Instantaneous smoothing and exponential decay of solutions for a degenerate evolution equation with application to Boltzmann's equation

TL;DR

This paper proves that solutions to certain degenerate kinetic equations, including the Boltzmann equation, become instantly smooth and decay exponentially, solving key open problems in the stability analysis of these equations.

Contribution

It establishes the instantaneous smoothing and exponential decay of solutions for degenerate evolution equations in kinetic theory, addressing open problems in the stability of the Boltzmann equation.

Findings

Solutions become infinitely differentiable for positive times.

Solutions decay exponentially over time.

Addresses open problems in the stability analysis of kinetic equations.

Abstract

We establish an instantaneous smoothing property for decaying solutions on the half-line $(0,+\infty)$ of certain degenerate Hilbert space-valued evolution equations arising in kinetic theory, including in particular the steady Boltzmann equation. Our results answer the two main open problems posed by Pogan and Zumbrun in their treatment of $H^1$ stable manifolds of such equations, showing that $L^2_{loc}$ solutions that remain sufficiently small in $L^\infty$ (i) decay exponentially, and (ii) are $C^\infty$ for $t>0$, hence lie eventually in the $H^1$ stable manifold constructed by Pogan and Zumbrun