Valid Until: 2027-01-31 23:59
Abstract: Three of the basic types of linear partial differential equations (PDEs) are elliptic, hyperbolic, and parabolic, following the standard classification for linear PDEs. Linear theories of PDEs of these types have been considerably better developed. On the other hand, many nonlinear PDEs arising in Mathematics and Science naturally are of mixed type. The solution of some longstanding fundamental problems greatly requires a deep understanding of such nonlinear PDEs of mixed type, especially mixed elliptic-hyperbolic 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 some old and new underlying connections of nonlinear PDEs of mixed type with the longstanding fundamental problems and will then discuss some recent developments in the analysis of these nonlinear PDEs through the examples with emphasis on developing/identifying unified approaches, ideas, and techniques for dealing with the mixed-type problems. Some most recent developments, further perspectives, and open problems in this direction will also be addressed.
Bio: Professor Gui-Qiang G. Chen is a leading expert in Partial Differential Equations, Nonlinear Analysis, Mathematical Physics, and related areas. He received his Ph.D. from the Chinese Academy of Sciences in 1987 and was a postdoctoral research fellow at the Courant Institute of Mathematical Sciences (NYU) in 1987-89. His recent research interests include nonlinear PDEs of mixed type, hyperbolic conservation laws, nonlinear waves, free boundary problems, geometric analysis, stochastic PDEs, measure-theoretical analysis, weak convergence methods, and entropy analysis. He has published more than 200 original research papers and more than 10 research books. Since 2000, he has delivered more than 300 invited lectures/talks around the world. He is a Member of the Academia Europaea, Member of the European Academy of Sciences, Fellow of the American Mathematical Society, Fellow of the Institute of Mathematics and its Applications, Fellow of the Society of Industrial and Applied Mathematics, was an Alexander von Humboldt Foundation Fellow (2003-07) and Alfred P. Sloan Foundation Fellow (1991-97), and received the 2011 SIAG/Analysis of Partial Differential Equations Prize (SIAM 2011), Royal Society Wolfson Research Merit Award (2009), Chinese National Natural Science Prize (1989), and Chinese Academy of Sciences Award in Mathematics (1988). Currently, he is Statutory Professor in the Analysis of Partial Differential Equations at the Mathematical Institute and Professorial Fellow of Keble College, University of Oxford.
More information can be found at his homepage: https://www.maths.ox.ac.uk/people/gui-qiang.chen