Multidimensional beyond-all-orders asymptotics for problems in fluid mechanics

近年来，超全阶渐近或指数渐近的技术在从水波和界面流动到图案形成和量子场论的应用领域都取得了重大突破。在这类感兴趣的问题中，目标是了解所有阶数(因此包括代数和指数修正)的渐近逼近的深层结构。例如，在水波理论中，这种超越所有阶数的贡献已被证明在奇异极限下支配自由表面波上的涟漪，例如在小表面张力极限下。然而，尽管取得了巨大的成功，这种技术主要局限于通过非线性常微分方程组(ODE)建模的应用。对于偏微分方程组和类似的多维时空系统的应用仍然非常有限。这位博士将研究一些流体机械问题，这些问题是此类工具迫切需要开发的。这类问题可包括：(I)一维和多维简化浅水(Korteweg-de Vries)模型；(Ii)具有波-结构相互作用的自由面流动模型；(Iii)欧拉方程中随时间变化的呼吸量。

In recent years, techniques in beyond-all-orders asymptotics or exponential asymptotics have led to significant breakthroughs in application areas ranging from water waves and interfacial flows, to pattern formation and quantum field theory. In such problems of interest, the objective is to understand the deep structure of asymptotic approximations beyond all orders (hence including algebraic and exponential corrections). In water-wave theories, for instance, such beyond-all-orders contributions have been shown to govern ripples on free-surface waves in singular limits, such as in the small surface-tension limit. However, while enormously successful, such techniques have been primarily limited to applications that are modeled via nonlinear ordinary differential equations (ODEs). Application to partial differential equations and similar multi-dimensional spatial and temporal systems remains very limited. This PhD will investigate a number of fluid mechanical problems where such tools are crucially in need of development. Such problems may include: (i) single and multi-dimensional reduced shallow-water (Korteweg-de Vries) models; (ii) models for free-surface flows with wave-structure interactions; (iii) time-dependent breathers in the Euler equations.