Well-posedness for the Linearized Motion of a Compressible Liquid with Free Surface Boundary

(1.2) (∂t + V ∂k)ρ+ ρdivV = 0, divV = ∂kV k in D, where V k = δvi = vk and we use the summation convention over repeated upper and lower indices. Here the velocity V = (V , ..., V ), the density ρ and the domain D = ∪0≤t≤T {t}× Dt, Dt ⊂ R are to be determined. The pressure p = p (ρ) is assumed to be a given strictly increasing smooth function of the density. The boundary ∂Dt moves with the velocity of the fluid particles at the boundary. The fluid body moves in vacuum so the pressure vanishes in the exterior and hence on the boundary. We therefore also require the boundary conditions on ∂D = ∪0≤t≤T {t}× ∂Dt: