Prandtl Conjecture and von Neumann Paradox for Shock Reflections

We plan to discuss the two fundamental problems, the Prandtl Conjecture and von Neumann Paradox, in the theory of supersonic flows with shock waves. These problems were raised in the first half of the 20th century and clearly formulated in the classical treaty of Courant-Friedrichs. There have been spetacular progresses made on one-dimensional shock wave theory during the second half od 20th century. For the past several years, serious efforts are made in the study of multi-dimensional shock wave theory, centered mostly around aforementioned two problems. These problems involve nonliear partial differential equations of mixed types and free boundaries. We plan to discuss the recent progresses and open research directions on these problems.