Lipschitz Metric for a Nonlinear Wave Equation

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

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.

- Download the Flyer -

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

Location & Details: 

To our visitors

  • RSVP may be required for this event.  Please check event details
  • Visitors will need to present a photo ID at the entrance
  • There is no public parking on campus
  • Entrance only through the South Lobby (1555 Century Avenue) 
  • Taxi card
  • Metro: Century Avenue Station, Metro Lines 2/4/6/9 Exit 6 in location B
  • Bus: Century Avenue at Pudian Road, Bus Lines 169/987