Abstract:
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.
Biography:
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.
Seminar by the NYU-ECNU Institute of Mathematical Sciences at NYU Shanghai
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