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: 2018
• 资助国家: 加拿大
• 起止时间: 2018-01-01 至 2019-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=668750
• 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由非线性偏微分方程（PDE）建模，其解经常显示复杂的时空动力学和模式形成。 观察到的行为往往来自许多不稳定自由度的非线性相互作用、空间尺度范围内的复杂动力学、能量级联以及空间和时间的混沌。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Navier-Stokes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动的整体性质的数学上严格的估计。 我特别感兴趣的瑞利-贝纳德问题的流体层之间的两个水平的板，从下面加热，作为一个模型的对流有应用在许多天体物理，地球物理和工程背景。 经过几十年的深入理论和实验研究，整体热传递在跨板温降方面的完全依赖性仍然没有完全理解。 数学定理，如果可用的话，可以帮助澄清这样的情况，通过集中在本质上，并建立对非严格的唯象理论的预测的约束。 我的工作主要关注板的热性质的影响：我以前已经展示了如何将这些系统地纳入数学形式，并建议继续建立在这些结果的基础上，以证明严格的边界，从而帮助理解各种流动情况下热传输的标度。我的建议的另一个主要主题是研究模型一维偏微分方程显示时空复杂和混乱的行为。 通过只包含与所研究的现象相关的基本项，这样的方程可以产生与更复杂和更高维的问题相关的见解，其中，特定对称性或不稳定性的影响可能难以隔离。 我的方法是将联合收割机分析工具--同样包括对平均数量的严格估计--与仔细的数值结合起来，以寻求对动态的详细理解。 我目前特别感兴趣的模式形成在一定的对称性（伽利略不变性）的存在。 我们以前的工作表明，在表面上看起来是一个相对“简单”的方程系统中，存在着异常的缩放和意外丰富的动力学;正在进行的研究旨在更深入地了解这种显然是新颖的时空混沌类型。** 在一个单独的方向，作为一个跨学科的研究小组（IMPACT-HIV）的一部分，隶属于卑诗省艾滋病毒/艾滋病卓越中心，我致力于与艾滋病毒有关的数学模型。 现代抗逆转录病毒疗法可以通过减少病毒载量，从而降低艾滋病毒阳性个体的传染性，帮助减缓艾滋病毒的传播;这激发了目前为建立“以治疗为预防”方案所做的密集努力：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，以模拟流行病的进展和治疗效果，目的是评估不同的方法，从而帮助设计最佳的干预策略。