The research supported in this proposal concerns the study of nonlinear dispersive and wave equations. These arise as fundamental models of a wide variety of physical systems, including the propagation of waves and the study of nonlinear optics, and are closely related to models of fluids, as well as a number of aspects of statistical and quantum mechanics. The development of mathematical tools to understand important issues such as existence and uniqueness of solutions, as well the behavior of the corresponding evolutions on a qualitative and quantitative level, is therefore an issue of fundamental scientific importance. As a particular example, several of the questions to be investigated in this research involve studying long-time properties of solutions to the nonlinear Schrodinger and nonlinear wave equations evolving from "generic" randomly chosen initial data. In addition to being of substantial mathematical interest, investigation into these questions addresses the broader scientific issue of whether possible singularities arising in the mathematical formulation can occur in physically relevant settings. Moreover, the topics to be studied are closely connected to a wide range of issues in partial differential equations, probability, and harmonic analysis, and the ideas and techniques developed will provide important contributions to the availability of mathematical tools for future research on related questions.
The particular scope of this research project is to investigate several problems concerning local and global well-posedness properties for nonlinear dispersive and wave equations, focusing on the nonlinear Schrodinger, nonlinear wave, and Korteweg-de Vries equations. The research encompasses four broad directions, which are often interrelated, and which each contribute to the wider theme of understanding the precise dynamical behavior of solutions both locally and globally in time. The first two directions of interest concern global in time existence of solutions, in particular focusing first on the development of global well-posedness results adapted to the energy-supercritical setting, and second on important issues surrounding probabilistic global well-posedness results in endpoint and limiting situations (in this probabilistic framework, initial data for the problem is chosen as a random Fourier series, and results are obtained by excluding sets occurring with small probability). The third direction of interest turns to the study of local in time stability properties of solutions, focusing in particular on initial data of low-regularity. The fourth direction of study investigates dispersive models with higher-order nonlocal terms, in which a key ingredient will be the adaptation of recent developments in the study of nonlinear elliptic and parabolic partial differential equations with similar nonlocal features. In each of these settings, the PI and collaborators will incorporate techniques from harmonic analysis, probability, and spectral theory to analyze the dynamical features involved in the evolution.
