American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Ground state homoclinic orbits for a class of asymptotically periodic second-order Hamiltonian systems
  • DCDS-B Home
  • This Issue
  • Next Article
    On a terminal value problem for a system of parabolic equations with nonlinear-nonlocal diffusion terms
March  2021, 26(3): 1615-1626. doi: 10.3934/dcdsb.2020175

Ergodicity of stochastic damped Ostrovsky equation driven by white noise

Shang Wu 1, , Pengfei Xu 1, , Jianhua Huang 1,, and  Wei Yan 2,

1. 

College of Liberal Arts and Science, National University of Defense Technology, Changsha 410073, China
2. 

College of Mathematics and Information Science, Henan Normal University, Xinxiang, 453007, China

* Corresponding author: Jianhua Huang

Received  December 2019 Revised  February 2020 Published  March 2021 Early access  May 2020

Fund Project: The authors are supported by the NSF of China(No.11771449)

The current paper is devoted to the stochastic damped Ostrovsky equation driven by white noise. By establishing the uniform estimates for the solution in $ H^1 $ norm, we prove the global well-posedness and the existence of invariant measure for stochastic damped Ostrovsky equation with random initial value. Moreover, we obtain the ergodicity of stochastic damped Ostrovsky equation with deterministic initial conditions.

Keywords: Invariant measure, damped Ostrovsky equation, white noises.
Mathematics Subject Classification: Primary: 60H15; Secondary: 37A25.
Citation: Shang Wu, Pengfei Xu, Jianhua Huang, Wei Yan. Ergodicity of stochastic damped Ostrovsky equation driven by white noise. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1615-1626. doi: 10.3934/dcdsb.2020175
References:
[1]

A. de Bouard and E. Hausenblas, The nonlinear Schrödinger equation driven by jump processes, J. Math. Anal. Appl., 475 (2019), 215-252.  doi: 10.1016/j.jmaa.2019.02.036.  Google Scholar

[2]

A. de Bouard and A. Debussche, On the stochastic Korteweg-de Vries equation, J. Funct. Anal., 154 (1998), 215-251.  doi: 10.1006/jfan.1997.3184.  Google Scholar

[3]

A. de Bouard, A. Debussche and Y. Tsutsumi, White noise driven Korteweg-de Vries equation, J. Funct. Anal., 169 (1999), 532-558.  doi: 10.1006/jfan.1999.3484.  Google Scholar

[4]

T. Dankel Jr., On the stochastic Korteweg-de Vries equation driven by white noise, Differential Integral Equations, 13 (2000), 827-836.   Google Scholar

[5]

I. Ekren, I. Kukavica and M. Ziane, Existence of invariant measure for the stochastic damped KdV equation, Indiana Univ. Math. J., 67 (2018), 1221-1254.  doi: 10.1512/iumj.2018.67.7365.  Google Scholar

[6]

I. Ekren, I. Kukavica and M. Ziane, Existence of invariant measures for the stochastic damped Schrödinger equation, Stoch. Partial Differ. Equ. Anal. Comput., 5 (2017), 343-367.  doi: 10.1007/s40072-016-0090-1.  Google Scholar

[7]

V. M. Galkin and Y. A. Stepan'yants, On the existence of stationary solitary waves in a rotating fluid, J. Appl. Math. Mech., 55 (1991), 939-943.  doi: 10.1016/0021-8928(91)90148-N.  Google Scholar

[8]

P. Isaza and J. Mejía, Cauchy problem for the Ostrovsky equation in spaces of low regularity, J. Differential Equations, 230 (2006), 661-681.  doi: 10.1016/j.jde.2006.04.007.  Google Scholar

[9]

P. Isaza and J. Mejía, Global Cauchy problem for the Ostrovsky equation, Nonlinear Anal., 67 (2007), 1482-1503.  doi: 10.1016/j.na.2006.07.031.  Google Scholar

[10]

P. Isazaa and J. Mejía, Local well-posedness and quantitative ill-posedness for the Ostrovsky equation, Nonlinear Anal., 70 (2009), 2306-2316.  doi: 10.1016/j.na.2008.03.010.  Google Scholar

[11]

S. Li, Well-Posedness and Asymptotic Behavior for Some Nonlinear Evolution Equations, Ph.D thesis, 2015. Google Scholar

[12]

