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Pettersson, Irina
Alternative names
Publications (10 of 12) Show all publications
Pettersson, I. & Piatnitski, A. (2018). Stationary convection-diffusion equation in an infinite cylinder. Journal of Differential Equations, 264(7), 4456-4487
Open this publication in new window or tab >>Stationary convection-diffusion equation in an infinite cylinder
2018 (English)In: Journal of Differential Equations, ISSN 0022-0396, E-ISSN 1090-2732, Vol. 264, no 7, p. 4456-4487Article in journal (Refereed) Published
Abstract [en]

We study the existence and uniqueness of a solution to a linear stationary convection–diffusion equation stated in an infinite cylinder, Neumann boundary condition being imposed on the boundary. We assume that the cylinder is a junction of two semi-infinite cylinders with two different periodic regimes. Depending on the direction of the effective convection in the two semi-infinite cylinders, we either get a unique solution, or one-parameter family of solutions, or even non-existence in the general case. In the latter case we provide necessary and sufficient conditions for the existence of a solution.

Keywords
Convection–diffusion equation, Infinite cylinder, Stabilization at infinity, Effective drift
National Category
Mathematics
Identifiers
urn:nbn:se:hig:diva-27148 (URN)10.1016/j.jde.2017.12.015 (DOI)
Available from: 2018-06-25 Created: 2018-06-25 Last updated: 2018-06-25Bibliographically approved
Pettersson, I. (2017). Two-scale convergence in thin domains with locally periodic rapidly oscillating boundary. Differential Equations & Applications, 9(3), 393-412
Open this publication in new window or tab >>Two-scale convergence in thin domains with locally periodic rapidly oscillating boundary
2017 (English)In: Differential Equations & Applications, ISSN 1847-120X, Vol. 9, no 3, p. 393-412Article in journal (Refereed) Published
Abstract [en]

The aim of this paper is to adapt the notion of two-scale convergence in Lp to the case of a measure converging to a singular one. We present a specific case when a thin cylinder with locally periodic rapidly oscillating boundary shrinks to a segment, and the corresponding measure charging the cylinder converges to a one-dimensional Lebegues measure of an interval. The method is then applied to the asymptotic analysis of linear elliptic operators with locally periodic coefficients and a p-Laplacian stated in thin cylinders with locally periodic rapidly varying thickness.

Keywords
Two-scale convergence, singular measure, homogenization, thin domain with varying thickness, oscillating boundary, dimension reduction, locally periodic operators, p-Laplacian
National Category
Mathematics
Identifiers
urn:nbn:se:hig:diva-27149 (URN)10.7153/dea-2017-09-28 (DOI)
Available from: 2018-06-25 Created: 2018-06-25 Last updated: 2018-06-25Bibliographically approved
Chechkina, A., Pankratova, I. & Pettersson, K. (2015). Spectral asymptotics for a singularly perturbed fourth order locally periodic elliptic operator. Asymptotic Analysis, 93(1-2), 141-160
Open this publication in new window or tab >>Spectral asymptotics for a singularly perturbed fourth order locally periodic elliptic operator
2015 (English)In: Asymptotic Analysis, ISSN 0921-7134, E-ISSN 1875-8576, Vol. 93, no 1-2, p. 141-160Article in journal (Refereed) Published
Abstract [en]

We consider the homogenization of a singularly perturbed self-adjoint fourth order elliptic operator with locally periodic coefficients, stated in a bounded domain. We impose Dirichlet boundary conditions on the boundary of the domain. The presence of large parameters in the lower order terms and the dependence of the coefficients on the slow variable lead to localization of the eigenfunctions. We show that the jth eigenfunction can be approximated by a rescaled function that is constructed in terms of the jth eigenfunction of fourth or second order effective operators with constant coefficients.

