Prelectura de tesis
Doctorando: Francisco Mengual
Directores: Daniel Faraco y Ángel Castro
Fecha y hora: Lunes 17 de mayo, 15:00
Lugar: ONLINE mediante una reunión virtual en el equipo de Microsoft Teams titulado “Prelectura de tesis – Francisco Mengual”
Title: Instabilities in fluid mechanics and convex integration
Despite turbulence can be observed in everyday phenomena (e.g. mixing coffee with milk) it is still one of the biggest challenges in mathematical physics.
In this talk we deal with 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 an interface where a hydrodynamic instability occurs. Experimentally, this instability triggers a laminar-turbulent transition in a neighborhood of the interface. Although its mathematical description seemed unattainable due to the wild nature of turbulence, De Lellis-Székelyhidi’s version of convex integration has enabled to describe several of these phenomena in the last years.
Following this approach, we construct infinitely many 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 Rayleigh-Taylor and smoothness breakdown. This is the first existence result for partially unstable data. In addition, we provide a h-principle for the IPM equation with density-viscosity jump.
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 solutions can behave wildly, but at a macroscopic scale they are almost indistinguishable from the subsolution.
This is joint work with Ángel Castro, Daniel Faraco and László Székelyhidi.