# Regularizing effect and local existence for non-cutoff Boltzmann equation - Mathematics > Analysis of PDEs

Abstract: The Boltzmann equation without Grad-s angular cutoff assumption is believedto have regularizing effect on the solution because of the non-integrableangular singularity of the cross-section. However, even though so far this hasbeen justified satisfactorily for the spatially homogeneous Boltzmann equation,it is still basically unsolved for the spatially inhomogeneous Boltzmannequation. In this paper, by sharpening the coercivity and upper bound estimatesfor the collision operator, establishing the hypo-ellipticity of the Boltzmannoperator based on a generalized version of the uncertainty principle, andanalyzing the commutators between the collision operator and some weightedpseudo differential operators, we prove the regularizing effect in all time,space and velocity variables on solutions when some mild regularity is imposedon these solutions. For completeness, we also show that when the initial datahas this mild regularity and Maxwellian type decay in velocity variable, thereexists a unique local solution with the same regularity, so that this solutionenjoys the $C^\infty$ regularity for positive time.

Author: Radjesvarane Alexandre IRENAV, Y. Morimoto, Seiji Ukai, Chao-Jiang Xu LMRS, Tong Yang

Source: https://arxiv.org/