Home > Academic Announcements > (Jan. 15) Partial Differential Equations of Mixed Elliptic-Hyperbolic Type--From Mechanics to Geometry

(Jan. 15) Partial Differential Equations of Mixed Elliptic-Hyperbolic Type--From Mechanics to Geometry

Last updated :2018-01-10

Topic: Partial Differential Equations of Mixed Elliptic-Hyperbolic Type--From Mechanics to Geometry
Speaker: Professor Gui-Qiang G. Chen
(University of Oxford)
Time: 16:30-17:30, Monday, January 15, 2018
Venue: Room 416, New Mathematics Building, Guangzhou South Campus, SYSU

Abstract:
As is well-known, two of the basic types of 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 fluid mechanics to differential geometry naturally are of mixed elliptic-hyperbolic type. The solution of some longstanding 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 connections 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.