Mathematical Sciences: Quasistatic Approximations for SecondOrder Evolution Equations

本文主要研究描述含时现象的二阶偏微分方程组或偏积分微分方程族的分析。在某些情况下，可以通过丢弃方程中的最高阶时间导数来获得对解的良好逼近。将通过考虑将最高时间导数发送到零的影响来证明这种近似是合理的。对于短时间行为，我们得到了一个奇异摄动问题，其中由相关方程控制的暂态现象应该在快速时间尺度上发生。对于长时间行为，期望时间渐近极限的规则依赖性，或者--在定态严重不唯一性的情况下--来自瞬时行为的选择效应。在这个项目中，将努力将这些启发式猜想中的一些放在数学上合理的基础上。还将就减少的问题是否存在解决办法的问题开展工作。所研究的方程的来源包括描述粘弹性材料的剪切流动和聚合物流变学的数学模型。

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.