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The one-dimensional compressible Navier-Stokes equations in critical regularity spaces

Raphaël Danchin

Abstract

We are concerned with the barotropic compressible Navier-Stokes equations on the real line. Our primary goal is to establish the global well-posedness in a critical regularity framework in the case where the initial data are small perturbations of a stable constant state. Surprisingly, even though the result in the multi-dimensional case is by now classical, the one-dimensional case has not been elucidated yet as far as we know. This is due to the fact that in the critical framework, the regularity of the velocity is so negative that some nonlinear terms are out of control. Here, we overcome the difficulty by considering the equations in the mass Lagrangian coordinates system. Granted with a global well-posedness statement, we then establish optimal time decay estimates and investigate the high viscosity limit, pointing out the convergence of the specific volume to the solution of some ordinary differential equation, after time and space rescaling.

The one-dimensional compressible Navier-Stokes equations in critical regularity spaces

Abstract

We are concerned with the barotropic compressible Navier-Stokes equations on the real line. Our primary goal is to establish the global well-posedness in a critical regularity framework in the case where the initial data are small perturbations of a stable constant state. Surprisingly, even though the result in the multi-dimensional case is by now classical, the one-dimensional case has not been elucidated yet as far as we know. This is due to the fact that in the critical framework, the regularity of the velocity is so negative that some nonlinear terms are out of control. Here, we overcome the difficulty by considering the equations in the mass Lagrangian coordinates system. Granted with a global well-posedness statement, we then establish optimal time decay estimates and investigate the high viscosity limit, pointing out the convergence of the specific volume to the solution of some ordinary differential equation, after time and space rescaling.
Paper Structure (10 sections, 5 theorems, 178 equations)

This paper contains 10 sections, 5 theorems, 178 equations.

Table of Contents

  1. Results
  2. The proof of global well-posedness
  3. A priori estimates
  4. Stability with respect to the data, and uniqueness
  5. The proof of Theorem \ref{['thm:main1']}
  6. Time decay estimates
  7. Decay of the low frequencies
  8. Decay of the high frequencies
  9. The diffusive limit
  10. Appendix

Key Result

Theorem 1.1

Fix some reference specific volume $\bar{\eta}>0$ and smooth pressure function $Q.$ Consider any data $(\eta_0,v_0)$ such that $a_0:=\eta_0-\bar{\eta}$ (resp. $v_0$) belongs to $\dot B^{1/p}_{p,1}$ (resp. $\dot B^{-1+1/p}_{p,1}$) for some $p\in[1,\infty).$ If, in addition, $\inf_x \eta_0(x)>0$ then

Theorems & Definitions (9)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.1
  • Theorem 1.3
  • Remark 1.2
  • Remark 1.3
  • Theorem 1.4
  • Proposition 2.1
  • proof