Weak Solutions and Turbulence in Fluid Dynamics

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In the field of fluid dynamics, a major goal is to obtain a detailed and quantitative understanding of turbulence, which is a type of fluid motion commonly observed in everyday experience with fluids that is characterized by rapid and irregular fluctuations in velocity. Turbulence plays a pivotal role in countless engineering and industrial applications such as the design of land and aircraft, where the production of turbulence must be understood deeply to achieve energy efficient and effective technology. It has long been believed that the inner workings of turbulence may be understood in terms of the differential equations that are being used to model fluid motion, including the Euler equations. However, the analysis of these equations has proved so difficult that much of the relationship between turbulence and solutions to the equations of fluid dynamics remains conjectural and unresolved. Recently through the work of the investigator and other mathematicians, a new approach has emerged to analyze solutions to fluid equations that have rapid fluctuations comparable to what is seen in turbulence. This approach is intimately tied to questions in the mathematical discipline of geometry. This research project aims to push this new approach towards its fullest potential to improve the understanding of turbulence and to expand the frontiers of what can be accomplished through mathematical analysis.At a technical level, the PI will prove rigorous results on the structure of "weak solutions" to the equations of fluid dynamics, including the incompressible Euler equations. The results will address the extent to which such weak solutions can be shown to exhibit the properties observed or predicted of turbulent flows. One direction of research will relate to the phenomenon of anomalous dissipation of energy and the extent to which this phenomenon may exist and be generic or stable under perturbation. The PI will consider analogues and extensions of Onsager's conjecture on anomalous dissipation, and will also investigate fine properties of the dynamical structure of general weak solutions to the Euler and related equations. The research will implement and expand upon the method of convex integration, as well as introduce new tools of geometric or harmonic analysis as needed.