Global weak solutions to a compressible Navier--Stokes/Cahn--Hilliard system with singular entropy of mixing

TL;DR

This paper proves the existence of global weak solutions for a compressible Navier-Stokes/Cahn-Hilliard system with a physically relevant singular entropy, advancing mathematical understanding of phase separation in viscous fluid mixtures.

Contribution

It introduces new estimates for the chemical potential and entropy in a density-dependent model with singular free energy, extending previous regular-potential results.

Findings

Existence of global weak solutions in 3D bounded domains.

Phase variable remains within physical bounds almost everywhere.

New estimates for chemical potential and entropy derived.

Abstract

We study a Navier-Stokes/Cahn-Hilliard system modeling the evolution of a compressible binary mixture of viscous fluids undergoing phase separation. The novelty of this work is a free energy potential including the physically relevant Flory-Huggins (logarithmic) entropy, as opposed to previous studies in the literature, which only consider regular potentials with polynomial growth. Our main result establishes the existence of global-in-time weak solutions in three-dimensional bounded domains for arbitrarily large initial data. The core contribution is the derivation of new estimates for the chemical potential and the Flory-Huggins entropy arising from a density-dependent Cahn-Hilliard equation under minimal assumptions: non-negative $\gamma$-integrable density with $\gamma>\frac32$. In addition, we prove that the phase variable, which represents the difference of the mass concentrations, takes value within the physical interval $[-1,1]$ almost everywhere on the set where the density is positive.