On some two phase problem for compressible and compressible viscous fluid flow separated by sharp interface

In this paper, we prove a local in time unique existence theorem for some two phase problem of compressible and compressible barotropic viscous fluid flow without surface tension in the L_p in time and the L_q in space framework with 2 < p < 1 and N < q < ∞ under the assumption that the initial domain is a uniform W_q^2-1/q domain in ℝ^N(N ≥ 2). After transforming a unknown time dependent domain to the initial domain by the Lagrangian transformation, we solve the problem by the contraction mapping principle with the maximal L_p-L_q regularity of the generalized Stokes operator for the compressible viscous fluid flow with free boundary condition. The key step of our method is to prove the existence of R-bounded solution operator to resolvent problem corresponding to linearized problem. The R-boundedness combined with Weis's operator valued Fourier multiplier theorem implies the generation of analytic semigroup and the maximal L_p-L_q regularity theorem.