Regularity of solutions to the spatially homogeneous Boltzmann equation without angular cutoff

Yoshinori Morimoto (1) , Seiji Ukai (2) , Chao-Jiang Xu (3) , Tong Yang (4)

1 Graduate School of Human and Environmental Studies
2 Mr. - retaite
3 LMRS - Laboratoire de Mathématiques Raphaël Salem
4 Pr. - Department of mathematics

Most of the work on the Boltzmann equation is based on the Grad's angular cutoff assumption. Even though the smoothing effect from the singular cross-section without the angular cutoff corresponding to the grazing collision is expected, there is no general mathematical theory especially for the spatially inhomogeneous case. As a further study on the problem in the spatially homogeneous situation, in this paper, we will prove the Gevrey smoothing property of the solutions to the Cauchy problem for Maxwellian molecules without angular cutoff by using pseudo-differential calculus. Furthermore, we apply similar analytic techniques for the Sobolev space regularity to the nonlinear equation, and prove the smoothing property of solutions for the spatially homogeneous nonlinear Boltzmann equation with the Debye-Yukawa potential.

Mots clés: Boltzmann equation; Debye-Yukawa potential; Gevrey hypoellipticity; non-cut0ff vross-sections

Domaines: Equations aux dérivées partielles [math.AP]

Yoshinori Morimoto, Seiji Ukai, Chao-Jiang Xu, Tong Yang. Regularity of solutions to the spatially homogeneous Boltzmann equation without angular cutoff. Discrete and Continuous Dynamical Systems - Series A, 2009, 24 (1), pp.187-212. ⟨hal-00434249⟩