Exponential asymptotics for integro-differential equations

许多物理问题涉及"小“效应的影响。一个例子是在两种不同粘度的粘性流体之间的界面处的表面张力的影响。虽然这种效应看起来“很小”，但在决定系统的全局动力学性质方面往往会有令人惊讶的影响。由于影响很小，在数学上很自然地试图在控制其对问题的影响的参数中进行渐近展开。然而，在许多情况下，标准的渐近分析失败，人们不得不检查指数小项以确定小扰动的全局影响。"指数渐近“这一主题已成为应用数学中一个非常活跃的领域。然而，最近的工作集中于使许多数学思想更加严谨。这项建议旨在继续把指数渐近的主题，特别是因为它涉及到积分微分方程的分析，在一个更严格的基础上的任务。最初的案例研究，包括泰勒泡沫上升的选择问题，将进行分析。

A large number of physical problems involve the influence of ``small'' effects. An example is the effect of surface tension at the interface between two viscous fluids of differing viscosities. While apparently ``small'', such effects can often be surprisingly influential in determining global dynamical properties of the system. Since the effect is small, mathematically it is natural to attempt an asymptotic expansion in the parameter governing its influence on the problem.However, in many cases, standard asymptotic analysis fails and one has to examine exponentially small terms in order to ascertain the global effect of the small perturbation. This subject of ``exponential asymptotics'' has become a very active field of applied mathematics.Recent work, however, has focussed on rendering many of the mathematical ideas more rigorous. This proposal seeks to continue with the task of putting the subject of exponential asymptotics, especially as it pertains to the analysis of integro-differential equations, on a more rigorous footing. An initial case study, consisting of the selection problem for the rising Taylor bubble, will be analysed.