Exponential asymptotics for multi-dimensional systems in fluid mechanics

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).