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是由非线性部分平等模型模型的），观察到的行为通常来自许多不稳定的自由度的N线性相互作用，在一系列空间尺度，能量级数，能量级数，能量级数，能量级数，，，能量级数，，能量级数，，能量级数，，能量级数，，能量级数，，能量级数，，，能量级数，能量cascades，cascade，，能量级数为和时空中的混乱。 对于大多数非线性，高维系统，例如Navier-Stokes方程式说明流体g仍然无法触及，即使是数学上严格的估计，我有时会特别感兴趣。在两种水平板之间的流体层的雷利 - 孔中，就像射血一样，在许多天体物理，地球物理和工程环境中都有应用。板的tems板不完全理解。对于系统的ithema tical形式主义，并提议继续进行这些结果，以证明严格的界限，并有助于在各种流动情况下有助于加热运输。 模型一维PDE的研究PDES PDES PDES PDES PDES PDES ANDING ANTONEMENA的另一个主要主题，这些方程式可以产生与更涵盖和不弥补的不符合性问题有关的IIELD见解，例如，某些特定的对称性或特定对称性或特定的对称性或效果不稳定性可能会对平均数量进行严格的估计 - 仔细的数字以寻求对动态的详细理解。表面上出现的动力学是方程式的相对“看起来简单的“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。 在一个单独的方向上，作为跨学科研究组（Impact-HIV）的一部分，与hiv/艾滋病中的卑诗氏症中心有关。因此，艾滋病毒阳性的个体）ention计划：扩大和早期研究的流行病，我们的流行病方法是通过人口调查数据来模拟治疗效果的进展，目的是评估不同的方法。