\({{\mathcal {R}}}\)-Bounded Operator Families Arising from a Compressible Fluid Model of Korteweg Type with Surface Tension in the Half-Space

In this paper, we consider a resolvent problem arising from the free boundary value problem for the compressible fluid model of Korteweg type, which is called as the Navier–Stokes–Korteweg system, with surface tension in the half-space. The Navier–Stokes–Korteweg system is known as a diffuse interface model for liquid–vapor two-phase flows. Our purpose is to show the \({{\mathcal {R}}}\) $\mathcal{R}$ -boundedness for the solution operator families of the resolvent problem, which gives us the maximal regularity estimates in the \(L_p\) $L_p$ -in-time and \(L_q\) $L_q$ -in-space setting by applying the Weis’s operator valued Fourier multiplier theorem (Weis in Math Ann 319:735–758, 2001).