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The uniform attractor for the 2D non-autonomous Navier-Stokes flow in some unbounded domain. (English) Zbl 1057.35031
The authors consider the Navier-Stokes equation on a region \(\Omega\subset\mathbb{R}^2\), where \(\Omega\) only has to satisfy the Poincaré inequality. The weak solutions define a semiprocess depending on an external force \(f\). The authors suppose \(f\in \mathcal{F}\), where \(\mathcal{F}\) is bounded (as a subset of \(L^\infty(\mathbb{R}^+,V')\) and \(V'\) is the dual of the space of divergence free functions in \(H_0^1(\Omega)\)) and positive invariant (i.e. \(s\mapsto f(s+h)\in \mathcal{F}\) for all \(h\geq 0\), \(f\in \mathcal{F}\)). Under these assumptions they prove the existence of a compact uniform (with respect to \(\mathcal{F}\)) attractor. In the case of \(f\) being a \(k\)-dimensional quasi-periodic external force, they also give an upper bound on the Hausdorff dimension of the attractor.

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
Full Text: DOI
[1] Abergel, F., Attractors for a navier – stokes flow in an unbounded domain, Math. modelling numer. anal., 23, 359-370, (1989) · Zbl 0676.76028
[2] Babin, A.V., The attractor of a navier – stokes system in an unbounded channel-like domain, J. dynam. differential equations, 4, 555-584, (1992) · Zbl 0762.35082
[3] Babin, A.V.; Vishik, V.B., Attractors of evolutionary partial differential equations and estimates of their attractors, Russian math. surveys, 38, 151-213, (1983), (English translation) · Zbl 0541.35038
[4] J. Ball, A proof of the existence of global attractors for damped semilinear wave equations, cited in [11]. · Zbl 1056.37084
[5] 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
[6] Constantin, P.; Foias, C.; Manley, O.; Temam, R., Determing modes and fractal dimension of turbulent flows, J. fluid mech., 150, 427-440, (1988) · Zbl 0607.76054
[7] Chepyzhov, V.V.; Efendiev, M.A., Hausdorff dimension estimation for attractors of non-autonomous dynamical systems in unbounded domainsan example, Comm. pure appl. math., LIII, 647-665, (2000) · Zbl 1022.37048
[8] Chepyzhov, V.V.; Vishik, M.I., Attractors of non-autonomous dynamical systems and their dimension, J. math. pures appl., 73, 279-333, (1994) · Zbl 0838.58021
[9] Foias, C.; Temam, R., Some analytic and geometric properties of the solutions of the evolution navier – stokes equations, J. math. pures appl., 58, 334-368, (1979) · Zbl 0454.35073
[10] Ghidaglia, J.M., Weakly damped forced korteweg – de Vries equations behave as a finite dimensional dynamical system in the long time, J. differential equations, 74, 369-390, (1988) · Zbl 0668.35084
[11] Ghidaglia, J.M., A note on the strong convergence towards attractors for damped forced KdV equations, J. differential equations, 110, 356-359, (1994) · Zbl 0805.35114
[12] Ghidaglia, J.M.; Marion, M.; Temam, R., Generalizations of the sobolev – lieb – thirring inequalities and application to the dimension of the attractor, Differential integral equations, 1, 1-21, (1988) · Zbl 0745.46037
[13] Hale, J.K., Asymptotic behavior of dissipative systems, Mathematical surveys and monographs, Vol. 25, (1988), American Mathematical Society Providence, RI · Zbl 0642.58013
[14] O. Ladyzhenskaya, On the dynamical system generated by the Navier-Stokes equations, Z. Nauch. Semin. LOMI 27:91-114. 1972 (English translation in J. Sov. Math. 1975, 3(4)).
[15] Moise, I.; Rosa, R.; Wang, X., Attractors for non-compact semigroups via energy equations, Nonlinearity, 11, 1369-1393, (1998) · Zbl 0914.35023
[16] Moise, I.; Rosa, R.; Wang, X., Attractors for noncompact non-autonomous systems via energy equations, Discrete continuous dynam. systems, 10, 473-496, (2004) · Zbl 1060.35023
[17] Rosa, R., The global attractor for the 2D navier – stokes flow on some unbounded domains, Nonlinear anal. TMA, 32, 71-85, (1998) · Zbl 0901.35070
[18] R. Temam, Navier-Stokes equations and nonlinear functional analysis, in: CBMS-NSF Regional Conference Series in Applied Mathematics, SIAM, Philadelphia, 1983. · Zbl 0522.35002
[19] Temam, R., Navier – stokes equations, theory and numerical analysis, (1984), North-Holland Amsterdam · Zbl 0568.35002
[20] Temam, R., Infinite dimensional dynamical systems in mechanics and physics, Applied mathematical sciences, Vol. 68, (1988), Springer New York · Zbl 0662.35001
[21] Wang, B.X., Attractors for reaction-diffusion equations in unbounded domains, Physica D, 128, 41-52, (1999) · Zbl 0953.35022
[22] Wang, X., An energy equation for the weakly damped driven nonlinear Schrödinger equations and its applications, Physica D, 88, 167-175, (1995) · Zbl 0900.35372
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