Local Gevrey Regularity for Linearized Homogeneous Boltzmann Equation

Shi-you Lin School of Mathematics and Statistics, Hainan Normal University, Hainan, Haikou 571158, China Correspondence should be addressed to Shi-you Lin, linsy1111@yahoo.cn Received 16 August 2012; Accepted 14 November 2012 Academic Editor: Ti-Jun Xiao Copyright q 2012 Shi-you Lin. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 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.


Introduction
This paper focuses on the Gevrey class smoothing property of solutions of the following linear Cauchy problems of the spatially homogeneous Boltzmann equation: where the initial datum f 0 / ≡ 0 satisfies the natural boundedness on mass, energy, and entropy: Q g, f is the Boltzmann quadratic operator which has the following form: where σ ∈ S 2 unit sphere of R 3 ; the post-and precollisional velocities are given as follows: 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 μ is called the normalized Maxwellian distribution in 1.1 .Notice that Q μ, μ ≡ 0. Recall that the inverse power law potential 1/ρ s , where s > 1, and ρ denotes the distance between two particles, has the form 1.5 with the corresponding kinetic factors: 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: 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: 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 datum f
For the definition of the Gevrey class functions, compare 1-5 .
or equivalently, where Notice that G 1 R 3 is the usual analytic function space.When 0 < s < 1, we call G s R 3 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: 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 factor Φ 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.
Then for any t ∈ 0, T , there exists a number s s t > 3 satisfying f t, • ∈ G s W .More precisely, for any fixed 0 < t 0 ≤ 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,

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.Suppose Φ v |v| γ 1 |v| 2 γ/2 where γ ∈ 0, 1 , v ∈ R n , and n ∈ N. Then the kth order derivative of Φ satisfies Proof.Without loss of generality, we only consider the case of n 1; the other cases are similar.By direct calculation, we have

2.2
In addition,

2.3
Thus we obtain and then we will prove the following inequality: The inequality is obviously true for m 1. Suppose it is valid for 1 ≤ m ≤ M, then Journal of Function Spaces and Applications which proves 2.5 by induction.Therefore, we have

2.7
The case of 2m 1 th order derivative is similar.This completes the proof of Lemma 2.1.
Setting M N ξ 1 |ξ| 2 Nt/2 for any ξ ∈ R 3 and N ∈ N, by using the similar technique of Lemma 2.1, we conclude the following. where

Lemma 2.3.
There exists a constant C such that for any k ∈ N, where μ is the absolute Maxwellian distribution in 1.1 .
Proof.Without loss of generality, we also only consider the case in the real space R 1 .Putting

2.11
Therefore, fixed a number m ≥ 0, together with the following assumption F m :

2.13
This completes the proof of Lemma 2.3 by induction. Setting where v is belong to a bounded set U. Then we state Lemma 2.4 as below.

Lemma 2.4.
There exists a constant C C U > 0, which satisfies that for any k ∈ N, , and the fact that when |v| ≥ 1, by using Lemma 2.3, we have

2.17
This completes the proof of Lemma 2.4.
By applying the Cauchy integral theorem, we will prove the helpful estimates as follows.
Lemma 2.5.Suppose the Fourier transform for v * , where μ is the absolute Maxwellian distribution in 1.1 .Then we have Proof.First we consider the case of n 1,

2.20
where z v * iξ, and C denotes the curve:

2.21
Now we turn to consider the case of n 3. Letting v v 1 , v 2 , v 3 , and v * v * 1 , v * 2 , v * 3 and using the previous result, we have dv * .

