# Compressible Navier-Stokes equations with ripped density

## TL;DR

This paper proves the global regularity and uniqueness of solutions for the 2D compressible Navier-Stokes equations with large bulk viscosity, and justifies the convergence to incompressible equations as viscosity increases.

## Contribution

It establishes the propagation of Sobolev regularity, solution uniqueness, and the rigorous limit from compressible to incompressible Navier-Stokes equations under large bulk viscosity.

## Key findings

- Global Sobolev regularity propagation for velocity.
- Uniqueness of solutions for perfect gas.
- Convergence to inhomogeneous incompressible Navier-Stokes as viscosity tends to infinity.

## Abstract

Here we prove the all-time propagation of the Sobolev regularity for the velocity field solution of the two-dimensional compressible Navier-Stokes equations, provided the volume (bulk) viscosity coefficient is large enough. The initial velocity can be arbitrarily large and the initial density is just required to be bounded. In particular, one can consider a characteristic function of a set as an initial density. Uniqueness of the solutions to the equations is shown, in the case of a perfect gas. As a by-product of our results, we give a rigorous justification of the convergence to the inhomogeneous incompressible Navier-Stokes equations when the volume viscosity tends to infinity. Similar results are proved in the three-dimensional case, under some scaling invariant smallness condition on the velocity field.