Daniel Faraco- Convex Integration and Mixing Flows

Explore the first part of a lecture on convex integration and mixing flows, delivered by Daniel Faraco as part of the Hausdorff Trimester Program: Evolution of Interfaces. Delve into the challenges of hydrodynamics, focusing on the Muskat problem as a prototype for ill-posed governing equations in certain parameter regimes. Examine the evolution of two fluids separated by an interface in a porous medium, with emphasis on the unstable situation where the heavier fluid is on top. Learn about the mixing zone, viscous fingering patterns, and the limitations of classical theory in establishing the existence of solutions. Compare the gradient flow approach by Otto with an alternative method based on convex integration schemes developed by De Lellis and Székelyhidi. Investigate the concept of a pseudo-interface and its associated nonlinear, nonlocal equation. Gain insights into the application of semiclassical calculus for non-smooth Fourier multipliers in addressing existence and uniqueness problems.