Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

High viscosity limit for the multi-dimensional compressible Navier-Stokes equations

Raphaël Danchin

Abstract

We investigate the high viscosity limit (also called inertial limit) of the barotropic compressible Navier-Stokes equations supplemented with initial data which are perturbations of a stable constant solution. In the case of constant viscosity coefficients, we establish that, after diffusive rescaling, the density tends to satisfy a transport equation with nonlinear damping which is globally well-posed, even for large data. Similar results are proved for variable viscosity coefficients. In this latter case, the damping term in the limit equation of the density is nonlocal.

High viscosity limit for the multi-dimensional compressible Navier-Stokes equations

Abstract

We investigate the high viscosity limit (also called inertial limit) of the barotropic compressible Navier-Stokes equations supplemented with initial data which are perturbations of a stable constant solution. In the case of constant viscosity coefficients, we establish that, after diffusive rescaling, the density tends to satisfy a transport equation with nonlinear damping which is globally well-posed, even for large data. Similar results are proved for variable viscosity coefficients. In this latter case, the damping term in the limit equation of the density is nonlocal.
Paper Structure (9 sections, 1 theorem, 206 equations)

This paper contains 9 sections, 1 theorem, 206 equations.

Table of Contents

  1. Results
  2. The proof of the main result
  3. Uniform bounds in critical norm
  4. Control of lower order norms
  5. The case of variable viscosity coefficients
  6. Inversion of the elliptic operator
  7. Control of lower order norms
  8. Solving the limit equation
  9. Passing to the limit

Key Result

Theorem 1.1

There exists a positive real number $\eta_0=\eta_0(d,\mu,P')$ such that for any family $(\rho_0^\nu,u_0^\nu)_{\nu>0}$ of data with $a_0^\nu:=\rho_0^\nu-1$ in $\dot B^{\frac{d}{2}-1}_{2,1}$ and $u_0^\nu$ in $\dot B^{\frac{d}{2}-1}_{2,1}$ satisfying Equations $(CNS_{\nu,c})$ supplemented with initial data $(\rho_0^\nu,u_0^\nu)$ admit a unique global solution $(\rho^\nu=1+a^\nu,u^\nu)$ such that Fu

Theorems & Definitions (6)

  • Theorem 1.1
  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5