Global well-posedness for the compressible Navier-Stokes equations with the highly oscillating initial velocity

TL;DR

This paper establishes the global well-posedness of the compressible Navier-Stokes equations with highly oscillating initial velocities, extending previous results from the incompressible case to the compressible setting.

Contribution

It introduces a new estimate for hyperbolic/parabolic systems with convection, enabling the analysis of highly oscillating initial data in the compressible case.

Findings

Proves global existence for compressible Navier-Stokes with oscillatory initial data

Develops new estimates for hyperbolic/parabolic systems with convection

Extends well-posedness results to a broader class of initial conditions

Abstract

Cannone \cite{Cannone} proved the global well-posedness of the incompressible Navier-Stokes equations for a class of highly oscillating data. In this paper, we prove the global well-posedness for the compressible Navier-Stokes equations in the critical functional framework with the initial data close to a stable equilibrium. Especially, this result allows us to construct global solutions for the highly oscillating initial velocity. The proof relies on a new estimate for the hyperbolic/parabolic system with convection terms.