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˖ڍመ᝶஠ڙጲ

http://www.paper.edu.cn

无截断空间齐次 Boltzmann 方程在软位势与

温和及临界奇异情形的 Gevrey 正则性

张腾飞，殷朝阳

中山大学数学与计算科学学院，广州，510275

摘要：本文研究了无截断的空间齐次 Boltzmann 方程于软位势下的 Gevrey 正则性, 考虑了温

和奇异性 s < 1/2 与临界奇异性 s = 1/2 的情形. 我们得到了一个修正的强制估计, 相较于我们

之前的工作, 推广了 γ 在软位势时的范围至 γ ∈ (−5/2, 0). 同时, 我们将 γ 与 s 的奇性作分离

讨论, 而不再将 γ + 2s 作为一个整体考虑.

关键词：无截断 Boltzmann 方程; 空间齐次; Gevrey 正则性; 软位势; 温和及临界奇异性.

中图分类号： O175.29

Gevrey Smoothing Eﬀect of Solutions to Non-Cutoﬀ

Boltzmann Equation for Soft Potential with Mild and

Critical Singularity

ZHANG Teng-Fei, YIN Zhaoyang

Department of Mathematics, Sun Yat-sen University, Guangzhou, 510275

Abstract: In this paper we study the Gevrey smoothing eﬀect of solutions to the non-cutoﬀ

spatially homogeneous Boltzmann equation for soft potential. We consider not only the mild

singularity case s < 1/2 as we did in the previous works for spatially homogeneous case and

for spatially inhomogeneous case, but also the critical singularity case s = 1/2 (with a

particular soft potential γ = −2). Besides, we try to extend the range of γ. We derive a new

coercivity estimate for collision operator, from which we obtain the propagation of Gevrey

regularity for γ ∈ (−5/2, 0), and Gevrey regularity for γ ∈ [−2, 0), which improve the previous

assumption γ ∈ (−1 − 2s, 0). In addition, we consider γ and s separately instead of viewing

γ + 2s as one untied quantity.

Key words: Non-cutoﬀ Boltzmann equation; Spatially homogeneous; Gevrey regularity; Soft

potential; Mild and critical singularity.

基金项目： NNSFC (No. 11271382 and No. 10971235), RFDP (No. 20120171110014), and the key project of Sun

Yat-sen University (No. c1185).

作者简介： ZHANG Teng-Fei (1986-), Male, Ph.D candidate. Major research direction: Nonlinear Partial D-

iﬀerential Equations. Corresponding author. YIN Zhaoyang (1968-), Male, Professor. Major research direction:

Nonlinear Partial Diﬀerential Equations.

- 1 -

˖ڍመ᝶஠ڙጲ

http://www.paper.edu.cn

0 Introduction

0.1 The Boltzmann equation

In this paper we consider the Cauchy problem of the non-cutoﬀ Boltzmann equation. First

we introduce the Cauchy problem of the full (or, spatially inhomogeneous) Boltzmann equation

without angular cutoﬀ, with a T > 0,







(t, x, v) + v · ∇

f(t, x, v) = Q(f, f), t ∈ (0, T ], x ∈ T

, v ∈ R

f(0, x, v) = f

(x, v).

(0.1)

Above, f = f (t, x, v) describes the density distribution function of particles located around

position x ∈ T

with velocity v ∈ R

at time t ≥ 0. The right-hand side of the ﬁrst equation is

the so-called Boltzmann bilinear collision operator acting only on the velocity variable v:

Q(g, f) =

B (v − v

∗

, σ) {g

∗

− g

∗

f} dσdv

∗

Note that we use the well-known shorthand f = f(t, x, v), f

∗

= f(t, x, v

∗

), f

= f(t, x, v

∗

= f(t, x, v

∗

) throughout this paper.

In this paper we consider the Cauchy problem of the Boltzmann equation in the spatially

homogeneous case, that is, for a T > 0,







(t, v) = Q(f, f)(v), t ∈ (0, T ], v ∈ R

f(0, v) = f

(v),

(0.2)

where “spatially homogeneous” means that f depends only on t and v.

By using the σ-representation, we can describe the relations between the post- and pre-

collisional velocities as follows, for σ ∈ S

v + v

∗

|v − v

∗

σ, v

∗

v + v

∗

−

|v − v

∗

σ.

