Local Solvability for a Compressible Fluid Model of Korteweg Type on General Domains

Abstract

In this paper, we consider a compressible fluid model of the Korteweg type on general domains in the N-dimensional Euclidean space for N≥2. The Korteweg-type model is employed to describe fluid capillarity effects or liquid–vapor two-phase flows with phase transition as a diffuse interface model. In the Korteweg-type model, the stress tensor is given by the sum of the standard viscous stress tensor and the so-called Korteweg stress tensor, including higher order derivatives of the fluid density. The local existence of strong solutions is proved in an Lp-in-time and Lq-in-space setting, p∈(1,∞) and q∈(N,∞), with additional regularity of the initial density on the basis of maximal regularity for the linearized system.

The existence of R$$ \mathcal{R} $$‐bounded solution operator for Navier–Stokes–Korteweg model with slip boundary conditions in half space

“…Inna et al [11] extensively examined system (1.1) and (1.2) in both Case I and Case II. In their collaborative work, Inna and Saito [12] further extended the Equations (1.1) and (1.2) into the time domain and explored the existence of local time-dependent solutions for this model.…”