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 RN<inline-graphic></inline-graphic>, 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<inline-graphic></inline-graphic>-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+<inline-graphic></inline-graphic> is Wq1<inline-graphic></inline-graphic> in space, but to solve the kinetic equation: uΓ·nt=[[ρu]]·nt/[[ρ]]<inline-graphic></inline-graphic> on Γt<inline-graphic></inline-graphic> we need Wq2-1/q<inline-graphic></inline-graphic> regularity of ρ+<inline-graphic></inline-graphic> on Γt<inline-graphic></inline-graphic>, which means the regularity loss. Since the regularity of ρ+<inline-graphic></inline-graphic> dominated by the Navier-Stokes-Korteweg equations is Wq3<inline-graphic></inline-graphic> in space, we eliminate the problem by using the Navier-Stokes-Korteweg equations instead of the compressible Navier-Stokes equations. [ABSTRACT FROM AUTHOR]