Inviscid Limits, Uniqueness, and Anomalous Dissipation in Hydrodynamics

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Turbulence, the complex, irregular and chaotic motion of fluids, is a ubiquitous and fundamental mechanism for the transfer of momentum and energy across spatio-temporal scales. Although the deterministic equations describing the dynamics of fluids are well-established and their numerical solution has revealed much about the turbulent behavior of fluids, their analysis is challenging, and basic results remain elusive. The use of probabilistic tools offer complementary approaches that naturally incorporate the uncertainties of the actual state of the fluid and random perturbations due to external forces. This project will leverage state-of-the-art tools in probability, stochastic analysis, and dynamical systems, to study the basic fluid dynamics models and some of its variants, which are relevant to physics, engineering, and atmosphere-ocean dynamics. Emphasis will be given to study the effect of random forcing on these systems, gain further understanding on the energy transfer across scales, and the possibility of obtaining basic existence and uniqueness results using the probabilistic framework, which remain elusive in the deterministic approach. The project will also provide opportunities for undergraduate and graduate students to participate in the research. This research program studies several mathematical problems stemming from the challenges of turbulence theory and involves stochastic analysis, dynamical systems, and partial differential equations. The goal is to understand the link between the Euler and Navier-Stokes equations, their stochastic versions, and the phenomenological laws of turbulence. The advantage of the stochastic approach is the major simplicity of balance laws between mean rates of energy injection, dissipation, and flux. Due to the rich structure of these stochastic models, some results on uniqueness and stability of some approximations of three-dimensional viscous and inviscid fluid flows could be proved while their deterministic counterpart is lacking. The smoothing effect of the noise on the associated dynamical system will be used. The study of the inviscid limit problem will be tackled for a better understanding of the direct energy cascade for the three-dimensional case and the inverse energy cascade for the two-dimensional case. Some inviscid limits for some geophysical models with anisotropic viscosity will be investigated as well. Theoretical issues, such as existence, uniqueness, and ergodicity of invariant measures, will be complemented by numerical simulations. This project is jointly funded by the Division of Mathematical Sciences Applied Mathematics program and the Established Program to Stimulate Competitive Research (EPSCoR).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.