Dynamics of partial differential equations

偏微分方程的动力学

• 批准号: 261892-2007
• 负责人: Wittenberg, Ralf
• 金额: $ 1.09万
• 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: 2007
• 资助国家: 加拿大
• 起止时间: 2007-01-01 至 2008-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=361723
• 关键词: Dynamics; partial; differential; equations

Complex spatial and temporal dynamics and pattern formation are ubiquitous features of spatially extended nonlinear dynamical systems modeled by partial differential equations (PDEs), with applications in physics, chemistry and biology.  Such systems frequently feature the nonlinear interactions of many unstable degrees of freedom, complicated dynamics over a range of scales, energy cascades, and chaos in space and time.    For most nonlinear, high-dimensional systems of interest, such as the celebrated problem of fluid turbulence, a detailed analytical understanding remains well beyond reach.  Nevertheless, even then one can sometimes obtain mathematically rigorous estimates for bulk properties.  The Rayleigh-Benard problem of a fluid layer heated from below attracts attention for its mathematical richness and its diversity of applications. For the problem of estimating bulk heat transport through the fluid in terms of the temperature drop across the boundaries, I have recently derived a reformulation for the realistic case of boundaries of finite conductivity.  My discovery that in the limit of large temperature difference, one should assume the boundaries to be insulating rather than conducting, has considerable implications which I propose to address.    The main theme of my proposal is the study of model one-dimensional PDEs displaying spatiotemporally complex and chaotic behaviour, with the goal of obtaining insights relevant to more realistic higher- dimensional problems.  In particular, I seek to identify the relevant statistics to measure, and to clarify potentially universal properties of spatiotemporal chaos.  My approach is to have careful analysis - including rigorous estimates on averaged quantities - suggest appropriate numerical investigations, and vice versa, to obtain a detailed scale-by-scale understanding of the dynamics.    The systems I am currently studying include a model for pattern formation with mean flow: the observed separation of scales and anomalous scaling seem to indicate a new type of spatiotemporal chaos.  A related model has recently found exciting applications to crystal growth and the formation of suncup patterns in snow.

复杂的时空动力学和模式形成是由偏微分方程 (PDE) 建模的空间扩展非线性动力系统的普遍特征，在物理、化学和生物学中都有应用，此类系统经常具有许多不稳定自由度的非线性相互作用。一系列尺度的动力学、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，例如著名的流体湍流问题，详细的分析然而，即使如此，人们有时也可以获得对流体层的体特性的严格数学估计，该问题因其数学丰富性和应用的多样性而引起人们的关注。根据跨越边界的温降来描述通过流体的大量热传输，我最近对有限电导率边界的实际情况进行了重新表述，我发现在大温差的限制下，人们应该假设边界。绝缘而不是传导具有相当大的影响，我建议解决这一问题。我的提案的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，目的是获得与更现实的相关的见解。特别是，我试图确定要测量的相关统计数据，并阐明时空混沌的潜在普遍属性。我的方法是进行仔细的分析 - 包括对平均数量的严格估计 - 提出适当的数值。调查，反之亦然，以获得对动力学的详细逐尺度理解​​我目前正在研究的系统包括一个具有平均流的模式形成模型：观察到的尺度分离和异常尺度似乎表明了一种新的现象。一种时空混沌。最近，一个相关模型在晶体生长和雪中太阳杯图案的形成中找到了令人兴奋的应用。