On classical solutions for viscous polytropic fluids with degenerate viscosities and vacuum

TL;DR

This paper establishes local well-posedness for three-dimensional viscous polytropic fluids with vacuum, introducing regular solutions and analyzing their finite-time breakdown, addressing longstanding degeneracy challenges in compressible Navier-Stokes equations.

Contribution

It proves the local-in-time existence of regular solutions with large initial data and vacuum, and demonstrates finite-time breakdown for certain initial conditions, advancing understanding of degenerate viscous flows.

Findings

Existence of regular solutions with large initial data and vacuum

Finite-time breakdown of solutions for specific initial data with vacuum

Addresses high degeneracy caused by vacuum in compressible fluid models

Abstract

In this paper, we consider the three-dimensional isentropic Navier-Stokes equations for compressible fluids with viscosities depending on density in a power law and allowing initial vacuum. We introduce the notion of regular solutions and prove the local-in-time well-posedness of solutions with arbitrarily large initial data and vacuum in this class, which is a long-standing open problem due to the very high degeneracy caused by vacuum. Moreover, for certain classes of initial data with local vacuum, we show that the regular solution that we obtained will break down in finite time, no matter how small and smooth the initial data are.