Dispersive and Wave Equations in the Presence of Background Geometry

This project seeks to further the understanding of solutions to the wave equation on geometric backgrounds. To achieve the requisite accuracy, GPS, for example, relies on general relativity, which in turn postulates that gravity is the result of a curved space-time on which waves travel rather than an external force. And in many applications involving nonlinear equations, the governing systems have geometry that depends on the solution but the solution in turn depends on the geometry. Einstein's equations, which are at the heart of general relativity and determine the evolution of a universe from a given starting state, can be realized as such a system of wave equations in certain coordinate systems. On background geometries, waves flow along special curves called geodesics rather than rays as is more familiar. A phenomenon called trapping occurs when some of these geodesics remain in a bounded set for all time. This occurs, for example, on known black hole space-times where there are indeed regions where light orbits the black hole rather than tending toward infinity. Trapping is a known obstruction to typical measures of dispersion, and a major focus of this project is to precisely quantify its effect in numerous scenarios. The project provides research training opportunities for both undergraduate and graduate students.

The problems to be examined largely focus on integrated local energy estimates, which are generalizations of the original estimates of Morawetz, and their application to nonlinear equations. Major initiatives include improving our understanding of such estimates in the presence of trapping and on non-stationary backgrounds. The construction of space-times with degenerate trapping provided the first examples where an algebraic loss of regularity is both necessary and sufficient for recovering local energy estimates. Numerous questions related to these examples remain unexplored, including the discovery of space-times with degenerate trapping outside of the highly symmetric warped product setting. Local energy estimates can be used to establish long-time existence for nonlinear equations and have particular benefits in the presence of background geometry. Planned work related to this include the examination of wave equations on half-spaces and applications of a related weighted estimate of Dafermos and Rodnianski to critically damped equations related to the Strauss conjecture.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.