F. Linares and A. Milanés, Local and global well-posedness for the Ostrovsky equation, J. Differential Equations, 222 (2006), 325-340.  doi: 10.1016/j.jde.2005.07.023.  Google Scholar

[13]

L. Ostrovsky, Nonlinear internal waves in a rotating ocean, Okeanologiya, 18 (1978), 181-191.   Google Scholar

[14] S. Peszat and J. Zabczyk, Stochatsic Partial Diffrential Equations with Lévy Noise. An Evolution Equation Approach, Encyclopedia of Mathematics and its Applications, 113, Cambridge University Press, Cambridge, 2007.  doi: 10.1017/CBO9780511721373.  Google Scholar
[15] G. Da Prato and J. Zabczyk, Ergodicity for Infinite-Dimensional Systems, London Mathematical Society Lecture Note Series, 229, Cambridge University Press, Cambridge, 1996.  doi: 10.1017/CBO9780511662829.  Google Scholar
[16] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Encyclopedia of Mathematics and its Applications, 44, Cambridge University Press, Cambridge, 1992.  doi: 10.1017/CBO9780511666223.  Google Scholar
[17]

W. Yan, Y. Li, J. Huang and J. Duan, The Cauchy problem for the Ostrovsky equation with positive dispersion, NoDEA Nonlinear Differential Equations Appl., 25 (2018), 37pp. doi: 10.1007/s00030-018-0514-x.  Google Scholar

[18]

W. Yan, M. Yang and J. Duan, White noise driven Ostrovsky equation, J. Differential Equations, 267 (2019), 5701-5735.  doi: 10.1016/j.jde.2019.06.003.  Google Scholar

show all references

References:
[1]

A. de Bouard and E. Hausenblas, The nonlinear Schrödinger equation driven by jump processes, J. Math. Anal. Appl., 475 (2019), 215-252.  doi: 10.1016/j.jmaa.2019.02.036.  Google Scholar

[2]

A. de Bouard and A. Debussche, On the stochastic Korteweg-de Vries equation, J. Funct. Anal., 154 (1998), 215-251.  doi: 10.1006/jfan.1997.3184.  Google Scholar

[3]

A. de Bouard, A. Debussche and Y. Tsutsumi, White noise driven Korteweg-de Vries equation, J. Funct. Anal., 169 (1999), 532-558.  doi: 10.1006/jfan.1999.3484.  Google Scholar

[4]

T. Dankel Jr., On the stochastic Korteweg-de Vries equation driven by white noise, Differential Integral Equations, 13 (2000), 827-836.   Google Scholar

[5]

I. Ekren, I. Kukavica and M. Ziane, Existence of invariant measure for the stochastic damped KdV equation, Indiana Univ. Math. J., 67 (2018), 1221-1254.  doi: 10.1512/iumj.2018.67.7365.  Google Scholar

[6]

I. Ekren, I. Kukavica and M. Ziane, Existence of invariant measures for the stochastic damped Schrödinger equation, Stoch. Partial Differ. Equ. Anal. Comput., 5 (2017), 343-367.  doi: 10.1007/s40072-016-0090-1.  Google Scholar

[7]

V. M. Galkin and Y. A. Stepan'yants, On the existence of stationary solitary waves in a rotating fluid, J. Appl. Math. Mech., 55 (1991), 939-943.  doi: 10.1016/0021-8928(91)90148-N.  Google Scholar

[8]

P. Isaza and J. Mejía, Cauchy problem for the Ostrovsky equation in spaces of low regularity, J. Differential Equations, 230 (2006), 661-681.  doi: 10.1016/j.jde.2006.04.007.  Google Scholar

[9]

P. Isaza and J. Mejía, Global Cauchy problem for the Ostrovsky equation, Nonlinear Anal., 67 (2007), 1482-1503.  doi: 10.1016/j.na.2006.07.031.  Google Scholar

[10]

P. Isazaa and J. Mejía, Local well-posedness and quantitative ill-posedness for the Ostrovsky equation, Nonlinear Anal., 70 (2009), 2306-2316.  doi: 10.1016/j.na.2008.03.010.  Google Scholar

[11]

S. Li, Well-Posedness and Asymptotic Behavior for Some Nonlinear Evolution Equations, Ph.D thesis, 2015. Google Scholar

[12]