We point out that the collision process satisﬁes the conservation of momentum and kinetic

energy, i.e.

v + v

∗

= v

+ v

∗

, |v|

+ |v

∗

= |v

+ |v

∗

The collision cross section B(z, σ) is a given non-negative function depending only on the

interaction law between particles. From a mathematical viewpoint, that means B(z, σ) depends

only on the relative velocity |z| = |v − v

∗

| and the deviation angle θ through the scalar product

cos θ =

|z|

· σ.

The cross section B is assumed here to be of the type:

B(v − v

∗

, cos θ) = Φ(|v − v

∗

|)b(cos θ), cos θ =

v − v

∗

|v − v

∗

· σ, 0 ≤ θ ≤

- 2 -

˖ڍመ᝶஠ڙጲ

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Above, Φ stands for the kinetic factor which is of the form:

Φ(|v − v

∗

|) = |v − v

∗

and b denotes the angular part with a singularity such that,

sin θb(cos θ) ∼ Kθ

−1−2s

, as θ → 0+,

for some positive constant K and 0 < s < 1.

We remark that if the inter-molecule potential satisﬁes the inverse-power law U(ρ) =

−(p−1)

(where p > 2), it holds γ =

p−5

p−1

, s =

p−1

. Generally, the cases γ > 0, γ = 0, and

γ < 0 correspond to so-called hard, Maxwellian, and soft potential respectively. And the cases

0 < s < 1/2, 1/2 ≤ s < 1 correspond to so-called mild singularity and strong singularity

respectively.

As is discussed by Desvillettes in [1] (also, by Villani in his handbook [2]), the most

interesting assumption for γ is included in (−3, 1). Further, we announce that we will focus on

the mild singularity case 0 < s < 1/2 for soft potential γ ∈ (−5/2, 0).

0.2 Review of related references

Now we give a brief review about some related researches. Firstly we refer the reader

to Villani’s review book [2] for the physical background and the mathematical theories of the

Boltzmann equation. And for more details about the non-cutoﬀ theories, we refer to Alexandre’s

review paper [3].

Before continuing the statement, we introduce the deﬁnition of Gevrey spaces G

(Ω) where

Ω is an open subset of R

. (It could be found in many references, e.g. [5, 18].)

Deﬁnition 1. For 0 < s < +∞, we say that f ∈ G

(Ω), if f ∈ C

∞

(Ω), and there exist

C > 0, N

> 0 such that

k∂

(Ω)

≤ C

|α|+1

{α!}

, ∀α ∈ N

, |α| ≥ N

If the boundary of Ω is smooth, by using the Sobolev embedding theorem, we have the same

type estimate with L

norm replaced by any L

norm for 2 < p ≤ +∞.

When s = 1, it is usual analytic function. If s > 1, it is Gevrey class function. And for

0 < s < 1, it is called ultra-analytic function.

In 1984 Ukai showed in [6] that there exists a unique local solution to the Cauchy problem

for the Boltzmann equation in Gevrey classes for both spatially homogeneous and inhomoge-

neous cases, under the assumption on the cross section:



B(|z|, cos θ)



≤ K(1 + |z|

−γ

+ |z|

)θ

−n+1−2s

, n is dimensionality,

- 3 -

˖ڍመ᝶஠ڙጲ

http://www.paper.edu.cn

(0 ≤ γ

< n, 0 ≤ γ < 2, 0 ≤ s < 1/2, γ + 6s < 2).

By introducing the norm of Gevrey space

kfk

δ,ρ,ν

|α|

{α!}

δhvi

∂

∞

)

Ukai proved that in the spatially homogeneous case, for instance, under some assumptions

for ν and the initial datum f

(v), the Cauchy problem (0.2) has a unique solution f (t, v) for

t ∈ (0, T ].

In [7] Desvillettes studied ﬁrstly the C

∞

smoothing eﬀect for solutions of Cauchy problem

in spatially homogeneous non-cutoﬀ case, and conjectured Gevrey smoothing eﬀect. And he

proved in [8] the propagation of Gevrey regularity for solutions without any assumptions on

the decay at inﬁnity in v variables.

In 2009 Morimoto et al. considered in [9] the Gevrey regularity for the linearized Boltzmann

equation around the absolute Maxwellian distribution, by virtue of the following molliﬁer:

(t, D

) =

thD

1/ν

1 + δe

thD

1/ν

, 0 < δ < 1.

