Research on Nonlinear Partial Differential Equations of Compressible Fluid Mechanics

可压缩流体力学非线性偏微分方程研究

• 批准号: 0647554
• 负责人: Marshall Slemrod
• 金额: $ 13.56万
• 依托单位: University of Wisconsin-Madison
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2007
• 资助国家: 美国
• 起止时间: 2007-07-01 至 2010-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0647554&HistoricalAwards=false
• 关键词: Research; Nonlinear; Partial; Differential; Equations

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.

Slemrod将继续他的研究到守恒定律的混合双曲椭圆型所产生的连续介质力学，特别是可压缩气体动力学。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正在开发各种流动几何形状的偏微分方程相关系统的解的存在性证明方法。一个令人惊讶的特点，他的做法是，它也导致了一个新的方法来证明存在的解决方案的高斯-Codazzi系统描述的问题等距嵌入的二维黎曼流形在三维欧氏空间与高斯曲率有正负sign. Implications这项研究是相当惊人的。首先，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，几何问题虽然在数学中有其自身的意义，但也出现在薄壳和纤维增强材料结构中出现的问题中。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。