2-2M)⋯(γ-4M)(γ-4M-2)|
which proves (2.5) by induction. Therefore, we have
(2.7)|Φ(2m)(v)|≤22m|γ(γ-2)⋯(γ-4m+2)|∑i=0m‍(1+v2)γ/2-i-mv2i≤(m+1)24m-1(2m-1)!Φ(v)〈|v|〉-2m≤42m(2m)!Φ(v)〈|v|〉-2m.
The case of (2m+1)th order derivative is similar. This completes the proof of Lemma 2.1.SettingMN(ξ)=(1+|ξ|2)Nt/2 for any ξ∈ℝ3 and N∈ℕ, by using the similar technique of Lemma 2.1, we conclude the following.Remark 2.2.
Fort∈(0,1],
(2.8)|∂ξkMN(ξ)|≤4k〈|ξ|〉Nt-k|N(N-1)⋯(N-k+1)|≤4k〈|ξ|〉(N-k)t|N(N-1)⋯(N-k+1)|,
where k∈ℕ,1≤k≤N.Lemma 2.3.
There exists a constantC such that for any k∈ℕ,
(2.9)|∂vkμ(v)|≤Ck·k!·max(1,|v|k)·μ(v),
where μ is the absolute Maxwellian distribution in (1.1).Proof.
Without loss of generality, we also only consider the case in the real spaceℝ1. Putting
(2.10)∂vkμ(v)=∂vk(e-v2/2)=∑j=0k‍aj,k′vje-v2/2,∂vkμ-1(v)=∂vk(ev2/2)=∑j=0k‍aj,kvjev2/2.
Evidently,
(2.11)0≤|aj,k′|≤aj,k,|ak,k′|=ak,k≡1,∑j=01‍aj,1=1≤81·1!,aj,k+1=aj-1,k+(j+1)aj+1,k.
Therefore,  fixed a number  m≥0, together with the following assumption (Fm):(2.12)∑j‍aj,m≤8m·m!,
we can obtain (Fm+1)(2.13)∑j‍aj,m+1=∑j‍(aj-1,m+(j+1)aj+1,m)≤(m+2)·(∑j‍aj-1,m+∑j‍aj+1,m)≤2(m+2)·8m·m!≤8m+1·(m+1)!.
This completes the proof of Lemma 2.3 by induction.Setting(2.14)H*(v)=(1+|v*|2)4μ*(v)=(1+|v*|2)4·μ(v+v*),
where v is belong to a bounded set U. Then we state Lemma 2.4 as below.Lemma 2.4.
There exists a constantC=C(U)>0, which satisfies that for any k∈ℕ,
(2.15)supv*|∂vkH*(v)|≤Ck·(k!)2.Proof.
Sincee-(v+v*)2/4≤ev2/4·e-v*2/8, and the fact that when |v|≥1,
(2.16)|vke-v2/4|=|v|k(∑n=0+∞‍|v|2n/22n·n!)≤|v|k(|v|k/2k·(k/2)!)≤2k·k!,
by using Lemma 2.3, we have
(2.17)|∂vkH*(v)|=|(1+|v*|2)4∂vkμ*(v)|≤(1+|v*|2)4·Ck·k!·max(1,|v+v*|k)·μ(v+v*)≤(1+|v*|2)4·Ck·k!·[max(1,|v+v*|k)·e-(v+v*)2/4]·e-(v+v*)2/4≤[(1+|v*|2)4e-v*2/8]·Ck·k!·[max(1,|v+v*|k)·e-(v+v*)2/4]·ev2/4≤(Ck·k!)2≤[C(U)]k·(k!)2.
This completes the proof of Lemma 2.4By applying the Cauchy integral theorem, we will prove the helpful estimates as follows.Lemma 2.5.
Suppose the Fourier transform forv*,
(2.18)ℱ(Φ(|v-v*|)μ(v*))(ξ)=h(v,ξ)μ^(ξ),
where μ is the absolute Maxwellian distribution in (1.1). Then we have
(2.19)h(v,ξ)=(2π)-3/2∫ℝ3‍e-|v*|2/2[1+|v-v*|2-|ξ|2+2i(v-v*)·ξ]γ/2dv*.Proof.
