Regularizing effect and local existence for non-cutoff Boltzmann equation

TL;DR

This paper demonstrates the regularizing effect of the non-cutoff Boltzmann equation in all variables and proves local existence of smooth solutions under mild initial regularity assumptions.

Contribution

It establishes the regularizing effect and local existence of smooth solutions for the spatially inhomogeneous non-cutoff Boltzmann equation, extending previous results.

Findings

Proved regularization in all variables for solutions.

Established local existence of smooth solutions.

Demonstrated regularity transfer from initial data to solutions.

Abstract

The Boltzmann equation without Grad's angular cutoff assumption is believed to have regularizing effect on the solution because of the non-integrable angular singularity of the cross-section. However, even though so far 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 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 Maxwellian type decay in velocity variable, there exists a unique local solution with the same regularity, so that this solution enjoys the $C^\infty$ regularity for positive time.