Compressible–Incompressible Two-Phase Flows with Phase Transition: Model Problem

Abstract

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 RN, 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: uΓ· nt= [[ρu]] · nt/ [[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.