First we consider the case ofn=1,
(2.20)ℱ(Φ(|v-v*|)μ(v*))(ξ)=∫ℝ1‍(1+|v-v*|2)γ/2(2π)-3/2e-|v*|2/2-iv*·ξdv*=(2π)-3/2e-ξ2/2∫ℝ1‍[1+(v-v*)2]γ/2e-(v*+iξ)2/2dv*=(2π)-3/2e-ξ2/2∫C‍e-z2/2[1+(v-z+iξ)2]γ/2dz,
where z=v*+iξ, and C denotes the curve: v*+iξ,-∞<v*<∞. By Cauchy integral theorem [9], it follows that
(2.21)∫C‍e-z2/2[1+(v-z+iξ)2]γ/2dz=∫ℝ1‍e-|v*|2/2[1+(v-v*+iξ)2]γ/2dv*.
Now we turn to consider the case of n=3. Letting v=(v1,v2,v3), and v*=(v*1,v*2,v*3) and using the previous result, we have
(2.22)ℱ(Φ(|v-v*|)μ(v*))(ξ)=∫ℝ3‍(1+|v-v*|2)γ/2(2π)-3/2e-|v*|2/2-iv*·ξdv*=(2π)-3/2∫ℝ2‍(∫ℝ1‍[1+(v1-v*1)2+(v2-v*2)2+(v3-v*3)2]γ/2e-v*12/2-iv*1ξ1dv*1)×e-(v*22+v*32)/2-i(v*2ξ2+v*3ξ3)dv*2dv*3=(2π)-3/2∫ℝ2‍(∫ℝ1‍[1+(v1-v*1+iξ1)2+(v2-v*2)2+(v3-v*3)2]γ/2e-v*12/2-ξ12/2dv*1)×e-(v*22+v*32)/2-i(v*2ξ2+v*3ξ3)dv*2dv*3=(2π)-3/2e-|ξ|2/2∫ℝ3‍e-|v*|2/2[1+∑j=13‍(vj-v*j+iξj)2]γ/2dv*=(2π)-3/2μ^(ξ)∫ℝ3‍e-|v*|2/2[1+|v-v*|2-|ξ|2+2i(v-v*)·ξ]γ/2dv*.
Thus we conclude the result of Lemma 2.5.Lemma 2.6.
For the expression ofh(v,ξ) in Lemma 2.5, we have
(2.23)|h(v,ξ)|≤C·〈|v|〉γ〈|ξ|〉γ,|∇ξ2h(v,ξ)|≤C·〈|v|〉γ〈|ξ|〉γ,|h(v,ξ+)-h(v,ξ)|≤C·〈|v|〉γ〈|ξ|〉1+γsinθ2,θ=arccos〈ξ|ξ|,σ〉,
where ξ+=(ξ+|ξ|σ)/2, and C is a constant independent of v and ξ.Proof.
The first inequality is obvious. To prove the third one, setξ=(ξ1,ξ2,ξ3). Since
(2.24)h(v,ξ)μ^(ξ)=ℱ(Φ(|v-v*|)μ(v*))(ξ)=∫ℝ3‍(1+|v-v*|2)γ/2(2π)-3/2e-|v*|2/2-iv*·ξdv*,
proceeding as in the proof of Lemma 2.5, we can get
(2.25)∂ξi(h(v,ξ)μ^(ξ))=μ^(ξ)[∂ξih(v,ξ)-ξih(v,ξ)]=∫ℝ3‍(1+|v-v*|2)γ/2(2π)-3/2e-|v*|2/2-iv*·ξ(-iv*i)dv*=(2π)-3/2μ^(ξ)∫ℝ3‍[1+|v-v*|2-|ξ|2+2i(v-v*)·ξ]γ/2·e-|v*|2/2·(-ξi-iv*i)dv*.
Therefore,
(2.26)∂ξih(v,ξ)=(2π)-3/2∫ℝ3‍e-|v*|2/2[1+|v-v*|2-|ξ|2+2i(v-v*)·ξ]γ/2·(-ξi-iv*i)dv*+ξih(v,ξ)=(2π)-3/2∫ℝ3‍e-|v*|2/2[1+|v-v*|2-|ξ|2+2i(v-v*)·ξ]γ/2·(-iv*i)dv*
which implies that
(2.27)|∇ξh(v,ξ)|≤C·〈|v|〉γ〈|ξ|〉γ.
