Abstract
The study of the asymptotic behavior of the weak solutions of the three-dimensional (3D for short) Navier–Stokes system is a challenging problem which is still far to be solved in a satisfactory way. In particular, the existence of a global attractor in the strong topology is an open problem for which only some partial or conditional results are given (see [3, 4, 6, 15, 17, 19, 20, 27, 38]). Concerning the existence of trajectory attractors, some results are proved in [13, 18, 36]. The main difficulty in this problem (but not the only one!) is to prove the asymptotic compactness of solutions (see [2], for a review on these questions).
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Anguiano M, Caraballo T, Real J, Valero J (2010) Pullback attractors for reaction-diffusion equations in some unbounded domains with an H − 1-valued non-autonomous forcing term and without uniqueness of solutions. Discrete Contin Dyn Syst B. doi:10.3934/dcdsb.2010.14.307
Balibrea F, Caraballo T, Kloeden P, Valero J (2010) Recent developments in dynamical systems: three perspective. Int J Bifur Chaos. doi:10.1142/S0218127410027246
Ball JM (2000) Continuity properties and global attractors of generalized semiflows and the Navier–Stokes equations. In: Mechanics: from theory to computation. Springer, New York, pp 447–474
Birnir B, Svanstedt N (2004) Existence theory and strong attractors for the Rayleigh–Bénard problem with a large aspect ratio. Discrete Contin Dyn Syst. doi:10.3934/dcds.2004.10.53
Caraballo T, Kloeden PE, Real J (2004) Pullback and forward attractors for a damped wave equation with delays. Stochast Dynam. doi:10.1142/S0219493704001139
Bondarevski VG (1997) Energetic systems and global attractors for the 3D Navier–Stokes equations. Nonlinear Anal 30:799–810
Caraballo T, Garrido-Atienza MJ, Schmalfuss B, Valero J (2008) Non-autonomous and random attractors for delay random semilinear equations without uniqueness. Discrete Contin Dyn Syst. doi:10.3934/dcds.2008.21.415
Caraballo T, Kloeden PE (2009) Non-autonomous attractors for integro-differential evolution equations. Discrete Contin Dyn Syst. doi:10.3934/dcdss.2009.2.17
Caraballo T, Langa JA, Melnik VS, Valero J (2003) Pullback Attractors of Nonautonomous and Stochastic Multivalued Dynamical Systems. Set-Valued Anal. doi:10.1023/A:1022902802385
Caraballo T, Langa JA, Valero J (2003) The dimension of attractors of nonautonomous partial differential equations. ANZIAM J 45:207–222
Caraballo T, Lukaszewicz G, Real J (2006) Pullback attractors for asymptotically compact nonautonomous dynamical systems. Nonlinear Anal. doi:10.1016/j.na.2005.03.111
Cheban DN, Kloeden PE, Schmalfuss B (2002) The relationship between pullback, forwards and global attractors of nonautonomous dynamical systems. Nonlinear Dynam Syst Theor 2:9–28
Chepyzhov VV, Vishik MI (1997) Evolution equations and their trajectory attractors. J Math Pures Appl. doi:10.1016/S0021-7824(97)89978-3
Chepyzhov VV, Vishik MI (2002) Attractors for equations of mathematical physics. American Mathematical Society, Providence RI
Cheskidov A, Foias C (2007) On global attractors of the 3D Navier–Stokes equations. J Differ Equat. doi:10.1016/j.jde.2006.08.021
Crauel H, Flandoli F (1994) Attractors for random dynamical systems. Probab Theor Relat Field. doi:10.1007/BF01193705
Cutland NJ (2005) Global attractors for small samples and germs of 3D Navier–Stokes equations. Nonlinear Anal. doi:10.1016/j.na.2005.02.114
Flandoli F, Schmalfuss B (1996) Random attractors for the 3D stochastic Navier–Stokes equation with multiplicative white noise. Stochast Stochast Rep. doi: 10.1080/17442509608834083
