Exponential asymptotics for integro-differential equations

大量的物理问题涉及“小”效应的影响。一个例子是不同粘度的两种粘性流体之间界面上的表面张力的影响。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产生令人惊讶的影响。由于影响很小，从数学上来说，尝试对控制其对问题的影响的参数进行渐近展开是很自然的。但是，在许多情况下，标准的渐近分析是失败的，为了确定小扰动的全局影响，人们必须检查指数级的小项。“指数渐近性”这门学科已经成为应用数学中一个非常活跃的领域。然而，最近的工作集中在使许多数学思想更加严谨。这项建议寻求继续将指数渐近这一主题放在更严格的基础上，特别是当它与积分-微分方程式的分析有关时。将分析一个初步的案例研究，包括泰勒泡沫上升的选择问题。

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.