By the mean value theorem of differentials, we have
(2.28)|h(v,ξ+)-h(v,ξ)|≤C·|∇ξh(v,η)|·|ξ+-ξ|≤C′·〈|v|〉γ〈|η|〉γ|ξ+-ξ|≤C′′·〈|v|〉γ〈|ξ|〉1+γsinθ2,
where θ=arccos〈ξ/|ξ|,σ〉. Thus the third inequality has been obtained.
Finally, the above way can also be used in estimating the second one. Similarly,(2.29)∂ξiξj2(h(v,ξ)μ^(ξ))=∂ξj(∂ξi(h(v,ξ)μ^(ξ)))=-∫ℝ3‍(1+|v-v*|2)γ/2(2π)-3/2e-|v*|2/2-iv*·ξv*iv*jdv*=-(2π)-3/2μ^(ξ)∫ℝ3‍[1+|v-v*|2-|ξ|2+2i(v-v*)·ξ]γ/2·e-|v*|2/2(v*i-iξi)(v*j-iξj)dv*.
On the other hand,
(2.30)∂ξiξj2(h(v,ξ)μ^(ξ))=∂ξj{μ^(ξ)[∂ξih(v,ξ)-ξih(v,ξ)]}=μ^(ξ)[ξjξih(v,ξ)+∂ξjξi2h(v,ξ)-ξj∂ξih(v,ξ)-ξi∂ξjh(v,ξ)].
Combining with the above expressions of h(v,ξ) and ∂ξih(v,ξ), we get
(2.31)∂ξjξi2h(v,ξ)=ξi∂ξjh(v,ξ)+ξj∂ξih(v,ξ)-ξjξih(v,ξ)-(2π)-3/2∫ℝ3‍[1+|v-v*|2-|ξ|2+2i(v-v*)·ξ]γ/2·e-|v*|2/2(v*i-iξi)(v*j-iξj)dv*=-(2π)-3/2∫ℝ3‍[1+|v-v*|2-|ξ|2+2i(v-v*)·ξ]γ/2·e-|v*|2/2v*iv*jdv*.
Therefore,
(2.32)|∇ξ2h(v,ξ)|≤C·〈|v|〉γ〈|ξ|〉γ.
This completes the proof of the second inequality.Lemma 2.7.
Suppose that0<ν<1 in (1.7). Then for any r>0,f∈L2+γ1(ℝ3)∩H+∞(ℝ3), there exists a constant C independent of r satisfying
(2.33)I0(τ)=(Q(f,μ),〈|D|〉rf)L2≤C∥f∥L2+γ1∥f∥L1(r+3)!.Proof.
Letξ±=(ξ±|ξ|σ)/2, from Bobylev’s formula (see [10]), we have
(2.34)I0(τ)=∫ℝ3‍〈|ξ|〉rf^(ξ)¯[∫ℝ6‍∫𝕊2‍Φ(|v-v*|)b(ξ|ξ|·σ)μ(v*)f(v)∫ℝ3‍〈|ξ|〉rf^(ξ)¯×(e-i(v*·ξ++v·ξ-)-e-iv*·ξ)dσdvdv*∫ℝ6‍∫𝕊2‍]dξ=∫ℝ3‍〈|ξ|〉rf^(ξ)¯∫ℝ3‍∫𝕊2‍[ℱ(Φ(|v-v*|)μ(v*))(ξ+)bf(v)e-iv·ξ-∫ℝ3‍〈|ξ|〉rf^(ξ)¯∫ℝ3‍∫𝕊2‍-ℱ(Φ(|v-v*|)μ(v*))(ξ)bf(v)(v)e-iv·ξ-]dσdvdξ=∫ℝ3‍〈|ξ|〉rf^(ξ)¯∫ℝ3‍∫𝕊2‍[h(v,ξ+)-h(v,ξ)]μ^(ξ+)bf(v)e-iv·ξ-dσdvdξ+∫ℝ3‍〈|ξ|〉rf^(ξ)¯∫ℝ3‍∫𝕊2‍h(v,ξ)[μ^(ξ+)-μ^(ξ)]bf(v)e-iv·ξ-dσdvdξ+∫ℝ3‍〈|ξ|〉rf^(ξ)¯∫ℝ3‍∫𝕊2‍h(v,ξ)μ^(ξ)bf(v)[e-iv·ξ--e0]dσdvdξ=I01+I02+I03.
