Questions in Wave Turbulence and Quantum Kinetic Theories

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This research is devoted to developing mathematical methods for studying two different classes of physical phenomena: wave turbulence and quantum kinetics. Wave turbulence is manifested in many physical systems, including familiar water waves, water surface gravity and capillary waves, and a great variety of waves in plasmas (in particular, in fusion devices). A Bose-Einstein condensate is a state of matter that was first predicted theoretically in 1924 and produced experimentally in 1995. While quantum phenomena are exhibited on micro-scales, and nature is well described on macro-scales by classical mechanics, for a Bose-Einstein condensate macroscopic quantum phenomena become apparent. The quantum kinetic theory that describes some physical properties of this state of matter is the second subject of this research. Although wave turbulence and quantum kinetics are dramatically different as physical phenomena, their mathematical description uses similar equations. Despite the widespread applications of these kinetic equations, little is known rigorously in this field. The aim of the project is to develop mathematical techniques and apply them to concrete physical situations, leading to a more profound understanding of both wave turbulence and quantum kinetics.The project is aimed at developing the theory of the existence, uniqueness, finite-time condensation, and relaxation to equilibrium for strong solutions of the kinetic equations. The principal investigator will address the local-in-time existence via techniques of harmonic analysis and novel Strichartz-type estimates. For important questions related to 3-wave wave turbulence, regularity theory for the kinetic equation with a particular broadening (regularization) of the resonance set will be used. This research project is informed by several sub-fields of mathematics, physics, and chemistry, and the work has potential applications in optical turbulence, oceanography and atmospheric sciences, and quantum physics.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.