×

Optimal bounds on the dimension of the attractor of the Navier-Stokes equations. (English) Zbl 0938.35127

Summary: We derive optimal upper bounds on the dimension of the attractor for the Navier-Stokes equations in two-dimensional domains, these bounds fully agree with the lower bounds obtained by Babin and Vishik (1983) (see also Ghidaglia and Temam, and Liu (1993)). As in Babin and Vishik (1983), we consider here elongated domains and leaving the density of volume forces and the viscosity fixed, we let the shape ratio of the domain become large so that the Grashof number is large. The estimates derived here are based on the general methods for estimating attractors dimensions as in Constantin et al. (1988), on a new version of the Lieb-Thirring inequalities for elongated domains and on techniques developed for such domains in Raugel and Sell (1993), and Temam and Ziane (1996).
At the end of the article, we also give some partial results in the three-dimensional case for which we need a physical assumption on the Reynolds number introduced in Ghidaglia and Temam.

MSC:

35Q30 Navier-Stokes equations
37L30 Attractors and their dimensions, Lyapunov exponents for infinite-dimensional dissipative dynamical systems
76D05 Navier-Stokes equations for incompressible viscous fluids
35B41 Attractors
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Babin, A. V.; Vishik, M. I., Attractors of partial differential equations and estimate of their dimensions, Russian Math. Survey, 38, 151-213 (1983) · Zbl 0541.35038
[2] Constantin, P.; Foias, C.; Manley, O.; Temam, R., Determining modes and fractal dimensions of turbulent flows, J. Fluid Mech., 150, 427-440 (1985) · Zbl 0607.76054
[3] Constantin, P.; Foias, C.; Temam, R., Attractors representing turbulent flows, Mem. AMS, 53, 314 (1985) · Zbl 0567.35070
[4] Constantin, P.; Foias, C.; Temam, R., On the dimension of the attractors in two-dimensional turbulence, Physica D, 30, 284-296 (1988) · Zbl 0658.58030
[5] C.R. Doering and X. Wang, in preparation.; C.R. Doering and X. Wang, in preparation.
[6] Foias, C.; Temam, R., Some analytic and geometric properties of the solutions to the Navier-Stokes equations, J. Math. Pures Appl., 58, 339-368 (1979) · Zbl 0454.35073
[7] Ghidaglia, J. M.; Temam, R., Lower bound on the dimension of the attractor for the Navier-Stokes equations in space dimension 3, (Francaviglia, M.; Holms, D., Mechanics, Analysis and Geometry: 200 Years after Lagrange (1991), Elsevier: Elsevier Amsterdam) · Zbl 0657.76029
[8] Kolmogrov, A. N., Dissipation of energy in locally isotropic turbulence, C.R. Acad. Sci. USSR, 32, 16 (1941) · Zbl 0063.03292
[9] Kraichnan, R. H., Inertial ranges in two-dimensional turbulence, Phys. Fluids, 10, 1417-1423 (1967)
[10] Liu, V. X., A sharp lower bound for the Hausdorff Dimension of the global attractors of the 2D Navier-Stokes equations, Comm. Math. Phys., 158, 327-339 (1993) · Zbl 0790.35085
[11] Liu, V. X., Remarks on the Navier-Stokes equations on the two and three dimensional torus, Comm. Partial Differential equations, 19, 873-900 (1994) · Zbl 0817.35072
[12] A. Miranville and M. Ziane, On the upper bound of the dimension of the attractor for the Bénard problem with free surfaces, Russ. J. Math. Phys., to appear.; A. Miranville and M. Ziane, On the upper bound of the dimension of the attractor for the Bénard problem with free surfaces, Russ. J. Math. Phys., to appear. · Zbl 0910.35148
[13] Raugel, G.; Sell, G., 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
[14] Temam, R., Infinite Dimensional Dynamical Systems in Mechanics and Physics (1988), Springer: Springer New York · Zbl 0662.35001
[15] Temam, R.; Ziane, M., Navier-Stokes equations in three-dimensional thin domains with various boundary conditions, Adv. in Differential Equations, 1, 499-546 (1996) · Zbl 0864.35083
[16] M. Ziane, On the 2D-Navier-Stokes equations with the free boundary condition, J. Appl. Math. and Optimization, to appear.; M. Ziane, On the 2D-Navier-Stokes equations with the free boundary condition, J. Appl. Math. and Optimization, to appear. · Zbl 0912.35127
[17] Ghidaglia, J. M.; Marion, M.; Temam, R., generalization of Sobolev-Lieb-Thirring inequalities and applications to the dimension of attractors, Diff. Integral Eqn, 1, 1-21 (1988) · Zbl 0745.46037
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.