Professor Taylor proposes to investigate problems in partial differential equations, with particular attention to two areas. The first area involves analysis on rough geometries. This includes analysis on domains with rough boundaries. Work with S. Hofmann and M. Mitrea has developed the theory of singular integral operators on a class of uniformly rectifiable domains that the authors call regular Semmes-Kenig-Toro domains, characterizing them as the class for which certain important families of layer potentials are compact, and has opened up what promises to be a very successful expansion of the theory of elliptic systems to this class of roughly bounded domains. Another topic in this area that Taylor will continue to investigate involves wave propagation on manifolds with rough metric tensor, arising as Gromov-Hausdorff limits of a class of smooth Riemannian manifolds with Ricci tensor bounded below. The second area involves evolution equations. Taylor has worked with A. Mazzucato, M. Lopes Filho, and H. Nussenzveig Filho and produced a detailed analysis of the vanishing viscosity limit of 2D circularly symmetric Navier-Stokes flows, with particular emphasis on the behavior of the boundary layer. Taylor and Mazzucato have been investigating a class of 3D channel flows, and have a program that is beginning to yield results on this technically more challenging problem. They plan to push further the results they have on these flows, variants such as pipe flows, and related singular perturbation problems. Taylor also proposes to continue to investigate analogues of the Gibbs phenomenon and the Pinsky phenomenon that arise in the short time behavior of solutions to nonlinear dispersive equations with discontinuous initial data, expanding the scope from nonlinear Schrodinger equations to other classes.
The analytical tools developed to tackle these problems have the potential to impact both the understanding of fundamental mathematical issues and the understanding of physical phenomena, such as wave phenomena and fluid flow phenomena, which these mathematical theories address. Taylor?s work on boundary problems has dealt with the problem of how to identify a region by how its boundary vibrates, and is part of a vigorous effort to identify hidden structures, be they behind a wall or underground. Many important fluids have very small viscosity, and understanding the role of this small viscosity in their behavior is an old and challenging problem, on which recent progress can produce new insights. The proposed work also has applications to the understanding of propagation of electromagnetic waves and elastic waves, which are vibrational modes in solid materials. The project will also play a role in training graduate students, including Taylor?s Ph.D. students.
