Dimension estimate for the global attractor of an evolution equation. (English) Zbl 1234.35039

Summary: We estimate the dimension of the global attractor of an evolution equation by the study of the evolution of the \(n\)-dimensional volumes under the flow. We compare these results with the estimate of the dimension of the inertial manifold.


35B41 Attractors
35B40 Asymptotic behavior of solutions to PDEs
37L30 Attractors and their dimensions, Lyapunov exponents for infinite-dimensional dissipative dynamical systems
Full Text: DOI


[1] R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, vol. 68, Springer, New York, NY, USA, 2nd edition, 1997. · Zbl 0871.35001
[2] J. C. Robinson, Infinite-Dimensional Dynamical Systems: An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, UK, 2001. · Zbl 0980.35001
[3] L. Yang and M.-H. Yang, “Attractors of the non-autonomous reaction-diffusion equation with nonlinear boundary condition,” Nonlinear Analysis, vol. 11, no. 5, pp. 3946-3954, 2010. · Zbl 1201.35054 · doi:10.1016/j.nonrwa.2010.03.002
[4] T. Caraballo, G. Łukaszewicz, and J. Real, “Pullback attractors for asymptotically compact non-autonomous dynamical systems,” Nonlinear Analysis, vol. 64, no. 3, pp. 484-498, 2006. · Zbl 1128.37019 · doi:10.1016/j.na.2005.03.111
[5] A. Tarasińska, “Pullback attractor for heat convection problem in a micropolar fluid,” Nonlinear Analysis, vol. 11, no. 3, pp. 1458-1471, 2010. · Zbl 1189.35030 · doi:10.1016/j.nonrwa.2009.03.003
[6] M. Yang and P. E. Kloeden, “Random attractors for stochastic semi-linear degenerate parabolic equations,” Nonlinear Analysis, vol. 12, no. 5, pp. 2811-2821, 2011. · Zbl 1222.35042 · doi:10.1016/j.nonrwa.2011.04.007
[7] Z. Wang, S. Zhou, and A. Gu, “Random attractor for a stochastic damped wave equation with multiplicative noise on unbounded domains,” Nonlinear Analysis, vol. 12, no. 6, pp. 3468-3482, 2011. · Zbl 1231.35025 · doi:10.1016/j.nonrwa.2011.06.008
[8] T. Caraballo, J. A. Langa, V. S. Melnik, and J. Valero, “Pullback attractors of nonautonomous and stochastic multivalued dynamical systems,” Set-Valued Analysis, vol. 11, no. 2, pp. 153-201, 2003. · Zbl 1018.37048 · doi:10.1023/A:1022902802385
[9] O. V. Kapustyan, P. O. Kasyanov, and J. Valero, “Pullback attractors for a class of extremal solutions of the 3D Navier-Stokes system,” Journal of Mathematical Analysis and Applications, vol. 373, no. 2, pp. 535-547, 2011. · Zbl 1206.35048 · doi:10.1016/j.jmaa.2010.07.040
[10] N. Saito, “Conservative upwind finite-element method for a simplified Keller-Segel system modelling chemotaxis,” IMA Journal of Numerical Analysis, vol. 27, no. 2, pp. 332-365, 2007. · Zbl 1119.65094 · doi:10.1093/imanum/drl018
[11] M. Efendiev, E. Nakaguchi, and W. L. Wendland, “Dimension estimate of the global attractor for a semi-discretized chemotaxis-growth system by conservative upwind finite-element scheme,” Journal of Mathematical Analysis and Applications, vol. 358, no. 1, pp. 136-147, 2009. · Zbl 1172.92311 · doi:10.1016/j.jmaa.2009.04.025
[12] A. Eden, C. Foias, B. Nicolaenko, and R. Temam, Exponential Attractors for Dissipative Evolution Equations, vol. 37, John Wiley & Sons, 1995. · Zbl 0842.58056
[13] M. Bulí\vcek and D. Pra, “A note on the dimension of the global attractor for an abstract semilinear hyperbolic problem,” Applied Mathematics Letters, vol. 22, no. 7, pp. 1025-1028, 2009. · Zbl 1179.37104 · doi:10.1016/j.aml.2009.01.027
[14] G. Bellettini, G. Fusco, and N. Guglielmi, “A concept of solution and numerical experiments for forward-backward diffusion equations,” Discrete and Continuous Dynamical Systems A, vol. 16, no. 4, pp. 783-842, 2006. · Zbl 1105.35007 · doi:10.3934/dcds.2006.16.783
[15] G. R. Chacón and R. Colucci, “Asymptotic behavior of a fourth order evolution equation,” submitted paper. · Zbl 1284.35040
[16] P. Constantin, C. Foias, B. Nicolaenko, and R. Temam, Integral Manifolds and Inertial Manifolds for Dissipative Partial Differential Equations, vol. 70 of Applied Mathematical Sciences, Springer, New York, NY, USA, 1st edition, 1988. · Zbl 0683.58002
[17] J. C. Robinson, Dimensions, Embeddings, and Attractors, vol. 186 of Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, UK, 2011. · Zbl 1222.37004
[18] M. Slemrod, “Dynamics of measure valued solutions to a backward-forward heat equation,” Journal of Dynamics and Differential Equations, vol. 3, no. 1, pp. 1-28, 1991. · Zbl 0747.35013 · doi:10.1007/BF01049487
[19] H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, New York, NY, USA, 2010. · Zbl 1220.46002
[20] R. Mañé, “On the dimension of the compact invariant sets of certain nonlinear maps,” in Dynamical Systems and Turbulence, vol. 898 of Lecture Notes in Mathematics, pp. 230-242, Springer, Berlin, Germany, 1981. · Zbl 0544.58014 · doi:10.1007/BFb0091916
[21] C. Foias and R. Temam, “Determination of the solutions of the Navier-Stokes equations by a set of nodal values,” Mathematics of Computation, vol. 43, no. 167, pp. 117-133, 1984. · Zbl 0563.35058 · doi:10.2307/2007402
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