\`x^2+y_1+z_12^34\`
Advanced Search
Article Contents
Article Contents

Invariant measures for complex-valued dissipative dynamical systems and applications

This work was partly supported by the NSFC (Grants No. 11471148,11522109).
Abstract Full Text(HTML) Related Papers Cited by
  • In this work, we extend the classical real-valued framework to deal with complex-valued dissipative dynamical systems. With our new complex-valued framework and using generalized complex Banach limits, we construct invariant measures for continuous complex semigroups possessing global attractors. In particular, for any given complex Banach limit and initial data $u_{0}$, we construct a unique complex invariant measure $\mu$ on a metric space which is acted by a continuous semigroup $\{S(t)\}_{t\geq 0}$ possessing a global attractor $\mathcal{A}$. Moreover, it is shown that the support of $\mu$ is not only contained in global attractor $\mathcal{A}$ but also in $\omega(u_{0})$. Next, the structure of the measure $\mu$ is studied. It is shown that both the real and imaginary parts of a complex invariant measure are invariant signed measures and that both the positive and negative variations of a signed measure are invariant measures. Finally, we illustrate the main results of this article on the model examples of a complex Ginzburg-Landau equation and a nonlinear Schrödinger equation and construct complex invariant measures for these two complex-valued equations.

    Mathematics Subject Classification: Primary:35B41, 35Q55, 35Q56, 37L40, 76F20.

    Citation:

    \begin{equation} \\ \end{equation}
  • 加载中
  • [1]

    M. Barton-Smith, Invariant measure for the stochastic Ginzburg-Landau equation, NoDEA Nonlinear Differential Equations Appl., 11 (2004), 29-52.

    [2]

    M. Bartuccelli, P. Constantin, C. R. Doering, J. D. Gibbon and M. Gisselfält, On the possibility of soft and hard turbulence in the complex Ginzburg-Landau equation, Phys. D, 44 (1990), 421-444.

    [3]

    J. Bourgain, Invariant measures for the 2D-defocusing nonlinear Schrödinger equation, Comm. Math. Phys., 176 (1996), 421-445.

    [4]

    P. Bechouche and A. Jüngel, Inviscid limits of the complex Ginzburg-Landau equation, Comm. Math. Phys., 214 (2000), 201-226.

    [5]

    F. Cacciafesta and A.-S. de Suzzoni, Invariant measure for the Schrödinger equation on the real line, J. Funct. Anal., 269 (2015), 271-324.

    [6]

    T. Caraballo, P. E. Kloeden and J. Real, Invariant measures and statistical solutions of the globally modified Navier-Stokes equations, Discrete Contin. Dyn. Syst. Ser. B, 10 (2008), 761-781.

    [7] T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics, 10. New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2003. xiv+323 pp.
    [8]

    M. D. Chekroun and N. E. Glatt-Holtz, Invariant measure for dissipative dynamical systems: Abstract results and applications, Commun. Math. Phys., 316 (2012), 723-761.

    [9]

    C. R. Doering, J. D. Gibbon and C. D. Levermore, Weak and strong solutions of the complex Ginzburg-Landau equation, Phys. D, 71 (1994), 285-318.

    [10]

    J. Q. Duan and P. Holmes, On the Cauchy problem of a generalized Ginzburg-Landau equation, Nonlinear Anal., 22 (1994), 1033-1040.

    [11]

    J. Q. Duan, P. Holmes and E. S. Titi, Global existence theory for a generalized GinzburgLandau equation, Nonlinearity, 5 (1992), 1303-1314.

    [12]

    G. Fibich, Self-focusing in the damped nonlinear Schrödinger equation, SIAM J. Appl. Math., 61 (2001), 1680-1705.

    [13] C. Foias, O. P. Manley, R. Rosa and R. Temam, Navier-Stokes Equations and Turbulence, Cambridge University Press, 2001.
    [14]

    T. Funaki and T. Nishikawa, Large deviations for the Ginzburg-Landau $\nabla \phi $ interface model, Probab. Theory Related Fields, 120 (2001), 535-568.

    [15]

    J. M. Ghidaglia, Finite-dimensional behavior for weakly damped driven Schrödinger equations, Ann. Inst. Henri Poincaré Anal. Non Linéaire, 5 (1988), 365-405.

    [16]

    B. L. Guo and H. J. Gao, Finite dimensional behavior of generalized Ginzburg-Landau equation (in Chinese), Progress in Natural Sciences, 4 (1994), 423-434.

