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

  • A. Piatnitski (a1), A. Rybalko (a2) and V. Rybalko (a3)
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.

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.

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G. Allaire and Y. Capdeboscq , Homogenization of a spectral problem in neutronic multigroup diffusion. Comput. Methods Appl. Mech. Engrg. 187 (2000) 91117.

G. Allaire and A. Piatnitski , Uniform spectral asymptotics for singularly perturbed locally periodic operators. Commun. Partial Differ. Eq. 27 (2002) 705725.

G. Allaire , I. Pankratova and A. Piatnitski , Homogenization and concentration for a diffusion equation with large convection in a bounded domain. J. Funct. Anal. 262 (2012) 300330.

G. Allaire and A.-L. Raphael , Homogenization of a convection-diffusion model with reaction in a porous medium. C. R. Math. Acad. Sci. Paris 344 (2007) 523528.

Y. Capdeboscq , Homogenization of a diffusion equation with drift. C. R. Acad. Sci. Paris Ser. I Math. 327 (1998) 807812.

I. Capuzzo-Dolcetta and P.-L. Lions , Hamilton−Jacobi equations with state constraints. Trans. Amer. Math. Soc. 318 (1990) 643683.

Yu. Kifer , On the principal eigenvalue in a singular perturbation problem with hyperbolic limit points and circles. J. Differ. Eqs. 37 (1980) 108139.

A. Piatnitski , Asymptotic Behaviour of the Ground State of Singularly Perturbed Elliptic Equations. Commun. Math. Phys. 197 (1998) 527551.

H. Mitake , Asymptotic solutions of Hamilton−Jacobi equations with state constraints. Appl. Math. Optim. 58 (2008) 393410.

H. Ishii and H. Mitake , Representation formulas for solutions of Hamilton−Jacobi equations with convex Hamiltonians. Indiana Univ. Math. J. 56 (2007) 21592183.

M.H. Protter and H.F. Weinberger , On the spectrum of general second order operators. Bull. Amer. Math. Soc. 72 (1966) 251255.

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ESAIM: Control, Optimisation and Calculus of Variations
  • ISSN: 1292-8119
  • EISSN: 1262-3377
  • URL: /core/journals/esaim-control-optimisation-and-calculus-of-variations
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