Regularizing Effect and Local Existence for the Non-Cutoff Boltzmann Equation |
| |
Authors: | Radjesvarane Alexandre Yoshinori Morimoto Seiji Ukai Chao-Jiang Xu Tong Yang |
| |
Institution: | 1. IRENAV Research Institute, French Naval Academy, 29290, Brest-Lanvéoc, France 2. Graduate School of Human and Environmental Studies, Kyoto University, Kyoto, 606-8501, Japan 3. 17-26 Iwasaki-cho, Hodogaya-ku, Yokohama, 240-0015, Japan 4. Université de Rouen, UMR 6085-CNRS Mathématiques, 76801, Saint-Etienne du Rouvray, France 5. School of Mathematics, Wuhan University, 430072, Wuhan, China 6. Department of Mathematics, City University of Hong Kong, Hong Kong, People’s Republic of China
|
| |
Abstract: | The Boltzmann equation without Grad’s angular cutoff assumption is believed to have a regularizing effect on the solutions
because of the non-integrable angular singularity of the cross-section. However, even though this has been justified satisfactorily
for the spatially homogeneous Boltzmann equation, it is still basically unsolved for the spatially inhomogeneous Boltzmann
equation. In this paper, by sharpening the coercivity and upper bound estimates for the collision operator, establishing the
hypo-ellipticity of the Boltzmann operator based on a generalized version of the uncertainty principle, and analyzing the
commutators between the collision operator and some weighted pseudo-differential operators, we prove the regularizing effect
in all (time, space and velocity) variables on the solutions when some mild regularity is imposed on these solutions. For
completeness, we also show that when the initial data has this mild regularity and a Maxwellian type decay in the velocity
variable, there exists a unique local solution with the same regularity, so that this solution acquires the C
∞ regularity for any positive time. |
| |
Keywords: | |
本文献已被 SpringerLink 等数据库收录! |
|