Local classical solutions to Navier-Stokes equations with degenerate viscosities and vacuum

TL;DR

This paper proves the local existence of classical solutions to 3D compressible Navier-Stokes equations with density-dependent viscosities that degenerate at vacuum, allowing initial vacuum states without compatibility conditions.

Contribution

It introduces a novel coupled hyperbolic-elliptic framework to handle degenerate viscosities and vacuum, establishing local well-posedness without initial compatibility requirements.

Findings

Established local well-posedness for degenerate viscosity Navier-Stokes with vacuum.

Developed a quasi-symmetric hyperbolic--degenerate elliptic structure.

Allowed initial vacuum in an open set without compatibility conditions.

Abstract

We consider the 3D isentropic compressible Navier-Stokes equations with degenerate viscousities and vacuum. The degenerate viscosities $\mu(\rho)$ and $\lambda(\rho)$ are proportional to some power of density, while the powers of density in $\mu(\rho)$ and $\lambda(\rho)$ are different(i.e., $\delta_1\neq \delta_2$). The local well-posedness of classical solution is established by introducing a ``quasi-symmetric hyperbolic''--``degenerate elliptic'' coupled structure to control the behavior of the velocity of the fluid near the vacuum and give some uniform estimates. In particular, the initial data allows vacuum in an open set and we do not need any initial compatibility conditions.