Approximations of the Resolvent for a Non--Self-Adjoint Diffusion Operator with Rapidly Oscillating Coefficients

S. E. Pastukhova

Matematičeskie zametki, Tome 94 (2013) no. 1, pp. 130-150

Résumé

A strongly inhomogeneous diffusion operator with drift depending on a small parameter $\varepsilon$ is studied in the space $L^2(\mathbb R^n)$. The strong inhomogeneity consists in that the coefficients of the operator are $\varepsilon$-periodic and, in addition, the drift vector is of the order of $\varepsilon^{-1}$. As $\varepsilon\to 0$, approximations in the operator $L^2$‑norm of order $\varepsilon$ and $\varepsilon^2$ are constructed for the resolvent of the operator. For each of these orders of approximation, an averaged diffusion operator is obtained. A spectral method based on the Bloch representation for an operator with periodic coefficients is used.

Détail

Keywords: diffusion operator with drift, resolvent of an operator, averaged diffusion operator, Bloch representation for an operator, Sobolev space, Gelfand transformation.