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

Abstract

We consider the 3D isentropic compressible Navier-Stokes equations with degenerate viscousities and vacuum. The degenerate viscosities $\hat{I}¼(\ddot{I})$ and $\hat{I}»(\ddot{I})$ are proportional to some power of density, while the powers of density in $\hat{I}¼(\ddot{I})$ and $\hat{I}»(\ddot{I})$ are different(i.e., $\hat{I}{´}_1\ne \hat{I}{´}_2$). 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.