On a thermodynamically consistent diffuse interface model for incompressible two-phase flows with unmatched densities: Energy equality and Lyapunov stability

TL;DR

This paper develops a thermodynamically consistent diffuse interface model for incompressible two-phase flows with unmatched densities, establishing energy conservation, existence, uniqueness, and stability of solutions.

Contribution

It introduces new conditions for energy equality in weak solutions and proves global existence, uniqueness, and Lyapunov stability of strong solutions under small perturbations.

Findings

Energy equality holds under specific regularity conditions.

Existence and uniqueness of global strong solutions are established.

Lyapunov stability is proven for steady states with small initial perturbations.

Abstract

We consider the initial-boundary value problem of a thermodynamically consistent diffuse interface model for incompressible two-phase flows with unmatched densities in a bounded domain $\Omega\subset\mathbb{R}^3$. Our first aim is to study the energy equality for global weak solutions by establishing mixed $L_t^qL_x^r$-regularity conditions on the velocity field and its gradient, under which the global weak solution conserves its energy for all times. The proof is based on the propagation of regularity for weak solutions to the convective Cahn-Hilliard equation with a physically relevant Flory-Huggins-type potential, combined with global mollification and boundary cut-off techniques. Next, we prove the existence and uniqueness of global strong solutions in the general setting with non-constant gradient energy coefficient and non-degenerate mobility, provided that the initial velocity is sufficiently small and the initial phase-field variable is a sufficiently small perturbation of a local minimizer of the free energy. This yields Lyapunov stability for each steady state consisting of a zero velocity together with a local energy minimizer. The proof relies on the energy equality for (local) strong solutions and the {\L}ojasiewicz-Simon approach.