Gevrey Regularity for the Noncutoff Nonlinear Homogeneous Boltzmann Equation with Strong Singularity

and Applied Analysis 3 A positive integer k is chosen such that kε−m > n/2. For any q > n/2, L1(Rn) ⊆ H(R). By combining this Lemma with (15), the following is obtained: 󵄩󵄩󵄩󵄩f 󵄩󵄩󵄩󵄩 1/2 k−1 H m−kε 2 k l ≤ C 󸀠 ⋅ 󵄩󵄩󵄩󵄩f 󵄩󵄩󵄩󵄩 1/2 k−1

Research on the Gevrey regularity of the Boltzmann equation can be traced back to the work of Ukai [6], who constructed a unique local solution in Gevrey space for both spatially homogeneous and inhomogeneous noncutoff Boltzmann equations.In 2004, Desvillettes and Wennberg [7] gave a conjecture of the Gevrey smoothing effect.Five years later, the propagation of Gevrey regularity for solutions of the nonlinear spatially homogeneous Boltzmann equation with Maxwellian molecules is obtained in [8].In that same year, Morimoto et al. [4] studied linearized cases and proved the Gevrey regularity of solutions without any extra assumption for the initial datum.They then considered the  ∞ solutions with Maxwellian decay in [9]; that is, a positive number  0 exists such that, for any  0 ∈ (0, ), Under the hypotheses of 0 <  < 1/2,  ≥ 0,  + 2 < 1, and the modified kinetic factor Φ(|V|) = (1 + |V| 2 ) /2 , they showed the Gevrey smooth property for this type of solutions to the Cauchy problem of the nonlinear homogeneous Boltzmann equation.By using the original definition of kinetic factor, Zhang and Yin [10] extended the above result in a general framework: 0 <  < 1/2 and −1 <  + 2 < 1.
In this paper, the same issue in the strong singularity case 1/2 <  < 1 is disussed.To discuss this issue properly, some notations are introduced.For any  = ( 1 ,  2 , . . .,   ) ∈ Z  + , V = (V 1 , V 2 , . . ., V  ) and  = ( 1 ,  2 , . . .,   ), the following expression is denoted: For any  ∈ R, let with a convention that ! = 1 if 0 ≥  ∈ Z.For any Instead of the assumption of Maxwellian decay, the smooth solutions (, ⋅) ∈ S(R  ) are considered to satisfy the following inequality (this type of solutions had been studied in some literature.E.g., cf.[11]): For any  ∈ R + , where S(R  ) is the standard Schwartz space and  1 is a fixed constant.For any  ∈ Z  + , A preliminary analysis in Section 2 is conducted and Theorem 2 is proved in Section 3.
The proof procedure of Theorem 3 is proved in Section 4.

Preliminary Analysis
In this section, the lemmas are stated and their proof process is provided.Lemma 5. Let  > 0,  > 0 be two given numbers.Assume that  is a function that satisfies (15).Then, for any fixed number  > 0, a constant  = () exists such that ‖‖ Moreover, if ] ≥ 1 and  > 1 + (2]/(] − 1)), then there exists a constant   depending on ] and  such that, for any  ∈ N, Proof.By using Proposition 3.1 in [9], the following is obtained: This completes the proof of the first inequality.Thereafter, the same analysis technique is applied as the proof of Proposition 3.1 in [9] to discuss the second inequality.Notice that This completes the proof of the second inequality.
By using this lemma with  =  and  =   , the following is obtained: Next it is planned to give an estimation of  2 .By using the conclusion in page 146 of [9] (see also page 1177 of [10]), one has Thus, One refers to the estimation from Proposition 3.6 in [13].

Proof of Theorem 3
In this section, the proof of Theorem 3 is provided.That is, for 1/2 <  < 1 and 0 <  < 1, considering the solution (, ⋅) of the Cauchy problem (1) that satisfies the hypotheses in Theorem 3, one shows that there is a positive number  0 that exists such that (, ⋅) ∈  (58) By integrating the above equation from zero to , the following expression is obtained: By writing one can get where 0 < 2 − 1 < 2 −  <   <  < 1. Considering that the analytical method is quite similar to the one in Section 3, the proof of the above inequality is omitted.Therefore, By using the conclusion in page 157 of [9], provided that  is sufficiently small.Thus, That is, which provides the Gevrey smoothing effect in (0,  0 ).This completes the proof of Theorem 3.