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解决的数学方程式所支配的。在现代，大规模的数值计算和数据分析是强有力的工具，但许多问题仍然难以解决。对于复杂的多尺度和多参数系统，往往需要根据某些参数的大小来应用关键的约简。这些约简的应用被称为渐近分析；这些方法有能力极大地简化复杂系统的显著特征，提取关键机制，并在数值和实验失败的区域提供细节。正如Cright ton[1]所指出的，“没有渐近信息指导的计算或实验方案的设计往好了说是浪费，往坏了说是危险的，因为可能无法识别关键的(僵硬的)特征……”一些最具挑战性的问题与预测指数级小的影响有关，这些影响对于传统的渐近分析是不可见的，并且经常被错误地认为可以忽略不计。在某些情况下，这些影响可能对应于某些可观察到的特征，例如系统中的振荡或波动；在其他情况下，它们可能在很大程度上是不可观察的，但相反地，它们用于确定某些解决方案是否允许。在过去的几十年里，人们已经意识到无处不在的问题，其中指数级的小效应是矛盾的重要的--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、地球物理等相关的研究中找到。对于指数级小项的研究，在数学和计算上都面临着巨大的挑战。例如，20世纪初发展起来的传统数学技术通常是不够的。指数渐近性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(I)重力驱动的水流通过缓慢移动的全船产生的水波；(Ii)有限深度流体中的孤立波，包括重力和毛细效应；以及(Iii)粘性流动，其中在界面上产生气泡或手指。这些问题都涉及到关键的指数小效应。尽管取得了上述成就，但在该领域的众多研究中出现了一个显著的瓶颈：现有的大多数指数渐近技术仅限于常微分方程组，例如，只考虑一维流体界面。在过去二十年中出现的许多指数渐近的惊人成功在更高维空间或依赖于时间的公式中都有类似之处，在这些公式中，系统由偏微分方程所支配。然而，指数渐近中的标准方法并不容易适用于研究这种情况。关于寻求理论扩展的最新初步工作表明，取得进展的可能途径在于将分析方法与计算和数据驱动的方法相结合-因此对指数渐近的混合数值-渐近方法。这些方法的发展，以及随后对流体力学中多维问题的应用，构成了本项目的主旨。[1]Cright ton，D.G.(1994)。渐近性--应用数学建模中思想、计算和实验不可或缺的补充。正在进行中。7欧元。电话会议关于数学。行业(ECMI)，蒙特卡蒂尼(第3-19页)。

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).