A Proof of Onsager’s Conjecture for the Incompressible Euler Equations
Abstract: In an effort to explain how anomalous dissipation of energy occurs in hydrodynamic turbulence, Onsager conjectured in 1949 that weak solutions to the incompressible Euler equations may fail to exhibit conservation of energy if their spatial regularity is below 1/3-Hölder. I will discuss a proof of this conjecture that shows that there are nonzero, (1/3-\epsilon)-Hölder Euler flows in 3D that have compact support in time. The construction is based on a method known as "convex integration," which has its origins in the work of Nash on isometric embeddings with low codimension and low regularity. A version of this method was first developed for the incompressible Euler equations by De Lellis and Székelyhidi to build Hölder-continuous Euler flows that fail to conserve energy, and was later improved by Isett and by Buckmaster-De Lellis-Székelyhidi to obtain further partial results towards Onsager's conjecture. The proof of the full conjecture combines convex integration using the “Mikado flows” introduced by Daneri-Székelyhidi with a new “gluing approximation” technique. The latter technique exploits a special structure in the linearization of the incompressible Euler equations.
Mean-Field Limits for Ginzburg-Landau vortices
Abstract: Ginzburg-Landau type equations are models for superconductivity, superfluidity, Bose-Einstein condensation. A crucial feature is the presence of quantized vortices, which are topological zeroes of the complex-valued solutions. This talk will review some results on the derivation of effective models to describe the statics and dynamics of these vortices, with particular attention to the situation where the number of vortices blows up with the parameters of the problem. In particular I will present new results on the derivation of mean field limits for the dynamics of many vortices starting from the parabolic Ginzburg-Landau equation or the Gross-Pitaevskii (=Schrodinger Ginzburg-Landau) equation. I will also discuss the situation with disorder and related homogenization questions.
Integro-differential equations and their applications.
Abstract: Integro-differential equations have been a very active research topic in recent years. In these talks we will start by explaining what they are and reviewing some basic results related to general problems. Then we will move on to some applications of these results to problems of mathematical physics. In particular, we will discuss some recent regularity estimates for the Boltzmann equation.
Singularity formation in critical parabolic problems
Abstract: We deal with construction and stability analysis of blow-up of solutions for parabolic equations that involve bubbling phenomena, corresponding to gradient flows of variational energies. The term bubbling refers to the presence of families of solutions which at main order look like scalings of a single stationary solution which in the limit become singular but at the same time have an approximately constant energy level. This phenomenon arises in various problems where critical loss of compactness for the underlying energy appears. Specifically, we present construction of threshold-dynamic solutions with infinite time blow-up in the Sobolev critical semilinear heat equation in $\R^n$, and finite time blow up for the harmonic map flow from a two-dimensional domain into $S^2$.