The Theory of Fast Diffusion Equations. Main features and recent news.

Juan Luis Vázquez (Universidad Autónoma de Madrid)

Abstract:   We will review the main mathematical features of the nonlinear heat °ow called the Fast Diffusion Equation δ_tu = Δu^m, m < 1. Much is known nowadays about this flow, posed in Rⁿ, in a bounded open subset or on a manifold. Surprising phenomena appear, like loss of regularity (solutions with L^p initial data may not be bounded), extinction in finite time, even lack of existence or lack of uniqueness for classes of small and smooth initial data.

As novelties, we will present geometrical results related to the 2-d Ricci °ow and the description of asymptotics using weighted functional inequali- ties of Hardy-Sobolev type.

Background Reference:   J. L. Vázquez. "Smoothing and Decay Estimates for Nonlinear Diffusion Equations. Equations of Porous Medium Type", Oxford Lecture Notes in Maths. and its Applications 33, Oxford University Press, 2006.

Pierre Raphaël (Princeton University and CNRS)

Abstract:   The focusing nonlinear Schrodinger equation iu_t +Δu+ u|u|^p-1 = 0 in R^N is a universal model for the self trapping of waves in a nonlinear medium. The cases p = 3 and N = 2,3 are of particular physical relevance but are quite different from the mathematical point of view. The phenomenon we aim at describing is the concentration of the nonlinear wave which mathematical counterpart is the finite time blow up of the solution to the Cauchy problem. A general understanding of the singularity formation for NLS is still widely open.

I will start by reviewing the series of results obtained in collaboration with Frank Merle for the L² critical NLS iu_t + Δu + u|u|^4/N = 0 in R^N and which prove the existence and stability of "log-log" type blow up scenario. I will then explan how these results may be extended to various situations and in particular provide tools to construct new blow up solutions in the L² super critical setting. I will also explain how the intuition based on the use of Liouville type classification theorem has recently allowed us to prove the blow up of the critical norm in super critical settings.

I will conclude by presenting a recent work joint with Mohammed Lemou anf Florian Mehats which concerns the quite unexpected adaptation of these techniques to nonlinear kinetic Vlasov-Poisson type problems emerging from astrophysiscs.

Symmetry of global solutions to certain fully nonlinear elliptic equations

Ovidiu Savin (Columbia University)

Abstract:   We consider bounded global solutions to fully nonlinear equations of the type $F(D^2u)=f(u)$. The main assumption on $F$ and $f$ is that there exists a one dimensional solutions $g$ that solves the equation in all directions. We show that solutions with Lipschitz level sets depend on one variable, that is the level sets are in fact hyperplanes. In the particular case when $F=\triangle$ and $f(u)=u3-u$ the same result was obtained by Barlow, Bass and Gui by probabilistic methods.

Vortices in the 2D Ginzburg-Landau model with magnetic field.

Sylvia Serfaty (Courant Institute)

Abstract:   We review some results obtained jointly with Etienne Sandier on the Ginzburg-Landau model of superconductivity in 2 dimensions, and described in our recent book. In a superconducting sample, according to the intensity of the applied field, phase transitions happen and vortices appear in certain regimes. By studying energy minimizers in a suitable parameter regime, we extract limiting problems that describe the optimal repartition of vortices according to the applied field. We will also describe some of our more recent results on the subject.

Background Reference:   Etienne Sandier, Sylvia Serfaty, Vortices in the Magnetic Ginzburg-Landau Model, Progress in Nonlinear Differential Equations 70, Birkhauser, 2007.

Equation of Monge-Ampère and Bellman Solutions for Certain Harmonic Analysis Problems (After Slavin-Stokolos)

Alexander Volberg (Michigan State University)

Abstract:   Bellman function method in Harmonic Analysis was introduced by Donald Burkholder for finding the norm in L^p of martingale transform. Later it became clear that scope of the method is relatively wide, in particular Nazarov-Treil-Volberg obtained by this method a necessary and sufficient condition for the two weight martingale transform to be bounded (which was used then in Bellman estimates of Ahlfors-Beurling transform by Petermichl and myself). Recently Slavin and Stokolos made an important observation how Monge-Ampère equation may help to find the exact Bellman function of certain Harmonic Analysis problems. They illustrated their idea by finding the elegant and short way to find the Bellman function of dyadic maximal operator (A. Melas' recent result). We will illustrate this approach by this and couple of other examples, where Monge-Ampère allows us to find exact Bellman function of a Harmonic Analysis problem. In particular, to find best constants and extremal functions (or extremal sequences).

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