Phase mixing in the fluid mechanics and kinetic theory

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Nonlinear mixing is a ubiquitous behavior observed in fluids and plasmas which plays an important role in understanding certain aspects of atmosphere and ocean sciences, turbulence, astrophysics and controlled fusion. For example, it is thought to be an important organizing mechanism in atmospheric dynamics that helps to explain why hurricanes retain their characteristic vortex shape for extended periods of time. Despite being a fundamental physical mechanism, the theoretical and mathematical analysis of nonlinear phase mixing has remained elusive. Moreover, many of these effects are difficult to isolate in computational or physical experiments and here mathematical analysis can provide important insights. This project specifically aims to expand our mathematical understanding by studying the effect in a variety of fundamental 'case studies' found in fluid mechanics and plasma physics.The purpose of this program is to further the mathematical analysis of nonlinear phase mixing, generally regarded as difficult due to the appearance of highly non-normal linear evolutions and unusual regularity issues intertwined with subtle nonlinear resonances. The PI will work to develop new tools necessary to approach these problems with a focus on working towards four specific settings: the vanishing viscosity limit of the incompressible Navier-Stokes equations (NSE) near the 2D Couette flow, the nonlinear instability of 3D shear flows, inviscid 'vortex axisymmetrization' in 2D Euler and Landau damping in Vlasov-Poisson (or Vlasov-Maxwell) in presence of external magnetic fields. Many related physically relevant problems exist, and may serve as intermediate steps or additional lines of research. The work in fluid mechanics will expand further on ideas employed by the PI and collaborator Nader Masmoudi in the recent work on the stability of 2D Couette flow [arXiv:1306.5028], especially the use of adaptive coordinate systems and the construction of norms which account for weakly nonlinear resonances and the non-normal transient linear behavior. The work in kinetic theory will further develop techniques used in the work of the PI and collaborators Nader Masmoudi and Clement Mouhot on Landau damping [arXiv:1311.2870], especially the efficient treatment of the plasma echoes. Most likely, in order to expand the range of kinetic theory applications, some ideas from the work in fluid mechanics may need to be adapted to the kinetic setting.