×

Attractors for Navier-Stokes equations in domains with finite measure. (English) Zbl 0859.35090

The Navier-Stokes system \[ \partial_tu+\sum^2_{i=1} u^i\partial_iu=\nu\Delta u-\nabla p+f,\;\text{div }u=0,\quad u|_{\partial\Omega}=0 \tag{1} \] is considered in a domain \(\Omega\subset\mathbb{R}^2\). The system (1) may be unbounded, but has finite measure \(|\Omega|<\infty\). The following lower bounds of the spectrum of the Stokes problem are proved \[ \sum^m_{k=1}\lambda_k\geq {\pi m^2\over |\Omega|},\quad \lambda_1\geq{2\pi\over |\Omega|}, \] where \(|\Omega|\) is the area of \(\Omega\). The author proves the existence of a global attractor, and derives the explicit estimates of its Hausdorff and fractal dimension by using these bounds.

MSC:

35Q30 Navier-Stokes equations
37C70 Attractors and repellers of smooth dynamical systems and their topological structure
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Ladyzhenskaya, O. A., J. Sov. Math., 3, 458-479 (1975) · Zbl 0336.35081
[2] Foias, C.; Temam, R., Some analytic and geometric proprties of the solutions of the evolution Navier-Stokes equations, J. Math. pures appl., 58, 339-368 (1979) · Zbl 0454.35073
[3] Temam, R., Attractors for Navier-Stokes equations, Res. Notes Math., 122, 272-292 (1985) · Zbl 0572.35083
[4] Temam, R., (Infinite Dimensional Dynamical Systems in Mechanics and Physics. (1988), Springer: Springer New York) · Zbl 0662.35001
[5] Ladyzhenskaya, O. A., (The Mathematical Theory of Viscous Incompressible Flow (1969), Gordon and Breach: Gordon and Breach New York) · Zbl 0184.52603
[6] Temam, R., (Navier-Stokes Equations. Theory and Numerical Analysis. (1984), North-Holland: North-Holland New York) · Zbl 0568.35002
[7] Li, P.; Yau, S.-T., On the Schrödinger equation and the eigenvalue problem, J. Communs math. Phys., 8, 309-318 (1983) · Zbl 0554.35029
[8] Jones, D. S., The eigenvalues of a cavity resonator, Quart. J. Mech. appl. Math., 41, 469-477 (1988) · Zbl 0701.35118
[9] Babin, A. V.; Vishik, M. I., (Attractors of Evolution Equations. (1992), North-Holland: North-Holland Moscow) · Zbl 0778.58002
[10] Ladyzhenskaya, O. A., First boundary value problem for Navier-Stokes equations in domain with non smooth boundaries, C. r. Acad. Sci. Paris, 314, 253-258 (1992) · Zbl 0744.35034
[11] Weinstein, M., Nonlinear Schrödinger equations and sharp interpolation estimates, Communs math. Phys., 87, 567-576 (1983) · Zbl 0527.35023
[12] Babin, A. V.; Vishik, M. I., Russ. Math. Survs, 38, 151-213 (1983) · Zbl 0541.35038
[13] Ilyin, A. A., Lieb-Thirring inequalities on the \(N\)-sphere and in the plane and some applications, (Proc. Lond. math. Soc., 67 (1993)), 159-182 · Zbl 0789.58079
[14] Metivier, G., Valeurs propres des opérateurs definis sur la restriction de systems variationnels à des sous-espases, J. Math, pures appl., 57, 133-156 (1978) · Zbl 0328.35029
[15] Kozhevnikov, A. N., On the linearized stationary Navier-Stokes problem, Mat. Sb., 125, 3-18 (1984), (In Russian.)
[16] Edmunds, D. E.; Ilyin, A. A., On some multiplicative inequalities and approximation numbers, Quart. J. Math. Oxford, 45, 159-179 (1994) · Zbl 0818.46027
[17] Ilyin, A. A., Russ. Acad Sci. Sb. Math., 78, 47-76 (1994) · Zbl 0840.35077
[18] Berger, M. S.; Schechter, M., Embedding theorems and quasi-linear elliptic boundary value problems for unbounded domains, Trans. Am. math. Soc., 172, 261-278 (1972) · Zbl 0253.35038
[19] Edmunds, D. E.; Evans, W. D., (Spectral Theory and Differential Operators (1987), Clarendon Press: Clarendon Press Amsterdam) · Zbl 0628.47017
[20] Brezis, H.; Gallouet, T., Nonlinear Schrödinger evolution equation, Nonlinear Analysis, 4, 677-681 (1980) · Zbl 0451.35023
[21] Lieb, E.; Thirring, W., Inequalities for the moments of the eigenvalues of the Schrödinger hamiltonian and their relation to Sobolev ineqalities, (Studies in Mathematical Physics (1976), Princeton University Press: Princeton University Press Oxford), 269-303
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.