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SPATIALLY NONDECAYING SOLUTIONS OF THE 2D NAVIER-STOKES EQUATION IN A STRIP

Published online by Cambridge University Press:  01 September 2007

SERGEY ZELIK
SERGEY ZELIK*
Affiliation:
Department of Mathematics, University of Surrey Guildford, GU2 7XH, United Kingdom e-mail: [email protected]
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Abstract

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The weighted energy theory for Navier-Stokes equations in 2D strips is developed. Based on this theory, the existence of a solution in the uniformly local phase space (without any spatial decaying assumptions), its uniqueness and the existence of a global attractor are verified. In particular, this phase space contains the 2D Poiseuille flows.

Keywords

35Q3037L3076D0376D05
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 49 , Issue 3 , September 2007 , pp. 525 - 588
Copyright
Copyright © Glasgow Mathematical Journal Trust 2007

References

REFERENCES

1.Afendikov, L. and Mielke, A., Multi-pulse solutions to Navier-Stokes problem between parallel planes, Z. Angew. Math. Phys. 52 (2001), 79–100.Google Scholar
2.Afendikov, L. and Mielke, A., Dynamical properties of spatially non-decaying 2D Navier-Stokes flows with Kolmogorov forcing in an infinite strip, J. Math. Fluid Mech 7, suppl. 1 (2005), 51–67.CrossRefGoogle Scholar
3.Amann, H., On the strong solvability of the Navier-Stokes equations, J. Math. Fluid. Mech. 2 (2000), 16–98.CrossRefGoogle Scholar
4.Babin, A., Asymptotic expansions at infinity of a strongly perturbed Poiseuille flow, Adv. Soviet Math. 10 (1992), 1–83.Google Scholar
5.Babin, A., The attractor of a Navier-Stokes system in an unbounded channel-like domain, J. Dynam. Differential Equations 4 (1992), 555–584.CrossRefGoogle Scholar
6.Babin, A. and Vishik, M., Attractors of evolutionary equations (North Holland, Amsterdam, 1992).Google Scholar
7.Babin, A. and Vishik, M., Attractors of partial differential evolution equations in an unbounded domain, Proc. Roy. Soc. Edinburgh Sect. A, 116 (1990), 221–243.Google Scholar
8.Efendiev, M., Miranville, A. and Zelik, S., Infinite-dimensional exponential attractors for nonlinear reaction-diffusion systems in unbounded domains and their approximation, Proc. R. Soc. London A. 460 (2004), 1107–1129.CrossRefGoogle Scholar
9.Efendiev, M. and Zelik, S., Upper and lower bounds for the Kolmogorov entropy of the attractor for an RDE in an unbounded domain, J. Dyn. Dif. Eqns, 14 (2002) 369–403.CrossRefGoogle Scholar
10.Efendiev, M., M. and Zelik, S., The attractor for a nonlinear reaction-diffusion system in an unbounded domain, Comm. Pure Appl. Math. 54 (2001), 625–688.CrossRefGoogle Scholar
11.Gallay, T. and Wayne, C., Invariant manifolds and the long-time asymptotics of the Navier-Stokes and vorticity equations in , Arch. Rat. Mech. Anal. 163 (2002), 209–258CrossRefGoogle Scholar
12.Giga, Y., Matsui, S. and Sawada, O., Global existence of two-dimensional Navier-Stokes flow with nondecaying initial velocity, J. Math. Fluid Mech. 3 (2001), 302–315.CrossRefGoogle Scholar
13.Iooss, G. and Kirchgässner, K., Water waves for small surface tension: an approach via normal form, Proc. Roy. Soc. Edinburgh Sect. A 122 (1992), 267–299.CrossRefGoogle Scholar
14.Iooss, G. and Mielke, A., Bifurcating time-periodic solutions of Navier-Stokes equations in infinite cylinders, J. Nonlinear Sci. 1 (1991), 107–146.CrossRefGoogle Scholar
15.Kirchgässner, K., Wave solutions of reversible systems and applications, J. Diff. Eqns. 45 (1982), 113–127.CrossRefGoogle Scholar
16.Ladyzhanskaya, O., Solonnikov, V. and Uraltseva, N., Linear and quasilinear equations of parabolic type (Moscow: Izdat “Nanka”, 1968).Google Scholar
17.Mielke, A., The Ginzburg-Landau equation in its role as a modulation equation, Handbook of dynamical systems (ed. by Fiedler, B.), Elsevier (2002), 759–834.Google Scholar
18.Mielke, A. and Schneider, G., Attractors for modulation equations on unbounded domains – existence and comparison, Nonlinearity 8 (1995), 743–768.CrossRefGoogle Scholar
19.Mielke, A. and Zelik, S., Infinite-dimensional hyperbolic sets and spatio-temporal chaos in reaction-diffusion systems in , J. Dynam. Diff. Eqns. 19 (2007), 333–389.CrossRefGoogle Scholar
20.Schaefer, H. and Wolff, M., Topological vector spaces. Second edition. Graduate Texts in Mathematics, No. 3 (Springer-Verlag 1999).CrossRefGoogle Scholar
21.Schneider, G., Global existence results for pattern formation processes in infinite cylindrical domains. Applications to 3D Navier-Stokes problem, J. Math. Pures. Appl. 78 (1999), 265–312.CrossRefGoogle Scholar
22.Schneider, G. and Uecker, H.Almost global existence and transient self-similar decay for Poiseuille flow at criticality for exponentially long times, Phys. D, 185 (2003) 209–226.CrossRefGoogle Scholar
23.Temam, R., Navier-Stokes equations, theory and numerical analysis (North-Holland, 1984).Google Scholar
24.Temam, R., Infinite-dimensional dynamical systems in mechanics and physics (Springer-Verlag 1997).CrossRefGoogle Scholar
25.Triebel, H., Interpolation theory, function space, differential operators (North-Holland, 1978).Google Scholar
26.Vishik, M., Hydrodynamics in Besov spaces, Arch. Rat. Mech. Anal. 145 (1998), 197–214.CrossRefGoogle Scholar
27.Zelik, S., Multiparametrical semigroups and attractors of reaction-diffusion systems in , Proc. Moscow Math. Soc. 65 (2004), 69–130.Google Scholar
28.Zelik, S., Attractors of reaction-diffusion systems in unbounded domains and their spatial complexity, Comm. Pure Appl. Math. 56 (2003), 584–637.CrossRefGoogle Scholar
29.Zelik, S., The attractor for a nonlinear reaction-diffusion system in an unbounded domain and Kolmogorov's epsilon-entropy, Math. Nachr. 232 (2001), 129–179.3.0.CO;2-T>CrossRefGoogle Scholar
30.Zelik, S., The attractor for a nonlinear hyperbolic equation in an unbounded domain, Discrete Contin. Dynam Systems 7 (2001), 593–641.CrossRefGoogle Scholar
31.Zelik, S., Spatial and temporal chaos in 2D Navier-Stokes equations in a strip, in preparation.Google Scholar
32.Zelik, S. Weak spatially non-decaying solutions of 3D Navier-Stokes equations in cylindrical domains, Instability in Models Connected with Fluids Flow, International Math. Series, to appear.Google Scholar