Wave Turbulence and Long-Time Dynamics of Dispersive Partial Differential Equations

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The great American physicist Richard Feynman described turbulence as the "the most important unsolved problem of classical physics." Here, he was referring to hydrodynamic turbulence, which is the phenomenon one observes on numerous occasions in daily life, particularly when one travels through fluids either in watercraft on the ocean or in airplanes in the atmosphere. Despite its intuitive manifestations, the scientific understanding of turbulence is far from satisfactory. A related phenomenon is "wave turbulence," which pertains to similar problems but for different physical systems involving wave interactions (e.g.,ocean or plasma waves). The aim of this project is to gain a better understanding of certain phenomena pertaining to wave turbulence from a rigorous mathematical viewpoint and thereby to take the first steps towards putting the theory on solid mathematical foundations.The project addresses two different regimes of long-time behavior for nonlinear dispersive and wave partial differential equations. The first regime can be characterized by "out-of-equilibrium dynamics," in which solutions do not exhibit long-time stability around equilibrium configurations. This is the typical behavior of nonlinear dispersive equations posed on compact domains, where dispersive effects do not translate to decay, and is the natural setting of wave turbulence theory. The other regime of long-time behavior concerns dispersive partial differential equations posed on Euclidean space, for which one can hope to have stable equilibrium points for the dynamics (stationary solutions). The "asymptotic stability" of trivial equilibria (zero solutions) is mostly well understood by now, but that of nontrivial stationary solutions is far from settled. The project suggests several avenues of research to improve our understanding in the two aforementioned directions.