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Lenya Ryzhik : How an incompressible flow helps diffusion to mix things

I will describe some recent results that concern various aspects of the mixing properties of a strong incompressible flow acting together with a diffusion. In particular, we will discuss the short-time decay of solutions of the corresponding initial value problem, asymptotics of the principle Dirichlet eigenvalue and the behavior of the explosion threshold in the Zeldovich problem when the incompressible flow is strong. When the flow is prescribed, the "enhancement" of these characteristics comes from the geometric properties of the flow. We will also show that flows arising from the Stokes-Bousisnesq problems possess these "enhancement" features.

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