Flows of Irregular Vector Fields in Fluid Dynamics - Lecture 5

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. Apply these concepts and techniques to the study of nonlinear PDEs, particularly in the context of fluid dynamics, focusing on the Euler and Navier-Stokes equations and their relation to the Kolmogorov theory of turbulence. Gain valuable insights from Maria Colombo of École polytechnique fédérale de Lausanne in this 1 hour and 55 minute presentation at the Institut des Hautes Etudes Scientifiques (IHES).