2.22
Thus we conclude the result of Lemma 2.5.
Lemma 2.6.For the expression of h v, ξ in Lemma 2.5, we have
Proof.The first inequality is obvious.To prove the third one, set ξ ξ 1 , ξ 2 , ξ 3 .Since proceeding as in the proof of Lemma 2.5, we can get

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Journal of Function Spaces and Applications Therefore, • −iv * i dv *

2.27
By the mean value theorem of differentials, we have

2.28
where θ arc cos 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
On the other hand,

2.30
Combining with the above expressions of h v, ξ and ∂ ξ i h v, ξ , we get

2.32
This completes the proof of the second inequality.

2.34
In 1 , it is shown that

2.35
Together with Lemma 2.6, we have

2.36
Now we turn to estimate the terms in I 01 and I 03 .For the case 0 < ν < 1 in 1.7 , it is easy to see that

2.37
Therefore, applying the above estimates and Lemma 2.6, we also conclude that for any i ∈ {1, 3}.This completes the proof of Lemma 2.7.

Related Analysis
Let f be the weak solution of the Cauchy problem 1.1 .For any k ∈ N, the compact support is also a weak solution of the following equation

3.3
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 , Thus, for any p ≥ 0,

3.5
Together with Lemma 2.6, we can get the fact that f ∈ H ∞ R 3 .This proof is similar as the proof of 11, Theorem 1.1 and hence omitted.Clearly, ||f|| H r R 3 ||f|| H r U .Moreover, without loss of generality, we restrict T ≤ 1, then for any k ∈ N, it is assumed that where C 0 is a sufficiently large constant satisfying In the following discussion, we will use C and C i , i ∈ N to denote the positive constants independent of k and t.
In order to prove Theorem 1.3, we need the propositions as below.

Proposition 3.1. One has
3.8 Proposition 3.2.One has where H * is the function which has the form 2.14 .
The proof of the above propositions will be given in Section 5.

Proof of Theorem 1.3
Now we will prove the main result in this section.For any t ∈ 0, T , we state the following identity from 11 : where

4.4
Here H * is the function which has the form 2.14 .It is clear that

4.5
By the hypothesis 3.4 , f has compact support in U, we obtain This, together with 4.5 -4.6 , implies The cancellation lemma gives cf. 10, 11 where S is a constant function.Therefore, whose Jacobian is bounded uniformly for v * , σ, τ see 11 , we have

4.13
Moreover, by 11, Lemma 2.2 and 11, page 467 , we have where C μ,1 and C μ,2 are the constants depending only on μ.Therefore, by 3.4 and 4.14 , we get

4.15
Together with 4.13 , we thus have

4.16
Let M 2 k f be the test function in the Cauchy problem 1.1 , for any t ∈ 0, T , we have

4.19
The Young's inequality gives

4.21
Taking 4.21 into 4.19 , and applying the assumption E k , we have

4.23
In other words, it follows from E k that Taking the same procedures as above, we can also gain E k 2 from E k 1 , which is described as below:

4.26
Let C 12 C 0 • C 11 , we thus conclude that for any k ∈ N,

4.27
For any fixed number 0 < t ≤ T ≤ 1, suppose that where 1/t denotes the smallest integer bigger than 1/t.With a convention that k!

4.29
This, together with 4.27 , implies that where k ∈ N, and C 16 is a constant only depending on t.Furthermore, for any fixed number t 0 > 0, put

4.31
Then for any k ∈ N, we can choose C 13 2 ss 1 C 12 and have the fact that This completes the proof of Theorem 1.3.

Proof of Propositions 3.1 and 3.2
Proof of Proposition 3.1.We first notice that e i v−y ξ M k ξ dξf y Φ * y − Φ * v dy.

5.1
Using the Taylor formula of order k 6, we get for some c ∈ y, v .Hence, where

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Journal of Function Spaces and Applications

5.5
Combining 5.5 , we complete the proof of Proposition 3.1.
Proof of Proposition 3.2.One has

5.6
Similar to the proof of Proposition 3.1, we obtain

5.7
Then 3.9 is obtained.The proof of 3.10 is similar so is omitted.This completes the proof of Proposition 3.2.

Theorem 1 . 3 .
Suppose Φ, b have the forms in 1.7 , 0 < ν < 1.Let W be a bounded open set of R 3 , and f t, v be the weak solution of the Cauchy problem 1.1 satisfying