On the behaviour at infinity of solutions to stationary convection-diffusion equation
in a cylinder

Pankratova, Iryna; Piatnitski, Andrey

doi:10.3934/dcdsb.2009.11.935

Cited by 11 publications

(18 citation statements)

References 16 publications

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“…This allows one to approximate the eigenpairs of the studied problem up to an arbitrary large power of the small parameter characterizing the microstructure period. The existence of exponentially decaying boundary layer correctors is assured by the results obtained in [14].…”

Section: Introductionmentioning

confidence: 89%

“…According to [14] there exists a unique bounded solution w ± ∈ H 1 loc (G ± ) ∩ C 1,α (G ± ) of problem (3.13). It stabilizes to some constantŵ ± , as |y 1 | → +∞:…”

Section: )mentioning

confidence: 96%

“…There exists a unique bounded solution w ± ∈ C 1,α (G ± ) of problem (4.23) stabilizing to some constantŵ ± , as |y 1 | → +∞ (see [14]):…”

Section: 2mentioning

confidence: 97%

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Homogenization of spectral problem for locally periodic elliptic operators with sign-changing density function

Pankratova

Piatnitski

2011

Journal of Differential Equations

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The paper deals with homogenization of a spectral problem for a second order self-adjoint elliptic operator stated in a thin cylinder with homogeneous Neumann boundary condition on the lateral boundary and Dirichlet condition on the bases of the cylinder. We assume that the operator coefficients and the spectral density function are locally periodic in the axial direction of the cylinder, and that the spectral density function changes sign. We show that the behavior of the spectrum depends essentially on whether the average of the density function is zero or not. In both cases we construct the effective 1-dimensional spectral problem and prove the convergence of spectra.

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Section: Introductionmentioning

confidence: 89%

“…According to [14] there exists a unique bounded solution w ± ∈ H 1 loc (G ± ) ∩ C 1,α (G ± ) of problem (3.13). It stabilizes to some constantŵ ± , as |y 1 | → +∞:…”

Section: )mentioning

confidence: 96%

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Homogenization of spectral problem for locally periodic elliptic operators with sign-changing density function

Pankratova

Piatnitski

2011

Journal of Differential Equations

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“…In the process of dealing with practical problems, for many mathematical models, especially partial differential equations, it is difficult to obtain their analytical solutions in general. On the one hand, some scholars consider the existence, uniqueness, or nonuniqueness of solutions for the convection-diffusion equations (for example, see [5][6][7][8][9][10][11][12][13]). On the other hand, others focus on the numerical solution of the convection-diffusion equation by all kinds of methods, for instance, the spectral element method [14], the finite element method [15][16][17], the finite difference method [18,19], and the Runge-Kutta method [20].…”

Section: Introductionmentioning

confidence: 99%

Flux Transport Characteristics of Free Boundary Value Problems for a Class of Generalized Convection-Diffusion Equation

Wei

2019

Advances in Mathematical Physics

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The similarity transformation is introduced for studying free boundary value problems for a class of generalized convection-diffusion equation. A class of singular nonlinear boundary value problems are obtained and solved by using Adomian decomposition method (ADM). The approximate solution can be expressed in terms of a rapid convergent power series with easily computable terms. The efficiency and reliability of the approximate solution are verified by numerical ones. The effects of the variable thermal conduction k(z), convection functional coefficient h(z), power law exponent n, and parameter α on the flux transport characteristics are presented graphically and analyzed in detail.

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“…Using the Krein-Rutman theorem one can show (see [17]) that this problem has a unique up to a multiplicative constant solution. Moreover, this solution does not change sign.…”

Section: Introductionmentioning

confidence: 99%

On the asymptotic behaviour of the kernel of an adjoint convection-diffusion operator in a long cylinder

Allaire¹,

Piatnitski²

2017

Rev. Mat. Iberoam.

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This paper studies the asymptotic behaviour of the principal eigenfunction of the adjoint Neumann problem for a convection diffusion operator defined in a long cylinder. The operator coefficients are 1periodic in the longitudinal variable. Depending on the sign of the so-called longitudinal drift (a weighted average of the coefficients), we prove that this principal eigenfunction is equal to the product of a specified periodic function and of an exponential, up to the addition of fast decaying boundary layer terms.

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On the behaviour at infinity of solutions to stationary convection-diffusion equation in a cylinder

Cited by 11 publications

References 16 publications

Homogenization of spectral problem for locally periodic elliptic operators with sign-changing density function

Homogenization of spectral problem for locally periodic elliptic operators with sign-changing density function

Flux Transport Characteristics of Free Boundary Value Problems for a Class of Generalized Convection-Diffusion Equation

On the asymptotic behaviour of the kernel of an adjoint convection-diffusion operator in a long cylinder

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