Journal of Function Spaces and Applications

Volume 2012 (2012), Article ID 121302, 23 pages

doi:10.1155/2012/121302

Research Article

Local Gevrey Regularity for Linearized Homogeneous Boltzmann Equation

Shi-you Lin

School of Mathematics and Statistics, Hainan Normal University, Hainan, Haikou 571158, China

Received 16 August 2012; Accepted 14 November 2012

Academic Editor: Ti-Jun  Xiao

Copyright © 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.

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.

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:

where the initial datum satisfies the natural boundedness on mass, energy, and entropy:

is the Boltzmann quadratic operator which has the following form:

where (unit sphere of ); the post- and precollisional velocities are given as follows:

The Boltzmann collision cross-section is a nonnegative function which depends only on and the scalar product . To capture its main properties, we usually assume

is called the normalized Maxwellian distribution in (1.1). Notice that .

Recall that the inverse power law potential , where , and denotes the distance between two particles, has the form (1.5) with the corresponding kinetic factors:

for a constant and . The cases , and 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 and the strong singularity when . 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 and 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 , is called a weak solution of the Cauchy problem (1.1) if it satisfies

for any test function .

For the definition of the Gevrey class functions, compare [1–5].

Definition 1.2. Suppose that is a bounded open set on , for , which is the Gevrey class function space with index , if and for any compact subset , there exists a constant such that for any ,

or equivalently,

where

Particularly, , that is, , is equivalent to the fact that there exists such that .

Notice that is the usual analytic function space. When , we call 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 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. Suppose , have the forms in (1.7), . Let be a bounded open set of , and be the weak solution of the Cauchy problem (1.1) satisfying

Then for any , there exists a number satisfying . More precisely, for any fixed and compact subset , there exists a constant and a number such that for any ,

From Theorem 1.3, we have the following remark.

Remark 1.4. Suppose that , have the forms in (1.7), . If the weak solution satisfies that

then for any , any bounded open set , there exists a constant satisfying .

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. Suppose where , and . Then the th order derivative of satisfies

Proof. Without loss of generality, we only consider the case of ; the other cases are similar. By direct calculation, we have

In addition,

Thus we obtain

and then we will prove the following inequality:

The inequality is obviously true for . Suppose it is valid for , then

which proves (2.5) by induction. Therefore, we have

The case of th order derivative is similar. This completes the proof of Lemma 2.1.

Setting for any and , by using the similar technique of Lemma 2.1, we conclude the following.

Remark 2.2. For ,

where .

Lemma 2.3. There exists a constant such that for any ,

where is the absolute Maxwellian distribution in (1.1).

Proof. Without loss of generality, we also only consider the case in the real space . Putting

Evidently,

Therefore, fixed a number  , together with the following assumption :

we can obtain

This completes the proof of Lemma 2.3 by induction.

Setting

where is belong to a bounded set . Then we state Lemma 2.4 as below.

Lemma 2.4. There exists a constant , which satisfies that for any ,

Proof. Since , and the fact that when ,

by using Lemma 2.3, we have

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 ,

where is the absolute Maxwellian distribution in (1.1). Then we have

Proof. First we consider the case of ,

where , and denotes the curve: . By Cauchy integral theorem [9], it follows that

Now we turn to consider the case of . Letting , and and using the previous result, we have

Thus we conclude the result of Lemma 2.5.

Lemma 2.6. For the expression of in Lemma 2.5, we have

where , and is a constant independent of and .

Proof. The first inequality is obvious. To prove the third one, set . Since

proceeding as in the proof of Lemma 2.5, we can get

Therefore,

which implies that

By the mean value theorem of differentials, we have

where . Thus the third inequality has been obtained.

Finally, the above way can also be used in estimating the second one. Similarly,

On the other hand,

Combining with the above expressions of and , we get

Therefore,

This completes the proof of the second inequality.

Lemma 2.7. Suppose that in (1.7). Then for any , there exists a constant independent of satisfying

Proof. Let , from Bobylev’s formula (see [10]), we have

In [1], it is shown that

Together with Lemma 2.6, we have

Now we turn to estimate the terms in and . For the case in (1.7), it is easy to see that

Therefore, applying the above estimates and Lemma 2.6, we also conclude that

for any . This completes the proof of Lemma 2.7.

3. Related Analysis

Let be the weak solution of the Cauchy problem (1.1). For any , the compact support

which implies that for any compact subset ,

is also a weak solution of the following equation

Since Theorem 1.3 is mainly concerned with the Gevrey smoothness property of the solution on , we need only to study the solution of the above equation on any fixed compact subset of . That is, we can suppose that has compact support in for any ,

Thus, for any ,

Together with Lemma 2.6, we can get the fact that . This proof is similar as the proof of [11, Theorem 1.1] and hence omitted. Clearly, . Moreover, without loss of generality, we restrict , then for any , it is assumed that

where is a sufficiently large constant satisfying

In the following discussion, we will use and to denote the positive constants independent of and . Let and . In order to prove Theorem 1.3, we need the propositions as below.

