|TITLE:||Why Are Multidimensional Conservation Laws So Difficult?|
|SPEAKER:||Barbara Lee Keyfitz (University of Houston)|
|DATE:||Vendredi, 10 Septembre 2004|
An outstanding open problem in the theory of partial differential equations is the well-posedness of initial-value problems for nonlinear hyperbolic equations in more than one space dimension. This talk will set a context for the problem: Why are mathematicians interested in partial differential equations, what are the differences between the way pure and applied mathematicians approach the subject, and how can different approaches reinforce each other?
The talk is intended for an audience which is not expert in partial differential equations, and will begin by explaining why the division of equations into "hyperbolic" and "elliptic" is natural mathematically as well as being grounded in applications. We will describe briefly the analysis used to prove existence theorems for linear equations of both types.
Generalizing the elliptic theory to quasilinear and nonlinear elliptic equations has been largely achieved, but the corresponding theory for hyperbolic equations is still being developed. Some simple examples serve to show the sorts of obstructions we may expect. A number of routes through these challenges seem ready to be explored. Finally, I will describe a new approach that I, along with co-workers and others, are pursuing, which exploits the better-developed theory of quasilinear elliptic equations to study multidimensional quasilinear hyperbolic equations.
About the speaker: Barbara Lee Keyfitz has made profound and original contributions to applied mathematics, particularly in the study of nonlinear partial differential equations such as those arising in the study of fluid flow or transonic shock waves. She has a distinguished record of research, mentoring, and service to her profession. Her undergraduate training took place at the University of Toronto, and she obtained her PhD from New York University's Courant Institute, prior to holding positions at Columbia University, Princeton University, and Arizona State University. She continues as the John and Rebecca Moores Professor at the University of Houston. She is a Fellow of the American Association for the Advancement of Science, and is chair elect of their Mathematics section. She is an officer of the International Council for Industrial and Applied Mathematics, and has made numerous contributions as a reviewer and advisor to the Canadian Natural Sciences and Engineering Council. She was recently awarded the Canadian Mathematical Society's Krieger-Nelson Prize (to be presented in 2005), has served as a Vice President of the Society for Industrial and Applied Mathematics, and is incoming President of the Association for Women in Mathematics. Barbara Lee Keyfitz, a native of Ottawa, started her term as Director of the Fields Institute for Research in Mathematical Sciences in July 2004.