On the asymptotic behaviour of the kernel of an adjoint convection-diffusion operator in a long cylinder

Grégoire Allaire (1, 2) , Andrey Piatnitski (3)

1 DeFI - Shape reconstruction and identification
2 CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique
3 LPI RAS - P. N. Lebedev Physical Institute of the Russian Academy of Sciences [Moscow]

This paper studies the asymptotic behaviour of the principal eigen-function of the adjoint Neumann problem for a convection diffusion operator defined in a long cylinder. The operator coefficients are 1-periodic in the longitudinal variable. Depending on the sign of the so-called longitudinal drift (a weighted average of the coefficients), we prove that this principal eigenfunction is equal to the product of a specified periodic function and of an exponential, up to the addition of fast decaying boundary layer terms.

Mots clés: homogenization; convection-diffusion; adjoint; boundary layers

Domaines: Equations aux dérivées partielles [math.AP]

Grégoire Allaire, Andrey Piatnitski. On the asymptotic behaviour of the kernel of an adjoint convection-diffusion operator in a long cylinder. Revista Matemática Iberoamericana, 2017, 33 (4), pp.1123-1148. ⟨10.4171/RMI/965⟩. ⟨hal-01258747⟩