Separation Property for the Nonlocal Cahn Hilliard Brinkman System with Singular Potential and Degenerate Mobility

TL;DR

This paper establishes the separation property for a nonlocal Cahn Hilliard Brinkman system with singular potential and degenerate mobility, ensuring solutions stay away from pure phases, using a novel De Giorgi iteration approach.

Contribution

It introduces a new method based on De Giorgi iteration to prove the separation property for complex nonlocal Cahn Hilliard systems with singular potentials and degenerate mobility.

Findings

Proves the separation property for the system.

Extends previous results to more general systems.

Provides a new analytical approach for similar models.

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.