Gevrey smoothing for weak solutions of the fully nonlinear homogeneous Boltzmann and Kac equations without cutoff for Maxwellian molecules

Abstract

It has long been suspected that the non-cutoff Boltzmann operator has similar coercivity properties as a fractional Laplacian. This has led to the hope that the homogenous Boltzmann equation enjoys similar regularity properties as the heat equation with a fractional Laplacian. In particular, the weak solution of the fully nonlinear non-cutoff homogenous Boltzmann equation with initial datum in $L_2^1({\mathbb{R}}^d)\cap L\log L({\mathbb{R}}^d)$, i.e., finite mass, energy and entropy, should immediately become Gevrey regular for strictly positive times. We prove this conjecture for Maxwellian molecules.

arXiv comment43 pages, 1 figure

References in corpus (5)

• Regularity theory for the spatially homogeneous Boltzmann equation with cut-off
• Regularizing effect and local existence for non-cutoff Boltzmann equation
• Smoothing effect of weak solutions for the spatially homogeneous Boltzmann Equation without angular cutoff
• Analytical regularizing effect for the radial and spatially homogeneous Boltzmann equation
• Strong smoothing for the non-cutoff homogeneous Boltzmann equation for Maxwellian molecules with Debye-Yukawa type interaction

Cited by in corpus (5)

• Regularity of the Vlasov-Poisson-Boltzmann System without angular cutoff
• Regularity of Non-cutoff Boltzmann Equation with Hard Potential
• Strong smoothing for the non-cutoff homogeneous Boltzmann equation for Maxwellian molecules with Debye-Yukawa type interaction
• Infinite order $\Psi$DOs: Composition with entire functions, new Shubin-Sobolev spaces, and index theorem
• Global Existence of Non-cutoff Boltzmann Equation in Weighted Sobolev Space