Compressible-incompressible two-phase flows with phase transition: Model problem

We study the compressible and incompressible two-phase flows separated by a sharp interface with a phase transition and a surface tension. In particular, we consider the problem in R^N, and the Navier-Stokes-Korteweg equations is used in the upper domain and the Navier-Stokes equations is used in the lower domain. We prove the existence of R-bounded solution operator families for a resolvent problem arising from its model problem. According to Shibata [13], the regularity of ρ₊ is W_q¹ in space, but to solve the kinetic equation: u_Γ · n_t = [[ρu]] · n_t/[[ρ]] on Γ_t we need W_q^2−1/q regularity of ρ₊ on Γ_t, which means the regularity loss. Since the regularity of ρ₊ dominated by the Navier-Stokes-Korteweg equations is W_q³ in space, we eliminate the problem by using the Navier-Stokes-Korteweg equations instead of the compressible Navier-Stokes equations.

MSC Codes 35Q30 (Primary), 76T10 (Secondary)