Nonlocal Cahn-Hilliard-Hele-Shaw systems with singular potential and degenerate mobility

TL;DR

This paper analyzes a complex fluid mixture model combining nonlocal Cahn-Hilliard equations with degenerate mobility and logarithmic potential, establishing existence, regularity, and uniqueness of solutions in multiple dimensions.

Contribution

It proves the existence of global weak and strong solutions for the nonlocal Cahn-Hilliard-Hele-Shaw system with singular potential and degenerate mobility, including weak-strong uniqueness results.

Findings

Existence of global weak solutions satisfying energy identity

Existence of strong solutions with additional regularity

Weak-strong uniqueness in 2D and under certain conditions in 3D

Abstract

We study a Cahn-Hilliard-Hele-Shaw (or Cahn-Hilliard-Darcy) system for an incompressible mixture of two fluids. The relative concentration difference $\varphi$ is governed by a convective nonlocal Cahn-Hilliard equation with degenerate mobility and logarithmic potential. The volume averaged fluid velocity $\mathbf{u}$ obeys a Darcy's law depending on the so-called Korteweg force $\mu\nabla \varphi$, where $\mu$ is the nonlocal chemical potential. In addition, the kinematic viscosity $\eta$ may depend on $\varphi$. We establish first the existence of a global weak solution which satisfies the energy identity. Then we prove the existence of a strong solution. Further regularity results on the pressure and on $\mathbf{u}$ are also obtained. Weak-strong uniqueness is demonstrated in the two dimensional case. In the three-dimensional case, uniqueness of weak solutions holds if $\eta$ is constant. Otherwise, weak-strong uniqueness is shown by assuming that the pressure of the strong solution is $\alpha$-H\"{o}lder continuous in space for $\alpha\in (1/5,1)$.