Local existence of classical solutions to the 3D isentropic compressible Navier-Stokes-Poisson equations with degenerate viscosities and vacuum

TL;DR

This paper proves the local existence of classical solutions to the 3D isentropic compressible Navier-Stokes-Poisson equations with degenerate viscosities and vacuum, using a novel coupled structure and uniform estimates without initial compatibility conditions.

Contribution

It introduces a new coupled hyperbolic-elliptic framework to handle vacuum states and degenerate viscosities in 3D Navier-Stokes-Poisson equations, establishing local well-posedness.

Findings

Established local existence of classical solutions with vacuum.

Developed a coupled structure controlling velocity near vacuum.

Achieved uniform estimates without initial compatibility conditions.

Abstract

We consider the isentropic compressible Navier-Stokes-Poisson equations with degenerate viscousities and vacuum in a three-dimensional torus. 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.