In [1], it is shown that
(2.35)|μ^(ξ+)-μ^(ξ)|≤μ^(ξ+)|ξ|2sin2θ2,e-|ξ|2/2=μ^(ξ)≤μ^(ξ+)≤e-|ξ|2/4,〈|ξ|〉r+γ+2≤(r+3)!e〈|ξ|〉(r+γ+2)/(r+3)≤(r+3)!e〈|ξ|〉.
Together with Lemma 2.6, we have
(2.36)|I02|≤C·∥f∥L1+γ1∫ℝ3‍〈|ξ|〉r+γ+2e-|ξ|2/4f^(ξ)¯dξ≤C·∥f∥L1+γ1∥f∥L1∫ℝ3‍〈|ξ|〉r+γ+2e-|ξ|2/4dξ≤C·∥f∥L1+γ1∥f∥L1(r+3)!∫ℝ3‍e〈|ξ|〉-|ξ|2/4dξ≤C′∥f∥L1+γ1∥f∥L1(r+3)!.
Now we turn to estimate the terms in I01 and I03. For the case 0<ν<1 in (1.7), it is easy to see that
(2.37)|e-iv·ξ--e0|=|-2sinv·ξ-2(sinv·ξ-2+icosv·ξ-2)|≤C|v||ξ-|≤C|v||ξ|sinθ2.
Therefore, applying the above estimates and Lemma 2.6, we also conclude that
(2.38)|I0i|≤C′∥f∥L1+γ1∥f∥L1(r+3)!
for any i∈{1,3}. This completes the proof of Lemma 2.7.


## 3. Related Analysis

Letf be the weak solution of the Cauchy problem (1.1). For any k∈ℕ, the compact support
(3.1)supp(Mkf)⊆supp(f),
which implies that for any compact subset U⊂W,
(3.2)f*={f,ifv∈U,0,ifv∉U,
is also a weak solution of the following equation
(3.3)(∂f∂t,Mk2f)L2(ℝ3)=(Q(μ,f)+Q(f,μ),Mk2f)L2(ℝ3).
Since Theorem 1.3 is mainly concerned with the Gevrey smoothness property of the solution f on W, we need only to study the solution of the above equation on any fixed compact subset of W. That is, we can suppose that f has compact support in U for any t∈[0,T],
(3.4)supp(f)⊆U,f(Uc)≡0.
Thus, for any p≥0,
(3.5)∥f∥Lp1(ℝ3)≤O(1)∥f∥L1(U)<+∞,∥f∥Hp(ℝ3)=∥f∥Hp(U).
Together with Lemma 2.6, we can get the fact that f∈H+∞(ℝ3). This proof is similar as the proof of [11, Theorem 1.1] and hence omitted. Clearly, ||f||Hr(ℝ3)=||f||Hr(U). Moreover, without loss of generality, we restrict T≤1, then for any k∈ℕ, it is assumed that
(3.6)(Ek):foranyi∈[0,k-1],supt∈(0,T]∥f(t,·)∥Hit≤C0i+1(i!)s,
where C0 is a sufficiently large constant satisfying
(3.7)C0≥166max(supt∈(0,T]∥f∥Li,i=1,2).In the following discussion, we will useC and Ci,i∈ℕ to denote the positive constants independent of k and t. Let Mk(Dv)=〈|Dv|〉kt and Φ*(v)=〈|v-v*|〉γ. In order to prove Theorem 1.3, we need the propositions as below.Proposition 3.1.
One has(3.8)supt∈(0,T]∥[Mk(Dv),Φ*]f(t,v)∥L2≤C·C0k+1(k!)s.Proposition 3.2.
