Flows of Irregular Vector Fields in Fluid Dynamics

Explore the mathematical theory of flows associated with irregular vector fields and their applications in fluid dynamics through this comprehensive advanced course. Delve into the classical Cauchy-Lipschitz theorem and understand how it breaks down for less regular vector fields, then discover the groundbreaking 1989 work of Di Perna and Lions who introduced a generalized notion of flow that extends existence, uniqueness, and stability results to much broader classes of vector fields. Master modern techniques including quantitative regularity estimates for flows of Sobolev vector fields, nonuniqueness phenomena through convex integration methods, similarity constructions, mixing theory, and enhanced and anomalous dissipation concepts. Apply these sophisticated mathematical tools to analyze nonlinear partial differential equations, particularly the Euler and Navier-Stokes equations that govern fluid motion. Connect theoretical developments to practical understanding of turbulence phenomena and explore connections to Kolmogorov's theory of turbulence. Examine open problems and recent advances in this active area of mathematical research that bridges differential equations, functional analysis, and mathematical fluid mechanics. Gain insight into how irregular vector field theory provides essential tools for understanding complex fluid dynamical systems where classical smooth theory fails to apply.