Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Kinetic shock profiles for the Landau equation

Dallas Albritton, Jacob Bedrossian, Matthew Novack

TL;DR

This work constructs small-amplitude kinetic shock profiles for the Landau equation by embedding a Navier–Stokes shock within the kinetic framework. The authors build an approximate solution via a Chapman-Enskog expansion, then perform a detailed linear and nonlinear analysis around the Navier–Stokes profile, employing macro–micro decomposition, Kawashima-type coercivity, and a κ-regularized collision operator to manage moment loss. They prove existence and uniqueness (up to translation) of traveling-wave solutions F(x,v) = F_NS(x,v) + μ0^{1/2} f with Maxwellian end states, and establish precise decay properties in x (stretched exponential for the remainder) and v (Gaussian moments). The approach extends prior Boltzmann-based results to long-range Landau interactions, providing sharp error control between kinetic and hydrodynamic descriptions and a robust fixed-point argument for the nonlinear problem. The findings have significant implications for understanding kinetic shock structure beyond hard-sphere models and highlight the viability of Chapman-Enskog–type constructions in the Landau setting with rigorous control of regularization and coercivity.

Abstract

The physical quantities in a gas should vary continuously across a shock. However, the physics inherent in the compressible Euler equations is insufficient to describe the width or structure of the shock. We demonstrate the existence of weak shock profiles to the kinetic Landau equation, that is, traveling wave solutions with Maxwellian asymptotic states whose hydrodynamic quantities satisfy the Rankine-Hugoniot conditions. These solutions serve to capture the structure of weak shocks at the kinetic level. Previous works considered only the Boltzmann equation with hard sphere and angular cut-off potentials.

Kinetic shock profiles for the Landau equation

TL;DR

This work constructs small-amplitude kinetic shock profiles for the Landau equation by embedding a Navier–Stokes shock within the kinetic framework. The authors build an approximate solution via a Chapman-Enskog expansion, then perform a detailed linear and nonlinear analysis around the Navier–Stokes profile, employing macro–micro decomposition, Kawashima-type coercivity, and a κ-regularized collision operator to manage moment loss. They prove existence and uniqueness (up to translation) of traveling-wave solutions F(x,v) = F_NS(x,v) + μ0^{1/2} f with Maxwellian end states, and establish precise decay properties in x (stretched exponential for the remainder) and v (Gaussian moments). The approach extends prior Boltzmann-based results to long-range Landau interactions, providing sharp error control between kinetic and hydrodynamic descriptions and a robust fixed-point argument for the nonlinear problem. The findings have significant implications for understanding kinetic shock structure beyond hard-sphere models and highlight the viability of Chapman-Enskog–type constructions in the Landau setting with rigorous control of regularization and coercivity.

Abstract

The physical quantities in a gas should vary continuously across a shock. However, the physics inherent in the compressible Euler equations is insufficient to describe the width or structure of the shock. We demonstrate the existence of weak shock profiles to the kinetic Landau equation, that is, traveling wave solutions with Maxwellian asymptotic states whose hydrodynamic quantities satisfy the Rankine-Hugoniot conditions. These solutions serve to capture the structure of weak shocks at the kinetic level. Previous works considered only the Boltzmann equation with hard sphere and angular cut-off potentials.
Paper Structure (36 sections, 37 theorems, 484 equations, 1 figure)

This paper contains 36 sections, 37 theorems, 484 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Outline
  3. The formal Chapman-Enskog expansion for the approximate solution
  4. Geometry of Maxwellian equilibria
  5. Construction of the approximate solution
  6. Solving for the correction
  7. Linearized Chapman-Enskog expansion
  8. Macroscopic a priori linear estimates
  9. Microscopic a priori linear estimates
  10. Existence
  11. The proof of Theorem \ref{['thm:main']}
  12. Error estimates for Chapman-Enskog expansion
  13. The $\sigma$ norms
  14. Properties of $L$ and $\Gamma$
  15. Linear estimates
  16. ...and 21 more sections

Key Result

Theorem 1.1

Let $N \geqslant 8$ and $0 \leqslant q_0 < 1$. Let $0 < \varepsilon \ll_{N,q_0} 1$. There exists a smooth shock profile solution $F$ to the Landau equations satisfying the decomposition where $F_{\rm NS}$, defined in def:fNS, is a 'lifting' to the kinetic setting of the unique, up to spatial translations, compressible Navier-Stokes shock profile $(\varrho_{\rm NS},u_{\rm NS},\theta_{\rm NS})$. F

Figures (1)

  • Figure :

Theorems & Definitions (96)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 2.1
  • Proposition 2.2: Linear estimate
  • Remark 2.3
  • Remark 2.4: Key new difficulties
  • Proposition 2.5: Macroscopic estimate
  • Proposition 2.6
  • Lemma 3.1: Norm control
  • ...and 86 more