On nonnegative solutions for the Functionalized Cahn-Hilliard equation with degenerate mobility

TL;DR

This paper investigates the existence of nonnegative weak solutions for a gradient flow of the Functionalized Cahn-Hilliard equation with degenerate mobility, relevant for modeling phase-separated mixtures of amphiphilic molecules.

Contribution

It establishes the existence of nonnegative weak solutions for the equation with degenerate mobility by approximation and energy analysis.

Findings

Constructed weak solutions as limits of nondegenerate cases

Verified energy dissipation inequality for solutions

Ensured solutions remain nonnegative under initial positivity

Abstract

The Functionalized Cahn-Hilliard equation has been proposed as a model for the interfacial energy of phase-separated mixtures of amphiphilic molecules. We study the existence of a nonnegative weak solutions of a gradient flow of the Functionalized Cahn-Hilliard equation subject to a degenerate mobility M(u) that is zero for u<=0. Assuming the initial data u0(x) is positive, we construct a weak solution as the limit of solutions corresponding to nondegenerate mobilities and verify that it satisfies an energy dissipation inequality.