STICKS - Stochastic Mikado Flows for Fluid Mechanics

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Fluid dynamics and turbulence play an extraordinarily important role in science and technology, with applications ranging from engineering to climatology and astrophysics. The partial differential equations (PDEs) that model the flow of fluids give rise to several paradoxical, weak solutions. For example, a weak solution could start at a complete rest, then suddenly start to move in a most irregular fashion, only to fall completely quiet again in the end. This kind of irregular behaviour clearly violates the fundamental conservation of energy and is conjectured to be important in the formation of turbulence, where it is known as anomalous dissipation. It has been the subject of intense research in the last two decades, leading recently to a mathematically rigorous proof of the so-called Onsager conjecture that gives a precise description of how smooth a flow has to be in order to be energy-preserving. The proof of this conjecture used a much older theory called convex integration, pioneered by Nobel and Abel laureate John F. Nash and Abel laureate Mikhael L. Gromov.More recently, these techniques have been applied by several groups to stochastic PDEs which model flows subjected to numerical, empirical and physical uncertainties, and which have recently become very popular in the wake of Martin Hairer's Fields Medal, where convex integration led to similarly spectacular results as in the deterministic case.In this project, we will investigate several stochastic PDE models from fluid dynamics and plasma physics and study their anomalous dissipation properties via the theory of convex integration. More precisely, we will study transport equations, Euler equations and magnetohydrodynamic (MHD) equations, subject to random perturbations.The project will be spent in Prof. Vlad Vicol's group at Courant Institute of Mathematical Sciences, New York. We will develop a mathematical theory of stochastic Mikado flows as key building blocks in convex integration and apply it to the above-mentioned systems, aiming to get nonuniqueness results for solutions to these equations.