Antonio Cordoba (Universidad Autónoma de Madrid)
Singular Integrals in Fluid Mechanics:
I) Blow up of solutions for a transport equation.
II) Interface evolution: The Muskat and Hele-Shaw problem.
Some new estimates for classical Singular Integrals will be introduced, discussing their applications to several problems in Fluid Mechanics.
Panagiotis Souganidis (University of Chicago)
Stochastic homogenization
In these talks I will describe recent advances to the theory of homogenization of first- and second-order partial differential equations set in general stationary ergodic environments.
Thomas Alazard (CNRS and Ecole Normale Supérieure, Paris)
On the Cauchy problem for the water-waves equations
The water-waves problem consists in describing the motion, under the influence of gravity, of a fluid occupying a domain delimited below by a fixed bottom and above by a free surface. We consider the Cauchy theory for low regularity solutions. In terms of Sobolev embeddings, the initial surfaces we consider turn out to be only of C3/2 class and consequently have unbounded curvature. Furthermore, no regularity assumption is assumed on the bottom. We also take benefit from an elementary observation to solve a question raised by Boussinesq on the water-wave equations in a canal.
Giuseppe Mingione (Università degli Studi di Parma)
Linear and nonlinear Calderon-Zygmund theories
Calderon-Zygmund theory deals with a fundamental problem in the theory of partial differential equations of elliptic and parabolic type: given a certain PDE, can we determine, in a possibly sharp way, the regularity and, especially, the integrability properties of the solution in terms of those of the assigned datum? In the linear case sharp answers are related to the theory of singular integrals, whose fundamentals have been established in the multidimensional case by Calderon and Zygmund more that fifty years ago. Recent years have witnessed a considerable activity towards establishing a series of analogous results for nonlinear equations, up to the stage that it appears to be possible to think about a nonlinear Calderon-Zygmund theory. I will give a survey of such results up to a few recent developments.
Gabriella Tarantello (Università di Roma `Tor Vergata')
Liouville–type systems in the study of non-topological solutions in
Chern Simons theory
We discuss elliptic systems of Liouville type in presence of singular sources, as derived from the study of non-abelian (selfdual) Chern-Simons vortices. We shall focus on the search of the so called non-topological vortex configurations. We present some known results and discuss many of the still open questions.
Rachel Ward (University of Texas at Austin)
Strengthened Sobolev inequalities for a random subspace of functions
We introduce some Sobolev inequalities for functions on the unit cube satisfying a random collection of linear constraints. We then explain how these inequalities provide near-optimal guarantees for accurate image recovery from under-sampled measurements using total variation minimization, with applications to medical imaging. We finish by discussing several open problems.