Foias C, Temam R (1987) The connection between the Navier–Stokes equations, dynamical systems, and turbulence theory. In: Directions in Partial Differential Equations, Academic Press, Boston, MA, pp 55–73
Foias C, Manley OP, Rosa R, Temam R (2001) Navier–Stokes equations and turbulence. Cambridge University Press, Cambridge
Gorban VA, Gorban IN (1996) On resonance properties for vortexes by the inequality on boundary. Reports NAS of Ukraine 2:44–47
Gorban V, Gorban I (1998) Dynamics of vortices in near-wall flows: eigenfrequencies, resonant properties, algorithms of control. AGARD Report. 827:15-1–15-11
Gorban VA, Gorban IN (2005) Vortical flow pattern past a square prism: numerical model and control algorithms. Appl Hydromechanics 6:23–48
Gorban VA, Gorban IN (2008) On interaction studing for square cylinders, ordered by tandem. Appl Hydromechanics 9:18–32
Gorban VA, Gorban IN (2009) On analysis of two square cylinders disposed side by side system streamline. Appl Hydromechanics 10:23–41
Kapustyan OV, Kasyanov PO, Valero J (2011) Pullback attractors for a class of extremal solutions of the 3D Navier–Stokes system. J Math Anal Appl. doi:10.1016/j.jmaa.2010.07.040
Kapustyan AV, Valero J (2007) Weak and strong attractors for the 3D Navier–Stokes system. J Differ Equat. doi:10.1016/j.jde.2007.06.008
Kloeden PE, Valero J (2007) The weak connectedness of the attainability set of weak solutions of the three-dimensional Navier–Stokes. Proc R Soc A. doi:10.1098/rspa.2007.1831
Kloeden PE, Valero J (2010) The Kneser property of the weak solutions of the three dimensional Navier–Stokes equations. Discrete Contin Dyn Syst. doi:10.3934/dcds.2010.28.161
Koumoutsakos PD (1993) Direct numerical simulations of unsteady separated flows using vortex methods. PhD Thesis. California Institute of Technology, Pasadena
Langa JA, Schmalfuss B (2004) Finite dimensionality of attractors for non-autonomous dynamical systems given by partial differential equations. Stoch Dyn. doi:10.1142/ S0219493704001127
Lighthill, MJ (1963) Laminar boundary layers. Oxford University Press, London
Marín-Rubio P, Real J (2009) On the relation between two different concepts of pullback attractors for non-autonomous dynamical systems. Nonlinear Anal. doi:10.1016/j.na.2009.02.065
Melnik VS, Valero J (2000) On global attractors of multivalued semiprocesess and nonautonomous evolution inclusions. Set-Valued Anal. doi:10.1023/A:1026514727329
Schmalfuss B (1999) Attractors for the non-autonomous dynamical systems. In: Fiedler B, Gröger K, Sprekels J (ed) In: Proceedings of Equadiff 99. World Scientific, Singapore, pp 684–689
Sell G (1996) Global attractors for the three-dimensional Navier–Stokes equations. J Dynam Differ Equat. doi:10.1007/BF02218613
Temam R (1979) Navier–Stokes equations. North-Holland, Amsterdam
Temam R (1988) Infinite-dimensional dynamical systems in mechanics and physics. Springer, New York
Wang B (2009) Pullback attractors for non-autonomous reaction-diffusion equations on \({\mathbf{R}}^{n}\). Front Math China. doi:10.1007/s11464-009-0033-5
Wu JC (1976) Numerical boundary conditions for viscous flow problems. AIAA J 14:1042–1049
Zgurovsky MZ, Mel’nik VS, Kasyanov PO (2010) Evolution inclusions and variation inequalities for earth data processing II. Heidelberg, Springer. doi:10.1007/978-3-642-13878-2
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Zgurovsky, M.Z., Kasyanov, P.O., Kapustyan, O.V., Valero, J., Zadoianchuk, N.V. (2012). Pullback Attractors for a Class of Extremal Solutions of the 3D Navier–Stokes System. In: Evolution Inclusions and Variation Inequalities for Earth Data Processing III. Advances in Mechanics and Mathematics, vol 27. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-28512-7_6
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