Stable structures and chaotic dynamics in fluid flows

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The proposal focuses on the theoretical understanding of long-time dynamics questions in fluid mechanics, from the point of hydrodynamic stability and turbulence theory. By putting on rigorous mathematical grounds several classical questions, such as the meta-stability of coherent structures and the ergodic/statistical properties of fluid flows, I aim to deepen the understanding of the long- time behavior of solutions to the Navier-Stokes and the Euler equations.The main objective is to establish a research group at Imperial College to develop novel mathematical techniques in the theory of PDEs, harmonic/stochastic analysis, and dynamical systems, that would allow to move beyond the current limits of the field. These techniques will be devised and mature in the context of two fundamental problems.- Coherent structures and stability of fluids: due to its stabilizing and infinite-dimensional nature, fluid mixing causes a transfer of enstrophy to small scales, in a manner which is reversible and conservative for finite times, but results in an irreversible loss of information at infinite times. With no viscosity, it generates inviscid damping, while its interaction with diffusion creates dissipation time-scales responsible for meta-stable behavior. Rigorous results on these stability problems constitute a milestone towards the resolution of classical long-standing questions related to the transition from laminar to turbulent states and the formation of coherent vortex-like structures in large-scale atmospheric dynamics.- Cascades in stochastic fluid mechanics: the rigorous derivation of scaling laws stemming from the phenomenological theory of turbulence is a central problem in fluid mechanics. I aim to understand how randomness affects foundational aspects of the theory, such as ergodicity and anomalous dissipation, in a highly innovative context that relies on the quantification with respect to relevant parameters of classical objects such as Lyapunov exponents.