Anomalous diffusion in two homogenization problems

Wednesday, June 27, 2018 - 10:00am - 11:00am
Lind 409
Gautam Iyer (Carnegie Mellon University)
In the first part of this talk, we study scaling limits of a passive tracers diffusing in the presence of an incompressible flow. If the flow is cellular (i.e.\ has a periodic Hamiltonian with no unbounded trajectories), then classical homogenization results show that the long time behaviour is an effective Brownian motion. The time scales involved for this behaviour to be observed, however, are often too long to be of practical interest. We thus study the behaviour on intermediate time scales. Surprisingly, the effective behaviour is now described by a fractional kinetic process instead of a Brownian motion. At the PDE level this means that while the long time scaling limit is the heat equation, the intermediate time scaling limit is a time fractional heat equation.

In the second part of this talk we study scaling limits of passive scalars in comb like domains. If the size and spacing of the teeth is scaled appropriately, then one again observes an anomalous diffusive behaviour described by a fractional kinetic process.