One has(3.9)supt∈(0,T]∥∇v[Mk(Dv),Φ*]f(t,v)∥L2≤C{(k+1)∥Mkf(t,v)∥L2+C0k+1(k!)s},(3.10)supt∈(0,T]∥[Mk(Dv),H*]f(t,v)∥L2≤C·C0k+1(k!)s,
whereH* is the function which has the form (2.14).The proof of the above propositions will be given in Section5.


## 4. Proof of Theorem1.3

Now we will prove the main result in this section. For anyt∈(0,T], we state the following identity from [11]:
(4.1)(Q(μ,f),Mk2f)L2-(Q(μ,Mkf),Mkf)L2=I1+I2+I3,
where
(4.2)I1=∫ℝ6‍∫𝕊2‍b(cosθ)μ(v*)(Mk(ξ)-Mk(ξ+))Φ*f^(ξ+)e-iv*ξ-Mk(ξ)f^(ξ)¯dσdv*dξ,I2=∫ℝ6‍∫𝕊2‍b(cosθ)μ(v*){[Mk,Φ*]f(v′)·Mkf(v′)-[Mk,Φ*]f(v)·Mkf(v)}dσdv*dv,I3=∫ℝ6‍∫𝕊2‍b(cosθ)μ(v*)([Mk,Φ*]f(v)-[Mk,Φ*]f(v′))Mkf(v′)dσdv*dv.Our purpose is to obtain the estimations ofI1, I2 and I3. Setting η=|ξ|2 and η+=|ξ+|2, since |ξ+|=|ξ|cos(θ/2) and |ξ+|2-|ξ|2=|ξ|2sin2(θ/2), applying the mean value theorem and the fact that 0<t≤T≤1, we have
(4.3)|Mk(ξ)-Mk(ξ+)|=|(1+η)kt/2-(1+η+)kt/2|≤C·kt·(η-η+)(1+η0)kt/2-1≤C·kt·sin2θ2(1+η)kt/2≤C′·k·sin2θMk(ξ),
where η0 is a number between η an η+. Therefore,
(4.4)|I1|≤C∫ℝ3‍∫𝕊2‍bsin2θMk(ξ)|∫ℝ6‍Φ(|v-v*|)μ(v*)f(v)e-iv*·ξ--iv·ξ+dv*dv|×|kMk(ξ)f^(ξ)|dσdξ=C∫ℝ3‍∫𝕊2‍bsin2θMk(ξ)|∫ℝ6‍Φ(|v*|)μ(v*+v)f(v)e-iv*·ξ--iv·ξdv*dv|×|kMk(ξ)f^(ξ)|dσdξ≤C∫ℝ6‍∫𝕊2‍bsin2θ〈|v*|〉γ-8·|Mk(ξ)H*f^(ξ)|·|kMk(ξ)f^(ξ)|dσdv*dξ≤C∫ℝ6‍∫𝕊2‍bsin2θ〈|v*|〉γ-8·|Mk(ξ)H*f^(ξ)|2dσdv*dξ+C∫ℝ6‍∫𝕊2‍bsin2θ〈|v*|〉γ-8·|kMk(ξ)f^(ξ)|2dσdv*dξ=I11+I12.
Here H* is the function which has the form (2.14). It is clear that
(4.5)I12≤C′k2∥Mkf∥L22,I11≤C∫ℝ3‍〈|v*|〉γ-8·∥Mk(Dv)H*f∥L22dv*≤C∫ℝ3‍〈|v*|〉γ-8∥[Mk,H*]f∥L22dv*+C∫ℝ3‍〈|v*|〉γ-8∥H*Mkf∥L22dv*=I111+I112.
By the hypothesis (3.4), f has compact support in U¯, we obtain
(4.6)I112≤C∫ℝ3‍〈|v*|〉γ+8·∥μ(v+v*)Mkf∥L22dv*≤C∫ℝ3‍〈|v*|〉γ+8e-|v*|2/4·∥e|v|2/2Mkf∥L22dv*≤C′∥Mkf∥L22.
Here we use the fact that e-|v+v*|2/2≤e|v|2/2·e-|v*|2/4. By (3.10) of Proposition 3.2, we get
(4.7)I111≤C′·[C0k+1(k!)s]2.
