Mathematical Sciences: Studies in Composite Media and Turbulent Transport

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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.