Regularity of a bulk-surface Cahn-Hilliard model driven by Leray velocity fields

TL;DR

This paper extends the well-posedness results of a bulk-surface Cahn-Hilliard model to less regular velocity fields, including physically relevant Leray type, using new elliptic system regularity theory.

Contribution

It proves well-posedness for weak and strong solutions under broader velocity regularity assumptions, especially for Leray type fields, with a novel elliptic system analysis.

Findings

Well-posedness of weak solutions under general velocity regularity.

Existence of strong solutions for Leray velocity fields.

New regularity results for a bulk-surface elliptic system.

Abstract

We consider a convective bulk-surface Cahn--Hilliard system with dynamic boundary conditions and singular potentials. For this model, well-posedness results concerning weak and strong solutions have already been established in the literature. However, they require the prescribed velocity fields to belong to function spaces with high time regularity. In this paper, we prove that the well-posedness of weak solutions holds true under more general regularity assumptions on the velocity fields. Next, via an alternative proof for higher regularity, we show the well-posedness of strong solutions for velocity fields of Leray type, which is a more relevant assumption for physical applications. Our approach hinges upon a new well-posedness and regularity theory for a bulk-surface elliptic system with singular nonlinearities, which may be of independent interest.