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Dimension of the attractors associated to the Ginzburg-Landau partial differential equation. (English) Zbl 0623.58049
We study the long-time behavior of solutions to the Ginzburg-Landau partial differential equation. It is shown that a finite-dimensional attractor captures all the solutions. An upper bound on this dimension is given in terms of physical quantities, by estimating the Lyapunov exponents on the trajectories. Finally, using the well-known side-band instabilities of an exact, time-dependent solution (Stokes solution) we derive lower bounds on the dimension of the universal attractor. Moreover the lower and upper bounds agree.

MSC:
58Z05 Applications of global analysis to the sciences
35Q99 Partial differential equations of mathematical physics and other areas of application
37C70 Attractors and repellers of smooth dynamical systems and their topological structure
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[1] Babin, A.V.; Vishik, M.I., Regular attractors of semi-groups and evolution equations, J. math. pures et appl., 62, 441-491, (1983) · Zbl 0565.47045
[2] Babin, A.V.; Vishik, M.I., Attractors of partial differential equations and estimates of their dimension, Russian math. surveys, 38, 151-213, (1983) · Zbl 0541.35038
[3] Blennerhassett, P.J., On the generation of waves by wind, Phil. trans. R. soc. lond. A, 298, 451-494, (1980) · Zbl 0445.76013
[4] Bourbaki, N., Espaces vectoriels topologiques, (1981), Masson Paris · Zbl 0482.46001
[5] Constantin, P.; Foias, C.; Manley, O.; Temam, R., Determining modes and fractal dimension of turbulent flows, J. fluid. mech., 150, 427-440, (1985) · Zbl 0607.76054
[6] Constantin, P.; Foias, C.; Temam, R., Attractors representing turbulent flows, Mem. amer. math. soc., 53, 314, (1985) · Zbl 0567.35070
[7] Courant, R.; Hilbert, D., ()
[8] Foias, C.; Manley, O.; Temam, R., Attractors for the Bénard problem: existence and physical bounds on their dimension, J. nonlinear analysis-T.M.A., (1987) · Zbl 0646.76098
[9] Foias, C.; Temam, R., Some analytic and geometric properties of the solutions of the evolution Navier-Stokes equations, J. math. pures et appl., 58, 339-368, (1979) · Zbl 0454.35073
[10] Frederickson, P., The Liapunov dimension of strange attractors, J. diff. equations, 49, 185-207, (1983) · Zbl 0515.34040
[11] Ghidaglia, J.M., Some backward uniqueness results, J. nonlinear analysis-T.M.A., 10, 777-790, (1986) · Zbl 0622.35029
[12] J.M. Ghidaglia, Invariant manifolds and nonlinear instabilities for semigroups generated by partial differential equations, to appear.
[13] J.M. Ghidaglia and D. Hilhorst, to appear.
[14] Generalization of the Sobolev-Lieb-Thirring inequalities and applications, to appear. · Zbl 0615.35006
[15] Hirsch, M.W.; Pugh, C.; Shub, M., Invariant manifolds, () · Zbl 0355.58009
[16] P. Huerre, Instabilités hydrodynamiques non linéaires et chaos déterministe, in: Equations aux Dérivées Partielles Non Linéaires et Systèmes Dynamiques, J.M. Ghidaglia and J.C. Saut, eds. (Hermann, Paris, to appear). · Zbl 0646.35064
[17] Hurewicz, W.; Wallman, H., Dimension theory, (1941), Princeton Univ. Press Princeton · JFM 67.1092.03
[18] Keefe, L., Integrability and structural stability of solutions to the Ginzburg-Landau equation, Phys. fluids, 29, 3135-3141, (1986) · Zbl 0602.76061
[19] Keefe, L., Dynamics of perturbed wavetrain solutions to the Ginzburg-Landau equation, Stud. in appl. math., 73, 91-153, (1985), submitted to · Zbl 0575.76055
[20] Lieb, E.; Thirring, W., Inequalities for the moments of the eigenvalues of the Schrödinger Hamiltonian and their relation to Sobolev inequalities, () · Zbl 0342.35044
[21] Lions, J.L., Quelques Méthodes de Résolution des problémes aux limites non linéaires, (1969), Dunod Paris · Zbl 0189.40603
[22] Mandelbrot, B., Fractals: form, chance and dimension, (1977), Freeman San Francisco · Zbl 0376.28020
[23] Moon, H.T.; Huerre, P.; Redekopp, L.G., Three-frequency motion and chaos in the Ginzburg-Landau equation, Phys. rev. lett., 49, 458-460, (1982)
[24] Moon, H.T.; Huerre, P.; Redekopp, L.G., Transitions to chaos in the Ginzburg-Landau equation, Physica, 7D, 135-150, (1983) · Zbl 0558.58030
[25] Newell, A.C.; Whitehead, J.A., Finite bandwidth, finite amplitude convection, J. fluid. mech., 38, 279-304, (1969) · Zbl 0187.25102
[26] Newton, P.K.; Sirovich, L., Instabilities of the Ginzburg-Landau equation: periodic solutions, Quart. appl. math., 43, 535-542, (1986)
[27] Newton, P.K.; Sirovich, L., Periodic solutions of the Ginzburg-Landau equation, Physica, 21D, 115-125, (1986) · Zbl 0619.35011
[28] Nirenberg, L., On elliptic partial differential equations, Ann. sc. norm. sup. Pisa, 13, 116-162, (1959) · Zbl 0088.07601
[29] Stuart, J.T.; DiPrima, F.R.S.; DiPrima, R.C., The eckhaus and Benjamin-feir resonance mechanisms, (), 27-41
[30] Temam, R., Behaviour at time t = 0 of the solutions of semi-linear evolution equations, J. diff. equations, 43, 73-92, (1982) · Zbl 0446.35057
[31] R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics (Springer, Berlin, to appear). · Zbl 0871.35001
[32] Wells, J.C., Invariant manifolds of nonlinear operators, Pacific J. of math., 62, 285-293, (1976) · Zbl 0343.58010
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