Lipschitz Metric for a Nonlinear Wave Equation

Lipschitz Metric for a Nonlinear Wave Equation
Date & Time: 
Monday, December 18, 2017 -
13:30 to 14:30
Geng Chen, University of Kansas
Room 1200, NYU Shanghai | 1555 Century Avenue, Pudong New Area, Shanghai

In this talk, we will discuss a recent breakthrough addressing the Lipschitz continuous dependence of solutions on initial data for a quasi-linear wave equation u_{tt} - c(u)[c(u)u_x]_x = 0. Our earlier results showed that this equation determines a unique flow of conservative solution within the natural energy space H^1(R). However, this flow is not Lipschitz continuous with respect to the H^1 distance, due to the formation of singularity first found by Glassy-Hunter-Zheng. To prove the desired Lipschitz continuous property, we construct a new Finsler type metric, where the norm of tangent vectors is defined in terms of an optimal transportation problem. For paths of piece-wise smooth solutions, we carefully estimate how the distance grows in time. To complete the construction, we prove that the family of piece-wise smooth solutions is dense, following by an application of the Thom's transversality theorem. This is a collaboration work with Alberto Bressan.

Geng Chen is Assistant Professor in Department of Mathematics at University of Kansas. His research interests are analysis, partial differential equations, fluid dynamics, mathematical physics, especially the well-posedness and behaviors of solutions for the Euler equations and nonlinear variational wave equations.

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Seminar by the NYU-ECNU Institute of Mathematical Sciences at NYU Shanghai

Location & Details: 

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