Gevrey Regularity for Solutions of the Non-Cutoff Boltzmann Equation: Spatially Inhomogeneous Case

TL;DR

This paper proves that solutions to the spatially inhomogeneous non-cutoff Boltzmann equation maintain Gevrey regularity in both space and velocity variables, extending previous homogeneous results with more complex analysis.

Contribution

It establishes the propagation of Gevrey regularity for solutions of the inhomogeneous Boltzmann equation, linking velocity and space regularity in a new, comprehensive way.

Findings

Gevrey regularity of order 1/(2s) in velocity

Gevrey regularity of order 1 in space

Extension of homogeneous case results to inhomogeneous case

Abstract

In this paper we consider the non-cutoff Boltzmann equation in spatially inhomogeneous case. We prove the propagation of Gevrey regularity for the so-called smooth Maxwellian decay solutions to the Cauchy problem of spatially inhomogeneous Boltzmann equation, and obtain Gevrey regularity of order $1/(2s)$ in the velocity variable $v$ and order 1 in the space variable $x$. The strategy relies on our recent results for spatially homogeneous case (J. Diff. Equ. 253(4) (2012), 1172-1190. DOI: 10.1016/j.jde.2012.04.023). Rather, we need much more intricate analysis additionally in order to handle with the coupling of the double variables. Combining with the previous result mentioned above, it gives a whole characterization of the Gevrey regularity of the particular kind of solutions to the non-cutoff Boltzmann.