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

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