Separation Property for the Nonlocal Cahn Hilliard Brinkman System with Singular Potential and Degenerate Mobility
Sheetal Dharmatti, Greeshma K
TL;DR
The work addresses the separation property for the nonlocal Cahn-Hilliard-Brinkman system with a singular logarithmic potential and degenerate mobility in a 2D bounded domain. It develops a De Giorgi iteration–based framework tailored to handle nonlocal coupling and mobility degeneracy, leveraging a global $L^1$ bound on $F'(\varphi)$ and a blow-up condition on $F''$ near the pure phases to close the argument. The main result is that for any $\tau>0$ there exists a $\delta>0$ such that $\|\varphi(t)\|_{L^\infty}\le 1-\delta$ for all $t\ge\tau$, providing an explicit separation from pure phases. The approach extends to a broader class of potentials satisfying assumptions [A1]-[A4] and confirms the separation property under the singular or near-singular settings, contributing a robust method for similar nonlocal, degenerate systems with singular potentials.
Abstract
This work studies the nonlocal Cahn Hilliard Brinkman system, which models the phase separation of a binary fluid in a bounded domain and porous media. We focus on a system with a singular potential namely logarithmic form and a degenerate mobility function. The singular potential introduces challenges due to the blow up of its derivatives near pure phases, while the degenerate mobility complicates the analysis. Our main result is the separation property, which ensures that the solution eventually stays away from the pure phases. We adopt a new method, inspired by the De Giorgi iteration, introduced for the two dimensional Cahn Hilliard equation with constant mobility. This work extends previous results and provides a general approach for proving the separation property for similar systems.