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Infinite dimensional attractors for porous medium equations in heterogeneous medium
Author(s) -
Efendiev Messoud
Publication year - 2012
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.2619
Subject(s) - attractor , degenerate energy levels , mathematics , porous medium , curse of dimensionality , entropy (arrow of time) , mathematical analysis , porosity , physics , thermodynamics , statistics , geotechnical engineering , quantum mechanics , engineering
In this paper, we give a detailed study of the global attractors for porous medium equations in a heterogeneous medium. Not only the existence but also the infinite dimensionality of the global attractors is obtained by showing that their ϵ ‐Kolmogorov entropy behaves as a polynomial of the variable 1 ∕ ϵ as ϵ tends to zero, which is not observed for non‐degenerate parabolic equations. The upper and lower bounds for the Kolmogorov ϵ ‐entropy of infinite‐dimensional attractors are also obtained. We believe that the method developed in this paper has a general nature and can be applied to other classes of degenerate evolution equations. Copyright © 2012 John Wiley & Sons, Ltd.