Lipschitz metric for a nonlinear wave equation

Title:Lipschitz metric for a nonlinear wave equation

报告人:Geng Chen,Assistant Professor,Department of Mathematics,University of Kansas

时间:2018年7月24日下午 15:30-16:30

地点:知新楼 B1032

邀请人:胡锡俊教授

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.