Weak solutions to the stationary Cahn-Hillard/Navier-Stokes equations for compressible fluids

TL;DR

This paper establishes the existence of weak solutions for stationary compressible fluid flows described by coupled Cahn-Hilliard and Navier-Stokes equations in three dimensions, using advanced energy estimates.

Contribution

It proves the existence of weak solutions for the stationary Cahn-Hilliard/Navier-Stokes system with $\gamma > 4/3$, introducing new techniques to handle capillary stress effects.

Findings

Existence of weak solutions for $\gamma > 4/3$

Development of weighted energy estimates

Overcoming difficulties from capillary stress

Abstract

We are concerned with the Cahn-Hilliard/Navier-Stokes equations for the stationary compressible flows in a three-dimensional bounded domain. The governing equations consist of the stationary Navier-Stokes equations describing the compressible fluid flows and the stationary Cahn-Hilliard type diffuse equation for the mass concentration difference. We prove the existence of weak solutions when the adiabatic exponent $\gamma$ satisfies $\gamma>\frac{4}{3}$. The proof is based on the weighted total energy estimates and the new techniques developed to overcome the difficulties from the capillary stress.