Cavitation and Concentration in the Solutions of the Compressible Euler Equations and Related Nonlinear PDEs in Fluid Dynamics

In this talk, we will discuss the intrinsic phenomena of cavitation/decavitation and concentration/deconcentration in the entropy solutions of the compressible Euler equations, the compressible Euler-Poisson equations, and related nonlinear PDEs, which are fundamental to the analysis of entropy solutions for nonlinear PDEs. We will start to discuss the formation process of cavitation and concentration in the entropy solutions of the isentropic Euler equations with respect to the initial data and the vanishing pressure limit. Then we will analyse a longstanding fundamental problem in fluid dynamics: Does the concentration occur generically so that the density develops into a Dirac measure at the origin in spherically symmetric entropy solutions of the multi-dimensional compressible Euler equations and related nonlinear PDEs? We will report our recent results and approaches developed for solving this problem for the Euler equations, the Euler-Poisson equations, and related nonlinear PDEs, and discuss its close connections with entropy methods and the theory of divergence-measure fields. Further related topics, perspectives, and open problems will also be addressed.