Compressible Navier-Stokes system : large solutions and incompressible limit

TL;DR

This paper establishes the global existence of regular solutions for the 2D compressible Navier-Stokes equations with large initial velocities and near-constant density, and extends results to higher dimensions under certain conditions.

Contribution

It proves the existence of large solutions for the compressible Navier-Stokes system in 2D and generalizes to higher dimensions with additional assumptions on the incompressible limit.

Findings

Global regular solutions exist for large initial velocities in 2D.

Extension of results to higher dimensions under incompressible limit assumptions.

Solutions are analyzed in critical Besov spaces.

Abstract

Here we prove the existence of global in time regular solutions to the two-dimensional compressible Navier-Stokes equations supplemented with arbitrary large initial velocity $v\_0$ and almost constantdensity $\varrho\_0$, for large volume (bulk) viscosity. The result is generalized to the higher dimensional case under the additional assumption that the strong solution of the classical incompressible Navier-Stokes equations supplemented with the divergence-freeprojection of $v\_0,$ is global. The systems are examined in $R^d$ with $d \geq 2$, in the critical $\dot B^s\_{2,1}$ Besov spaces framework.