Stationary Cahn-Hilliard-Navier-Stokes equations for the diffuse interface model of compressible flows

TL;DR

This paper establishes the existence of weak solutions for a stationary diffuse interface model combining Navier-Stokes and Cahn-Hilliard equations for compressible immiscible fluids in three dimensions.

Contribution

It introduces a novel approach with new estimates to prove weak solution existence for the coupled system with a broad range of adiabatic exponents.

Findings

Existence of weak solutions for the stationary system.

Development of a two-level approximation procedure.

Application of weak convergence methods to the coupled PDE system.

Abstract

A system of partial differential equations for a diffusion interface model is considered for the stationary motion of two macroscopically immiscible, viscous Newtonian fluids in a three-dimensional bounded domain. The governing equations consist of the stationary Navier-Stokes equations for compressible fluids and a stationary Cahn-Hilliard type equation for the mass concentration difference. Approximate solutions are constructed through a two-level approximation procedure, and the limit of the sequence of approximate solutions is obtained by a weak convergence method. New ideas and estimates are developed to establish the existence of weak solutions with a wide range of adiabatic exponent.