Harmonic Analysis and Geometric Partial Differential Equations

University Of Massachusetts Amherst, Amherst MA

Investigators

Abstract

Nahmod's research lies in the overlap of harmonic analysis, geometry and partial differential equations. It aims at studying the behavior of nonlinear waves arising in geometry, ferromagnetism and gauge field theories; and that of functions along vector fields whose integral curves lack sufficient curvature. In the first part the focus is in geometric partial differential equations. Of special interest are Schroedinger maps, Wave maps and other gauge field theories such as the Yang Mills equations in Minkowski space-time. All of these equations model `wave like phenomena'. Their solutions arise as minimizers of the corresponding energy functionals. To conform with natural physical situations, it is of interest to study their existence, uniqueness under minimal regularity assumptions. These are difficult issues because the nonlinearities of these equations involve not just the solutions but also their derivatives. Nahmod will address these questions and plans to show that in the scale invariant set up solutions to the Cauchy initial value problem exist globally provided that the data is sufficiently small when measured relative to the critical regularity norm. She also plans stability issues; e.g. whether such a system remains close to its initial state as time evolves when the data has small energy. From a physical viewpoint the latter models whether such systems are close to equilibrium. The techniques exploit geometric aspects of these equations to extract crucial information -such as special structures in the nonlinearity- which is then used in the analysis. The method combines deep Fourier analysis with gauge theoretic geometric tools. The goal of the second part is the study of the Hilbert transform along vector fields and its associated maximal operator in two dimensions. Their treatment departs from the classical study of singular integrals for in the present situation, the singularity lives on some variety that is changing at each point. Nahmod will investigate how to develop time frequency techniques to study operators under no curvature assumptions. This is the case, for example, in studying differentiability properties of functions along vector fields. Partial differential equations are the mathematical models to the laws governing much of the phenomena in our physical world. The wave equation models the propagation of different kind of waves -such as light waves- in homogeneous media. Nonlinear models of conservative type arise in quantum mechanics while other variants appear for example in the study of vibrating systems and semiconductors. The nonlinear Schroedinger equation arises in various physical contexts in the description of nonlinear waves- such as propagation of a laser beam in a medium whose index of refraction is sensitive to the wave amplitude, water waves at the free surface of an ideal fluid as well as in plasma waves. Some of the interesting questions are those about local and global existence of solutions, uniqueness as well as long time behavior of global solutions. The role of mathematical analysis is to understand the behavior of the solutions to these equations, provide the tools to extract their quantitative and qualitative information and lay the foundations upon which methods to accurately approximate the solutions are developed. Fourier analysis and more generalized adapted frequency decompositions such as time-frequency analysis' consists in decomposing complex objects via `modulated waveforms' into basic building blocks which are localized and easy to understand, and then piecing them back together in a straightforward manner. It works very similarly to a musical score. The modulated waveforms have four attributes: amplitude (loudness), scale (duration), frequency (pitch) and position (instant it is played). The objects could be speech, radar signals, as well as oscillatory expressions arising in optics, AC ousting scattering, wave propagation and other phenomena of nonlocal nature.