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Shock profiles for the non-cutoff Boltzmann equation with hard potentials

Dominic Wynter

Abstract

The Boltzmann equation models gas dynamics in the low density or high Mach number regime, using a statistical description of molecular interactions. Planar shock wave solutions have been constructed for the Boltzmann equation for hard potentials with angular cutoff, and more recently for the Landau equation of plasma dynamics. In this work, we construct shock profile solutions for the Boltzmann equation where the molecular interactions are long-range, and we show these solutions to be smooth and well approximated by compressible Navier Stokes shock profiles. Our proof procedes by standard energy estimates and a quantitative Chapman-Enskog approximation.

Shock profiles for the non-cutoff Boltzmann equation with hard potentials

Abstract

The Boltzmann equation models gas dynamics in the low density or high Mach number regime, using a statistical description of molecular interactions. Planar shock wave solutions have been constructed for the Boltzmann equation for hard potentials with angular cutoff, and more recently for the Landau equation of plasma dynamics. In this work, we construct shock profile solutions for the Boltzmann equation where the molecular interactions are long-range, and we show these solutions to be smooth and well approximated by compressible Navier Stokes shock profiles. Our proof procedes by standard energy estimates and a quantitative Chapman-Enskog approximation.

Paper Structure

This paper contains 17 sections, 48 theorems, 497 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Maxwellian equilibrium manifold
  4. Strategy of the proof
  5. Estimates on the collision operator
  6. Kawashima compensators
  7. The Chapman-Enskog approximation
  8. Fixed point argument
  9. Microscopic energy estimates
  10. Macroscopic stability estimates
  11. Galerkin approximation
  12. Discretizing the Maxwellian manifold
  13. Discretized stability estimates
  14. Eigenvalue computations
  15. Existence for the linearized problem
  16. ...and 2 more sections

Key Result

Theorem 1.1

Let $\epsilon\in(0,\epsilon_0)$ for $\epsilon_0=\epsilon_0(k)>0$ sufficiently small, let assumptions Hypothesis1, Hypothesis2, and Hypothesis3 hold on the collision kernel, and let $(\bar{\rho}_{\mathrm{NS}},\bar{u}_{\mathrm{NS}},\bar{T}_{\mathrm{NS}})(x-\mathsf{s} t)$ be a shock solution to the com and shock speed $\mathsf{s} = \mathsf{s}_3(\epsilon)$. Write $M_{\mathrm{NS}} = M_{(\bar{\rho}_{\ma

Theorems & Definitions (94)

  • Theorem 1.1
  • Lemma 2.1
  • proof
  • Proposition 2.2
  • Lemma 2.3
  • proof
  • Corollary 2.4
  • proof
  • Lemma 2.5
  • proof
  • ...and 84 more