Exponential asymptotics for multi-dimensional systems in fluid mechanics

绝大多数处于科学前沿的问题都是由无法精确求解的数学方程控制的。在现代，大规模的数值计算和数据分析是强大的工具，但许多问题仍然逃避蛮力计算。对于复杂的多尺度和多参数系统，通常需要根据某些参数的大小来应用关键约简。这些简化的应用被称为渐近分析;这些方法有能力将复杂系统简化为显着特征，提取关键机制，并在数值和实验失败的区域提供细节。正如Crighton [1]所指出的那样：“没有渐近信息指导的计算或实验方案的设计充其量是浪费，最坏的情况是危险的，因为可能无法识别关键（刚性）特征。“一些最具挑战性的问题涉及到预测指数级小的影响，这些影响是传统渐近分析不可见的，并且经常被错误地认为是可以忽略的。在某些情况下，这些效应可能对应于一些可观察到的特征，例如系统中的振荡或波;在其他情况下，它们可能在很大程度上是不可观察的，而是用来确定某些解决方案是否是允许的。在过去的几十年里，人们已经认识到指数小效应是矛盾重要的问题的普遍性-这些问题可以在与树枝状晶体生长，粘性流体流动，水波，量子隧道，量子物理学等相关的研究中找到。对于指数小项的研究，存在显著的数学和计算挑战。例如，现有的、发展于世纪早期的传统数学技术通常是不够的。指数渐近是过去二十年来为解决这些问题而开发的一套专门技术的名称。在过去的几年里，指数渐近性的一些最重要的应用与自由表面流动理论的发展有关。这包括研究（一）由重力驱动的水流通过缓慢移动的完整船舶产生的水波;（二）包括重力和毛细效应在内的有限深度流体中的孤立波;以及（三）在界面处产生气泡或手指的粘性流动。这些问题都涉及到关键的指数小effects.Despite上述成功，一个显着的瓶颈已经出现在许多研究领域：大多数现有的指数渐近技术是有限的常微分方程，例如，只有一个一维的流体界面被认为是。在过去的二十年里，指数渐近的许多惊人的成功在高维空间或依赖于时间的公式中都有类似物，其中系统由偏微分方程控制。然而，指数渐近的标准技术不容易适应研究这种情况。最近的初步工作，寻求扩展的理论表明，可能的途径进展在于结合分析方法与计算和数据驱动的方法-因此，混合数值渐近方法的指数渐近。这些方法的发展，以及随后在流体力学多维问题中的应用，形成了这个项目的主要推力。[1]克赖顿湾G.（1994年）。渐近性--应用数学模型中思想、计算和实验的不可缺少的补充。在Proc.7th Eur. Conf. on Math. Industry（ECMI），Montecatini（pp. 3-19）。

The vast majority of problems that lie at the forefront of science are governed by mathematical equations that cannot be solved exactly. In the modern era, large-scale numerical computation and data analysis are powerful tools, but many questions still elude brute-force computation. For complex multi-scale and multi-parameter systems, it is often necessary to apply key reductions dependent on the smallness or largeness of certain parameters. The application of these reductions is called asymptotic analysis; these methods have the power to dramatically simplify complex systems to their salient features, extract key mechanisms, and provide details in regions where numerics and experiments fail. As noted by Crighton [1] "[the] design of computational or experimental schemes without the guidance of asymptotic information is wasteful at best, and dangerous at worst, because of the possible failure to identify crucial (stiff) features..."Some of the most challenging problems relate to the prediction of exponentially small effects that are invisible to traditional asymptotic analysis and often mistakenly considered as negligible. In some cases, these effects may correspond to some observable feature, such as an oscillation or wave in the system; in other cases, they may be largely non-observable, but instead serve to determine whether certain solutions are permissible. Over the last few decades, there has been an appreciation for the ubiquity of problems where exponentially-small effects are paradoxically important -- these problems can be found in studies related to dendritic crystal growth, viscous fluid flow, water waves, quantum tunneling, geophysics, and more. There are significant mathematical and computational challenges for the study of exponentially small terms. For example, the traditional mathematical techniques that exist, developed in the early 20th century, are usually insufficient. Exponential asymptotics is the name given to the set of specialised techniques that have been developed over the last two decades for these problems. In the last few years, some of the most significant applications of exponential asymptotics have related to the development of theory for free-surface flows. This includes the study of (i) water waves produced by gravity-driven flows past slow-moving full-bodied ships; (ii) solitary waves in a fluid of finite depth including both gravity and capillary effects; and (iii) viscous flows where bubbles or fingers are produced at an interface. These problems all involve crucial exponentially small effects.Despite the above successes, a significant bottleneck has emerged in numerous studies in the area: the majority of existing exponential asymptotic techniques are limited to ordinary differential equations where, for instance, only a one-dimensional fluid interface is considered. Many of the spectacular successes of exponential asymptotics that have emerged in the last two decades have analogues in higher-dimensional space or in time-dependent formulations, where the system is governed by partial differential equations. However, the standard techniques in exponential asymptotics are not easily adapted to study such situations. The most recent preliminary work on seeking extensions of the theory has shown that the likely avenue for progress lies with combining analytical methods with computational and data-driven approaches---hence a hybrid numerical-asymptotic approach to exponential asymptotics. The development of these methodologies, and the subsequent applications to multi-dimensional problems in fluid mechanics forms the main thrust of this project. [1] Crighton, D. G. (1994). Asymptotics--an indispensable complement to thought, computation and experiment in applied mathematical modelling. In Proc. 7th Eur. Conf. on Math. Industry (ECMI), Montecatini (pp. 3-19).