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On the continuity of global attractors

Authors: Luan T. Hoang, Eric J. Olson and James C. Robinson
Journal: Proc. Amer. Math. Soc. 143 (2015), 4389-4395
MSC (2010): Primary 35B41
Published electronically: April 6, 2015
MathSciNet review: 3373937
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Abstract: Let $ \Lambda $ be a complete metric space, and let $ \{S_\lambda (\cdot ):\ \lambda \in \Lambda \}$ be a parametrised family of semigroups with global attractors $ \mathscr {A}_\lambda $. We assume that there exists a fixed bounded set $ D$ such that $ \mathscr {A}_\lambda \subset D$ for every $ \lambda \in \Lambda $. By viewing the attractors as the limit as $ t\to \infty $ of the sets $ S_\lambda (t)D$, we give simple proofs of the equivalence of `equi-attraction' to continuity (when this convergence is uniform in $ \lambda $) and show that the attractors $ \mathscr {A}_\lambda $ are continuous in $ \lambda $ at a residual set of parameters in the sense of Baire Category (when the convergence is only pointwise).

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  • [1] A. V. Babin and M. I. Vishik, Attractors of evolution equations, Studies in Mathematics and its Applications, vol. 25, North-Holland Publishing Co., Amsterdam, 1992. Translated and revised from the 1989 Russian original by Babin. MR 1156492 (93d:58090)
  • [2] A. V. Babin and S. Yu. Pilyugin, Continuous dependence of attractors on the shape of domain, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 221 (1995), no. Kraev. Zadachi Mat. Fiz. i Smezh. Voprosy Teor. Funktsii. 26, 58-66, 254 (English, with English and Russian summaries); English transl., J. Math. Sci. (New York) 87 (1997), no. 2, 3304-3310. MR 1359748 (96k:34128),
  • [3] J. E. Billotti and J. P. LaSalle, Dissipative periodic processes, Bull. Amer. Math. Soc. 77 (1971), 1082-1088. MR 0284682 (44 #1906)
  • [4] Alexandre N. Carvalho, José A. Langa, and James C. Robinson, Attractors for infinite-dimensional non-autonomous dynamical systems, Applied Mathematical Sciences, vol. 182, Springer, New York, 2013. MR 2976449
  • [5] Vladimir V. Chepyzhov and Mark I. Vishik, Attractors for equations of mathematical physics, American Mathematical Society Colloquium Publications, vol. 49, American Mathematical Society, Providence, RI, 2002. MR 1868930 (2003f:37001c)
  • [6] I. D. Chueshov, Vvedenie v teoriyu beskonechnomernykh dissipativnykh sistem, Universitetskie Lektsii po SovremennoĭMatematike. [University Lectures in Contemporary Mathematics], AKTA, Kharkiv, 1999 (Russian, with English and Russian summaries). MR 1788405 (2001k:37126)
  • [7] P. Constantin and C. Foias, Global Lyapunov exponents, Kaplan-Yorke formulas and the dimension of the attractors for $ 2$D Navier-Stokes equations, Comm. Pure Appl. Math. 38 (1985), no. 1, 1-27. MR 768102 (87g:35186),
  • [8] Charles R. Doering and J. D. Gibbon, Applied analysis of the Navier-Stokes equations, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 1995. MR 1325465 (96a:76024)
  • [9] Jack K. Hale, Asymptotic behavior of dissipative systems, Mathematical Surveys and Monographs, vol. 25, American Mathematical Society, Providence, RI, 1988. MR 941371 (89g:58059)
  • [10] Jack K. Hale, Xiao-Biao Lin, and Geneviève Raugel, Upper semicontinuity of attractors for approximations of semigroups and partial differential equations, Math. Comp. 50 (1988), no. 181, 89-123. MR 917820 (89a:47093),
  • [11] Jack K. Hale and Geneviève Raugel, Lower semicontinuity of attractors of gradient systems and applications, Ann. Mat. Pura Appl. (4) 154 (1989), 281-326. MR 1043076 (91f:58087),
  • [12] James R. Munkres, Topology: a first course, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1975. MR 0464128 (57 #4063)
  • [13] Olga Ladyzhenskaya, Attractors for semigroups and evolution equations, Lezioni Lincee. [Lincei Lectures], Cambridge University Press, Cambridge, 1991. MR 1133627 (92k:58040)
  • [14] Desheng Li and P. E. Kloeden, Equi-attraction and the continuous dependence of attractors on parameters, Glasg. Math. J. 46 (2004), no. 1, 131-141. MR 2034840 (2004m:37025),
  • [15] John C. Oxtoby, Measure and category, 2nd ed., Graduate Texts in Mathematics, vol. 2, Springer-Verlag, New York-Berlin, 1980. A survey of the analogies between topological and measure spaces. MR 584443 (81j:28003)
  • [16] James C. Robinson, Infinite-dimensional dynamical systems, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 2001. An introduction to dissipative parabolic PDEs and the theory of global attractors. MR 1881888 (2003f:37001a)
  • [17] A. M. Stuart and A. R. Humphries, Dynamical systems and numerical analysis, Cambridge Monographs on Applied and Computational Mathematics, vol. 2, Cambridge University Press, Cambridge, 1996. MR 1402909 (97g:65009)
  • [18] Roger Temam, Infinite-dimensional dynamical systems in mechanics and physics, 2nd ed., Applied Mathematical Sciences, vol. 68, Springer-Verlag, New York, 1997. MR 1441312 (98b:58056)

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Additional Information

Luan T. Hoang
Affiliation: Department of Mathematics and Statistics, Texas Tech University, Box 41042, Lubbock, Texas 79409-1042

Eric J. Olson
Affiliation: Department of Mathematics/084, University of Nevada, Reno, Nevada 89557

James C. Robinson
Affiliation: Mathematics Institute, University of Warwick, Coventry CV4 7AL, United Kingdom

Keywords: Global attractor, continuity, Baire one function, equi-attraction, Dini's Theorem
Received by editor(s): July 11, 2014
Received by editor(s) in revised form: July 15, 2014
Published electronically: April 6, 2015
Additional Notes: The third author was supported by an EPSRC Leadership Fellowship EP/G007470/1, which supported the time spent in Warwick by the first and second authors
Communicated by: Yingefi Yi
Article copyright: © Copyright 2015 American Mathematical Society

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