Conservation Laws: Transonic Flow and Differential Geometry

The connection between gas dynamics and differential geometry is discussed. Some history of boundary value problems for systems of conservation laws is first given. Then the mathematical formulation of compressible gas dynamics, especially the subsonic and transonic flows past an obstacle (such as an airfoil), is provided. Some recent results on transonic flow from viscous approximation and compensated compactness are presented. Finally, a fluid dynamic formulation of the Gauss-Codazzi system for the isometric embedding problem in R3 is discussed. 1. History of Boundary Value Problems for Systems of Conservation Laws A quick glance at post-World War II literature may surprise the reader. The subject of systems of conservation laws was dominated not by initial value problems but by boundary value problems. A little thought of course makes this state even clear: It was the underlying military role of gas dynamics, and the flow of compressible fluids over boundaries that was crucial in wartime and postwar research. However, taking a more civilian view, it is pleasant to quote A. Jameson [17] who wrote: “The most important requirement for aeronautical applications of computational methods in fluid mechanics is the capability to predict the steady flow past a proposed configuration, so that key performance parameters such as the lift to drag ratio can be estimated. Even in maneuvering flight the time scales of the motion are large compared with those of the flow, so that unsteady effects are secondary”. In slightly less technical terms, we civilian fliers should note that, in most of the time we spend on our flight, we are flying at steady (not accelerating or decelerating) flow, and the engineer and mathematician need be concerned only with boundary 2000 Mathematics Subject Classification. 76H05,35M10,35A35,76N10,76L05, 53C42.