Collaborative Research: Analysis of incompressible high Reynolds number flows

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The purpose of this project is to conduct international collaborative research, between two teams of scientists from the USA and Brazil, in the mathematical analysis and modeling of turbulent incompressible fluids. The topics to be investigated are: small viscosity regime of second grade fluids; uniqueness of weak solutions for certain linear perturbations of the two-dimensional Euler equations; the vanishing viscosity limit of the three-dimensional Navier-Stokes equations with helical symmetry; the search for hypotheses on the structure of invariant measures or stationary statistical solutions of the two and three-dimensional Navier-Stokes equations with anomalous dissipation; the search for energy cascade for flows in domains with physical boundaries; two-dimensional cascades with large gap bimodal forcing. Turbulence is a common phenomenon in fluid motion, in which macroscopic quantities (velocity, pressure, temperatures, etc.) no longer have a deterministic relation with global parameters of the flow. Direct numerical simulations of turbulent flows at large Reynolds numbers, that occur in practical applications, such as in geophysical modeling and mixing in industrial processes, are out of reach even for the state-of-the-art computer power. Therefore, there is an urgent need to pursue this challenging problem analytically, by developing rigorous mathematical and statistical tools to investigate it, and to test these tools computationally. While turbulence is an everyday occurrence, our understanding is still lacking in many aspects. Quantifying the effect of small scales on the dynamics of large scales is fundamental in modern multiscale science. The goal of the project is to enable a predictive analytical study of turbulent flows. This study will impact wide-ranging applications, from geophysical modeling, such as dispersion of pollutants in the ocean, to biological and industrial modeling, such as design of polymeric materials. The project will consolidate the well-established collaborative efforts of the principal investigators with their Brazilian counterparts, and may lead to new collaboration, especially among the junior research personnel. International collaboration among scientists is a key to economic competitiveness in global markets. Four US and three Brazilian academic institutions are involved in the project. The international dimension of the project is further emphasized through two planned workshops. Training and supervision of at least six Ph.D. students and postdoctoral fellows is also achieved through planned summer schools and scientific workshops. Students and postdoctoral fellows from the US will travel to Brazil to participate in the workshops and the summer schools and interact with the US and Brazilian researchers.This project is co-funded with the Americas Program of the Office of International Science and Engineering.