Strong well-posedness and separation properties for a bulk-surface convective Cahn--Hilliard system with singular potentials

TL;DR

This paper proves well-posedness, regularity, and separation properties for a bulk-surface convective Cahn--Hilliard system with singular potentials, advancing mathematical understanding of these complex models.

Contribution

It introduces a novel approach to establish existence, uniqueness, and regularity of solutions for the system with singular potentials, including separation properties.

Findings

Existence of global weak solutions via Yosida regularization

Uniqueness and continuous dependence of solutions

Higher regularity and separation properties for logarithmic potentials

Abstract

This paper addresses the well-posedness of a general class of bulk-surface convective Cahn--Hilliard systems with singular potentials. For this model, we first prove the existence of a global-in-time weak solution by approximating the singular potentials via a Yosida regularization, applying the corresponding results for regular potentials, and eventually passing to the limit in this approximation scheme. Then, we prove the uniqueness of weak solutions and their continuous dependence on the velocity fields and the initial data. Afterwards, assuming additional regularity of the domain as well as the velocity fields, we establish higher regularity properties of weak solutions and eventually the existence of strong solutions. In the end, we discuss strict separation properties for logarithmic type potentials in both two and three dimensions.