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Analysis and Applied Mathematics Seminar

Francisco Mengual
Institute for Advanced Study
Instabilities in Fluid Mechanics and Convex Integration
Abstract: In this talk we consider two problems related to turbulence: The vortex sheet problem for the incompressible Euler equation (Kelvin-Helmholtz instability) and the unstable Muskat problem for the incompressible porous media equation (Saffman-Taylor instability). In both cases the fluid is smooth but at a curve where a hydrodynamic instability occurs. Experimentally, this instability triggers a laminar-turbulent transition in a neighborhood of the interface. Although unstable configurations in Hydrodynamics are very difficult to model, De Lellis-Székelyhidi’s version of convex integration have successfully describe several of these phenomena in the last years. Following this approach, we construct weak solutions for the two problems mentioned above. In the first one, we construct dissipative Euler flows for a large class of non-analytic vortex sheets without fixed sign. The mixed sign case was an open problem from the celebrated work of Delort. In the second one, we construct mixing flows after the Saffman-Taylor and smoothness breakdown. This is the first existence result for partially unstable data. Furthermore, we present a quantitative h-principle which shows that: Outside the “turbulence zone” these solutions are smooth and equal to a “subsolution”. Inside the turbulence zone these (infinitely many) solutions can behave wildly, but at a macroscopic scale they are almost indistinguishable from the subsolution.
Monday April 25, 2022 at 4:00 PM in 636 SEO
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