R. F. Barry Jr. Seminar: Geng Chen, University of Kansas - "Lipschitz metric for a nonlinear wave equation"
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- Date/Time
- 06/28/2016 12:30 PM EST - 1:30 PM EST
- Location
- Engineering & Computational Sciences Building - 2120
- Fee
- Free
- Description
- ABSTRACT: The nonlinear wave equation: u_{tt} - c(u)[c(u)u_x]_x = 0 is a natural generalization of the linear wave equation, which has application on the nematic liquid crystal. In this talk, we will discuss a recent breakthrough addressing the Lipschitz continuous dependence of solutions on initial data for this quasi-linear wave equation. 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. To prove the desired Lipschitz continuous property, we constructed a new Finsler type metric, where the norm of tangent vectors is defined in terms of an optimal transportation problem. For paths of piecewise smooth solutions, we carefully estimated how the distance grows in time. This is a collaboration work with Alberto Bressan.