RESOLVENT PROBLEM FOR COMPRESSIBLE NAVIER-STOKES EQUATIONS WITH FREE SURFACE

. The motion of a compressible, viscous and heat-conducting fluid occupying an infinite layer, which is bounded below by a rigid bottom and above by a free surface, is described by a free boundary problem for the Navier-Stokes equations. We consider the associated resolvent equations with non-homogeneous boundary conditions arising from this free boundary problem. Different from the argument for the Cauchy problem where the solution can be obtained by utilizing the reflection symmetry of the Navier-Stokes equations directly, we derive an explicit representation of solutions to this boundary value problem, and the unknown boundary values are also well-characterized. Moreover, since the entropy variation and heat transfer effects make the problem much more intricate and challenging, the resolvent parameter needs to be treated more carefully than that for the isentropic Navier-Stokes equations. By analyzing each individual term in the solution formula, we establish the Lp estimates of solutions to the resolvent problem. Our result is the key and crucial step towards a further investigation of the free boundary problem for the full compressible Navier-Stokes equations by the semigroup approach.