Low Mach number limit for the global large solutions to the $2$D Navier--Stokes--Korteweg system in the critical $\widehat{L^p}$ framework

TL;DR

This paper proves the existence of unique global solutions for the 2D compressible Navier--Stokes--Korteweg system in critical Fourier--Besov spaces when the Mach number is small, and shows convergence to incompressible flow as Mach number approaches zero.

Contribution

It establishes the global well-posedness and zero Mach number limit for large initial data in critical Fourier--Besov spaces, a novel result in this setting.

Findings

Global solutions exist for small Mach number.

Solutions converge to incompressible flow as Mach number tends to zero.

Uses nonlinear stability and Strichartz estimates in Fourier--Besov spaces.

Abstract

In the present paper, we consider the compressible Navier--Stokes--Korteweg system on the $2$D whole plane and show that a unique global solution exists in the scaling critical Fourier--Besov spaces for arbitrary large initial data provided that the Mach number is sufficiently small. Moreover, we also show that the global solution converges to the $2$D incompressible Navier--Stokes flow in the singular limit of zero Mach number. The key ingredient of the proof lies in the nonlinear stability estimates around the large incompressible flow via the Strichartz estimate for the linearized equations in Fourier--Besov spaces.