Mixed Type Problems for Transonic Flow in Gas Dynamics and Isometric Embedding in Geometry

The mixed type problems arising in transonic flows in gas dynamics and isometric embeddings in geometry are considered. The two-dimensional steady Euler equations for the transonic flow past an obstacle such as an airfoil are first discussed. A viscous approximation to the steady transonic flow problem is presented, and its convergence is obtained by the method of compensated compactness. Then the isometric embedding problem in geometry ∗Also, Department of Mathematics, Northwestern University, Evanston, IL 60208, USA; School of Mathematical Sciences, Fudan University, Shanghai 200433, China. G.-Q. Chen’s research was supported in part by the National Science Foundation under Grants DMS-0935967, DMS-0807551, the Natural Science Foundation of China under Grant NSFC-10728101, the Royal Society-Wolfson Research Merit Award (UK), and the UK EPSRC Science and Innovation Award to the Oxford Centre for Nonlinear PDE (EP/E035027/1) †Also, Department of Mathematics, University of Pittsburgh, Pittsburgh, PA 15260, USA; Department of Mathematical Sciences, Korean Advanced Institute for Science and Technology, Daejeon, Republic of Korea. M. Slemrod’s research was supported in part by the National Science Foundation under Grant DMS-0647554. ‡D. Wang’s research was supported in part by the National Science Foundation under Grants DMS-0604362 and DMS-0906160, and by the Office of Naval Research under Grant N00014-07-1-0668. 2 G.-Q. Chen, M. Slemrod, D. Wang is discussed. A fluid dynamic formulation of the Gauss-Codazzi system for the isometric embedding of two-dimensional surfaces is provided, and an existence result of isometric immersions with negative Gauss curvature is given. The Div-Curl structure and the weak continuity of the Gauss-Coddazi-Ricci system for the isometric embedding of higher dimensional manifolds are also studied.