Speaker: Tim Roberts (U Chicago)
Title: Existence of spots in a Cahn-Hillard model using multiple spatial scales
Abstract: A striking feature of many partial differential equations are are localized patterns: self-sustaining structures that show large variation only in small regions of the spatial domain. Such patterns appear throughout applications in the physical biological sciences ranging from vegetation spots in arid ecosystems, to diffraction patterns in nonlinear optics. They also present a substantial mathematical challenge, as they often cannot be described by simple linear instability, instead requiring genuinely nonlinear effects to emerge. In this talk, I will describe recent work providing rigorous existence results in a Cahn-Hilliard model for amphiphilic copolymer molecules. We are able to show that radially symmetric patterns exist, and find asymptotic expansions for their radius depending on the physical parameters of the model. This is joint work with Paul Carter (UCI).