Cooper, S;
Savostianov, A;
(2019)
Homogenisation with error estimates of attractors for damped semi-linear anisotropic wave equations.
Advances in Nonlinear Analysis
, 9
(1)
pp. 745-787.
10.1515/anona-2020-0024.
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%5B2191950X%20-%20Advances%20in%20Nonlinear%20Analysis%5D%20Homogenisation%20with%20error%20estimates%20of%20attractors%20for%20damped%20semi-linear%20anisotropic%20wave%20equations.pdf - Published Version Download (789kB) | Preview |
Abstract
Homogenisation of global A ε and exponential Mε attractors for the damped semi-linear anisotropic wave equation ∂ 2 t u ε + γ∂tu ε − div a x ε � ∇u ε � + f(u ε ) = g, on a bounded domain Ω ⊂ R 3 , is performed. Order-sharp estimates between trajectories u ε (t) and their homogenised trajectories u 0 (t) are established. These estimates are given in terms of the operator-norm di�erence between resolvents of the elliptic operator div a x ε � ∇ � and its homogenised limit div � a h∇ � . Consequently, norm-resolvent estimates on the Hausdor� distance between the anisotropic attractors and their homogenised counter-parts A 0 and M0 are established. These results imply error estimates of the form distX(A ε , A 0 ) ≤ Cεκ and dists X(Mε , M0 ) ≤ Cεκ in the spaces X = L 2 (Ω) × H −1 (Ω) and X = (C β (Ω))2 . In the natural energy space E := H 1 0 (Ω) × L 2 (Ω), error estimates distE(A ε , TεA 0 ) ≤ C √ ε κ and dists E(Mε , TεM0 ) ≤ C √ ε κ are established where Tε is �rst-order correction for the homogenised attractors suggested by asymptotic expansions. Our results are applied to Dirchlet, Neumann and periodic boundary conditio
Type: | Article |
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Title: | Homogenisation with error estimates of attractors for damped semi-linear anisotropic wave equations |
Open access status: | An open access version is available from UCL Discovery |
DOI: | 10.1515/anona-2020-0024 |
Publisher version: | https://doi.org/10.1515/anona-2020-0024 |
Language: | English |
Additional information: | This work is licensed under the Creative Commons Attribution alone 4.0 License. http://creativecommons.org/licenses/by/4.0/ |
Keywords: | damped wave equation, global attractor, exponential attractor, homogenisation, homogenization, error estimates |
UCL classification: | UCL UCL > Provost and Vice Provost Offices > UCL BEAMS UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences > Dept of Mathematics |
URI: | https://discovery.ucl.ac.uk/id/eprint/10120077 |
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