Vortices, phase boundaries and defects arising in nonlinear PDE and variational models

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In this project the PI will investigate a collection of mathematical models for important phenomena related to continuum mechanics. These include the study of certain energy-conserving flows, in the plane or on a curved surface such as a sphere, that contain vortices-points on the surface around which the flow rotates. A second project involves a model for nematic liquid crystals. These are rod-like molecules that behave in part like a liquid and in other ways like solid crystals. Recently there has been much activity in the materials science community aimed at designing liquid crystals to take on desirable material properties when deposited on a curved surface and the PI will explore how to adapt the known models for nematics to this relatively unexplored setting of liquid crystals on surfaces. A final project concerns a geometric problem that stands as a paradigm for periodic pattern formation, that is for physical systems where one expects two different states of matter to be separated by interfaces that on the one hand tend to minimize interfacial surface area (like a soap bubble does) and on the other hand tend to develop patterns that replicate themselves on a small scale throughout a large sample.This proposal concerns the study of nonlinear partial differential equations and variational problems drawn from continuum mechanics and related fields. These include the Gross-Pitaevskii system, the Landau-deGennes model for nematic liquid crystals and a nonlocal variant of the classic isoperimetric problem related to models for diblock co-polymers. The goal in these investigations is to describe the behavior of solutions to these systems in terms of lower-dimensional objects--vortices, defects or phase boundaries within appropriate asymptotic regimes for the parameters arising in the models. A major theme is to analyze how solutions to some of these models behave when posed on curved surfaces. These objects--vortices, defects or phase boundaries--largely characterize the state of the overall system. For the project on Gross-Pitaevskii, an additional purpose is to draw a deeper connection between this important nonlinear Schrodinger equation and the very well-studied point-vortex problem. The point-vortex problem is more commonly associated with incompressible Euler flow in the context of fluid mechanics but here, focusing on periodic solutions, we intend to strengthen a bridge to a setting in quantum mechanics. For the problem of nematics on surfaces, we hope to give rigorous mathematical support to the growing body of work by materials scientists studying liquid crystals deposited on curved surfaces. For the nonlocal isoperimetric problem our goal is in part to illuminate how fine scale periodic structures emerge as the strength of the nonlocality grows. The methods to be employed in this program include a combination of constrained minimization techniques, Gamma-convergence and geometric measure theory. The educational component will include the involvement of doctoral students on many of the projects.