This, together with (4.5)-(4.6), implies
(4.8)|I1|≤C1·(k2∥Mkf∥L22+[C0k+1(k!)s]2).
The cancellation lemma gives (cf. [10, 11])
(4.9)I2=S∫ℝ6‍μ(v*)[Mk,Φ*]f(v)·Mkf(v)dvdv*,
where S is a constant function. Therefore,
(4.10)|I2|≤C∥μ∥L1·∥[Mk,Φ*]f∥L2·∥Mkf∥L2≤C·C0k+1(k!)s∥μ∥L1·∥Mkf∥L2≤C2{[C0k+1(k!)s]2+∥Mkf∥L22}.
Since |v′-v|≤C〈|v′|〉〈|v*|〉sin(θ/2), by using (3.4), Proposition 3.2, and the change of variables
(4.11)v→z=v′+τ(v-v′)
whose Jacobian is bounded uniformly for v*,σ,τ (see [11]), we have
(4.12)|I3|≤C∫01‍∫ℝ6‍∫𝕊2‍b(cosθ)|v′-v|·|Mkf(v′)|·|∇v[Mk,Φ*]f(v′+τ(v-v′))|×|μ(v*)|dσdvdv*dτ≤C′∫01‍∫ℝ6‍∫𝕊2‍b(cosθ)sinθ2|Mkf(v′)|·|∇v[Mk,Φ*]f(v′+τ(v-v′))|×〈|v*|〉|μ(v*)|dσdvdv*dτ≤C′∫01‍∫ℝ6‍∫𝕊2‍b(cosθ)sinθ2〈|v*|〉μ(v*)×{|Mkf(v′)|2+|∇v[Mk,Φ*]f(v′+τ(v-v′))|2}dσdvdv*dτ≤C′′(∥Mkf∥L22+∥∇v[Mk,Φ*]f∥L22)≤C3{[C0k+1(k!)s]2+k2∥Mkf∥L22}.
Combining (4.8), (4.10), and (4.12), we obtain
(4.13)(Q(μ,f),Mk2f)L2-(Q(μ,Mkf),Mkf)L2≤C4{[C0k+1(k!)s]2+k2∥Mkf∥L22}.
Moreover, by [11, Lemma 2.2] and [11, page 467], we have
(4.14)∥〈·〉γ/2Mkf∥Hν/22=O(1)∥〈|D|〉ν/2Mkf∥Lγ/222,(Q(μ,Mkf),Mkf)L2≤-Cμ,1∥〈·〉γ/2Mkf∥Hν/22+Cμ,2∥Mkf∥Lγ/222,
where Cμ,1 and Cμ,2 are the constants depending only on μ. Therefore, by (3.4) and (4.14), we get
(4.15)(Q(μ,Mkf),Mkf)L2≤-C5∥Mkf∥Hν/22+C6∥Mkf∥L22.
Together with (4.13), we thus have
(4.16)(Q(μ,f),Mk2f)L2≤C7·[C0k+1(k!)s]2+C8·k2∥Mkf∥L22-C5∥Mkf∥Hν/22.
Let Mk2f be the test function in the Cauchy problem (1.1), for any t∈(0,T], we have
(4.17)∥Mkf(t,v)∥L22=∥f0(v)∥L22+2∫0t‍∫ℝ3‍L(f)(τ,v)Mk2f(τ,v)dvdτ+∫0t‍∫ℝ3‍f(τ,v)(∂τMk2(τ))f(τ,v)dvdτ=2∫0t‍{(Q(μ,f),Mk2f)L2+(Q(f,μ),Mk2f)L2}dτ+∫0t‍∫ℝ3‍f(τ,v)(∂τMk2(τ))f(τ,v)dvdτ+∥f0(v)∥L22.
Since
(4.18)O(1)(k!)2s≥22(k+2)[(k+2)!]2≥(2k+3)!,∂tMk2(t,ξ)=2kMk2(t,ξ)log〈ξ〉,
by Lemma 2.7 and (4.16), it holds that
(4.19)∥Mkf(t,v)∥L22+C5∫0t‍∥Mkf∥Hν/22dτ≤2k∫0t‍∥(log〈Dv〉)1/2(Mkf)(τ)∥L22dτ+C8∫0t‍k2∥Mkf∥L22dτ+C9[C0k+1(k!)s]2+∥f0(v)∥L22.
