Singular Perturbation & Riemann Problems

奇异扰动

• 批准号: 9501255
• 负责人: Stephen Schecter
• 金额: $ 10.13万
• 依托单位: North Carolina State University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1995
• 资助国家: 美国
• 起止时间: 1995-11-01 至 1999-04-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9501255&HistoricalAwards=false
• 关键词: Singular; Perturbation; & Riemann; Problems

9501255 Schecter and Lin The investigators propose to continue their research on composite wave-front solutions for singularly perturbed partial differential equations and on Riemann problems for systems of conservation laws. Lin proposes to extend his earlier work on the construction of asymptotic expansions for composite wave-front solutions of reaction-diffusion equations, and on a shadowing lemma approach to proving that there is a true solution near such an expansion, to more general singularly perturbed partial differential equations and to partially singularly perturbed systems. He also proposes to extend to higher dimensional systems the SLEP method for studying the stability of such solutions in a special case. Schecter proposes to build on earlier work that characterized structurally stable strictly hyperbolic Riemann solutions for systems of two conservation laws in one space dimension. He proposes in particular to extend this work to the non-strictly-hyperbolic case, to find all codimension one Riemann solutions by examining the violation of each of the conditions for structural stability, and to study each codimension one bifurcation in detail. Schecter and Lin together propose to look at whether Lin's approach to singular perturbation problems will enable one to regard a Riemann problem solution as the start of an asymptotic expansion of a solution to an associated parabolic problem. %%% Sharp wave fronts occur in many areas of science. From the mathematical viewpoint, they arise as solutions of partial differential equations that model various physical situations. The investigators propose to continue their research on wave fronts. For one type of partial differential equation, reaction-diffusion equations, Lin has developed a method of calculating formal solutions in which several sharp fronts separate more slowly changing portions of the solution. He has also developed a rigorous method, modeled on the "shadowing lemma " of dynamical systems theory, of showing that there is a true solution near the calculated formal solution. He proposes to extend this work to more general partial differential equations. He also proposes to use his approach to extend a technique that has been used in a special case to study the stability of such solutions. For another type of partial differential equation, systems of two conservation laws in one space dimension, Schecter has studied solutions in which jump discontinuities separate slowly changing portions. These arise as solutions of Riemann problems, in which the initial state of the system consists of two constant values separated by a single jump. Schecter has characterized the Riemann solutions that are structurally stable, in the sense that the basic character of the solution does not change when the initial states are varied slightly. He proposes to extend this work in a number of directions, for example, by identifying the simplest ways in which structural stability can break down. Schecter and Lin together propose to look at whether Lin's approach to calculating formal solutions with sharp wave fronts will enable one to a regard a Riemann solution as the start of such a formal solution to a more realistic partial differential equation. ***

9501255 Schecter和Lin研究人员建议继续研究奇摄动偏微分方程的复合波前解和守恒律系统的黎曼问题。 林建议延长他早期的工作建设的渐近展开复合波前解决方案的反应扩散方程，并在一个阴影引理的方法来证明，有一个真正的解决方案附近的这种扩张，更一般的奇摄动偏微分方程和部分奇摄动系统。 他还建议扩展到高维系统的SLEP方法研究的稳定性，这种解决方案在特殊情况下。 Schecter建议建立在早期的工作，其特点是结构稳定的严格双曲黎曼解系统的两个守恒律在一个空间维度。 他建议特别是延长这项工作的非严格双曲的情况下，找到所有余维一黎曼解决方案，通过检查违反每个条件的结构稳定性，并研究每个余维一分歧的详细情况。 Schecter和林一起提出看看是否林的方法奇异摄动问题将使人们能够把黎曼问题的解决方案作为开始的渐近展开的解决方案，以相关的抛物问题。 尖锐的波前出现在许多科学领域。 从数学的角度来看，它们是模拟各种物理情况的偏微分方程的解。 研究人员建议继续他们对波阵面的研究。 对于一种类型的偏微分方程，反应扩散方程，林已经开发了一种计算形式解的方法，其中几个尖锐的前沿分离更缓慢变化的部分的解决方案。 他还开发了一种严格的方法，仿照“阴影引理“的动力系统理论，显示有一个真正的解决方案附近的计算正式解决方案。 他建议将这项工作扩展到更一般的偏微分方程。 他还建议使用他的方法来扩展一种在特殊情况下使用的技术，以研究这种解决方案的稳定性。 对于另一种类型的偏微分方程，系统的两个守恒律在一个空间维，谢克特研究的解决方案，其中跳跃间断分离缓慢变化的部分。 这些都是黎曼问题的解，在黎曼问题中，系统的初始状态由两个被单个跳跃分开的常数值组成。 Schecter已经描述了黎曼解的结构稳定性，即当初始状态稍微改变时，解的基本特征不会改变。 他建议在许多方向上扩展这项工作，例如，通过确定结构稳定性可能崩溃的最简单方式。 Schecter和林一起建议看看是否林的方法来计算正式的解决方案与尖锐的波前将使一个方面的黎曼解开始这样一个正式的解决方案，以一个更现实的偏微分方程。 ***