Ground states of singularly perturbed convection-diffusion equation with oscillating coefficients. (4th August 2014)

• Record Type: Journal Article
• Authors: Piatnitski, A.; Rybalko, A.; Rybalko, V.
• Is Part Of: ESAIM. Volume 20:Number 4(2014:Oct.)
• Page Start: 1059
• Page End: 1077
• Publication Date: 2014-08-04
• Subjects: Singularly perturbed operators, -- eigenpair asymptotics, -- homogenization; System analysis -- Periodicals; Calculus of variations -- Periodicals; Mathematical analysis -- Periodicals; Mathematical optimization -- Periodicals; Control theory -- Periodicals; 515.64
• DOI: 10.1051/cocv/2014007
• Languages: English
• ISSNs: 1292-8119

Abstract

Abstract : We study the first eigenpair of a Dirichlet spectral problem for singularly perturbed convection-diffusion operators with oscillating locally periodic coefficients. It follows from the results of [A. Piatnitski and V. Rybalko, On the first eigenpair of singularly perturbed operators with oscillating coefficients. Preprintwww.arxiv.org, arXiv:1206.3754] that the first eigenvalue remains bounded only if the integral curves of the so-called effective drift have a nonempty ω -limit set. Here we consider the case when the integral curves can have both hyperbolic fixed points and hyperbolic limit cycles. One of the main goals of this work is to determine a fixed point or a limit cycle responsible for the first eigenpair asymptotics. Here we focus on the case of limit cycles that was left open in [A. Piatnitski and V. Rybalko, Preprint.