The Young’s inequality gives
(4.20)C8k2≤[4C8C5·(ν+2)]2/ν·C8νν+2·k2+4/ν〈|ξ|〉-2+C52〈|ξ|〉ν,2klog〈|ξ|〉=4kνlog〈|ξ|〉ν/2≤4kν·〈|ξ|〉ν/2≤[4·(4+ν)C8·ν(ν+2)](4+ν)/ν2ν+2·k2+4/ν〈|ξ|〉-2+C52〈|ξ|〉ν,
which implies
(4.21)2k∫0t‍∥(log〈Dv〉)1/2(Mkf)(τ)∥L22dτ+C8∫0t‍∥kMkf∥L22dτ≤C5∫0t‍∥Mkf∥Hν/22dτ+C10·k2+4/ν·∫0t‍∥Mk-1f∥L22dτ.
Taking (4.21) into (4.19), and applying the assumption (Ek), we have
(4.22)∥Mkf(t,v)∥L22≤∥f0(v)∥L22+C9[C0k+1(k!)s]2+C10·k2+4/ν∫0t‍∥Mk-1f∥L22dτ≤∥f0(v)∥L22+C9[C0k+1(k!)s]2+C10·k2+4/ν{C0k·[(k-1)!]s}2≤C11[C0k+1(k!)s]2,
which implies that
(4.23)supt∈(0,T]∥Mkf(t,v)∥L2=supt∈(0,T]∥f(t,·)∥Hkt≤C11·C0k+1(k!)s.
In other words, it follows from (Ek) that
(4.24)(Ek+1):foranyi∈[0,k],supt∈(0,T]∥f(t,·)∥Hit≤C11·C0i+1(i!)s.
Taking the same procedures as above, we can also gain (Ek+2) from (Ek+1), which is described as below:
(4.25)(Ek+2):foranyi∈[0,k+1],supt∈(0,T]∥f(t,·)∥Hit≤C112·C0i+1(i!)s,
that is,(4.26)supt∈(0,T]∥f(t,·)∥H0t≤C01(0!)s⇒supt∈(0,T]∥f(t,·)∥H1t≤C111·C02(1!)s⇒supt∈(0,T]∥f(t,·)∥H2t≤C112·C03(2!)s⋮⇒supt∈(0,T]∥f(t,·)∥Hkt≤C11k·C0k+1(k!)s.
Let C12=C0·C11, we thus conclude that for any k∈ℕ,
(4.27)supt∈(0,T]∥f(t,·)∥Hkt(U)≤C12k+1(k!)s.
For any fixed number 0<t≤T≤1, suppose that
(4.28)s0=(⌈1t⌉+1)s,s1=∑i=0⌈1/t⌉‍(1-it),
where ⌈1/t⌉ denotes the smallest integer bigger than 1/t. With a convention that k!=1 if 0≥k∈ℤ, we have
(4.29)2ks1=2k·2k(1-t)·2k(1-2t)⋯2k(1-{1/t}t)≥k!(kt)![(1-t)k]!·[(1-t)k]!(kt)![(1-2t)k]!⋯[(1-⌈1/t⌉t)k]!(kt)![(1-(⌈1/t⌉+1)t)k]!≥k![(kt)!](⌈1/t⌉+1)≥k![(kt)!]s0/s.
This, together with (4.27), implies that
(4.30)supt∈(0,T]∥f(t,·)∥Hkt(U)≤C12k+1(k!)s≤2kss1C12k+1[(kt!)]s0≤C13k+1[(kt!)]s0,
where k∈ℕ, and C16 is a constant only depending on t. Furthermore, for any fixed number t0>0, put
(4.31)s0′=(⌈1t0⌉+1)s,s1′=∑i=0⌈1/t0⌉‍(1-it0).
Then for any k∈ℕ, we can choose C13′=2ss1′C12 and have the fact that
(4.32)supt∈[t0,T]∥f(t,·)∥Hkt(U)≤(C13′)k+1[(k!)]s0′.
This completes the proof of Theorem 1.3.


