  Some of recent developments, including recent results, ideas, techniques, and
approaches, in the study of degenerate partial differential equations are
surveyed and analyzed. Several examples of nonlinear degenerate, even mixed,
partial differential equations, are presented, which arise naturally in some
longstanding, fundamental problems in fluid mechanics and differential
geometry. The solution to these fundamental problems greatly requires a deep
understanding of nonlinear degenerate partial differential equations. Our
emphasis is on exploring and/or developing unified mathematical approaches, as
well as new ideas and techniques. The potential approaches we have identified
and/or developed through these examples include kinetic approaches, free
boundary approaches, weak convergence approaches, and related nonlinear ideas
and techniques. We remark that most of the important problems for nonlinear
degenerate partial differential equations are truly challenging and still
widely open, which require further new ideas, techniques, and approaches, and
deserve our special attention and further efforts.
