Navier-Stokes-Cahn-Hilliard equations on evolving surfaces

TL;DR

The work develops a Navier-Stokes-Cahn-Hilliard system on evolving surfaces as a diffuse-interface counterpart to Model H on a moving manifold, deriving two equivalent surface formulations via balance laws and thin-film limits. It establishes existence and uniqueness of weak solutions under prescribed normal velocity for both a smooth polynomial-type potential and a singular logarithmic potential, using a robust framework of evolving Bochner spaces and the Piola transform. Central to the analysis are energy estimates, compactness arguments, and careful handling of nonlinear terms to pass to the limit, as well as a reintroduction of surface pressure in a mixed formulation. The results provide a rigorous foundation for hydrodynamic phase-field models on evolving surfaces and suggest avenues for numerical schemes and sharp-interface limits in future work.

Abstract

We derive a system of equations which can be seen as an evolving surface version of the diffuse interface "Model H" of Hohenberg and Halperin (1977). We then consider the well-posedness for the corresponding (tangential) system when one prescribes the evolution of the surface. Well-posedness is proved for smooth potentials in the Cahn-Hilliard equation with polynomial growth, and also for a thermodynamically relevant singular potential.

Theorems & Definitions (50)

• Remark 2.1
• Lemma 2.2: miura2018singular, Lemma 2.7
• Lemma 2.3: miura2018singular, Lemma 2.8
• Theorem 2.4
• proof
• Remark 2.5
• Lemma 3.1: OlsReuZhi22, Lemma 3.1
• Lemma 3.2: OlsReuZhi22, Lemma 3.6
• Lemma 3.3: Transport theorem
• Theorem 3.4