Theses
PRELECTURA DE TESIS -- Instabilities in Fluid Mechanics and Convex Integration
Speaker: Francisco Mengual, ICMAT-UAM ()Advisor/s: Daniel Faraco, ICMAT-UAM, y Ángel Castro, ICMAT-CSICDate: Monday, 17 May 2021 - 15:00Place: Online - Microsoft Teams: “Prelectura de tesis-Francisco Mengual”
Abstract:
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.