Quantitative Homogenization of Attractors of Non-Newtonian Filtration Equations

Emil Novruzov

Mathematical Problems in Engineering, 2010, vol. 2010, 1-25

Abstract:

For a rapidly spatially oscillating nonlinearity ð ‘” we compare solutions ð ‘¢ 𠜖 of non-Newtonian filtration equation 𠜕 ð ‘¡ ð ›½ ( ð ‘¢ 𠜖 ) − ð · ( | ð · ð ‘¢ 𠜖 | ð ‘ âˆ’ 2 ð · ð ‘¢ 𠜖 + 𠜑 ( ð ‘¢ 𠜖 ) ð · ð ‘¢ 𠜖 ) + ð ‘” ( ð ‘¥ , ð ‘¥ / 𠜖 , ð ‘¢ 𠜖 ) = ð ‘“ ( ð ‘¥ , ð ‘¥ / 𠜖 ) with solutions ð ‘¢ 0 of the homogenized, spatially averaged equation 𠜕 ð ‘¡ ð ›½ ( ð ‘¢ 0 ) − ð · ( | ð · ð ‘¢ 0 | ð ‘ âˆ’ 2 ð · ð ‘¢ 0 + 𠜑 ( ð ‘¢ 0 ) ð · ð ‘¢ 0 ) + ð ‘” 0 ( ð ‘¥ , ð ‘¢ 0 ) = ð ‘“ 0 ( ð ‘¥ ) . Based on an 𠜀 -independent a priori estimate, we prove that | | ð ›½ ( ð ‘¢ 𠜖 ) − ð ›½ ( ð ‘¢ 0 ) | | ð ¿ 1 ( Ω ) ≤ ð ¶ ð œ– ð ‘’ 𠜌 ð ‘¡ uniformly for all ð ‘¡ ≥ 0 . Besides, we give explicit estimate for the distance between the nonhomogenized ð ´ ð œ– and the homogenized ð ´ 0 attractors in terms of the parameter 𠜖 ; precisely, we show fractional-order semicontinuity of the global attractors for 𠜖 ↘ 0 ∶ d i s t ð ¿ 1 ( Ω ) ( ð ´ ð œ– , ð ´ 0 ) ≤ ð ¶ ð œ– ð ›¾ .

Date: 2010
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jnlmpe:173408

DOI: 10.1155/2010/173408

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