Integral and Differential Systems in Function Spaces and Related ProblemsView this Special Issue
Local Gevrey Regularity for Linearized Homogeneous Boltzmann Equation
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.
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 .
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 . 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  that, for the Maxwellian molecules model, the Gevrey regularity can propagate globally in time. Other results for the nonlinear case can be found in , where the Gevrey regularity of the radially symmetric weak solutions has been proved. Meanwhile, this issue is also considered in  for the Maxwellian decay solutions. For the linear case, the best result so far is obtained by the work of Morimoto et al. in ; 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 , 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 , 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
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 ), we have In , 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 : 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 ), 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