## 5. Proof of Propositions3.1 and 3.2

Proof of Proposition3.1.
We first notice that(5.1)[Mk(Dv),Φ*]f(v)=Mk(Dv)(Φ*f)(v)-Φ*(v)Mk(Dv)f(v)=(ℱ-1Mk(ξ)*Φ*f)(v)-Φ*(v)(ℱ-1Mk(ξ)*f)(v)=∫ℝ6‍ei(v-y)ξMk(ξ)dξf(y)(Φ*(y)-Φ*(v))dy.
Using the Taylor formula of order k+6, we get
(5.2)Φ*(y)-Φ*(v)=∑j=1k+5‍(y-v)jj!∂vjΦ*(v)+(y-v)k+6(k+6)!∂vk+6Φ*(c)
for some c∈(y,v). Hence,
(5.3)[Mk(Dv),Φ*]f(v)=∑j=1k+5‍Γjf(v)+Γk+6f,
where
(5.4)Γjf(v)=∫ℝ6‍ei(v-y)ξMk(ξ)dξf(y)(y-v)jj!∂vjΦ*(v)dy=(-i)jj!∫ℝ6‍ei(v-y)ξ∂ξjMk(ξ)dξf(y)∂vjΦ*(v)dy=(-i)jj!(ℱ-1∂ξjMk*f)(v)·∂vjΦ*(v),Γk+6f=∫ℝ6‍ei(v-y)ξMk(ξ)dξf(y)(y-v)k+6(k+6)!∂vk+6Φ*(c)dy=(-i)k+6(k+6)!∫ℝ6‍ei(v-y)ξ∂ξk+6Mk(ξ)dξf(y)∂vk+6Φ*(c)dy.
From Lemma 2.1, Remark 2.2, (3.6), and (3.7), it follows that
(5.5)∑j=1k‍∥Γjf(v)∥L2=∑j=1k‍∥(-i)jj!(ℱ-1∂ξjMk*f)(v)·∂vjΦ*(v)∥L2≤∑j=1k‍1j!∥∂ξjMk(ξ)ℱf(ξ)∥L2·∥∂vjΦ*(v)∥L∞≤C·∑j=1k‍16jk⋯(k-j+1){(k-j)!}s·C0k-j+1≤C·C0k+1{(k-1)!}sk2≤C1·C0k+1(k!)s,∑j=k+1k+5‍∥Γjf(v)∥L2≤∑j=k+1k+5‍1j!∥∂ξjMk(ξ)ℱf(ξ)∥L2·∥∂vjΦ*(v)∥L∞≤C·(k+5)!supt∈(0,T]∥f(t,v)∥L2≤C2·C0k+1(k!)s,∥Γk+6f∥L2=∥(-i)k+6(k+6)!∫ℝ6‍ei(v-y)ξ∂ξk+6Mk(ξ)dξf(y)∂vk+6Φ*(c)dy∥L2≤1(k+6)!∥∫ℝ6‍|∂ξk+6Mk(ξ)|dξ·|f(y)|·|∂vk+6Φ*(c)|dy∥L2≤C·C0k+1(k!)s∫ℝ3‍(1+|ξ|2)-3dξ≤C3·C0k+1(k!)s.
Combining (5.5), we complete the proof of Proposition 3.1.Proof of Proposition3.2.
One has(5.6)∥∇v[Mk(Dv),Φ*]f(t,v)∥L2≤C·∥〈|ξ|〉ℱ([Mk,Φ*]f)(t,ξ)∥L2≤C·∥[〈Dv〉Mk,Φ*]f(t,v)-[〈Dv〉,Φ*]Mkf(t,v)∥L2.
Similar to the proof of Proposition3.1, we obtain
(5.7)∥[〈Dv〉Mk,Φ*]f(t,v)∥L2≤C{(k+1)∥Mkf(t,v)∥L2+C0k+1(k!)s},∥[〈Dv〉,Φ*]Mkf(t,v)∥L2≤C{∥Mkf(t,v)∥L2+C0k+1(k!)s}.
Then (3.9) is obtained. The proof of (3.10) is similar so is omitted. This completes the proof of Proposition 3.2.



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*Source: 121302-2012-12-13.xml*