August  2017, 22(6): 2427-2446. doi: 10.3934/dcdsb.2017124

Invariant measures for complex-valued dissipative dynamical systems and applications

1. 

School of Mathematical and Statistics, Lanzhou University, Lanzhou, Gansu, China

2. 

Department of Mathematics and Statistics, Auburn University, Auburn, AL, USA

E-mail address: sunchy@lzu.edu.cn

Received  June 2016 Revised  February 2017 Published  March 2017

Fund Project: This work was partly supported by the NSFC (Grants No. 11471148,11522109)

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.

Citation: Xin Li, Wenxian Shen, Chunyou Sun. Invariant measures for complex-valued dissipative dynamical systems and applications. Discrete & Continuous Dynamical Systems - B, 2017, 22 (6) : 2427-2446. doi: 10.3934/dcdsb.2017124
References:
[1]

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

[2]

M. BartuccelliP. ConstantinC. R. DoeringJ. 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. Google Scholar

[3]

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

[4]

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

[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. Google Scholar

[6]

T. CaraballoP. 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. Google Scholar

[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.Google Scholar

[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. Google Scholar

[9]

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

[10]

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

[11]

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

[12]

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

[13]

C. Foias, O. P. Manley, R. Rosa and R. Temam, Navier-Stokes Equations and Turbulence, Cambridge University Press, 2001.Google Scholar

[14]

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

[15]

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

[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. Google Scholar

[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. Google Scholar

[18]

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

[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. Google Scholar

[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. Google Scholar

[21]

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

[22]

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

[23]

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

[24]

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

[25]

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

[26]

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

[27]

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

[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. Google Scholar

[29]

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

[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. Google Scholar

[31]

L. E. Reichl, A Modern Course in Statistical Physics, John Wiley & Sons, Inc, New York, 1998. xx+822 pp.Google Scholar

[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.Google Scholar

[33]

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

[34]

W. Rudin, Real and Complex Analysis, McGraw-Hill Book Co. , New York-Toronto, Ont. -London 1966. xi+412 pp.Google Scholar

[35]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, SpringerVerlag, New York, 1997. xxii+648 pp.Google Scholar

[36]

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

[37]

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

[38]

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

[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. Google Scholar

[40]

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

[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. Google Scholar

show all references

References:
[1]

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

[2]

M. BartuccelliP. ConstantinC. R. DoeringJ. 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. Google Scholar

[3]

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

[4]

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

[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. Google Scholar

[6]

T. CaraballoP. 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. Google Scholar

[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.Google Scholar

[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. Google Scholar

[9]

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

[10]

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

[11]

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

[12]

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

[13]

C. Foias, O. P. Manley, R. Rosa and R. Temam, Navier-Stokes Equations and Turbulence, Cambridge University Press, 2001.Google Scholar

[14]

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

[15]

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

[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. Google Scholar

[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. Google Scholar

[18]

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

[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. Google Scholar

[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. Google Scholar

[21]

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

[22]

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

[23]

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

[24]

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

[25]

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

[26]

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

[27]

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

[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. Google Scholar

[29]

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

[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. Google Scholar

[31]

L. E. Reichl, A Modern Course in Statistical Physics, John Wiley & Sons, Inc, New York, 1998. xx+822 pp.Google Scholar

[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.Google Scholar

[33]

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

[34]

W. Rudin, Real and Complex Analysis, McGraw-Hill Book Co. , New York-Toronto, Ont. -London 1966. xi+412 pp.Google Scholar

[35]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, SpringerVerlag, New York, 1997. xxii+648 pp.Google Scholar

[36]

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

[37]

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

[38]

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

[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. Google Scholar

[40]

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

[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. Google Scholar

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