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Explore the intricacies of flows of irregular vector fields in fluid dynamics through this comprehensive lecture. Delve into the limitations of the classical Cauchy-Lipschitz theorem and discover the groundbreaking work of Di Perna and Lions, who introduced a generalized notion of flow for less regular vector fields. Examine modern perspectives, recent advancements, and open problems in this field. Investigate quantitative regularity estimates on the flow of Sobolev vector fields, nonuniqueness of solutions via convex integration, similarity constructions, mixing, and enhanced and anomalous dissipation. Learn how these concepts apply to nonlinear PDEs and their significance in understanding fluid dynamics phenomena, particularly in relation to the Euler and Navier-Stokes equations and the Kolmogorov theory of turbulence. Presented by Maria Colombo from École polytechnique fédérale de Lausanne, this 1 hour and 54-minute lecture offers a deep dive into the mathematical foundations of fluid dynamics and their practical applications.
