Global regularity and asymptotic stabilization for the incompressible Navier-Stokes-Cahn-Hilliard model with unmatched densities

TL;DR

This paper proves global regularity and asymptotic stability for the incompressible Navier-Stokes-Cahn-Hilliard system with unmatched densities, extending understanding of fluid mixture models and their long-term behavior.

Contribution

It introduces a novel global regularity result for the Cahn-Hilliard equation with divergence-free velocity and establishes stability and regularity properties of solutions in three and two dimensions.

Findings

Global weak solutions in 3D become regular over time and stabilize.

Existence and uniqueness of strong solutions in 2D for the full system.

Global weak solutions exist for the double obstacle potential case.

Abstract

We study an initial-boundary value problem for the incompressible Navier-Stokes-Cahn-Hilliard system with non-constant density proposed by Abels, Garcke and Gr\"{u}n in 2012. This model arises in the diffuse interface theory for binary mixtures of viscous incompressible fluids. This system is a generalization of the well-known model H in the case of fluids with unmatched densities. In three dimensions, we prove that any global weak solution (for which uniqueness is not known) exhibits a propagation of regularity in time and stabilizes towards an equilibrium state as $t \rightarrow \infty$. More precisely, the concentration function $\phi$ is a strong solution of the Cahn-Hilliard equation for (arbitrary) positive times, whereas the velocity field $\mathbf{u}$ becomes a strong solution of the momentum equation for large times. Our analysis hinges upon the following key points: a novel global regularity result (with explicit bounds) for the Cahn-Hilliard equation with divergence-free velocity belonging only to $L^2(0,\infty; \mathbf{H}^1_{0,\sigma}(\Omega))$, the energy dissipation of the system, the separation property for large times, a weak-strong uniqueness type result, and the Lojasiewicz-Simon inequality. Additionally, in two dimensions, we show the existence and uniqueness of global strong solutions for the full system. Finally, we discuss the existence of global weak solutions for the case of the double obstacle potential.