Global-in-time well-posedness of the compressible Navier-Stokes equations with striated density

TL;DR

This paper proves the global-in-time well-posedness of the compressible Navier-Stokes equations with striated density regularity, allowing vacuum and large bulk viscosity, and explores the incompressible limit.

Contribution

It establishes global well-posedness under minimal regularity assumptions and large viscosity, solving the density-patch problem and analyzing the incompressible limit.

Findings

Global well-posedness with striated density regularity.

Solution existence for vacuum initial data with large viscosity.

Incompressible limit as viscosity tends to infinity.

Abstract

We first show local-in-time well-posedness of the compressible Navier-Stokes equations, assuming striated regularity while no other smoothness or smallness conditions on the initial density. With these local-in-time solutions served as blocks, for \textit{less} regular initial data where the vacuum is permitted, the global-in-time well-posedness follows from the energy estimates and the propagated striated regularity of the density function, if the bulk viscosity coefficient is large enough in the two dimensional case. The global-in-time well-posedness holds also true in the three dimensional case, provided with large bulk viscosity coefficient together with small initial energy. This solves the density-patch problem in the exterior domain for the compressible model with $W^{2,p}$-Interfaces. Finally, the singular incompressible limit toward the inhomogenous incompressible model when the bulk viscosity coefficient tends to infinity is obtained.