Homogenization of periodic non self-adjoint problems with large drift and potential

Allaire, Grégoire; Orive, Rafael

ESAIM: Control, Optimisation and Calculus of Variations, Tome 13 (2007) no. 4, pp. 735-749.

We consider the homogenization of both the parabolic and eigenvalue problems for a singularly perturbed convection-diffusion equation in a periodic medium. All coefficients of the equation may vary both on the macroscopic scale and on the periodic microscopic scale. Denoting by $\epsilon$ the period, the potential or zero-order term is scaled as ${\epsilon }^{-2}$ and the drift or first-order term is scaled as ${\epsilon }^{-1}$. Under a structural hypothesis on the first cell eigenvalue, which is assumed to admit a unique minimum in the domain with non-degenerate quadratic behavior, we prove an exponential localization at this minimum point. The homogenized problem features a diffusion equation with quadratic potential in the whole space.

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