On the Evolution of Compressible and Incompressible Viscous Fluids with a Sharp Interface

In this paper, we consider some two phase problems of compressible and incompressible viscous fluids’ flow without surface tension under the assumption that the initial domain is a uniform W q 2 − 1 / q domain in R N ( N ≥ 2 ). We prove the local in the time unique existence theorem for our problem in the L p in time and L q in space framework with 2 < p < ∞ and N < q < ∞ under our assumption. In our proof, we first transform an unknown time-dependent domain into the initial domain by using the Lagrangian transformation. Secondly, we solve the problem by the contraction mapping theorem with the maximal L p - L q regularity of the generalized Stokes operator for the compressible and incompressible viscous fluids’ flow with the free boundary condition. The key step of our proof is to prove the existence of an R -bounded solution operator to resolve the corresponding linearized problem. The Weis operator-valued Fourier multiplier theorem with R -boundedness implies the generation of a continuous analytic semigroup and the maximal L p - L q regularity theorem.