Shock Wave Theory

冲击波理论

• 批准号: 9803323
• 负责人: Tai-Ping Liu
• 金额: $ 22.6万
• 依托单位: Stanford University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1998
• 资助国家: 美国
• 起止时间: 1998-08-01 至 2003-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9803323&HistoricalAwards=false
• 关键词: Shock; Wave; Theory

DMS-9803323 Tai-Ping Liu We propose to study two fundamental questions for general systems of hyperbolic conservation laws: The first is the well-posedness problem. This is done with Tong Yang. For this, we already constructed a nonlinear functional for the two conservation laws and are in the process of doing so for the general systems. The second problem is to study multi-dimensional gas flows. Lien and the author have succeeded in showing that three-dimensional self-similar gas flow past a cone is nonlinearly stable. We are generalizing the new approach of superpositioning local self-similar flows to other problems. We expect this approach to yield new qualitative information on nonlinear stability and instability of multi-dimensional gas flows. Our second project is to study the viscous conservation laws. With Zeng, we are in the process of studying the stability of nonlinear waves for hyperbolic-parabolic systems such as the compressible Navier-Stokes equations. Other problems in combustion and MHD, and numerical analysis of shock calculations are being considered. Of particular interest is the role of dissipative parameters on the structure and stability of nonlinear waves. The pointwise approach the author introduced is effective for these purposes. The author plans to study basic questions concerning the invisicd and viscous solutions for gas dynamics and mechanics. One of the fundamental questions concerns the validity of the Euler equations for the gas dynamics. The equations were derived by Euler in the eighteenth century. Only until recently the author and his collaborator are able to show that the equations are mathematically valid in that small errors in the data yield small errors in the solutions. This is of obvious importance in the applications because there is always small error either in the experimental data or numerical computations. Other problems of interest to the author include the sensitive role of di ssipation parameters. In combustion, magneto-hydrodynamics, nonlinear elasticity and other continuum physics, dissipative mechanisms, such as viscosity, heat conductivity, and species diffusions are important. The geometric properties and stability of the nonlinear waves may depend sensitively on these parameters. The author has recently introduced a pointwise approach, which is effective in studying the nonlinear interactions of waves. The role of dissipation parameters are being studied using the basic conservation laws in physics.

我们提出研究一般双曲守恒律系统的两个基本问题：第一个是适定性问题。这是和童阳一起做的。为此，我们已经为这两个守恒定律构造了一个非线性泛函，并且正在为一般系统这样做。第二个问题是研究多维气体流动。Lien和作者已经成功地证明了三维自相似气体流过锥体是非线性稳定的。我们将局部自相似流叠加的新方法推广到其他问题。我们期望这种方法能够产生关于多维气体流动的非线性稳定性和不稳定性的新的定性信息。我们的第二个课题是研究粘性守恒定律。与Zeng一起，我们正在研究双曲抛物系统的非线性波的稳定性，如可压缩的Navier-Stokes方程。燃烧和MHD的其他问题，以及激波计算的数值分析也在考虑之中。特别令人感兴趣的是耗散参数对非线性波的结构和稳定性的作用。作者介绍的点式方法对这些目的是有效的。作者计划研究气体动力学和力学的无形解和粘性解的基本问题。欧拉方程在气体动力学中的有效性是最基本的问题之一。这些方程是欧拉在18世纪推导出来的。直到最近，作者和他的合作者才能够证明这些方程在数学上是有效的，因为数据中的小误差会导致解的小误差。这在实际应用中具有明显的重要性，因为无论是实验数据还是数值计算，都存在很小的误差。作者感兴趣的其他问题包括耗散参数的敏感作用。在燃烧、磁流体力学、非线性弹性和其他连续介质物理中，耗散机制，如粘度、导热性和物质扩散是重要的。非线性波的几何性质和稳定性可能敏感地依赖于这些参数。作者最近介绍了一种有效地研究波的非线性相互作用的逐点方法。利用物理学中的基本守恒定律研究了耗散参数的作用。