Mathematical Questions in Kinetic Theory

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This research project will address fundamental problems concerning solutions to partial differential equations that are used to model the dynamics of plasmas and fluids in physics. The focus will be on studying the fundamental law of physics to predict mixing and relaxation of macroscopic quantities in the large time. The project will contribute new mathematical techniques to the theory of partial differential equations and the field of mathematical physics, dynamical systems, and applied mathematics. In addition, it will advance our understanding of turbulence in plasma physics and fluid dynamics. The project provides training opportunities for graduate students and other early-career researchers. The project seeks to advance beyond the study of mixing and relaxation, or Landau damping, in plasma physics and fluid dynamics, and to address fundamental stability problems concerning the behavior of solutions with limited regularity. The fundamental equations to be studied include the classical Vlasov models in plasma physics and the Euler and Navier-Stokes equations in fluid dynamics. The goal is to provide new insights into Landau damping and mixing in plasmas and fluids. The main approaches will involve mathematical techniques from spectral theory, resolvent analysis, Fourier analysis, dispersive PDEs, probability, and statistical 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.