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Classical solutions to a BGK-type model relaxing to the isentropic gas dynamics

Byung-Hoon Hwang

Abstract

In this paper, we consider a BGK-type kinetic model relaxing to the isentropic gas dynamics in the hydrodynamic limit. We introduce a linearization of the equation around the global equilibrium. Then we prove the global existence of classical solutions with an exponential convergence rate toward the equilibrium state in the periodic domain when the initial data is a small perturbation of the global equilibrium.

Classical solutions to a BGK-type model relaxing to the isentropic gas dynamics

Abstract

In this paper, we consider a BGK-type kinetic model relaxing to the isentropic gas dynamics in the hydrodynamic limit. We introduce a linearization of the equation around the global equilibrium. Then we prove the global existence of classical solutions with an exponential convergence rate toward the equilibrium state in the periodic domain when the initial data is a small perturbation of the global equilibrium.
Paper Structure (8 sections, 11 theorems, 107 equations)

This paper contains 8 sections, 11 theorems, 107 equations.

Table of Contents

  1. Introduction
  2. Model equation
  3. Main result
  4. Linearization around the global equilibrium
  5. Local-in-time existence
  6. Proof of Theorem \ref{['main_thm']}: global existence and large-time behavior
  7. Macro-micro decomposition
  8. Global existence and large-time behavior

Key Result

Theorem 1.1

Let $N\ge 3$ and $\gamma \in (1,1+\frac{2}{4N+6+d}]$. Suppose that the initial data $F_0$ is compactly supported in a way that If $E(f_0)$ is sufficiently small, then there exists a unique global-in-time solution $f$ to perturbation satisfying

Theorems & Definitions (21)

  • Theorem 1.1
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Proposition 2.1
  • proof
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • ...and 11 more