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.

复杂的空间和临时动力学以及模式形成是由部分微分方程（PDE）建模的空间扩展的非线性动态系统的普遍特征，并在物理，化学和生物学中应用。这样的系统经常以许多不稳定程度的自由度，复杂的动力学，能量级联和时空中的混乱的非线性相互作用。对于大多数非线性，高维感兴趣的系统，例如流体湍流的著名问题，详细的分析理解仍然无法实现。然而，即使是这样，有时也可以获得数学上严格的批量估计。从景点下方加热的流体层的瑞利 - 贝纳德问题因其数学丰富度和应用多样性而引起了人们的注意。对于在边界上的温度下降方面估计通过流体估算大量热传输的问题，我最近对有限电导率的界限进行了改革。我的发现，在较大的温度差的极限下，应该假设边界是在隔热而不是进行隔离，这考虑了我建议解决的含义。我的提案的主题是研究模型一维PDE的研究，该模型表现出空间暂时复杂和混乱的行为，目的是获得与更现实的高维问题相关的见解。特别是，我寻求确定衡量的相关统计数据，并阐明空间暂时混乱的潜在普遍特性。我的方法是进行仔细的分析 - 包括对平均数量的严格估计 - 建议适当的数值投资，反之亦然，以获得对动态的详细规模的理解。我目前正在研究的系统包括一个平均流量的模型形成模型：观察到的尺度和异常尺度的分离似乎表明了一种新型的空间颞叶混乱。一个相关的模型最近发现了令人兴奋的晶体生长应用程序和雪中的日光浴模式的形成。