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Coherent structures in non-local active-dissipative equations: theory and computations

非局部主动耗散方程中的相干结构：理论与计算

基本信息

批准号：
EP/J001740/1
负责人：
Dmitri Tseluiko
金额：
$ 12.83万
依托单位：
Loughborough University
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2012
资助国家：
英国
起止时间：
2012 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FJ001740%2F1
关键词：
Coherent structures non local active

项目摘要

The aim of the proposed research is to analyse coherent structures for non-local active-dissipative partial differential equations (PDEs). Such equations are characterised by the presence of mechanisms of instability (energy production) and stability (energy dissipation). In addition, such equations contain non-local terms that cannot be expressed as polynomials or functions of the solution to be found and its derivatives. The solutions of active-dissipative equations are often characterised by a particular spatial and temporal behaviour, e.g. by space, time or space-time localised structures, the so-called coherent structures. The study of such structures has received a lot of attention over recent years due to their importance in hydrodynamics, nonlinear optics, chemical-reaction theory and mathematical biology. Despite recent developments in the theory of such structures for 'local' equations, coherent-structures interaction in non-local active-dissipative equations has not been developed as of yet and new mathematical techniques need to be introduced. The proposed research is a decisive step towards the understanding of the formation and interaction of coherent structures for such non-local equations with the ultimate goal to advance the understanding of non-local effects in PDEs, which is important both from the theoretical point of view and for numerous practical applications. The research is particularly timely because of its relevance to a wide range of industrial devices that utilise thin liquid films in the presence of various external effects and complexities. Examples include cooling systems, falling-film chemical reactors, compact reflux condensers and microfluidic devices.

本研究的目的是分析非局部主动耗散偏微分方程（PDE）的相干结构。这种方程的特点是存在不稳定性（能量产生）和稳定性（能量耗散）的机制。此外，这些方程包含非局部项，这些非局部项不能表示为多项式或待求解的函数及其导数。主动耗散方程的解通常以特定的空间和时间行为为特征，例如空间、时间或时空局域结构，即所谓的相干结构。由于其在流体力学、非线性光学、化学反应理论和数学生物学中的重要性，近年来对这种结构的研究受到了广泛的关注。尽管最近的事态发展，这种结构的理论“本地”方程，相干结构的相互作用，在非本地主动耗散方程尚未开发，新的数学技术需要引入。拟议的研究是一个决定性的一步，对这种非局部方程的相干结构的形成和相互作用的理解，最终目标是推进对偏微分方程中的非局部效应的理解，这是重要的，无论是从理论的角度来看，并为众多的实际应用。这项研究特别及时，因为它与各种工业设备相关，这些设备在存在各种外部效应和复杂性的情况下利用薄液膜。例子包括冷却系统、降膜化学反应器、紧凑型回流冷凝器和微流体装置。