Singular Perturbation & Riemann Problems

9501255谢克特和林研究人员建议继续研究奇摄动偏微分方程组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推广到反应扩散方程复合波前解的渐近展开式的构造，以及证明在这样的展开式附近存在真解的跟踪引理方法，推广到更一般的奇摄动偏微分方程组和部分奇摄动系统。他还建议将SLEP方法推广到高维系统，在特殊情况下研究此类解的稳定性。谢克特建议建立在早期工作的基础上，这些工作刻画了一维空间中两个守恒律系统的结构稳定、严格双曲的黎曼解。他特别建议将这项工作推广到非严格双曲情形，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Riemann解，并详细地研究每个余维一分叉。谢克特和林一起提出了一个问题，即林处理奇异摄动问题的方法是否使人能够将黎曼问题的解视为相关抛物问题解的渐近展开的开始。在科学的许多领域都会出现尖锐的波前。从数学的观点来看，它们是以偏微分方程解的形式出现的，这些偏微分方程会模拟各种物理情况。研究人员建议继续他们对波前的研究。对于一类偏微分方程，反应扩散方程，林发展了一种计算形式解的方法，在这种方法中，几个尖锐的前锋将解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“影子引理”为模型，证明在计算出的形式解附近存在真正的解。他建议将这项工作扩展到更一般的偏微分方程式。他还建议使用他的方法来扩展一种在特殊情况下用于研究此类解的稳定性的技术。对于另一类偏微分方程，即一维空间中的两个守恒律系统，谢克特研究了跳跃间断将缓慢变化的部分分开的解。这些都是黎曼问题的解，其中系统的初始状态由两个被一次跳跃隔开的常量组成。谢克特刻画了结构上稳定的黎曼解，即当初始状态稍有变化时，解的基本性质不变。他建议将这项工作扩展到多个方向，例如，通过确定可能破坏结构稳定性的最简单方式。谢克特和林一起建议研究林计算具有尖锐波前的形式解的方法是否使人能够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. ***