On the $\mathcal{R}$-boundedness of solution operators for a compressible fluid model of Korteweg type in general domains

In this paper, we consider a resolvent problem arising from the free boundary problem for the compressible fluid model of the Korteweg type, which is called the Navier-Stokes-Korteweg system, with surface tension in general domains. The Navier-Stokes-Korteweg system describes the liquid-vapor two-phase flow with non-zero thickness phase boundaries, which is often called the diffuse interface model. Our purpose is to show the solution operator families of the resolvent problem are $\mathcal{R}$-bounded, which gives us the generation of analytic semigroup and the maximal regularity in the $L_p$-in-time and $L_q$-in-space setting by applying the Weis operator valued Fourier multiplier theorem.