Convergence along mean flows

(Joint work with Harsha Hutridurga (Imperial College London) and Jeffery Rauch (University of Michigan). We consider the problem of homogenising convection dominated parabolic equations with rapidly oscillating locally periodic coefficients. Here the drift appears as (1/eps)b(x,x/eps). In the case that the drift vector field b(x,y) is purely periodic, i.e. b(x,y)=b(y), and of zero mean in y, homogenisation of such problems is classical and well understood, for example through the methods of asymptotic expansions and two-scale convergence. More recently, this has been extended to the case where b is purely periodic but not of zero mean. The general case, where b(x,y) is locally periodic, has remained open. We give a partial answer to this question, that under assumptions on the coefficients, the equation homogenises in the Lagrangian frame given by the flow associated to the mean of the drift vector field b. To achieve this we introduce a new notion of two-scale convergence along flows. Time permitting, I will discuss work in preparation on how this new notion of convergence can be applied to the Friedlin-Wentzell averaging principle in 2d Hamiltonian flows.