Surface Waves for a Compressible Viscous Fluid

In this paper, we are concerned with free boundary problem for compressible viscous isotropic Newtonian fluid. Our problem is to find the three-dimensional domain occupied by the fluid which is bounded below by the fixed bottom and above by the free surface together with the density, the velocity vector field and the absolute temperature of the fluid satisfying the system of Navier-Stokes equations and the initial-boundary conditions. The Navier-Stokes equations consist of the conservations of mass, momentum under the gravitational field in a downward direction and energy. The effect of the surface tension on the free surface is taken into account. The purpose of this paper is to establish two existence theorems to the problem mentioned above: the first concerns with the temporary local solvability in anisotropic Sobolev-Slobodetskii spaces and the second the global solvability near the equilibrium rest state. Here the equilibrium rest state (heat conductive state) means that the temperature distribution is a linear function with respect to a vertical direction and the density is determined by an ordinary differential equation which involves equation of state. For the proof, we rely on the methods due to Solonnikov in the case of incompressible fluid with some modifications, since our problem is hyperbolic-parabolic coupled system.