×

The global attractor for the 2D Navier-Stokes flow on some unbounded domains. (English) Zbl 0901.35070

The author studies the Navier-Stokes equations in a domain \(\Omega \subset \mathbb{R}^2\), i.e. \[ u_t- \nu\Delta u+(u \cdot \nabla) u+\nabla p=f, \quad \text{div} u =0\quad\text{in } \Omega \times (0,\infty), \]
\[ u=0\quad\text{on }\partial \Omega \times (0,\infty), \quad u(0)= u_0\quad\text{in } \Omega. \] Here, \(\Omega\) is an arbitrary open set in \(\mathbb{R}^2\) having the property that the Poincaré inequality holds on it and \(f\) is a time independent external force. Assuming that \(f\in V'\) (the dual of \(V= \overline {\{v\in C^\infty_0 (\Omega) \mid \text{div} v= 0\}}^{\| \nabla \cdot \|_{L^2}})\) the author proves the existence of the global attractor improving on earlier work by Abergel and Babin. Furthermore he gives an estimate for its dimension in terms of the usual physical parameters. The proof uses a weak continuity property of the underlying semigroup \(S(t)\) which is employed to show that \(S(t)\) is asymptotically compact. The existence of the global attractor then follows from an abstract result.

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., On the dynamical system generated by the Navier-Stokes equations, Zapiskii of nauchnish seminarovs LOMI. Zapiskii of nauchnish seminarovs LOMI, J. of Soviet Math., 3, 4, 91-114 (1975), English translation
[2] Foias, C.; Temam, R., Some analytic and geometric properties of the solutions of the evolution Navier-Stokes equations, J. Math. Pures et Appl., 58, 334-368 (1979) · Zbl 0454.35073
[3] Constantin, P.; Foias, C., Global Lyapunov exponents, Kaplan-Yorke formulas and the dimension of the attractor for the 2D Navier-Stokes equations, Comm. Pure Appl. Math., XXXVIII, 1-27 (1985) · Zbl 0582.35092
[4] Constantin, P.; Foias, C.; Manley, O.; Temam, R., Determining modes and fractal dimension of turbulent flows, J. Fluid Mech., 150, 427-440 (1988) · Zbl 0607.76054
[5] Temam, R., (Infinite Dimensional Dynamical Systems in Mechanics and Physics, Vol. 68 of Applied Mathematical Sciences (1988), Springer-Verlag: Springer-Verlag New York) · Zbl 0662.35001
[6] Abergel, F., Attractors for a Navier-Stokes flow in an unbounded domain, Math. Mod. and Num. Anal., 23, 3, 359-370 (1989) · Zbl 0676.76028
[7] Babin, A. V., The attractor of a Navier-Stokes system in an unbounded channel-like domain, J. Dynamics and Diff. Eqs., 4, 4, 555-584 (1992) · Zbl 0762.35082
[8] Abergel, F., Existence and finite dimensionality of the global attractor for evolution equations on unbounded domains, J. Diff. Equations, 83, 1, 85-108 (1990) · Zbl 0706.35058
[9] Ladyzhenskaya, O., First boundary value problem for the Navier-Stokes equations in domains with nonsmooth boundaries, C. R. Acad. Sci. Paris, Serie I, 314, 253-258 (1992) · Zbl 0744.35034
[10] Babin, A. V.; Vishik, M. I., Attractors of partial differential equations in an unbounded domain, (Proc. Roy. Soc. Edinburgh, 116A (1990)), 221-243 · Zbl 0721.35029
[11] Feireisl, E.; Laurençot, P.; Simondon, F.; Touré, H., Compact attractors for reaction-diffusion equations in \(R^N\), C. R. Acad. Sci. Paris, Série I, 319, 147-151 (1994) · Zbl 0806.35075
[12] Ladyzhenskaya, O., Attractors for Semigroups and Evolution Equations (1991), Cambridge University Press, Lezioni Lincei · Zbl 0755.47049
[13] Hale, J. K., Asymptotic behavior and dynamics in infinite dimensions, (Hale, J. K.; Martínez-Amores, P., Research Notes in Mathematics, Vol. 132 (1985), Pitman: Pitman New York), 1-420
[14] Haraux, A., Two remarks on hyperbolic dissipative problems, (Brezis, H.; Lions, J. L., Nonlinear Partial Differential Equations and Their Applications. Nonlinear Partial Differential Equations and Their Applications, College de France Seminar, Vol. VII (1985), Pitman), 161-179 · Zbl 0579.35057
[15] Ball, J. M., A proof of the existence of global attractors for damped semilinear wave equations. (To appear.) Cited in [16].; Ball, J. M., A proof of the existence of global attractors for damped semilinear wave equations. (To appear.) Cited in [16]. · Zbl 1056.37084
[16] Ghidaglia, J. M., A note on the strong convergence towards the attractors for damped forced KdV equations, J. Diff. Equations, 110, 356-359 (1994) · Zbl 0805.35114
[17] Wang, X., An energy equation for the weakly damped driven nonlinear Schrödinger equations and its applications to their attractors, Physica D, 88, 167-175 (1995) · Zbl 0900.35372
[18] Temam, R., Navier-Stokes Equations (1979), North-Holland: North-Holland Amsterdam · Zbl 0454.35073
[19] Marion, M. and Temam, R., Navier-Stokes equation-theory and approximation. Handbook of Numerical Analysis.; Marion, M. and Temam, R., Navier-Stokes equation-theory and approximation. Handbook of Numerical Analysis. · Zbl 0921.76040
[20] Temam, R., Navier-Stokes equations and nonlinear functional analysis, (CBMS-NSF Regional Conference Series in Applied Mathematics, Vol. 66 (1983), SIAM: SIAM Philadelphia), (2nd edition, 1995) · Zbl 0833.35110
[21] Constantin, P.; Foias, C.; Temam, R., Attractors representing turbulent flows, Mem. Amer. Math. Soc., 53, 314 (1985) · Zbl 0567.35070
[22] Ghidaglia, J. M.; Temam, R., Attractors for damped nonlinear hyperbolic equations, J. Math. Pures Appl., 66, 273-319 (1987) · Zbl 0572.35071
[23] Ghidaglia, J. M.; Marion, M.; Temam, R., Generalizations of the Sobolev-Lieb-Thirring inequalities and application to the dimension of the attractor, Differential and Integral Equations, 1, 1-21 (1988) · Zbl 0745.46037
[24] Temam, R., Infinite dimensional dynamical systems in fluid mechanics, (Browder, F., Nonlinear Functional Analysis and its Applications. Nonlinear Functional Analysis and its Applications, Vol. 45 of Proceedings of Symposia in Pure Mathematics (1986), American Mathematical Society), 413-445 · Zbl 0598.35095
[25] Métivier, G., Valeurs propres d’opérateaurs définis par la restriction de systèmes variationelles a des sous-espaces, J. Math. Pures Appl., 57, 133-156 (1978) · Zbl 0328.35029
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.