Speaker

　　Title

　　Time

　　Venue

N613

　　Abstract

In one space dimension, it is well known that hyperbolic conservation laws have unique entropy-admissible solutions, depending continuously on the initial data.  
Moreover, these solutions can be obtained as limits of vanishing viscosity approximations.

For many years it was expected that similar results would hold in several space dimensions. However, fundamental work by De Lellis and Szekelyhidi, and later by collaborators, has shown that multidimensional hyperbolic Cauchy problems usually have infinitely many weak solutions. Moreover, all known entropy criteria fail to select a single admissible one.

In the first part of this talk I shall outline this approach based on a Baire category argument, yielding the existence of infinitely many weak solutions. 

I then wish to discuss an alternative research program,aimed at constructing multiple solutions to some specific Cauchy problems.
Starting with some numerical simulations, here the eventual goal is to achieve rigorous, computer-aided proofs of the existence of two distinct self-similar solutions with the same initial data.While solutions obtained via Baire category have turbulent nature, these self-similar solutions are smooth, with the exception of one or two points of singularity. They are thus much easier to visualize and understand. 

　　Affiliation

PRESENT POSITION
Eberly Family Chair Professor of Mathematics, Penn State University

HONORS
-Plenary speaker at the International Congress of Mathematicians, Beijing, 2002.
-"Antonio Feltrinelli" prize for Mathematics, Mechanics and Applications, Accademia Nazionale dei Lincei. Rome, 2006.
-"Analysis of Partial Differential Equations" prize, Society for Industrial and Applied Mathematics. Phoenix, 2007 (with Stefano Bianchini).
-"Maxim Bocher" prize, American Mathematical Society. San Diego, 2008.
-"Luigi and Wanda Amerio" prize, Accademia di Scienze e Lettere, Istituto Lombardo. Milan, 2010.
-Member of the Royal Norwegian Society of Sciences and Letters, Trondheim, 2011.
-Member of the Accademia di Scienze e Lettere, Istituto Lombardo. Milan, 2011.
-"Gaetano Fichera" prize for mathematical analysis, Unione Matematica Italiana. Bologna, 2011.
-Fellow of the American Mathematical Society, 2012.