Evolution equations displaying complex spatiotemporal behaviour

显示复杂时空行为的演化方程

• 批准号: RGPIN-2014-06691
• 负责人: Wittenberg, Ralf
• 金额: $ 0.8万
• 依托单位: Simon Fraser University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2014
• 资助国家: 加拿大
• 起止时间: 2014-01-01 至 2015-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=567973
• 关键词: Evolution; equations; displaying; complex; spatiotemporal

The time evolution of spatially extended physical, chemical or biological systems is commonly modelled by nonlinear partial differential equations (PDEs), whose solutions frequently display complex spatial and temporal dynamics and pattern formation. The observed behaviour often arises from the nonlinear interaction of many unstable degrees of freedom, complicated dynamics over a range of spatial scales, energy cascades, and chaos in space and time. For most nonlinear, high-dimensional systems of interest, such as the Navier-Stokes equations describing fluid flow, a detailed analytical understanding remains well out of reach. Nevertheless, even then mathematically rigorous estimates on bulk properties of the flows may sometimes be obtained. I am particularly interested in the Rayleigh-Bénard problem of a fluid layer between two horizontal plates, heated from below, which as a model for convection has applications in many astrophysical, geophysical and engineering contexts. After decades of intensive theoretical and experimental study, the full dependence of the bulk heat transfer in terms of the temperature drop across the plates remains incompletely understood. Mathematical theorems, when available, can help clarify such a situation by focussing on the essentials and establishing constraints on the predictions of nonrigorous phenomenological theories. My work mainly concerns the influence of the thermal properties of the plates: I have previously shown how to incorporate these systematically into the mathematical formalism, and propose to continue building on those results to prove rigorous bounds and thereby help understand the scaling of heat transport in a variety of flow situations. An additional major theme of my proposal is the study of model one-dimensional PDEs displaying spatiotemporally complex and chaotic behaviour. By containing only the essential terms relevant to the phenomena being investigated, such equations can yield insights relevant to more complicated and higher-dimensional problems in which the effects of, say, a particular symmetry or instability may be difficult to isolate. My approach is to combine analytical tools - including, again, rigorous estimates on averaged quantities - with careful numerics to seek a detailed understanding of the dynamics. I am currently particularly interested in a model for pattern formation in the presence of a certain symmetry (Galilean invariance). Our previous work demonstrated anomalous scaling and unexpectedly rich dynamics in what appeared, on the surface, to be a relatively "simple-looking" system of equations; and ongoing research is aimed at a deeper understanding of this apparently novel type of spatiotemporal chaos. In a separate direction, as part of an interdisciplinary research group (IMPACT-HIV) affiliated with the BC Centre for Excellence in HIV/AIDS, I work on mathematical models related to HIV. Modern antiretroviral treatments can help slow the spread of HIV, by reducing viral loads and hence the infectivity of HIV-positive individuals; this inspires current intensive efforts to build Treatment as Prevention programs: expanded testing and early treatment to combat the epidemic. Our epidemiological modelling research combines analytical and computational approaches with population survey data to model the progression of the epidemic and the effects of treatment, with the goal of assessing different approaches and thereby helping to design optimal intervention strategies.

空间扩展的物理、化学或生物系统的时间演化通常通过非线性偏微分方程 (PDE) 进行建模，其解经常显示复杂的空间和时间动力学和模式形成。观察到的行为通常源于许多不稳定自由度的非线性相互作用、一系列空间尺度上的复杂动力学、能量级联以及时空混沌。对于大多数感兴趣的非线性、高维系统，例如描述流体流动的纳维-斯托克斯方程，详细的分析理解仍然遥不可及。然而，即使这样，有时也可以获得对流的整体特性的数学上严格的估计。我对两个水平板之间的流体层（从下方加热）的瑞利-贝纳德问题特别感兴趣，该问题作为对流模型在许多天体物理、地球物理和工程领域都有应用。经过数十年的深入理论和实验研究，整体传热对板间温降的完全依赖性仍未完全了解。数学定理（如果可用）可以通过关注本质并为非严格现象学理论的预测建立约束来帮助澄清这种情况。我的工作主要涉及板的热特性的影响：我之前已经展示了如何将这些系统地纳入数学形式主义中，并建议继续以这些结果为基础来证明严格的界限，从而帮助理解各种流动情况下的热传输缩放。我的提案的另一个主要主题是研究显示时空复杂和混沌行为的模型一维偏微分方程。通过仅包含与所研究的现象相关的基本项，此类方程可以产生与更复杂和更高维问题相关的见解，在这些问题中，例如特定对称性或不稳定性的影响可能难以分离。我的方法是将分析工具（再次包括对平均数量的严格估计）与仔细的数字相结合，以寻求对动态的详细了解。我目前对存在某种对称性（伽利略不变性）的模式形成模型特别感兴趣。我们之前的工作证明了表面上看起来相对“简单”的方程组的反常尺度和出乎意料的丰富动力学；正在进行的研究旨在更深入地了解这种明显新颖的时空混沌类型。在另一个方向上，作为隶属于不列颠哥伦比亚省艾滋病毒/艾滋病卓越中心的跨学科研究小组 (IMPACT-HIV) 的一部分，我致力于研究与艾滋病毒相关的数学模型。现代抗逆转录病毒治疗可以减少病毒载量，从而降低艾滋病毒阳性个体的传染性，从而有助于减缓艾滋病毒的传播；这激发了当前大力制定“治疗即预防”计划：扩大检测和早期治疗以抗击这一流行病。我们的流行病学建模研究将分析和计算方法与人口调查数据相结合，对流行病的进展和治疗效果进行建模，目的是评估不同的方法，从而帮助设计最佳的干预策略。