# Ground states of singularly perturbed convection-diffusion equation with oscillating coefficients

A. Piatnitski; A. Rybalko; V. Rybalko

ESAIM: Control, Optimisation and Calculus of Variations (2014)

- Volume: 20, Issue: 4, page 1059-1077
- ISSN: 1292-8119

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topPiatnitski, A., Rybalko, A., and Rybalko, V.. "Ground states of singularly perturbed convection-diffusion equation with oscillating coefficients." ESAIM: Control, Optimisation and Calculus of Variations 20.4 (2014): 1059-1077. <http://eudml.org/doc/272831>.

@article{Piatnitski2014,

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. Preprint www.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.},

author = {Piatnitski, A., Rybalko, A., Rybalko, V.},

journal = {ESAIM: Control, Optimisation and Calculus of Variations},

keywords = {singularly perturbed operators; eigenpair asymptotics; homogenization; Dirichlet spectral problem; elliptic convection-diffusion operator},

language = {eng},

number = {4},

pages = {1059-1077},

publisher = {EDP-Sciences},

title = {Ground states of singularly perturbed convection-diffusion equation with oscillating coefficients},

url = {http://eudml.org/doc/272831},

volume = {20},

year = {2014},

}

TY - JOUR

AU - Piatnitski, A.

AU - Rybalko, A.

AU - Rybalko, V.

TI - Ground states of singularly perturbed convection-diffusion equation with oscillating coefficients

JO - ESAIM: Control, Optimisation and Calculus of Variations

PY - 2014

PB - EDP-Sciences

VL - 20

IS - 4

SP - 1059

EP - 1077

AB - 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. Preprint www.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.

LA - eng

KW - singularly perturbed operators; eigenpair asymptotics; homogenization; Dirichlet spectral problem; elliptic convection-diffusion operator

UR - http://eudml.org/doc/272831

ER -

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