On weak solutions for the stationary Cahn-Hillard-Navier-Stokes equations with singular potential

TL;DR

This paper proves the existence of weak solutions for the stationary compressible Navier-Stokes-Cahn-Hilliard system with a singular free energy, addressing mathematical challenges posed by vacuum states and the singular potential.

Contribution

It introduces a novel regularization and analytical approach to establish weak solutions for the steady system with singular energy and vacuum, a first in the field.

Findings

Existence of weak solutions under certain conditions.

Regularization method effectively handles singular potential.

Uniform estimates enable the limiting process to obtain solutions.

Abstract

The stationary Navier--Stokes--Cahn--Hilliard equations are considered, governing the motion of a compressible, two-phase fluid mixture with a diffuse interface. The free energy density in this paper has a singular logarithmic (Flory-uggins) form, ensuring that the mass fraction remains in the physical range and allowing for vacuum states. We prove the existence of weak solutions in a three-dimensional bounded domain under structural assumptions on the adiabatic exponent. The stationary setting poses two main mathematical challenges: the absence of an energy inequality driven by the evolution process to control the singular potential, and the degeneracy of the density near the vacuum. To address these issues, we introduce a specialized regularization of the logarithmic term that eliminates the quadratic growth induced by anti-diffusion, thereby restoring compactness. Uniform estimates are obtained through a special choice of artificial pressure and an interpolation argument that controls the desired norm of the density. A two-level limiting process then yields a weak solution that satisfies the physical bounds almost everywhere on the support of the density.To our knowledge, this is the first existence result for the steady compressible Navier--Stokes--Cahn--Hilliard system that incorporates both a singular free energy and vacuum regions.