Well-posedness for the Linearized Motion of the Free Surface of a Compressible Fluid

(1.2) (∂t + V ∂k)ρ+ ρdivV = 0, divV = ∂kV k in D where the velocity v = (v1, ..., vn), the density ρ and the domain D ⊂ [0, T ]×Rn are to be determined. Here the pressure p = p(ρ) is assumed to be a given function of the density which is strictly increasing at the boundary where the density is assume to be constant. Furthermore V k = δvi = vk and we have used the summation convention that repeated upper and lower indices are summed over. We also require the boundary conditions on the free boundary ∂D;