Long-Term Dynamics of Nonlinear Dispersive and Hyperbolic Equations: Deterministic and Probabilistic Methods

This project focuses on several problems connected with the long-term dynamics of solutions of partial differential evolution equations. Equations of this type are central to the modeling and simulation of a wide range of phenomena, including water waves, propagation of light, behavior of plasmas, and gravitation. The investigators are senior researchers with experience in different aspects of the study of partial differential equations; their complementary expertise will be brought to bear on four main research directions in the study of solutions of dispersive and hyperbolic equations. The four aspects of nonlinear partial differential equations are all manifestations of the general goal of understanding the long time dynamics of solutions to important nonlinear dispersive and hyperbolic equations.

This project is a collective effort of a group of five senior researchers from four institutions with research experience in various areas of partial differential equations. The investigators have been interested in both dispersive and hyperbolic partial differential equations and will collaborate to approach the problems under study from several different directions. The project centers on four main research directions in the study of solutions of dispersive and hyperbolic equations: (1) the analysis of critical semilinear evolutions, with special emphasis on the "soliton resolution conjecture" for extended type II solutions, (2) construction of global solutions of certain quasilinear dispersive models, such as water wave models, plasma models, and crystal optics, (3) stability problems in General Relativity, motivated by the outstanding Kerr nonlinear stability conjecture, and (4) long-term dynamics of solutions corresponding to randomized data. These four aspects of nonlinear partial differential equations are all manifestations of the general goal of understanding the long time dynamics of solutions. While specific major problems are identified for the project, this is a very active field, and it is anticipated that this project will identify and study other important problems during the course of the investigation. The project will also enhance links and collaborations among researchers at leading mathematical centers in the U.S., promoting training through active research involvement of students and postdoctoral researchers. Research resulting from this project will be disseminated widely in advanced graduate courses, survey articles, and monographs.