On Degenerate Partial Differential Equations

TL;DR

This survey reviews recent advances in the study of nonlinear degenerate partial differential equations, highlighting new approaches and techniques crucial for solving longstanding problems in fluid mechanics and differential geometry.

Contribution

It provides a comprehensive analysis of unified mathematical approaches, including kinetic, free boundary, and weak convergence methods, for nonlinear degenerate PDEs.

Findings

Identification of key mathematical approaches

Application to fundamental problems in fluid mechanics

Highlighting open challenges and future directions

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.