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

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 non-degenerate mobilities and verify that it satisfies an energy dissipation inequality. Our approach is a combination of Galerkin approximation, energy estimates, and weak convergence methods.

Keywords

• Weak solutions
• Nonnegative solutions
• The Functionalized Cahn–Hilliard equation
• Degenerate mobility