Center manifolds for a class of degenerate evolution equations and existence of small amplitude kinetic shocks

TL;DR

This paper constructs center manifolds for degenerate evolution equations like the Boltzmann equation, demonstrating the existence and decay properties of small-amplitude kinetic shocks and boundary layers using dynamical systems techniques.

Contribution

It introduces a novel approach to center manifold construction for degenerate kinetic equations, including decay analysis and boundary layer existence, with modifications for ill-posedness.

Findings

Center manifolds exist for the steady Boltzmann and related kinetic equations.

Elements of the center manifold decay in velocity at near-Maxwellian rates.

The analysis aligns with the Chapman-Enskog and Grad moment approximations.

Abstract

We construct center manifolds for a class of degenerate evolution equations including the steady Boltzmann equation and related kinetic models, establishing in the process existence and behavior of small-amplitude kinetic shock and boundary layers. Notably, for Boltzmann's equation, we show that elements of the center manifold decay in velocity at near-Maxwellian rate, in accord with the formal Chapman-Enskog picture of near-equilibrium ow as evolution along the manifold of Maxwellian states, or Grad moment approximation via Hermite polynomials in velocity. Our analysis is from a classical dynamical systems point of view, with a number of interesting modifications to accommodate ill-posedness of the underlying evolution equation.