Mathematical Analysis of Fluid Flow at High Reynolds Number from the Point of View of Turbulence

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While the equations describing the motion of incompressible fluids have been around for more than two centuries, the underlying mathematics is still not fully understood. Do the solutions of the Navier-Stokes equations correctly describe what we see in fluid experiments? Can we use these equations to make precise predictions about the flows, to predict tomorrow's weather, or to predict the macro-scale evolution of the Earth's climate? These questions are conjecturally related through the phenomenological theories of turbulence and the statistical properties of solutions to the Euler and Navier-Stokes equations. When attempting to give rigorous answers to these questions we are faced with new frontiers in mathematical analysis, and fundamental new ideas are needed to understand the underlying phenomena. In this project the investigator studies the relation between turbulence and the equations that are used to describe fluid flows. The project focuses on furthering our understanding of the hypothesized link between fluid turbulence and the Navier-Stokes equations. The complexity of turbulent flows observed in experiments translates into fundamental mathematical issues, chief among which are the problems of singularities, uniqueness, and stability in the fluid equations. The investigator and colleagues attack these problems from two intimately related angles: by analyzing the statistical properties of solutions to stochastic partial differential equations; and by studying the emergence of singularities in the Euler and related active scalar equations. In tackling these problems the investigator appeals to ideas from hypoellipticity (in the sense of Hormander), convex integration, Lagrangian particle adapted methods, and Landau damping. The goal is to develop nonlinear, solution-adapted methods.