National Category
Mathematics
Identifiers
urn:nbn:se:hig:diva-27150 (URN)10.3233/ASY-151291 (DOI)
Available from: 2018-06-25 Created: 2018-06-25 Last updated: 2018-06-25Bibliographically approved
Pankratova, I. & Pettersson, K. (2015). Spectral asymptotics for an elliptic operator in a locally periodic perforated domain. Applicable Analysis, 94(6), 1207-1234
Open this publication in new window or tab >>Spectral asymptotics for an elliptic operator in a locally periodic perforated domain
2015 (English)In: Applicable Analysis, ISSN 0003-6811, E-ISSN 1563-504X, Vol. 94, no 6, p. 1207-1234Article in journal (Refereed) Published
Abstract [en]

We consider the homogenization of an elliptic spectral problem with a large potential stated in a thin cylinder with a locally periodic perforation. The size of the perforation gradually varies from point to point. We impose homogeneous Neumann boundary conditions on the boundary of perforation and on the lateral boundary of the cylinder. The presence of a large parameter 1/ε in front of the potential and the dependence of the perforation on the slow variable give rise to the effect of localization of the eigenfunctions. We show that the jth eigenfunction can be approximated by a scaled exponentially decaying function that is constructed in terms of the jth eigenfunction of a one-dimensional harmonic oscillator operator.

Keywords
homogenization, spectral problem, localization of eigenfunctions, locally periodic perforated domain, dimension reduction
National Category
Mathematics
Identifiers
urn:nbn:se:hig:diva-27151 (URN)10.1080/00036811.2014.924110 (DOI)
Available from: 2018-06-25 Created: 2018-06-25 Last updated: 2018-06-25Bibliographically approved
Chiadò Piat, V., Pankratova, I. & Piatnitski, A. (2013). Localization effect for a spectral problem in a perforated domain with Fourier boundary conditions. SIAM Journal on Mathematical Analysis, 45(3), 1302-1327
Open this publication in new window or tab >>Localization effect for a spectral problem in a perforated domain with Fourier boundary conditions
2013 (English)In: SIAM Journal on Mathematical Analysis, ISSN 0036-1410, E-ISSN 1095-7154, Vol. 45, no 3, p. 1302-1327Article in journal (Refereed) Published
Abstract [en]

This paper is aimed at homogenization of an elliptic spectral problem stated in a perforated domain, Fourier boundary conditions being imposed on the boundary of perforation. The presence of a locally periodic coefficient in the boundary operator gives rise to the effect of localization of the eigenfunctions. Moreover, the limit behavior of the lower part of the spectrum can be described in terms of an auxiliary harmonic oscillator operator. We describe the asymptotics of the eigenpairs and derive estimates for the rate of convergence. 

National Category
Mathematics
Identifiers
urn:nbn:se:hig:diva-27152 (URN)10.1137/120868724 (DOI)
Available from: 2018-06-25 Created: 2018-06-25 Last updated: 2018-06-25Bibliographically approved
Allaire, G., Pankratova, I. & Piatnitski, A. (2012). Homogenization and concentration for a diffusion equation with large convection in a bounded domain. Journal of Functional Analysis, 262(1), 300-330
Open this publication in new window or tab >>Homogenization and concentration for a diffusion equation with large convection in a bounded domain
2012 (English)In: Journal of Functional Analysis, ISSN 0022-1236, E-ISSN 1096-0783, Vol. 262, no 1, p. 300-330Article in journal (Refereed) Published
Abstract [en]

We consider the homogenization of a non-stationary convection–diffusion equation posed in a bounded domain with periodically oscillating coefficients and homogeneous Dirichlet boundary conditions. Assuming that the convection term is large, we give the asymptotic profile of the solution and determine its rate of decay. In particular, it allows us to characterize the “hot spot”, i.e., the precise asymptotic location of the solution maximum which lies close to the domain boundary and is also the point of concentration. Due to the competition between convection and diffusion, the position of the “hot spot” is not always intuitive as exemplified in some numerical tests.

Keywords
Homogenization, Convection–diffusion, Localization
National Category
Mathematics
Identifiers
urn:nbn:se:hig:diva-27154 (URN)10.1016/j.jfa.2011.09.014 (DOI)
Available from: 2018-06-21 Created: 2018-06-21 Last updated: 2018-06-21Bibliographically approved
Allaire, G., Pankratova, I. & Piatnitski, A. (2012). Homogenization of a nonstationary convection-diffusion equation in a thin rod and in a layer. SeMA Journal, 58(1), 53-95
Open this publication in new window or tab >>Homogenization of a nonstationary convection-diffusion equation in a thin rod and in a layer
2012 (English)In: SeMA Journal, ISSN 1575-9822, Vol. 58, no 1, p. 53-95Article in journal (Refereed) Published
Abstract [en]

The paper deals with the homogenization of a non-stationary convection-diffusion equation defined in a thin rod or in a layer with Dirichlet boundary condition. Under the assumption that the convection term is large, we describe the evolution of the solution’s profile and determine the rate of its decay. The main feature of our analysis is that we make no assumption on the support of the initial data which may touch the domain’s boundary. This requires the construction of boundary layer correctors in the homogenization process which, surprisingly, play a crucial role in the definition of the leading order term at the limit. Therefore we have to restrict our attention to simple geometries like a rod or a layer for which the definition of boundary layers is easy and explicit.