    [17]

    B. L. Guo and Y. Q. Han, Attractors of derivative complex Ginzburg-Landau equation in unbounded domains, Front. Math. China, 2 (2007), 383-416.

    [18]

    N. Hayashi, K. Nakamita and M. Tsutsumi, On solution of the initial value problem for the nonlinear Schrödinger equations, J. Funct. Anal., 71 (1987), 218-245.

    [19]

    N. Hayashi and M. Tsutsumi, L($\mathbb{R}^N$)-decay of classical solution of nonlinear Schrödinger equations, Proc. Roy. Soc. Edinburgh, A, 104 (1986), 309-327.

    [20]

    N. I. Karachalios and N. M. Stavrakakis, Global attractor for the weakly damped driven Schrödinger equation in H2(Ω), NoDEA Nonlinear Differential Equations Appl., 9 (2002), 347-360.

    [21]

    T. Kato, On nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Phys, Théor., 46 (1987), 113-129.

    [22]

    G. R. Kent, A Riesz representation theorem, Proc. Amer. Math. Soc., 24 (1970), 629-636.

    [23]

    J. U. Kim, Invariant measures for a stochastic nonlinear Schrödinger equation, Indiana Univ. Math. J., 55 (2006), 687-717.

    [24]

    J. L. Lebowitz, H. A. Rose and E. R. Speer, Statistical mechanics of the nonlinear Schrödinger equations, J. Stat. Phys., 50 (1988), 657-687.

    [25]

    F. Li and B. you, Global attractors for the complex Ginzburg-Landau equation, J. Math. Anal. Appl., 415 (2014), 14-24.

    [26]

    G. Łukaszewicz, J. Real and J. C. Robinson, Invariant measures for dissipative systems and generalized banach limits, J. Dynam. Differential Equations, 23 (2011), 225-250.

    [27]

    G. Łukaszewicz and J. C. Robinson, Invariant measures for non autonomous dissipative systems, Discrete Contin. Dyn. Syst., 34 (2014), 4211-4222.

    [28]

    N. Okazawa and T. Yokota, Monotonicity method applied to the complex Ginzburg-Landau and related equations, J. Math. Anal. Appl., 267 (2002), 247-263.

    [29]

    E. Pereira, Relaxation to stationary nonequilibrium states in stochastic Ginzburg-Landau models, Lett. Math. Phys., 64 (2003), 129-135.

    [30]

    X. K. Pu and B. L. Guo, Momentum estimates and ergodicity for the 3D stochastic cubic Ginzburg-Landau equation with degenerate noise, J. Differential Equations, 251 (2011), 1747-1777.

    [31] L. E. Reichl, A Modern Course in Statistical Physics, John Wiley & Sons, Inc, New York, 1998. xx+822 pp.
    [32] J. C. Robinson, Infinite-Dimensional Dynamical Systems, An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors, Cambridge University Press, Cambridge, 2001. xviii+461 pp.
    [33]

    J. Rougemont, Space-time invariant measures, entropy, and dimension for stochastic Ginzburg-Landau equations, Comm. Math. Phys., 225 (2002), 423-448.

    [34] W. Rudin, Real and Complex Analysis, McGraw-Hill Book Co. , New York-Toronto, Ont. -London 1966. xi+412 pp.
    [35] R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, SpringerVerlag, New York, 1997. xxii+648 pp.
    [36]

    M. Tsutsumi and N. Hayashi, Classical solution of nonlinear Schrödinger equations in higher dimensions, Math. Z., 177 (1981), 217-234.

    [37]

    N. Tzvetkov, Invariant measures for the defocusing nonlinear Schrödinger equation, Ann. Inst. Fourier (Grenoble), 58 (2008), 2543-2604.

    [38]

    N. Tzvetkov, Invariant measures for the nonlinear Schrödinger equation on the disc, Dyn. Partial Differ. Equ., 3 (2006), 111-160.

    [39]

    B. X. Wang, The limit behavior of solutions for the Cauchy problem of the complex GinzburgLandau equation, Comm. Pure Appl. Math., 55 (2002), 481-508.

    [40]

    L. H. Xu, Ergodicity of the stochastic real Ginzburg-Landau equation driven by α-stable noises, Stochastic Process. Appl., 123 (2013), 3710-3736.

    [41]

    P. E. Zhidkov, On an infinite sequence of invariant measures for the cubic nonlinear Schrödinger equation, Int. J. Math. Math. Sci., 28 (2001), 375-394.

  • 加载中
SHARE

Article Metrics

HTML views(137) PDF downloads(194) Cited by(0)

Access History

Other Articles By Authors

Catalog

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return