Dynamics of partial differential equations

复杂的时空动力学和图案形成是由偏微分方程(PDE)模拟的空间扩展的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。然而，这类系统经常具有许多不稳定自由度的非线性相互作用、一系列尺度上的复杂动力学、能量级联以及空间和时间上的混沌。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，详细的分析理解仍然是遥不可及的。然而，即使这样，人们有时也可以获得对块体性质的数学上的严格估计。从下面加热的流体层的瑞利-贝纳德问题因其数学上的丰富和应用的多样性而引起人们的注意。对于用边界温降来估计流体的整体热传输的问题，我最近得到了有限导电性边界的实际情况的重新公式。我的发现是，在大温差的限制下，人们应该假设边界是绝缘的，而不是导电的，这具有相当大的影响，我建议解决这一问题。我建议的主要主题是研究显示时空复杂和混乱行为的一维模型偏微分方程组，目的是获得与更现实的更高维问题相关的见解。尤其是，我试图确定要测量的相关统计数据，为了阐明时空混沌的潜在普遍性质。我的方法是进行仔细的分析--包括对平均量的严格估计--建议进行适当的数值研究，反之亦然，以获得对动力学的详细了解。此外，我目前正在研究的系统包括一个具有平均流的图案形成模型：观察到的尺度分离和异常尺度似乎表明了一种新型的时空混沌。最近，一个相关的模型在晶体生长和雪中形成密集图案方面发现了令人兴奋的应用。

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.