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.
Mots clés : singularly perturbed operators, eigenpair asymptotics, homogenization
@article{COCV_2014__20_4_1059_0,
author = {Piatnitski, A. and Rybalko, A. and Rybalko, V.},
title = {Ground states of singularly perturbed convection-diffusion equation with oscillating coefficients},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {1059--1077},
publisher = {EDP-Sciences},
volume = {20},
number = {4},
year = {2014},
doi = {10.1051/cocv/2014007},
mrnumber = {3264234},
language = {en},
url = {http://www.numdam.org/articles/10.1051/cocv/2014007/}
}
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 SP - 1059 EP - 1077 VL - 20 IS - 4 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv/2014007/ DO - 10.1051/cocv/2014007 LA - en ID - COCV_2014__20_4_1059_0 ER -
%0 Journal Article %A Piatnitski, A. %A Rybalko, A. %A Rybalko, V. %T Ground states of singularly perturbed convection-diffusion equation with oscillating coefficients %J ESAIM: Control, Optimisation and Calculus of Variations %D 2014 %P 1059-1077 %V 20 %N 4 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv/2014007/ %R 10.1051/cocv/2014007 %G en %F COCV_2014__20_4_1059_0
Piatnitski, A.; Rybalko, A.; Rybalko, V. Ground states of singularly perturbed convection-diffusion equation with oscillating coefficients. ESAIM: Control, Optimisation and Calculus of Variations, Tome 20 (2014) no. 4, pp. 1059-1077. doi : 10.1051/cocv/2014007. http://www.numdam.org/articles/10.1051/cocv/2014007/
[1] , , and , Matrix Riccati equations in control and systems theory. Systems and Control: Foundations and Applications. Birkhäuser Verlag, Basel (2003). | MR | Zbl
[2] and , Homogenization of a spectral problem in neutronic multigroup diffusion. Comput. Methods Appl. Mech. Engrg. 187 (2000) 91-117. | MR | Zbl
[3] and , Uniform spectral asymptotics for singularly perturbed locally periodic operators. Commun. Partial Differ. Eq. 27 (2002) 705-725. | MR | Zbl
[4] , and , Homogenization and concentration for a diffusion equation with large convection in a bounded domain. J. Funct. Anal. 262 (2012) 300-330. | MR | Zbl
[5] and , Homogenization of a convection-diffusion model with reaction in a porous medium. C. R. Math. Acad. Sci. Paris 344 (2007) 523-528. | MR | Zbl
[6] , Non-negative solutions of linear parabolic equations. Annal. Scuola Norm. Sup. Pisa 22 (1968) 607-694. | Numdam | MR | Zbl
[7] , Homogenization of a diffusion equation with drift. C. R. Acad. Sci. Paris Ser. I Math. 327 (1998) 807-812. | MR | Zbl
[8] and , Hamilton−Jacobi equations with state constraints. Trans. Amer. Math. Soc. 318 (1990) 643-683. | MR | Zbl
[9] and , Averaging of nonstationary parabolic operators with large lower order terms. Multi scale problems and asymptotic analysis. Vol. 24, GAKUTO Int. Ser. Math. Sci. Appl. Gakkotosho, Tokyo (2006) 153-165. | MR | Zbl
[10] , On the principal eigenvalue in a singular perturbation problem with hyperbolic limit points and circles. J. Differ. Eqs. 37 (1980) 108-139. | MR | Zbl
[11] , and , Linear and Quasi-linear Equations of Parabolic Type. AMS (1988). | Zbl
[12] , Asymptotic Behaviour of the Ground State of Singularly Perturbed Elliptic Equations. Commun. Math. Phys. 197 (1998) 527-551. | MR | Zbl
[13] and , On the first eigenpair of singularly perturbed operators with oscillating coefficients. Preprint available at www.arxiv.org, arXiv:1206.3754.
[14] and , A PDE approach to some asymptotic problems concerning random differential equation with small noise intensities. Ann. Inst. Henri Poincaré 2 (1985) 1-20. | Numdam | MR | Zbl
[15] , Asymptotic solutions of Hamilton−Jacobi equations with state constraints. Appl. Math. Optim. 58 (2008) 393-410. | MR | Zbl
[16] and , Representation formulas for solutions of Hamilton−Jacobi equations with convex Hamiltonians. Indiana Univ. Math. J. 56 (2007) 2159-2183. | MR | Zbl
[17] and , On the spectrum of general second order operators. Bull. Amer. Math. Soc. 72 (1966) 251-255. | MR | Zbl
[18] and , On the asymptotic behavior of the eigenvalues and eigenfunctions of a nonselfadjoint operator in Rn. (Russian) Tr. Semin. Im. I.G. Petrovskogo 23 (2003) 287-308, 412; translation in J. Math. Sci. 120 (2004) 1411-1423. | MR | Zbl
[19] and , Random perturbations of dynamical systems, vol. 260. Fundamental Principles Math. Sci. Springer-Verlag, New York (1984). | MR | Zbl
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