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 Göts and Shibata (Asymptot Anal 90(3–4):207–236, 2014), the regularity of ρ₊ is Wq1 in space, but to solve the kinetic equation: u_Γ· n_t= [[ρu]] · n_t/ [[ρ]] on Γ _t we need Wq2-1/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 Wq3 in space, we eliminate the problem by using the Navier–Stokes–Korteweg equations instead of the compressible Navier–Stokes equations.