seminar: Instabilities in the Cahn-Hilliard equation and the Willmore functional
speaker: Carola-Bibiane Schoenlieb (Department of Applied Mathematics and Theoretical Physics (DAMTP), University of Cambridge)abstract: The Cahn-Hilliard equation is a fourth order reaction diffusion equation and is a classical model for phase separation and subsequent phase coarsening of binary alloys. In this talk we are interested in the finite-time stability of transition solutions of the Cahn-Hilliard equation and its connection to the Willmore functional. The Willmore functional has its origin in differential geometry where it appears as a phase field approximation for solutions of the so called Willmore problem. We show that the Willmore functional locally decreases or increases in time in the linearly stable or unstable case of the Cahn-Hilliard equation respectively. This linear analysis explains the behaviour near stationary solutions of the Cahn-Hilliard equation. We perform numerical examples in one and two dimensions and show that in the neighbourhood of transition solutions local instabilities occur in finite time. Beside that we show convergence of solutions of the Cahn-Hilliard equation for arbitrary dimension to a stationary state by proving asymptotic decay of the Willmore functional in time.
timetable:
Wed 28 May, 15:00 - 16:00, Aula Dini
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