Abstract: Many nonlinear partial differential equations arising
naturally in mechanics and geometry are of mixed type, including
nonlinear conservation laws of mixed hyperbolic-parabolic type and mixed
hyperbolic-elliptic type. The solution of some fundamental issues in these
areas greatly requires a deep understanding of such nonlinear partial
differential equations of mixed type. Important examples include
nonlinear degenerate diffusion-convection equations and transonic flow
equations in fluid mechanics and the Gauss-Codazzi system for isometric
immersion in differential geometry. In this talk we will discuss some
recent developments in the study of nonlinear conservation laws of mixed
type through these examples with emphasis on identifying/developing
unified mathematical approaches, ideas, and techniques to deal with the
mixed-type problems. Further trends, perspectives, and open problems
in this direction will be also addressed. This talk will be mainly
based on the joint works with M. Feldman, B. Perthame, M. Slemrod,
and D. Wang, respectively.