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

TL;DR

This paper proves global well-posedness for the one-dimensional compressible Navier-Stokes equations in critical regularity spaces, overcoming nonlinear control issues by using mass Lagrangian coordinates, and analyzes decay and high viscosity limits.

Contribution

It establishes the first global well-posedness result in critical spaces for the 1D case and introduces a novel approach using mass Lagrangian coordinates.

Findings

Global well-posedness in critical spaces for 1D case

Optimal decay estimates for solutions

Convergence of specific volume in high viscosity limit

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.