Keywords
Homogenization, convection-diffusion, localization, thin cylinder, layer
National Category
Mathematics
Identifiers
urn:nbn:se:hig:diva-27153 (URN)
Available from: 2018-06-21 Created: 2018-06-21 Last updated: 2018-06-21Bibliographically approved
Pankratova, I. & Piatnitski, A. (2011). Homogenization of convection-diffusion equation in infinite cylinder. Networks and Heterogeneous Media, 6(1), 111-126
Open this publication in new window or tab >>Homogenization of convection-diffusion equation in infinite cylinder
2011 (English)In: Networks and Heterogeneous Media, ISSN 1556-1801, E-ISSN 1556-181X, Vol. 6, no 1, p. 111-126Article in journal (Refereed) Published
Abstract [en]

The paper deals with a periodic homogenization problem for a non-stationary convection-diffusion equation stated in a thin infinite cylindrical domain with homogeneous Neumann boundary condition on the lateral boundary. It is shown that homogenization result holds in moving coordinates, and that the solution admits an asymptotic expansion which consists of the interior expansion being regular in time, and an initial layer.

Keywords
infinite cylinder, asymptotic behaviour., Homogenization, convection-diffusion equation
National Category
Mathematics
Identifiers
urn:nbn:se:hig:diva-27156 (URN)10.3934/nhm.2011.6.111 (DOI)
Available from: 2018-06-21 Created: 2018-06-21 Last updated: 2018-06-21Bibliographically approved
Pankratova, I. & Piatnitski, A. (2011). Homogenization of spectral problem for locally periodic elliptic operators with sign-changing density function. Journal of Differential Equations, 250(7), 3088-3134
Open this publication in new window or tab >>Homogenization of spectral problem for locally periodic elliptic operators with sign-changing density function
2011 (English)In: Journal of Differential Equations, ISSN 0022-0396, E-ISSN 1090-2732, Vol. 250, no 7, p. 3088-3134Article in journal (Refereed) Published
Abstract [en]

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.

Keywords
Spectral problem, Sign-changing density, Homogenization, Thin cylinder
National Category
Mathematics
Identifiers
urn:nbn:se:hig:diva-27157 (URN)10.1016/j.jde.2011.01.022 (DOI)
Available from: 2018-06-21 Created: 2018-06-21 Last updated: 2018-06-21Bibliographically approved
Nazarov, S. A., Pankratova, I. L. & Piatnitski, A. L. (2011). Homogenization of the spectral problem for periodic elliptic operators with sign-changing density function. Archive for Rational Mechanics and Analysis, 200(3), 747-788
Open this publication in new window or tab >>Homogenization of the spectral problem for periodic elliptic operators with sign-changing density function
2011 (English)In: Archive for Rational Mechanics and Analysis, ISSN 0003-9527, E-ISSN 1432-0673, Vol. 200, no 3, p. 747-788Article in journal (Refereed) Published
Abstract [en]

The paper deals with the asymptotic behaviour of spectra of second order self-adjoint elliptic operators with periodic rapidly oscillating coefficients in the case when the density function (the factor on the spectral parameter) changes sign. We study the Dirichlet problem in a regular bounded domain and show that the spectrum of this problem is discrete and consists of two series, one of them tending towards +∞ and another towards −∞. The asymptotic behaviour of positive and negative eigenvalues and their corresponding eigenfunctions depends crucially on whether the average of the weight function is positive, negative or equal to zero. We construct the asymptotics of eigenpairs in all three cases.

National Category
Mathematics
Identifiers
urn:nbn:se:hig:diva-27155 (URN)10.1007/s00205-010-0370-2 (DOI)
Available from: 2018-06-25 Created: 2018-06-25 Last updated: 2018-06-25Bibliographically approved
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