Conservation Laws with Partial Dissipation in Continuum Mechanics

连续介质力学中的部分耗散守恒定律

• 批准号: 0207154
• 负责人: Yanni Zeng
• 金额: $ 7.39万
• 依托单位: University of Alabama at Birmingham
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2002
• 资助国家: 美国
• 起止时间: 2002-07-01 至 2006-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0207154&HistoricalAwards=false
• 关键词: Conservation; Laws; Partial; Dissipation; Continuum

Proposal #0207154PI: Yanni ZengInstitution: University of Alabama BirminghamTitle: Conservation laws with partial dissipation in continuum mechanicsThis project is concerned with systems of conservation laws (balance laws), a broad class of nonlinear partial differential equations (PDEs) arising from continuum mechanics. Of special interest are equations with partial dissipation, including hyperbolic-parabolic systems, hyperbolic systems with fading memory, and hyperbolic systems with relaxation. The purpose of the research is to obtain a better understanding of the underlying physical phenomena through the study of the qualitative behavior of the solutions of the PDEs. The first objective is to obtain the nonlinear stability of shock profiles for hyperbolic systems with relaxation, including fully dissipative systems and those containing one non-decaying wave in a linearly degenerate field (systems of composite type). The second objective is to obtain a similar stability result for initial boundary-value problems for hyperbolic-parabolic systems, especially to understand the effect of the boundary. The third objective is to extend the study to contact discontinuities and rarefaction waves. The research conducted under this award is relevant for the study of compressible viscous flows, the motion of viscoelastic materials, gas dynamics in thermal nonequilibrium, and magnetohydrodynamics. By including viscosity, heat conduction, electrical resistivity, excited internal structures of molecules, chemical reactions, and so on we focus on mathematical models that are closer to the real world than the traditional ideal ones. The project has three specific objectives. The first objective is to understand the stability properties of shock profiles in, for example, high-temperature gas flows when the internal structure of molecules and chemical reactions are no longer negligible. This problem arises, for example, at the reentry of a space shuttle into the atmosphere and is quite different from the traditional supersonic flight of an airplane. The second objective is to better understand gas flows near a solid surface. The third objective is to study more complete wave patterns in several cases.

提案#0207154 PI：Yanni Zeng机构：亚拉巴马伯明翰大学标题：连续介质力学中的部分耗散守恒律本项目涉及守恒律系统（平衡律），这是一个广泛的非线性偏微分方程（PDE），源于连续介质力学。 特别感兴趣的是具有部分耗散的方程，包括双曲抛物方程组，具有衰减记忆的双曲方程组和具有松弛的双曲方程组。 研究的目的是通过对偏微分方程解的定性行为的研究，更好地理解潜在的物理现象。 第一个目标是获得具有松弛的双曲系统的激波剖面的非线性稳定性，包括完全耗散系统和在线性退化场中包含一个非衰减波的系统（复合型系统）。 第二个目标是获得双曲抛物方程组初边值问题的类似稳定性结果，特别是了解边界的影响。 第三个目标是将研究扩展到接触不连续和稀疏波。在该奖项下进行的研究是相关的可压缩粘性流动，粘弹性材料的运动，热不平衡气体动力学和磁流体力学的研究。 通过包括粘度，热传导，电阻率，分子的激发内部结构，化学反应，等等，我们专注于数学模型，更接近真实的世界比传统的理想。 该项目有三个具体目标。 第一个目标是了解激波剖面的稳定性，例如，当分子的内部结构和化学反应不再可以忽略不计时，高温气流中的激波剖面。 例如，在航天飞机重返大气层时就会出现这个问题，这与传统的飞机超音速飞行完全不同。 第二个目标是更好地理解固体表面附近的气体流动。 第三个目标是在几种情况下研究更完整的波型。