UCLA

Degenerate diffusion equations and their interface dynamics have received a lot of attention in the past couple of decades. In particular, surfaces moving with curvature dependent velocities and discontinuous diffusion intensities naturally appear in physical and biological models. In this dissertation, we study global well-posedness and geometry of two equations: mean curvature flows with forcing and degenerate nonlinear parabolic equations with discontinuous diffusion coefficients. Both problems have gradient flow structures in the space of sets and the Wasserstein space, respectively, which are useful to study the global-time behavior.

In Chapter 1, we develop a parabolic version of the Aleksandrov and Serrin's moving plane methods for mean curvature flow with forcing. With the class of forcing which bounds the volume of evolving sets away from zero and infinity, we show that a strong version of star-shapedness is preserved over time. Based on this geometric property, we prove that volume preserving mean curvature flow starting from a star-shaped set converges to a ball.

Chapter 2 is devoted to the study of degenerate parabolic equations with discontinuous diffusion intensities. We show the existence and uniqueness of the solutions in the sense of distributions. Our notion of solutions allows us to give a fine characterization of the emerging critical regions, observed previously in numerical experiments.