Research on Nonlinear Partial Differential Equations of Compressible Fluid Mechanics

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Slemrod will continue his research into conservation laws of mixed hyperbolic- elliptic type arising in continuum mechanics and especially compressible gas dynamics. The classical example of such flows occurs in transonic gas dynamics for flight near the speed of sound. Slemrod is developing methods for proving existence of solutions to the relevant system of partial differential equations in various flow geometries. A surprising feature of his approach is that it also leads to a new approach for proving existence of solutions to the Gauss-Codazzi system describing the problem of isometric embedding of a two dimensional Riemannian manifold in three dimensional Euclidean space with Gauss curvature having both positive and negative sign.The implications of this research are quite striking. First of all the methods given provide a new way for engineers to solve problems of transonic flight on computers. Secondly the geometry problem while of interest in its own right in mathematics also occurs in problems arising in the structure of thin shells and and fiber re-enforced materials. Again the research will provide engineers new tools for solving such problems on computers.