Hostname: page-component-cd9895bd7-hc48f Total loading time: 0 Render date: 2024-12-24T00:41:46.839Z Has data issue: false hasContentIssue false
  • English
  • Français

Ground states of singularly perturbed convection-diffusionequation with oscillating coefficients

Published online by Cambridge University Press:  04 August 2014

A. Piatnitski ,
A. Rybalko  and
V. Rybalko
A. Piatnitski
Affiliation:
Narvik University College, Postboks 385, 8505 Narvik, Norway, and P.N. Lebedev Physical Institute of RAS, 53, Leninski pr., 119991 Moscow, Russia. [email protected]
A. Rybalko
Affiliation:
Kharkiv National University of Economics, 9a Lenin ave., 61166 Kharkiv, Ukraine; [email protected]
V. Rybalko
Affiliation:
Mathematical Department, B.Verkin Institute for Low Temperature Physics and Engineering of the NASU, 47 Lenin ave., 61103 Kharkiv, Ukraine; [email protected] ,
Get access

Abstract

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

Keywords

Singularly perturbed operatorseigenpair asymptoticshomogenization
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 20 , Issue 4 , October 2014 , pp. 1059 - 1077
Copyright
© EDP Sciences, SMAI, 2014

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

H. Abou-Kandil, G. Freiling, V. Ionescu and G. Jank, Matrix Riccati equations in control and systems theory. Systems and Control: Foundations and Applications. Birkhäuser Verlag, Basel (2003).
Allaire, G. and Capdeboscq, Y., Homogenization of a spectral problem in neutronic multigroup diffusion. Comput. Methods Appl. Mech. Engrg. 187 (2000) 91–117. Google Scholar
Allaire, G. and Piatnitski, A., Uniform spectral asymptotics for singularly perturbed locally periodic operators. Commun. Partial Differ. Eq. 27 (2002) 705–725. Google Scholar
Allaire, G., Pankratova, I. and Piatnitski, A., Homogenization and concentration for a diffusion equation with large convection in a bounded domain. J. Funct. Anal. 262 (2012) 300–330. Google Scholar
Allaire, G. and Raphael, A.-L., Homogenization of a convection-diffusion model with reaction in a porous medium. C. R. Math. Acad. Sci. Paris 344 (2007) 523–528. Google Scholar
Aronson, D.G., Non-negative solutions of linear parabolic equations. Annal. Scuola Norm. Sup. Pisa 22 (1968) 607–694. Google Scholar
Capdeboscq, Y., Homogenization of a diffusion equation with drift. C. R. Acad. Sci. Paris Ser. I Math. 327 (1998) 807–812. Google Scholar
Capuzzo-Dolcetta, I. and Lions, P.-L., Hamilton−Jacobi equations with state constraints. Trans. Amer. Math. Soc. 318 (1990) 643–683. Google Scholar
P. Donato and A. Piatnitski, 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.
Kifer, Yu., On the principal eigenvalue in a singular perturbation problem with hyperbolic limit points and circles. J. Differ. Eqs. 37 (1980) 108–139. Google Scholar
O.A. Ladyzhenskaya, V.A.Solonnikov and N.N. Uraltzeva, Linear and Quasi-linear Equations of Parabolic Type. AMS (1988).
Piatnitski, A., Asymptotic Behaviour of the Ground State of Singularly Perturbed Elliptic Equations. Commun. Math. Phys. 197 (1998) 527–551. Google Scholar
A. Piatnitski and V. Rybalko, On the first eigenpair of singularly perturbed operators with oscillating coefficients. Preprint available at www.arxiv.org, arXiv:1206.3754.
Evans, L.C. and Ishii, H., A PDE approach to some asymptotic problems concerning random differential equation with small noise intensities. Ann. Inst. Henri Poincaré 2 (1985) 1–20. Google Scholar
Mitake, H., Asymptotic solutions of Hamilton−Jacobi equations with state constraints. Appl. Math. Optim. 58 (2008) 393–410. Google Scholar
Ishii, H. and Mitake, H., Representation formulas for solutions of Hamilton−Jacobi equations with convex Hamiltonians. Indiana Univ. Math. J. 56 (2007) 2159–2183. Google Scholar
Protter, M.H. and Weinberger, H.F., On the spectrum of general second order operators. Bull. Amer. Math. Soc. 72 (1966) 251–255. Google Scholar
A.L. Pyatnitskii and A.S. Shamaev, 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.
M.I. Freidlin and A.D. Wentzell, Random perturbations of dynamical systems, vol. 260. Fundamental Principles Math. Sci. Springer-Verlag, New York (1984).