Homogenization of a nonstationary convection-diffusion equation in a thin rod and in a layer
- 50 Downloads
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.
Key wordsHomogenization convection-diffusion localization thin cylinder layer
Unable to display preview. Download preview PDF.
- (0797571) N.S. Bakhvalov, G.P. Panasenko, Homogenization: Averaging processes in periodic media, Nauka, Moscow, 1984(Russian); English transl.,Kluwer, Dordrecht/Boston/London, 1989.Google Scholar
- (0503330) A. Bensoussan, J.L. Lions, G. Papanicolaou, Asymptotic Analysis for Periodic Structure, Studies in Mathematics and its Applications, 5. North-Holland Publishing Co., Amsterdam-New York, 1978.Google Scholar
- D. Gérard-Varet, N. Masmoudi, Homogenization and boundary layer, to appear in Acta Mathematica.Google Scholar
- (1814364) D. Gilbarg and N.S. Trudinger, “Elliptic Partial Differential Equations of Second Order,” Reprint of the 1998 edition. Classics in Mathematics. Springer-Verlag, Berlin, 2001.Google Scholar
- (0241822) Ladyzenskaja, O. A.; Solonnikov, V. A.; Ural’ceva, N. N.. Linear and quasilinear equations of parabolic type. Translations of Mathematical Monographs, Vol. 23 American Mathematical Society, Providence, R.I. 1967.Google Scholar
- (2133084) G. Panasenko, Multi-Scale Modelling for Structures and Composites, Springer, Dordrecht, 2005.Google Scholar
- (2505654) I. Pankratova, A. Piatnitski, On the behaviour at infinity of solutions to stationary convection-diffusion equation in a cylinder, DCDS-B, 11(4) (2009).Google Scholar
- (1366209) G.C. Papanicolaou, Diffusion in random media, Surveys in applied mathematics, Vol. 1, 205–253, Plenum, New York, 1995.Google Scholar