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Explore recent developments in the unstable Muskat problem for incompressible porous media equations and the vortex sheet problem for incompressible Euler equations in this advanced mathematics lecture. Delve into the h-principle in IPM for two-phase flows with different densities and mobilities, and discover how the connection between the convex integration method and gradient flux approach is extended to cases of different mobilities. Examine the construction of globally dissipative weak solutions to incompressible Euler equations with vortex sheet initial data, focusing on C^(k,α)-regularity and non-fixed sign. Learn how local energy inequality bounds the turbulence zone in these complex fluid dynamics problems. This in-depth talk, part of the Hausdorff Trimester Program on Evolution of Interfaces, presents collaborative research findings that push the boundaries of our understanding in fluid mechanics and partial differential equations.
