1:00 pm Wednesday, September 25, 2019
Analysis Seminar: Finsler type optimal transport Lipschitz metric for a nonlinear wave equation by
Geng Chen [mail] (The University of Kansas) in RLM 10.176
In this talk, I will discuss the collaboration work with Alberto Bressan on the Lipschitz metric for a quasi-linear wave equation u_{tt} - c(u)[c(u)u_x]_x = 0, modelling elasticity and nematic liquid crystal. 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 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. The recent global existence result on the Poiseuille flow of nematic liquid crystals via full Ericksen-Leslie model might be briefly introduced if time is permitted. Submitted by
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