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

• Published in: Results in applied mathematics Vol. 12; p. 100195
• Main Authors: Dai, Shibin, Liu, Qiang, Luong, Toai, Promislow, Keith
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 01.11.2021; Elsevier
• Subjects: Degenerate mobility; Nonnegative solutions; The Functionalized Cahn–Hilliard equation; Weak solutions; Degenerate mobility; Weak solutions; Nonnegative solutions; The Functionalized Cahn–Hilliard equation
• ISSN: 2590-0374; 2590-0374
• DOI: 10.1016/j.rinam.2021.100195

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.