Convergence along mean flows

Harsha Hutridurga (Cambridge, UK)

This talk shall address the homogenization of parabolic equations of convection-diffusion type with large drift and with locally periodic rapidly oscillating coefficients. We answer an outstanding open problem in the theory of homogenization of parabolic problems. We shall develop a technique of multiple scale asymptotic expansions along mean flows and a corresponding notion of weak multiple scale convergence. Crucial to our analysis is the introduction of a fast time variable. We shall prove that the solution family taken along a particularly chosen rapidly moving coordinate system converges to the solution of a diffusion equation. The effective diffusion coefficient is expressed in terms of the average of Eulerian cell solutions along the orbits of the mean flow in the fast time variable. To make this notion rigorous, we use the theory of ergodic algebras with mean value. This is a joint work with Thomas Holding (Cambridge) and Jeffrey Rauch (Michigan).

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