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Perturbation theory of the compressible Navier-Stokes equations and its application

Kazuyuki Tsuda

Abstract

In this article, a perturbation theory of the compressible Navier-Stokes equations in $\mathbb{R}^n$ $(n \geq 3)$ is studied to investigate decay estimate of solutions around a non-constant state. As a concrete problem, stability is considered for a perturbation system from a stationary solution $u_ω$ belonging to the weak $L^n$ space. Decay rates of the perturbation including $L^\infty$ norm are obtained which coincide with those of the heat kernel except a bit loss. The proof is based on deriving suitable resolvent estimates with perturbation terms in the low frequency part having a parabolic spectral curve. Our method can be applicable to dispersive hyperbolic systems like wave equations with strong damping. Indeed, a parabolic type decay rate of a solution is obtained for a damped wave equation including variable coefficients which satisfy spatial decay conditions.

Perturbation theory of the compressible Navier-Stokes equations and its application

Abstract

In this article, a perturbation theory of the compressible Navier-Stokes equations in is studied to investigate decay estimate of solutions around a non-constant state. As a concrete problem, stability is considered for a perturbation system from a stationary solution belonging to the weak space. Decay rates of the perturbation including norm are obtained which coincide with those of the heat kernel except a bit loss. The proof is based on deriving suitable resolvent estimates with perturbation terms in the low frequency part having a parabolic spectral curve. Our method can be applicable to dispersive hyperbolic systems like wave equations with strong damping. Indeed, a parabolic type decay rate of a solution is obtained for a damped wave equation including variable coefficients which satisfy spatial decay conditions.

Paper Structure

This paper contains 12 sections, 45 theorems, 343 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Formulation
  4. Estimates for the low frequency part
  5. Estimates for the high frequency part
  6. Existence of global solutions with decay rates
  7. Application to the damped wave equations
  8. Formulation
  9. Estimates of the low frequency part
  10. Estimates of the high frequency part
  11. Deriving an a priori estimate
  12. Appendix

Key Result

Theorem 1.3

There exists a solution $u$ globally in time to (1.1) and (1.2) for $u={}^\top(\rho-\rho_s, v-v_s)$. Moreover, if it holds that for $0<t$, where $\widehat{H}^1$ is the homogeneous $L^2$ Sobolev space and $\epsilon>0$ is any small positive number. This together with some space-time integral stated in Shibata-Tanaka2 yields

Theorems & Definitions (65)

  • Remark 1.2
  • Theorem 1.3
  • Remark 1.4
  • Theorem 1.5
  • Remark 1.6
  • Theorem 1.8
  • Definition 2.1: Distribution Function
  • Definition 2.2: Non-increasing Rearrangement
  • Definition 2.3: Lorentz Space
  • Lemma 2.4
  • ...and 55 more