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

Abstract

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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.

How to cite

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Piatnitski, 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 -

References

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