| 1. |
- Engström, Jonas, et al.
(författare)
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Homogenization of random degenerated nonlinear monotone operators
- 2006
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Ingår i: Glasnik Matematicki - Serija III. - : University of Zagreb, Faculty of Science, Department of Mathematics. - 0017-095X .- 1846-7989. ; 41:1, s. 101-114
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Tidskriftsartikel (refereegranskat)abstract
- This paper deals with homogenization of random nonlinear monotone operators in divergence form. We assume that the structure conditions (strict monotonicity and continuity conditions) degenerate and are given in terms of a weight function. Under proper integrability assumptions on the weight function we construct the effective operator and prove the homogenization result.
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| 2. |
- Miroshnikova, Elena
(författare)
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Some new results in homogenization of flow in porous media with mixed boundary condition
- 2016
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Licentiatavhandling (övrigt vetenskapligt/konstnärligt)abstract
- The present thesis is devoted to derivation of Darcy’s Law for incompressible Newtonian fluid in perforated domains by means of homogenization techniques.The problem of describing flow in porous media occurs in the study of various physical phenomena such as filtration in sandy soils, blood circulation in capillaries etc. In all such cases physical quantities (e.g. velocity, pressure) are dependent of the characteristic size ε 1 of the microstructure of the fluid domain. However in most practical applications the significant role is played by averaged characteristics, such as permeability, average velocity etc., which do not depend on the microstructure of the domain. In order to obtain such quantities there exist several mathematical techniques collectively referred to as homogenization theory.This thesis consists of two papers (A and B) and complementary appendices. We assume that the flow is governed by the Stokes equation and that global normal stress boundary condition and local no-slip boundary condition are satisfied. Such mixed boundary condition is natural for many applications and here we develop the rigorous mathematical theory connected to it. The assumption of mixed boundary condition affects on corresponding forms of Darcy’s law in both papers and raises some essential difficulties in analysis in Paper A.In both papers the perforated domain is supposed to have periodical structure and the fluid to be incompressible and Newtonian. In Paper A the situation described above is considered in a framework of rigorous functional analysis, more precisely the theorem concerning the existence and uniqueness of weak solutions for the Stokes equation is proved and Darcy’s law is obtained by using two-scale convergence procedure. As it was mentioned, vast part of this paper is devoted to adaptation of classical results of functional analysis to the case of mixed boundary condition.In Paper B the Navier–Stokes system with mixed boundary condition is studied in thin perforated domain. In such cases it is natural to introduce another small parameter δ which corresponds to the thickness of the domain (in addition to the perforation parameter ε). For the case of thin porous medium the asymptotic behavior as both the film thickness δ and the perforation period ε tend to zero at different rates is investigated. The results are obtained by using the formal method of asymptotic expansions. Depending on how fast the two small parameters δ and ε go to zero relative to each other, different forms of Darcy’s law are obtained in all three limit cases — very thin porous medium (δ ε), proportionallythin porous medium (δ ∼ λε, λ ∈ (0,∞)) and homogeneously thin porous medium (δ ε).
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| 3. |
- Nazarov, Sergey A., et al.
(författare)
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Homogenization of the spectral problem for periodic elliptic operators with sign-changing density function
- 2011
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Ingår i: Archive for Rational Mechanics and Analysis. - : Springer Science and Business Media LLC. - 0003-9527 .- 1432-0673. ; 200:3, s. 747-788
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Tidskriftsartikel (refereegranskat)abstract
- 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.
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| 4. |
- Pankratova, Iryna, et al.
(författare)
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Homogenization of convection-diffusion equation in infinite cylinder
- 2011
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Ingår i: Networks and Heterogeneous Media. - : American Institute of Mathematical Sciences (AIMS). - 1556-1801 .- 1556-181X. ; 6:1, s. 111-126
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Tidskriftsartikel (refereegranskat)abstract
- 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.
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| 5. |
- Pankratova, Iryna, et al.
(författare)
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On the behaviour at infinity of solutions to stationary convection-diffusion equations in a cylinder
- 2009
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Ingår i: Discrete and continuous dynamical systems. Series B. - : American Institute of Mathematical Sciences (AIMS). - 1531-3492 .- 1553-524X. ; 11:4, s. 935-970
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Tidskriftsartikel (refereegranskat)abstract
- The work focuses on the behaviour at infinity of solutions to second order elliptic equation with first order terms in a semi-infinite cylinder. Neumann's boundary condition is imposed on the lateral boundary of the cylinder and Dirichlet condition on its base. Under the assumption that the coefficients stabilize to a periodic regime, we prove the existence of a bounded solution, its stabilization to a constant, and provide necessary and sufficient condition for the uniqueness.
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| 6. |
- Pettersson, Irina, et al.
(författare)
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Stationary convection-diffusion equation in an infinite cylinder
- 2018
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Ingår i: Journal of Differential Equations. - : Elsevier BV. - 0022-0396 .- 1090-2732. ; 264:7, s. 4456-4487
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Tidskriftsartikel (refereegranskat)abstract
- 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.
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