Evolution Equations and Elliptic Boundary Problems in PDE

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This mathematics research project will investigate problems in partial differential equations, including both evolution equations and stationary problems. Evolution equations to be investigated include nonlinear wave equations on non-Euclidean spaces, nonlinear Schrodinger equations, and Euler and Navier-Stokes equations (NLS). Specific goals include investigation of strong dispersive estimates for nonlinear wave equations, local and global solutions to NLS whose initial data satisfy conditions not given purely in terms of Sobolev space estimates, standing wave and traveling wave solutions to nonlinear wave and NLS equations and their stability, small viscosity limits of solutions to Navier-Stokes equations, and Euler and Navier-Stokes equations on regions with rough boundary. Stationary problems to be investigated include elliptic boundary problems on rough domains and analysis on manifolds with weakly bounded geometry. Specific goals include investigation of Fredholm theory for as large a class of uniformly rectifiable domains as can be managed, looking for an appropriately general version of the notion of a regular elliptic boundary problem in this nonsmooth context. This builds on work with Hofmann and Mitrea on classes of chord-arc domains (also called SKT domains), and looks forward to more general classes, including domains with inclusions. Investigations of these topics will result in the development of useful analytical tools for partial differential equations. These will likely include new results in the areas of singular integral operators, harmonic analysis of elliptic operators, geometric measure theory, and singular perturbation theory.This mathematics research project in the area of partial differential equations makes contact with a range of scientific issues. The Navier-Stokes equations bear on the study of fluid flows, both how a fluid will flow through a pipe and how that flow will transport solvents, and what sorts of different behaviors govern flows in pipes with smooth or rough walls. Nonlinear Schrodinger equations model various real-life phenomena and have applications to subjects such as optical cables, surface waves in shallow water, and a delicate state of low temperature matter known as a Bose-Einstein condensate. Potential applications of the study of equations on rough domains include the investigation of hidden structures, such as sudden transitions in the nature of materials in the earth, often across rough, unknown boundaries, the knowledge of whose locations can be of substantial value. The mathematical techniques developed in the proposed project will have the potential to impact these areas. Taylor has directed three Ph.D. students in the past four years, who have begun to make their own contributions to such problems, and he is now beginning to train a new student.