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CNA Seminar/Colloquium/Joint Pitt-CNA Colloquium
Colloquium
Alberto Bressan
Pennsylvania State University
Title: Lipschitz Metrics for Nonlinear Wave Equations

Abstract: The talk is concerned with some classes of nonlinear wave equations: of first order, such as the Camassa-Holm equation, or of second order, as the variational wave equation u_tt - c(u)(c(u)u_x)_x = 0. In both cases, it is known that the equations determine a unique flow of conservative solutions within the natural energy" space H^1(R). However, this flow is not continuous w.r.t. the H^1 distance.

Our goal is to construct a new metric, which renders this flow uniformly Lipschitz continuous on bounded subsets of H^1. For this purpose, H^1 is given the structure of a Finsler manifold, where the norm of tangent vectors is defined in terms of an optimal transportation problem. For paths of piecewise smooth solutions, one can carefully estimate how the weighted length grows in time.

To complete the construction, one needs an additional argument showing that the family of piecewise smooth solutions is dense. This generic regularity property can be proved using a variable transformation that reduces the equations to a semilinear system, followed by an application of Thom's transversality theorem.

Recording: http://mm.math.cmu.edu/recordings/cna/alberto_bressan_small.mp4
Date: Tuesday, September 15, 2015
Time: 4:30 pm
Location: Wean Hall 7218
Submitted by:  David Kinderlehrer
Note: Refreshments at 4:00 pm.