Mathematical Sciences: Quasistatic Approximations for SecondOrder Evolution Equations

数学科学：二阶演化方程的准静态近似

• 批准号: 9003543
• 负责人: Hans Engler
• 金额: $ 3.82万
• 依托单位: Georgetown University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1990
• 资助国家: 美国
• 起止时间: 1990-07-15 至 1993-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9003543&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Quasistatic; Approximations; SecondOrder

This work will be concerned principally with the analysis of classes of second order partial differential equations or partial integrodifferential equations describing time-dependent phenomena. In certain cases, good approximations to the solutions can be obtained by discarding the highest order time derivative in the equation. Work will be done in justifying such approximations by considering the effects of sending the highest time derivative to zero. For the short time behavior, one obtains a singular perturbation problem in which transient phenomena governed by a related equation should occur on a fast time scale. For the long time behavior, regular dependence of the time-asymptotic limit or - in cases of severe non-uniqueness of stationary states - selection effects from the transient behavior are expected. In this project, efforts will be made to place some of these heuristic conjectures on a mathematically sound basis. Work will also be done on questions of existence of solutions for the reduced problems. Sources for the equations under study include mathematical models describing shear flows for viscoelastic materials and polymerrheology.

这项工作将主要涉及分析 一类二阶偏微分方程或偏微分方程 描述时间相关的积分微分方程 现象。 在某些情况下， 可以通过丢弃最高阶时间来获得解 方程中的导数。 将努力证明这种做法是正当的。 通过考虑发送最高值的影响， 时间导数为零。 对于短时间行为， 得到一个奇异摄动问题，其中瞬态 由相关方程控制的现象应该发生在一个快速的 时间尺度 对于长时间行为， 时间渐近极限或-在严重非唯一性的情况下 稳定状态-选择效应从瞬态 行为是预期的。 在该项目中，将努力 把这些启发式的算法放在一个数学上， 健全的基础。 还将就存在的问题开展工作， 减少问题的解决方案。 研究中的方程的来源包括数学 描述粘弹性材料剪切流的模型， 聚合物流变学