Dynamics of partial differential equations

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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.