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Mathematics Colloquium

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Date:  Tuesday, March 28, 2006

Title:  Why are Multidimensional Conservation Laws so Difficult?

Abstract:  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 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.


Speaker:  Barbara Lee Keyfitz
Institution:  Fields Institute/University of Houston

 

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