Linear and nonlinear problems in dispersive Partial Differential Equations

University Of California-Los Angeles, Los Angeles CA

Investigators

Abstract

Wave dispersion, the phenomenon of waves with differing frequencies traveling with different speeds, is the norm for waves in media. Nonlinear effects are precisely those that generate interactions between such waves. The principal objective of this project is to study models that combine both phenomena. However, it has become evident that further progress on these problems requires a deeper understanding of the purely dispersive effects; correspondingly, a significant fraction of the project is directed toward this goal. In particular, the principal investigator will focus on situations where the dispersive behavior is complicated either by taking place in a region of finite extent or in a heterogeneous medium. Although the project focuses on simple models, these embody fundamental hurdles appearing much more broadly in the theory of wave motion. All the models under consideration can arise as mechanical systems. It has recently been discovered that, at least for finite systems of particles, this places considerable restrictions on the possible behavior, far beyond those discovered in the nineteenth century; however, the full nature of these restrictions is poorly understood. Part of the project is to exhibit (for the first time) such restrictions for equations modeling infinite systems of particles, spread over an infinite volume. The project concerns several topics in dispersive partial differential equations, both linear and nonlinear. Strichartz estimates encapsulate much about the linear flow in a manner well-adapted to the nonlinear theory. Further study of such estimates in the setting of Schrodinger equations on compact manifolds will be undertaken, building on recent dramatic advances, including those of Bourgain and Demeter. The treatment of scaling-critical nonlinear equations requires one to understand not just estimates for the linear flow, but also the defects of compactness therein. Toward this direction, the principal investigator seeks to obtain mass-critical profile decompositions for models with nonconstant coefficients, building on some initial successes in this direction. Turning now to truly nonlinear questions, nonsqueezing results will be sought in two distinct infinite-volume settings. Recall that nonsqueezing is a peculiar property of general finite-dimensional Hamiltonian flows uncovered by Gromov saying that no ball can flow wholly into a cylinder whose (symplectic) cross-section has lesser radius. Past work has been restricted to tori, which aids significantly in the development of finite dimensional approximations. Due to substantial differences in the nature of complete integrability, the low-regularity theory of the Korteweg-de Vries equation is further advanced on the torus than on the whole line. The project seeks to make some inroads on the low-regularity problem for the line, building on a recently developed method for global control of Sobolev/Besov norms that works simultaneously in both geometries.