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: 2015
• 资助国家: 加拿大
• 起止时间: 2015-01-01 至 2016-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=589158
• 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变通常由非线性偏微分方程（PDE）建模，其溶液经常显示复杂的空间和临时动力学以及模式形成。观察到的行为通常源于许多不稳定的自由度的非线性相互作用，在空间尺度，能量级联和时空中混乱的各个方面的复杂动力学。 对于大多数非线性，高维感兴趣的系统，例如描述流体流的Navier-Stokes方程，详细的分析理解仍然无法触及。然而，即使如此，有时仍可以获得数学上严格的对流量的散装特性的估计。我对两个水平板之间的流体层的雷利 - 贝纳德问题特别感兴趣，从下方加热，作为连接的模型，在许多天体物理，地球物理和工程环境中都有应用。经过数十年的深入理论和实验研究，在整个板上的温度下降方面，整体热传递的完全依赖性尚未完全理解。数学定理（如果可用）可以通过重点关注必需品并建立对非棘手现象学理论的预测的约束来帮助阐明这种情况。我的工作主要涉及板块的热性能的影响：我以前已经展示了如何系统地纳入数学形式，并建议继续建立这些结果以证明严格的界限，从而有助于理解在各种流动情况下的热传输的缩放。 我的提案的另一个主要主题是研究一维PDE的研究，该模型显示出空间暂时复杂和混乱的行为。通过仅包含与所研究现象相关的基本术语，这种方程可以产生与更复杂和更高维度有关的见解，例如，特定对称性或不稳定的影响可能很难分离。我的方法是将分析工具（包括对平均数量的严格估计）结合起来，以仔细的数字来寻求对动态的详细理解。目前，我对在某种对称性（伽利亚不变性）存在下形成模型的模型特别感兴趣。我们以前的工作表明了异常的缩放和出乎意料的丰富动态，表面上出现的方程式相对“简单”系统。持续的研究旨在更深入地了解这种显而易见的时空混乱类型。 作为跨学科研究小组（Impact-HIV）的一部分，我从事与艾滋病毒相关的数学模型的一部分。现代抗逆转录病毒治疗可以通过减少病毒载量并因此感染HIV阳性个体的感染来帮助减慢HIV的传播。这激发了当前的强化努力作为预防计划：扩大测试和早期治疗以应对流行病。我们的流行病学建模研究将分析和计算方法与人群调查数据结合在一起，以模拟流行病的进展和治疗的影响，并目的是评估不同的方法，从而有助于设计最佳的干预策略。