Mathematical Sciences: Studies in Composite Media and Turbulent Transport

这个项目涉及两个具有重要应用的偏微分方程求平均值的问题。第一种是研究响应张量，即不同过程相互耦合的复合材料的有效性质的张量，例如热电或热弹性介质。二是研究扩散-平流方程的长时间、大尺度的渐近性，其中速度是一个随机的、不可压缩的场，具有激发空间和时间尺度的连续体。众所周知，这个方程呈现出有趣的交叉现象，这在多维情况下并没有得到很好的理解。首席研究员希望在齐化理论的基础上，发展出对这些问题的新的数学见解。特别是，平流-扩散问题涉及理解平均场理论有效的区域之外具有随机系数的方程的重整化。

This project concerns two problems involving the averaging of partial differential equations with important applications go Physics. The first one consists of studying the response tensor, i.e. the tensor of effective properties of a composite material in which different processes are coupled, such as thermoelectric or thermoelastic media. The second is to study the long time, large-scale asymptotics of diffusion-advection equations in which the velocity is a random, incompressible field, with a continuum of excited space and time scales. This equation is known to exhibit interesting crossover phenomena, which are not well understood in the multidimensional case. The principal investigator hopes to develop new mathematical insight to such problems, based on the theory of homogenization. In particular, the advection-diffusion problem involves understanding renormalization of equations with random coefficient outside the regime in which mean-field theory is valid.