Global well-posedness of strong solutions to a bulk-surface Navier-Stokes-Cahn-Hilliard model with non-degenerate mobilities in two dimensions

TL;DR

This paper proves the global existence, uniqueness, and continuous dependence of strong solutions for a coupled bulk-surface Navier-Stokes-Cahn-Hilliard model in two dimensions, including physically relevant non-degenerate mobilities.

Contribution

It establishes the first global well-posedness results for this complex bulk-surface fluid mixture model with non-degenerate mobilities.

Findings

Proved global well-posedness of strong solutions in 2D.

Established uniqueness and continuous dependence on initial data.

Developed new regularity theory for coupled bulk-surface equations.

Abstract

We examine a thermodynamically consistent diffuse interface model for bulk-surface viscous fluid mixtures. This model consists of a Navier--Stokes--Cahn--Hilliard model in the bulk coupled to a surface Navier--Stokes--Cahn--Hilliard system on the boundary. In this paper, we address the global well-posedness of strong solutions in the two-dimensional setting, also covering the physically meaningful case of non-degenerate mobility functions. Lastly, we prove the uniqueness of the corresponding strong solutions and their continuous dependence on the initial data. Our approach hinges upon new well-posedness and regularity theory for a convective bulk-surface Cahn--Hilliard equation with non-degenerate mobilities, as well as a bulk-surface Stokes equation with non-constant coefficients.