Low Mach number limit of the global solution to the compressible Navier-Stokes system for large data in the critical Besov space

TL;DR

This paper proves the global existence of solutions to the compressible Navier-Stokes system with large initial data in critical Besov spaces under low Mach number conditions and demonstrates convergence to incompressible solutions.

Contribution

It establishes the global well-posedness for large data in critical Besov spaces and analyzes the low Mach number limit for the compressible Navier-Stokes system.

Findings

Global solutions exist for large initial data at low Mach numbers.

Compressible solutions converge to incompressible solutions in space-time norms.

The results extend understanding of low Mach number limits in critical function spaces.

Abstract

In this paper, we consider the compressible Navier--Stokes system around the constant equilibrium states and prove the unique existence of a global solution for arbitrarily large initial data in the scaling critical Besov space provided that the Mach number is sufficiently small and the incompressible part of the initial velocity generates the global solution of the incompressible Navier--Stokes equation. Moreover, we consider the low Mach number limit and show that the compressible solution converges to the solution of the incompressible Navier--Stokes equation in some space-time norms.