A drift homogenization problem revisited

TL;DR

This paper revisits a homogenization problem involving Stokes equations with oscillating drift, proposing a new method based on Laplace operator parametrices to handle cases with limited integrability.

Contribution

A novel homogenization approach using Laplace parametrices is introduced to address equi-integrability limitations in drift oscillations.

Findings

The new method effectively handles equi-integrable drifts.

Counter-examples demonstrate the necessity of equi-integrability.

Limit equations include an additional zero-order term.

Abstract

This paper revisits a homogenization problem studied by L. Tartar related to a tridimensional Stokes equation perturbed by a drift (connected to the Coriolis force). Here, a scalar equation and a two-dimensional Stokes equation with a $L^2$-bounded oscillating drift are considered. Under higher integrability conditions the Tartar approach based on the oscillations test functions method applies and leads to a limit equation with an extra zero-order term. When the drift is only assumed to be equi-integrable in $L^2$, the same limit behaviour is obtained. However, the lack of integrability makes difficult the direct use of the Tartar method. A new method in the context of homogenization theory is proposed. It is based on a parametrix of the Laplace operator which permits to write the solution of the equation as a solution of a fixed point problem, and to use truncated functions even in the vector-valued case. On the other hand, two counter-examples which induce different homogenized zero-order terms actually show the optimality of the equi-integrability assumption.