Proposition 3.1. One has

Proposition 3.2. One has

where is the function which has the form (2.14).

The proof of the above propositions will be given in Section 5.

4. Proof of Theorem 1.3

Now we will prove the main result in this section. For any , we state the following identity from [11]:

where

Our purpose is to obtain the estimations of , and . Setting and , since and , applying the mean value theorem and the fact that , we have

where is a number between an . Therefore,

Here is the function which has the form (2.14). It is clear that

By the hypothesis (3.4), has compact support in , we obtain

Here we use the fact that . By (3.10) of Proposition 3.2, we get

This, together with (4.5)-(4.6), implies

The cancellation lemma gives (cf. [10, 11])

where is a constant function. Therefore,

Since , by using (3.4), Proposition 3.2, and the change of variables

whose Jacobian is bounded uniformly for (see [11]), we have

Combining (4.8), (4.10), and (4.12), we obtain

Moreover, by [11, Lemma 2.2] and [11, page 467], we have

where and are the constants depending only on . Therefore, by (3.4) and (4.14), we get

Together with (4.13), we thus have

Let be the test function in the Cauchy problem (1.1), for any , we have

Since

by Lemma 2.7 and (4.16), it holds that

The Young’s inequality gives

which implies

Taking (4.21) into (4.19), and applying the assumption , we have

which implies that

In other words, it follows from that

Taking the same procedures as above, we can also gain from , which is described as below:

that is,

Let , we thus conclude that for any ,

For any fixed number , suppose that

where denotes the smallest integer bigger than . With a convention that if , we have

This, together with (4.27), implies that

where , and is a constant only depending on . Furthermore, for any fixed number , put

Then for any , we can choose and have the fact that

This completes the proof of Theorem 1.3.

5. Proof of Propositions 3.1 and 3.2

Proof of Proposition 3.1. We first notice that

Using the Taylor formula of order , we get

for some . Hence,

where

From Lemma 2.1, Remark 2.2, (3.6), and (3.7), it follows that

Combining (5.5), we complete the proof of Proposition 3.1.

Proof of Proposition 3.2. One has

Similar to the proof of Proposition 3.1, we obtain

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

Acknowledgments

This work was supported by the Tian Yuan Specialized Research Fund for Mathematics (no. 11226167), the Natural Science Foundation of Hainan Province (no. 111005), the Scientific Research Foundation of Hainan Province Education Bureau (no. Hjkj2011-19), and the Ph.D. Scientific Research Starting Foundation of Hainan Normal University (no. HSBS1016).

References

Y. Morimoto, S. Ukai, C.-J. Xu, and T. Yang, “Regularity of solutions to the spatially homogeneous Boltzmann equation without angular cutoff,” Discrete and Continuous Dynamical Systems A, vol. 24, no. 1, pp. 187–212, 2009.

H. Chen, W.-X. Li, and C.-J. Xu, “Gevrey hypoellipticity for linear and non-linear Fokker-Planck equations,” Journal of Differential Equations, vol. 246, no. 1, pp. 320–339, 2009.

H. Chen, W. Li, and C. Xu, “Gevrey regularity for solution of the spatially homogeneous Landau equation,” Acta Mathematica Scientia B, vol. 29, no. 3, pp. 673–686, 2009.

N. Lekrine and C.-J. Xu, “Gevrey regularizing effect of the Cauchy problem for non-cutoff homogeneous Kac's equation,” Kinetic and Related Models, vol. 2, no. 4, pp. 647–666, 2009.

Y. Morimoto and C.-J. Xu, “Ultra-analytic effect of Cauchy problem for a class of kinetic equations,” Journal of Differential Equations, vol. 247, no. 2, pp. 596–617, 2009.

L. Desvillettes, G. Furioli, and E. Terraneo, “Propagation of Gevrey regularity for solutions of the Boltzmann equation for Maxwellian molecules,” Transactions of the American Mathematical Society, vol. 361, no. 4, pp. 1731–1747, 2009.

Y. Morimoto and S. Ukai, “Gevrey smoothing effect of solutions for spatially homogeneous nonlinear Boltzmann equation without angular cutoff,” Journal of Pseudo-Differential Operators and Applications, vol. 1, no. 1, pp. 139–159, 2010.

S. Ukai, “Local solutions in Gevrey classes to the nonlinear Boltzmann equation without cutoff,” Japan Journal of Applied Mathematics, vol. 1, no. 1, pp. 141–156, 1984.

W. Rudin, Real and Complex Analysis, McGraw-Hill Book Co., Singapore, 3rd edition, 1987.

R. Alexandre, L. Desvillettes, C. Villani, and B. Wennberg, “Entropy dissipation and long-range interactions,” Archive for Rational Mechanics and Analysis, vol. 152, no. 4, pp. 327–355, 2000.

Z. Huo, Y. Morimoto, S. Ukai, and T. Yang, “Regularity of solutions for spatially homogeneous Boltzmann equation without angular cutoff,” Kinetic and Related Models, vol. 1, no. 3, pp. 453–489, 2008.