On degenerate partial differential equations

Abstract

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.

Exact and Numerical Solutions of the Riemann Problem for a Conservative Model of Compressible Two-Phase Flows

“…Systems with missing eigenvectors are often called weakly hyperbolic and systems with coinciding eigenvalues are sometimes also called hyperbolic degenerate. For further reading and a more detailed survey we recommend [10]. Coinciding eigenvalues may lead to difficult situations, but as long as there is a full set of eigenvectors which span the complete space the system is still diagonisable.…”