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On global attractors of the 3D Navier--Stokes equations. (English) Zbl 1113.35140
For certain dissipative partial differential equations unique or regular solutions in a given state space and for all time are not available, either because solutions cease to exist in the relevant topology or because it is not known whether such solutions exist in the first place. A famous example is furnished by strong solutions of the three-dimensional Navier-Stokes equations. In this situation the concept of a global attractor, the minimal closed set in the state space that uniformly attracts (in the topology of the state space) the trajectories starting from any bounded subset of initial states, is not suitable to describe the asymptotic behavior of solutions. The work under review introduces an abstract framework which allows studying the long-term dynamics of possibly multi-valued evolution systems with respect to two related metric topologies of the state space, referred to as the weak and strong topology. While in this context the dissipative system always possesses a global attractor with respect to the weak topology (“weak global attractor”), this need not hold true in the strong topology (“strong global attractor”). However, if the strong global attractor exists and is closed in the weak topology, then it coincides with the weak global attractor. The authors give a sufficient condition for the existence of the strong global attractor and demonstrate that this condition is satisfied for the three-dimensional space-periodic Navier-Stokes equations when all solutions on the weak global attractor are continuous in the strong topology. They extend their analysis to so-called tridiagonal models for the Navier-Stokes equations: a two-parameter family of simple models for the Navier-Stokes equations with a similar nonlinearity. These model equations fall within the framework introduced here and have weak global attractors. Solutions for certain parameter values, however, are shown to blow up in finite time (with respect to the appropriate norm). Some open questions on blow-up of solutions and strong global attractors for the tridiagonal models of the Navier-Stokes are posed and corresponding results for tridiagonal models of the Euler equations are discussed.

35Q30Stokes and Navier-Stokes equations
35B41Attractors (PDE)
Full Text: DOI
[1] Ball, J. M.: J. nonlinear sci.. 8, 233 (1998)
[2] Caraballo, T.; Marn-Rubio, P.; Robinson, J. C.: A comparison between two theories for multi-valued semiflows and their asymptotic behavior. Set-valued anal. 11, 297-322 (2003) · Zbl 1053.47050
[3] Chepyzhov, V. V.; Vishik, M. I.: Attractors for equations of mathematical physics. Amer. math. Soc. colloq. Publ. 49 (2002) · Zbl 0986.35001
[4] A. Cheskidov, Blow-up in finite time for the dyadic model of the Navier -- Stokes equations, Trans. Amer. Math. Soc., in press · Zbl 1156.35073
[5] Constantin, P.; Foias, C.: Navier -- Stokes equation. (1989)
[6] Flandoli, F.; Schmalfuß, B.: Weak solutions and attractors for three-dimensional Navier -- Stokes equations with nonregular force. J. dynam. Differential equations 11, 355-398 (1999) · Zbl 0931.35124
[7] Foias, C.; Manley, O. P.; Rosa, R.; Temam, R.: Navier -- Stokes equation and turbulence. Encyclopedia of mathematics and its applications 83 (2001)
[8] Foias, C.; Temam, R.: The connection between the Navier -- Stokes equations, and turbulence theory. Directions in partial differential equations, 55-73 (1985)
[9] Friedlander, S.; Pavlović, N.: Blowup in a three-dimensional vector model for the Euler equations. Comm. pure appl. Math. 57, 705-725 (2004) · Zbl 1060.35100
[10] Hale, J. K.: Asymptotic behavior of dissipative systems. (1988) · Zbl 0642.58013
[11] Hale, J. K.; Lasalle, J. P.; Slemrod, M.: Theory of a general class of dissipative processes. J. math. Anal. appl. 39, 177-191 (1972) · Zbl 0238.34098
[12] Kukavica, I.: Role of the pressure for validity of the energy equality for solutions of the Navier -- Stokes equation. J. dynam. Differential equations 18, 461-482 (2006) · Zbl 1105.35081
[13] Katz, N. H.; Pavlović, N.: Finite time blow-up for a dyadic model of the Euler equations. Trans. amer. Math. soc. 357, 695-708 (2005) · Zbl 1059.35096
[14] Ladyzhenskaya, O. A.: On the dynamical system generated by the Navier -- Stokes equations. J. sov. Math. 3, 458-479 (1975) · Zbl 0336.35081
[15] Ladyzhenskaya, O. A.: Attractors for semigroups and evolution equations. (1991) · Zbl 0755.47049
[16] Melnik, V. S.; Valero, J.: On attractors of multivalued semi-flows and differential inclusions. Set-valued anal. 6, 83-111 (1998) · Zbl 0915.58063
[17] Raugel, G.; Sell, G. R.: Navier -- Stokes equations on thin 3D domains, I: Global attractors and global regularity of solutions. J. amer. Math. soc. 6, 503-568 (1993) · Zbl 0787.34039
[18] R.M.S. Rosa, Asymptotic regularity condition for the strong convergence towards weak limit sets and weak attractors of the 3D Navier -- Stokes equations, J. Differential Equations, in press · Zbl 1111.35039
[19] Sell, G. R.: Global attractors for the three-dimensional Navier -- Stokes equations. J. dynam. Differential equations 8, 1-33 (1996) · Zbl 0855.35100
[20] Sell, G. R.; You, Y.: Dynamics of evolutionary equations. Appl. math. Sci. 143 (2002) · Zbl 1254.37002
[21] Temam, R.: Infinite dimensional dynamical systems in mechanics and physics. Appl. math. Sci. 68 (1988) · Zbl 0662.35001
[22] Temam, R.; Ziane, M.: Navier -- Stokes equations in three-dimensional thin domains with various boundary conditions. Adv. differential equations 1, 499-546 (1996) · Zbl 0864.35083