Invariant measures for complex-valued dissipative dynamical systems and applications

Xin Li; Wenxian Shen; Chunyou Sun

doi:10.3934/dcdsb.2017124

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

Invariant measures for complex-valued dissipative dynamical systems and applications

Xin Li ^1,2, , Wenxian Shen ^2, and Chunyou Sun ^1,,

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.

Keywords: Complex-valued dynamical systems, complex invariant measure, global attractor, Ginzburg-Landau equation, nonlinear Schrödinger equation.

Mathematics Subject Classification: Primary: 35B41, 35Q55, 35Q56, 37L40, 76F2.

Citation: Xin Li, Wenxian Shen, Chunyou Sun. Invariant measures for complex-valued dissipative dynamical systems and applications. Discrete and 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.
[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.

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.
[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.

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