Abstract Taylor 9600065 Prof. Taylor will investigate problems in partial differential equations, with particular attention to four areas. One involves the further development of microlocal analysis as a tool in nonlinear partial differential equations. A basic component of this theory is the paradifferential operator calculus, which allows one to represent a nonlinear function F(u) as the sum of a linear operator applied to u and a remainder. Recent developments highlight the desirability of extending the scope of this theory, in ways which bear on the study of solutions to nonlinear partial differential equations with low regularity. The second involves the use of analysis on Morrey spaces. Created originally to analyze solutions to nonlinear elliptic equations, Morrey spaces have proved useful for other sorts of equations, such as the Navier-Stokes equations for viscous fluid flow. Taylor plans to pursue a systematic study of the applicability of Morrey spaces to nonlinear PDE. A particular example is the extension of the Morrey space analysis to Navier-Stokes equations on a region with boundary. The third area concerns an inverse scattering problem: the problem of describing an unknown obstacle in terms of observations of how it scatters waves. Microlocal analysis is brought to bear on the problem of describing features of such an obstacle, on a length scale comparable to the wave length used to illuminate the obstacle. The fourth area concerns analysis of diffraction of waves by a boundary. In particular, various Airy operators arise in this study, and while many basic properties are well understood, a number of natural problems from mathematical physics, including propagation of electromagnetic waves, give rise to problems about the Airy operator calculus which need to be resolved. A central theme in these studies is the propagation of waves, both linear and nonlinear. A number of types of waves are considered, including sound waves, waves in water, light waves and other electromagnetic waves, and vibrations in the ground and in other solids. How these waves propagate through space and interact with objects, often in the form of boundaries, is a rich source of problems. Of equal interest are inverse problems, how to tell about the space in which waves propagate, from a knowledge of the nature of their propagation. Understanding these inverse problems is helpful in a variety of practical problems, ranging from nondestructive medical imaging to finding special structures under ground and under water.
