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Explore the advanced mathematical concepts of flows of irregular vector fields in fluid dynamics through this comprehensive lecture. Delve into the modern view, recent developments, and open problems surrounding the generalized notion of flow introduced by Di Perna and Lions in 1989. Examine quantitative regularity estimates on the flow of Sobolev vector fields, nonuniqueness of solutions via convex integration, similarity constructions, mixing, and enhanced and anomalous dissipation. Discover how these techniques are applied to study nonlinear PDEs and their relevance in understanding phenomena in fluid dynamics, particularly for the Euler and Navier-Stokes equations and in relation to the Kolmogorov theory of turbulence. Presented by Maria Colombo from École polytechnique fédérale de Lausanne, this 1 hour and 45 minute lecture offers an in-depth exploration of cutting-edge research in the field of mathematical fluid dynamics.
