Research on Nonlinear Partial Differential Equations of Compressible Fluid Mechanics

斯莱姆罗德将继续研究连续介质力学特别是可压缩气体动力学中出现的混合双曲-椭圆型守恒定律。这种流动的经典例子发生在接近音速飞行的跨音速气体动力学中。Slemrod正在开发各种方法来证明各种流动几何中相关偏微分方程组的解的存在性。他的方法的一个令人惊讶的特点是，它还导致了一个新的方法来证明高斯-Codazzi系统解的存在性，该系统描述了三维欧氏空间中高斯曲率既有正号又有负号的二维黎曼流形的等距嵌入问题。首先，给出的方法为工程技术人员在计算机上解决跨音速飞行问题提供了一种新的途径。其次，几何问题虽然本身在数学上很有趣，但也出现在薄壳和纤维增强材料的结构问题中。这项研究将再次为工程师提供在计算机上解决此类问题的新工具。

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.