Dynamics of partial differential equations

复杂的时空动力学和模式形成是偏微分方程（PDE）建模的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。 这样的系统经常具有许多不稳定自由度的非线性相互作用，在一系列尺度上的复杂动力学，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，如著名的流体湍流问题，详细的分析理解仍然遥不可及。 然而，即使这样，人们有时也可以获得对整体性质的数学上严格的估计。 流体层自下而上加热的Rayleigh-Benard问题因其数学上的丰富性和应用上的多样性而备受关注。对于问题的估计散装热输运通过流体中的温度下降跨越边界，我最近推导出一个reformulation的有限传导率的边界的现实情况。 我的发现，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，这一发现有相当大的影响，我打算加以解决。 我的建议的主要主题是模型一维偏微分方程显示时空复杂和混沌行为的研究，获得更现实的高维问题相关的见解的目标。 特别是，我试图确定相关的统计数据来衡量，并澄清潜在的时空混沌的普遍属性。 我的方法是进行仔细的分析--包括对平均数量的严格估计--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模型：观察到的尺度分离和异常尺度似乎表明一种新型的时空混沌。 一个相关的模型最近发现了令人兴奋的应用，晶体生长和雪中suncup图案的形成。

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.