New methods for the study of supercritical wave equations

Suny At Albany, Albany NY

Investigators

Abstract

In this project the principal investigator will study dispersive critical and supercritical partial differential equations, further developing several recently introduced methods. Partial differential equations provide a rigorous mathematical description of many important physical, ecological, and economical theories (e.g., Einstein's theory of general relativity, the Black-Scholes model of option pricing in finance). They are studied both analytically and numerically in order to determine the long-term behavior of solutions and their dependence on initial data. In many cases, physically meaningful concepts, such as energy, mass, and momentum, play important roles, since having good control of these quantities may enable one to prove that solutions exist globally in time and to study their behavior. However, it is of great interest to study equations (called energy-supercritical) for which such quantities are insufficient to govern the solution. This class of highly nonlinear equations includes many famous and important ones that arise from modeling physical phenomena (e.g., Einstein's equations, the Navier-Stokes equation, the Yang-Mills equations, and Euler's equations). The goal of the project is to understand, as far as possible, the behavior of "large" solutions to wave equations of supercrticial type, about which little is currently known. The principal investigator's potential contribution to the subject is based on two methods: (1) splitting solutions into incoming and outgoing parts and (2) a comparison principle for the wave equation. His preliminary results apply only to some (canonical) model cases, but seem promising. It will be interesting to generalize them to the nonradial case, to all dimensions, and to other types of nonlinearities and equations. One objective of the project is to take techniques (comparison principles, sub- and supersolutions) specific to the study of elliptic and parabolic equations and apply them in a meaningful way to the supercritical wave equation. The author's approach is different from those of Kenig and Merle and of others (Tao, Krieger and Schlag, Li, Wang and Yu). It leads to unconditional results and to sharp scattering criteria in several cases. Another avenue of investigation is that of linear and nonlinear evolution equations driven by time-dependent potentials. The potential can be deterministic or random (e.g., Brownian motion). This has clear applications to the study of soliton stability and some physical problems. The methods that the principal investigator has introduced, such as a structure formula for wave operators and the use of an abstract Wiener theorem to prove dispersive estimates, provide an improved way of looking at such equations, by allowing a more general time dependence of the potential, and permit stronger conclusions to be reached.