Abstracts for Rivière-Fabes Symposium 2020

Panagiota Daskalopoulos (Columbia University)

Ancient solutions to Geometric flows I and II

Some of the most important problems in partial differential equations are related to the understanding of singularities. This usually happens through a blow up procedure near the potential singularity which uses the scaling properties of the equation. In the case of a parabolic equation the blow up analysis often leads to special solutions which are defined for all time −∞ < t ≤ T, for some T ≤ +∞. We refer to them as ancient solutions. The classification of such solutions often sheds new insight to the singularity analysis. In some flows it is also important for performing surgery near a singularity. In these lectures we will give an overview of Uniqueness Theorems for ancient solutions to geometric flows, with emphasis on recent results for the Mean curvature flow and the Ricci flow. 

 

Pierre Germain (Courant Institute)

On the derivation of the kinetic wave equation

The kinetic wave equation (KWE) is conjectured to describe (weakly) nonlinear dispersive equations in a turbulent regime. It is used to describe and simulate a number of physical situations, from waves on the the ocean to plasma physics and nonlinear optics. In a first lecture, I will present the KWE, and explain how it is related to various questions in mathematical physics, such as the Boltzmann equation, the turbulent spectrum, and the growth of solutions of nonlinear dispersive equations. In a second lecture, I will present recent progress (obtained in collaboration with Charles Collot) regarding its derivation. 

 

Rowan Killip (University of California, Los Angeles)

Well-posedness for integrable PDE

I will describe a suite of techniques developed recently to prove well-posedness of completely integrable Hamiltonian PDE. These have lead to sharp results for a number of well-know problems: KdV, mKdV, and cubic NLS in one dimension. This touches on joint work with Bjoern Bringmann, Benjamin Harrop-Griffiths, and Monica Visan. 

 

Ovidiu Savin (Columbia University)

Free boundary regularity for the 3 membranes problem

For a positive integer N, the N-membranes problem describes the equilibrium position of N ordered elastic membranes subject to forcing and boundary conditions. If the heights of the membranes are described by real functions u1, u2, . . . , uN , then the problem can be understood as a system of N − 1 coupled obstacle problems with interacting free boundaries which can cross each other. When N = 2 there is only one free boundary and the problem is equivalent to the classical obstacle problem. In my first lecture I will review some of the regularity theory for the standard obstacle problem, and in my second lecture I will discuss some recent work in collaboration with Hui Yu about the case when N = 3 and there are two interacting free boundaries.