On the existence of local classical solutions to the Navier-Stokes equations with degenerate viscosities

Abstract

In this paper, we consider the three-dimensional Cauchy problem of compressible Navier-Stokes equations with degenerate viscosities depending on density. With the initial data containing vacuum at far fields, we prove the local existence of regular solutions with some compatibility conditions. The regular solution is classical due to the smoothing effect of the velocity u.