Shock Wave Theory, History and Possible Future
There have been continuing progresses in shock wave theory for the past half century. Deep results are obtained for one-dimensional conservation laws. Some of the central issues that have been raised and resolved are the entropy condition, Riemann problem, N-waves, compactness and regularity due to nonlinearity, well-posedness, and zero dissipation limits. In recent years, there are focused and intensive efforts on the study of multi-dimensional shock waves in gas flows. This is to go back to the basic physical concerns of people in the first half of 20th century, as recorded in the classical book of Courant-Friedrichs. However, the works of Prandtl, von Neumann and others are mostly about shock polar and simple waves and are algebraic in nature. The recent efforts aim at solving nonlinear partial differential equations of mixed types with free boundary. Because of the obvious mathematical difficulties, it is important to identify basic problems, in particular the role played by the solid boundary. We will survey the historical developments, discuss the recent efforts, and raise open problems.
Tai-Ping Liu, Stanford University