CAREER: Dynamics of Nonlinear Dispersive Partial Differential Equations

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From the dynamics of subatomic particles to electromagnetism, fluids, plasmas and gravity among astronomical bodies, nature is governed by nonlinear dispersive partial differential equations at all scales. The objective of this NSF CAREER project is to enhance the rigorous understanding of the long-term behavior and singularities of solutions to nonlinear dispersive partial differential equations of physical origin by attacking strategically chosen open problems for a wide array of such equations. The theoretical advances from this project will lead to a clearer and more effective understanding of highly nonlinear phenomena in physics, such as solitons, gravitational singularities in general relativity and magnetic reconnection in plasma physics. The project will train undergraduate, graduate, and post-doctoral researchers by developing accessible expositions of modern research topics, organizing summer workshops and mentoring research projects.Specifically, for three classes of nonlinear dispersive equations at different levels of complexity, the following goals will be pursued: (i) for the energy-critical wave maps and (hyperbolic) Yang–Mills equations, which are examples of relativistic field theories, to prove a new forward-in-time scattering criterion for “initially outgoing” solutions, with a view towards resolving the soliton resolution conjecture in the one-soliton regime, and along a sequence of times in general; (ii) for linear and nonlinear wave equations on a black hole background, to develop a Fourier-based approach that rigorously justifies the sharp late-time asymptotics along the event horizon, which will be a starting point for investigation of the linear and nonlinear instability of the Kerr Cauchy horizon, and ultimately the strong cosmic censorship conjecture in the vicinity of the Kerr spacetimes; (iii) for the Hall-magnetohydrodynamics equations in plasma physics, to establish a general local wellposedness theory, which is interesting in view of the recent work of the PI that proved illposedness of the Cauchy problem in the vicinity of the trivial solution. Insights from various disciplines of mathematics, such as differential geometry, calculus of variations, harmonic analysis, spectral theory and microlocal analysis, will naturally enter in attaining the above goals.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.