Singular Perturbation & Riemann Problems

[501255] 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。Lin建议将他早期的工作扩展到反应扩散方程复合波前解的渐近展开式的构造，以及证明在这种展开式附近存在真解的阴影引理方法，扩展到更一般的奇摄动偏微分方程和部分奇摄动系统。他还提出将SLEP方法推广到高维系统，用于研究这类解在特殊情况下的稳定性。Schecter建议建立在早期的工作基础上，该工作描述了一维空间中具有两个守恒定律的系统的结构稳定的严格双曲黎曼解。他特别建议将这项工作扩展到非严格双曲情况，通过检查结构稳定性的每个条件的违反来找到所有的余维一黎曼解，并详细研究每个余维一分岔。Schecter和Lin一起提出，看看Lin对奇异摄动问题的方法是否能够使人们将黎曼问题的解视为相关抛物问题解的渐近展开的开始。尖锐的波锋出现在许多科学领域。从数学角度来看，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锋的研究。对于一类偏微分方程，反应-扩散方程，Lin开发了一种计算形式解的方法，其中几个尖锐的锋面将溶液中变化较慢的部分分开。他还开发了一种严格的方法，以动力系统理论的“阴影引理”为模型，表明在计算的形式解附近存在一个真解。他建议将这项工作扩展到更一般的偏微分方程。他还建议用他的方法来扩展一种技术，这种技术已经在一个特殊情况下用于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，一个空间维度上的两个守恒律系统，Schecter研究了跳跃不连续分离缓慢变化部分的解。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常数值组成，它们被一个跳跃分开。谢克特描述了结构稳定的黎曼解，即当初始状态发生轻微变化时，解的基本特征不会改变。他建议将这项工作扩展到多个方向，例如，通过确定结构稳定性可能被破坏的最简单方式。Schecter和Lin一起提出，看看Lin计算尖锐波前形式解的方法是否能使人们将黎曼解视为更现实的偏微分方程形式解的开始。＊＊＊

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. ***