# Abstracts for Rivière-Fabes Symposium on Analysis and PDE

April 7-9, 2006

### Wilhelm Schlag

Lecture 1 - Spectral theory and applications to nonlinear PDE, I

Lecture 2 - Spectral theory and applications to nonlinear PDE, II

Abstract 1: We will discuss some recent work, most of it joint with Joachim Krieger at Harvard, concerning nonlinear PDE that allow for a family of nonlinear bound-states (e.g., standing waves). These families can be either stable or unstable under small perturbations. We will describe some new results on the unstable case and show that stability can be achieved provided the perturbations are chosen on suitable finite co-dimensional manifolds. Some conjectures and possible further work will be discussed.

### Charles Fefferman

Lecture 1 - Fitting a Smooth Function to Data, I

Lecture 2 - Fitting a Smooth Function to Data, II

Abstract 1: Fix positive integers

m,n, and suppose we are givenNpoints inRn+1. We compute a functionFinCm(Rn), whose graph passes through (or close to) all (or nearly all) of the given points, and whoseCmnorm has the smallest possible order of magnitude. Joint work with Bo'az Klartag.

### Isabelle Gallagher

Lecture 1 - Mathematical analysis of equatorial waves

Abstract 1: In this talk we will consider a model of rotating fluids, describing the motion of the ocean in the equatorial zone. This model is known as the Saint-Venant, or shallow-water type system, to which a rotation term is added whose amplitude is linear with respect to the latitude; in particular it vanishes at the equator. After a quick physical introduction to the model, we describe the various waves involved and the resonances associated with those waves. We then exhibit the limit system (as the rotation becomes large), obtained as usual by filtering out the waves, and study its wellposedness. Finally we present three types of convergence results: a weak convergence result towards a linear, geostrophic equation, a "hybrid" strong convergence result of the filtered solutions towards a weak solution to the limit system, and finally a strong convergence result of the filtered solutions towards the unique strong solution to the limit system for smooth enough initial data. In particular we show that there are no confined equatorial waves in the mean motion as the rotation becomes large.

### Alexandru Ionescu

Lecture 1 - Low-regularity solutions of nonlinear equations

Abstract 1: I will discuss some recent work with C. Kenig on local and global well-posedness of several nonlinear dispersive equations. The main models I will consider are the Benjamin-Ono equation (BO), the Kadomtsev-Petviashvili I equation (KP-I), and Schrödinger maps.

### Diego Maldonado

Lecture 1 - On the Monge-Ampere equation and its linearization

Abstract 1: During the 90's, Luis Caffarelli pioneered a geometric approach to the study of convex solutions *u* to the Monge-Ampere equation det*D*2*u* = μ. One ground-breaking consequence of that approach is the *C*1,α-regularity result for *u* when μ verifies a doubling property. In this talk, we will go over a new proof for this theorem that can be extended to the context of the Heisenberg group (and Carnot groups in general). Then, we will show the connections between the mentioned geometric approach and two topics of independent interest: the real analysis on spaces of homogeneous type and the theory of quasi-conformal mappings. Finally, we will prove a weak reverse-Holder inequality for non-negative solutions to the linearized Monge-Ampere equation that involves techniques developed by E. Fabes and D. Stroock in the 80's. The material of this talk is based on several collaborations with Luca Capogna, Liliana Forzani, and Leonid Kovalev.