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.

复杂时空动力学和模式形成是偏微分方程（PDE）空间扩展非线性动力学系统的普遍特征，在物理、化学和生物学中有着广泛的应用，这类系统通常具有多个不稳定自由度的非线性相互作用、多个尺度上的复杂动力学、能量级联和时空混沌等特征. 对于大多数的非线性、高维系统，如著名的流体湍流问题，详细的解析解仍然遥不可及。然而，即使这样，有时也可以得到对整体性质的数学严格估计。从下面加热的流体层的Rayleigh-Benard问题因其数学上的丰富性和应用的多样性而引起人们的注意。对于问题的估计散装热输运通过流体中的温度下降的边界，我最近推导出一个reformulation为现实情况下的边界的有限conductivity. My发现，在大温差的限制，应该假设的边界是绝缘的，而不是进行，有相当大的影响，我建议解决。 我的建议的主题是研究模型一维偏微分方程的时空复杂性和混沌行为，目标是获得与更现实的高维问题相关的见解。特别是，我试图确定相关的统计数据来测量，并阐明时空混沌潜在的普遍性质。我的方法是仔细分析-包括对平均数量的严格估计-建议进行适当的数值研究，反之亦然，以获得对动态的详细的逐尺度理解。 我目前正在研究的系统包括一个平均流模式形成的模型：观测到的尺度分离和异常尺度似乎表明了一种新型的时空混沌，一个相关的模型最近在晶体生长和雪中的太阳杯模式形成方面找到了令人兴奋的应用。