3:15pm - 4:15pm
Wednesday 23 May 2018

Partial differential equations of mixed elliptic-hyperbolic type - From mechanics to geometry

Professor Gui-Qiang Chen from the University of Oxford will be speaking.

Free

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Speakers
  • Prof Gui-Qiang Chen (University of Oxford)

Abstract

As is well-known, two of the basic types of linear partial differential equations (PDEs) are elliptic and hyperbolic types, following the classification for linear PDEs proposed by Jacques Hadamard in the 1920s; and linear theories of PDEs of these two types have been considerably established, respectively.

On the other hand, many nonlinear PDEs arising in many areas from mechanics to geometry naturally are of mixed elliptic-hyperbolic type. The solution of some long standing fundamental problems in these areas greatly requires a deep understanding of such nonlinear PDEs of mixed type.

Important examples include shock reflection-diffraction problems in fluid mechanics (the Euler equations) and isometric embedding problems in differential geometry (the Gauss-Codazzi-Ricci equations), among many others.

In this talk we will present natural interconnections of nonlinear PDEs of mixed elliptic-hyperbolic type with these longstanding problems and will then discuss some of the most recent developments in the analysis of these nonlinear PDEs through the examples with emphasis on developing and identifying mathematical approaches, ideas, and techniques for dealing with the mixed-type problems. Further trends, perspectives, and open problems in this direction will also be addressed.