We remark that the same operator was used in many related researches and models such as

the Kac’s equation (a simpliﬁcation of Boltzmann equation to one dimension case), the ultra-

analytic smoothing eﬀect for spatially homogeneous nonlinear Landau equation and the linear

and non-linear Fokker-Planck equations.

In the mild singularity case 0 < s < 1/2, Huo et al. proved in [10] that any weak solution

f(t, v) to the Cauchy problem (0.2) satisfying the natural boundedness on mass, energy and

entropy, namely,

f(v)[1 + |v|

+ log(1 + f(v))]dv < +∞, (0.3)

belongs to H

+∞

) for any 0 < t ≤ T , and moreover,

f ∈ L

∞



, T ]; H

+∞

)



, (0.4)

for any T > 0 and t

∈ (0, T ).

In the framework of small perturbation of an equilibrium state, Alexandre et al. studied the

Cauchy problem of the Boltzmann equation for soft and hard potential (see [11, 12]), and obtain

the global existence of solution in weighted Sobolev spaces. Some other results about existence

of perturbative solutions for the cutoﬀ Boltzmann equation are due to Guo (for instance, see

[13]) around a global Maxwellian, and Tai-Ping Liu (compare [14, 15], for example), and so on.

In [16] the ﬁve authors considered a kind of solution having the Maxwellian decay, based

on which we introduce the following deﬁnition:

- 4 -

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Deﬁnition 2. We say that f (t, v) is a smooth Maxwellian decay solution to the Cauchy problem

(0.2) if







f ≥ 0, 6≡ 0,

∃ δ

> 0 such that e

hvi

f ∈ L

∞

([0, T ]; H

+∞

)) .

(Note that the Theorem 1.2 of [16] shows the uniqueness of the smooth Maxwellian decay

solution to the Cauchy problem (0.2).)

The ﬁve authors also proved in [16] the smoothing eﬀect on the solutions with weight. In

detail, if the non-negative f belongs to H



, t

)×Ω×R



, solves the spatially inhomogeneous

Boltzmann equation (0.1) in this domain in the classic sense, and satisﬁes the non-vacuum

condition kf(t, x, v)k

)

> 0, then

f ∈ H

+∞



, t

) × Ω × R



hence it follows that,

f ∈ C

∞



, t

) × Ω; S(R

)



In 2010 Morimoto-Ukai considered in [17] the Gevrey regularity of C

∞

solutions with

the Maxwellian decay to the Cauchy problem of spatially homogeneous Boltzmann equation.

Motivated by their idea, we considered the problem in [18] for a more general case. More

precisely, we considered the general kinetic factor Φ(|v|) = |v|

instead of the moderate form

hvi

= (1 + |v|

)

γ/2

in [17], and a wider range of the parameter of γ (s.t. γ + 2s ∈ (−1, 1)) so

as to ﬁt for both hard potential and soft potential.

In the ensuing paper [19] we studied still the general case Φ(|v|) = |v|

with γ+2s ∈ (−1, 1)

in the mild singularity case, but the spatially inhomogeneous case. We obtain a corresponding

result about the Gevrey regularity.

In this present paper, we resume to the spatially homogeneous case, and try to extend

the range of γ for soft potential (in γ ≥ 0 we make no change). For the propagation of

Gevrey smoothing eﬀect, we consider the case γ ∈ (−5/2, 0) here to take place of the previous

assumption γ ∈ (−1 − 2s, 0). (Recall that in [18], we assume γ + 2s ∈ (−1, 1). Note that the

simple inequality −1 − 2s > −2 > −5/2 gives the range of extending.) To prove the order of

Gevrey regularity, we assume further that γ ∈ [−2, 0) (so as to make the extra weight γ/2 to

be no more than 1). We emphasize that we consider not only the mild singularity case s < 1/2,

but also the critical singularity case s = 1/2 (with a particular soft potential γ = −2). It is

generally known that there are few results concerning the strong singularity case s ≥ 1/2, even

if for the Maxwellian case. In addition, γ and s are considered separately instead of viewing

γ + 2s as one united quantity. To achieve the goal, we need a new coercivity estimate for

collision operator diﬀerent from that the authors proved in [20] with respect to the parameter

γ.

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