Nonlinear Hyperbolic and Dispersive Partial Differential Equations Arising from Physics

University Of California-Berkeley, Berkeley CA

Investigators

Abstract

Nonlinear hyperbolic and dispersive partial differential equations (PDE) are fundamental to describing natural phenomena across all scales, from subatomic particle dynamics to electromagnetism, fluid dynamics, and the behavior of astronomical bodies. This project aims to deepen our rigorous understanding of these equations by investigating a strategically chosen array of key problems concerning long-term dynamics, singularity formation, and the stability or instability of special solutions. The insights gained will clarify highly nonlinear phenomena in physics, including gravitational singularities in general relativity, soliton resolution in dispersive models, and small-scale formation in plasma physics. The project will train undergraduate, graduate, and postdoctoral researchers by mentoring research projects and organizing research-focused seminars. The main scientific goals of the project are as follows. First, for the Einstein vacuum equation, the Principal Investigator (PI) plans to make direct progress towards the Strong Cosmic Censorship conjecture for perturbations of the subextremal Kerr black hole spacetimes, based on the recently introduced technique for computing and justifying generic late-time tails. Second, for the Skyrme model, the goal is to prove the asymptotic stability of the equivariant B = 1 Skyrmion, based on the PI's recent work on the asymptotic stability of the 3-dimensional catenoid. Third, for critical semi-linear geometric dispersive equations, the PI plans to attack the Soliton Resolution conjecture along a sequence of times without any symmetry assumptions, based on the PI's work on critical geometric flows. Additionally, the project aims to provide a more precise description of generic dynamics under suitable symmetry assumptions, which goes beyond the Soliton Resolution conjecture. Finally, for the Hall-magnetohydrodynamics equations in plasma physics, the PI plans to utilize the instability mechanism of degenerate dispersion to establish enhanced (or even anomalous) dissipation, based on the PI's work on ill-posedness by the same mechanism in the non-dissipative case. 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.