The author surveys and analyzes some recent developments, including recent results, ideas, techniques, and approaches, in the study of degenerate partial differential equations arising mainly in fluid mechanics. The examples include nonlinear degenerate hyperbolic systems of conservation laws, the nonlinear degenerate diffusion-advection equation, the Euler equations for compressible flow, and the Gauss-Codazzi system for isometric embedding in differential geometry. The emphasis is on exploring and developing unified mathematical approaches, as well as new ideas and techniques. Among these, let us mention some of the author’s recent works. A compensated compactness approach to establish the existence of a weak solution of the Gauss-Codazzi system for isometric embedding was presented in [the author, M. Slemrod and D. Wang, Commun. Math. Phys. 294, No. 2, 411–437 (2010; Zbl 1208.53006)]; an approach with free-boundary techniques was developed in the shock reflection-diffraction problem for potential flow in [the author and M. Feldman, Ann. Math. (2) 171, No. 2, 1067–1182 (2010; Zbl 1277.35252)]; how the singular limits to nonlinear degenerate hyperbolic systems of conservation laws via weak convergence methods can be achieved through the vanishing viscosity limit problem for the Navier-Stokes equations to the isentropic Euler equations can be found in [the author and M. Perepelitsa, Commun. Pure Appl. Math. 63, No. 11, 1469–1504 (2010; Zbl 1205.35188)].
For the entire collection see [Zbl 1200.35002].