Singularities and Long Time Dynamics in Nonlinear Dispersive Flows

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The main goal of this project is to study the solutions to a wide class of equations that can best be described as nonlinear waves. These mathematical equations model a broad range of physical phenomena in our universe and even in our daily lives, including the water waves at the surface of our oceans and the quantum-mechanical interaction of multi-particle systems, as well as, on a broader scale, gravitational waves and their evolution in the proximity of black holes. The common feature in these models lies in the strength of the wave interactions, which drastically affects both their short-time and long-time behavior. Phenomena to be studied include both short-time breakdown of waves, as well as long-time persistence and/or dispersion and decay of waves. The objective of this project is to attack a broad array of challenging problems within the field of nonlinear dispersive equations. The equations to be considered range from semilinear to fully nonlinear, and the types of properties to be investigated span from short-range to long-range behavior, from scattering to blow-up questions. The research connects ideas and methods in partial differential equations to geometry, harmonic analysis, complex analysis, and microlocal analysis.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.