F. Linares and A. Milanés, Local and global well-posedness for the Ostrovsky equation, J. Differential Equations, 222 (2006), 325-340.  doi: 10.1016/j.jde.2005.07.023.  Google Scholar

[13]

L. Ostrovsky, Nonlinear internal waves in a rotating ocean, Okeanologiya, 18 (1978), 181-191.   Google Scholar

[14] S. Peszat and J. Zabczyk, Stochatsic Partial Diffrential Equations with Lévy Noise. An Evolution Equation Approach, Encyclopedia of Mathematics and its Applications, 113, Cambridge University Press, Cambridge, 2007.  doi: 10.1017/CBO9780511721373.  Google Scholar
[15] G. Da Prato and J. Zabczyk, Ergodicity for Infinite-Dimensional Systems, London Mathematical Society Lecture Note Series, 229, Cambridge University Press, Cambridge, 1996.  doi: 10.1017/CBO9780511662829.  Google Scholar
[16] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Encyclopedia of Mathematics and its Applications, 44, Cambridge University Press, Cambridge, 1992.  doi: 10.1017/CBO9780511666223.  Google Scholar
[17]

W. Yan, Y. Li, J. Huang and J. Duan, The Cauchy problem for the Ostrovsky equation with positive dispersion, NoDEA Nonlinear Differential Equations Appl., 25 (2018), 37pp. doi: 10.1007/s00030-018-0514-x.  Google Scholar

[18]

W. Yan, M. Yang and J. Duan, White noise driven Ostrovsky equation, J. Differential Equations, 267 (2019), 5701-5735.  doi: 10.1016/j.jde.2019.06.003.  Google Scholar

[1]

Davit Martirosyan. Exponential mixing for the white-forced damped nonlinear wave equation. Evolution Equations & Control Theory, 2014, 3 (4) : 645-670. doi: 10.3934/eect.2014.3.645
[2]

V. Varlamov, Yue Liu. Cauchy problem for the Ostrovsky equation. Discrete & Continuous Dynamical Systems, 2004, 10 (3) : 731-753. doi: 10.3934/dcds.2004.10.731
[3]

Shengfan Zhou, Min Zhao. Fractal dimension of random attractor for stochastic non-autonomous damped wave equation with linear multiplicative white noise. Discrete & Continuous Dynamical Systems, 2016, 36 (5) : 2887-2914. doi: 10.3934/dcds.2016.36.2887
[4]

Giuseppe Da Prato. An integral inequality for the invariant measure of some finite dimensional stochastic differential equation. Discrete & Continuous Dynamical Systems - B, 2016, 21 (9) : 3015-3027. doi: 10.3934/dcdsb.2016085
[5]

Yan Wang, Guanggan Chen. Invariant measure of stochastic fractional Burgers equation with degenerate noise on a bounded interval. Communications on Pure & Applied Analysis, 2019, 18 (6) : 3121-3135. doi: 10.3934/cpaa.2019140
[6]

Zhaojuan Wang, Shengfan Zhou. Random attractor and random exponential attractor for stochastic non-autonomous damped cubic wave equation with linear multiplicative white noise. Discrete & Continuous Dynamical Systems, 2018, 38 (9) : 4767-4817. doi: 10.3934/dcds.2018210
[7]

Min Zhao, Shengfan Zhou. Random attractor for stochastic Boissonade system with time-dependent deterministic forces and white noises. Discrete & Continuous Dynamical Systems - B, 2017, 22 (4) : 1683-1717. doi: 10.3934/dcdsb.2017081
[8]

Min Niu, Bin Xie. Comparison theorem and correlation for stochastic heat equations driven by Lévy space-time white noises. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 2989-3009. doi: 10.3934/dcdsb.2018296
[9]

Hongwei Wang, Amin Esfahani. Well-posedness and asymptotic behavior of the dissipative Ostrovsky equation. Evolution Equations & Control Theory, 2019, 8 (4) : 709-735. doi: 10.3934/eect.2019035
[10]

Roger Grimshaw, Dmitry Pelinovsky. Global existence of small-norm solutions in the reduced Ostrovsky equation. Discrete & Continuous Dynamical Systems, 2014, 34 (2) : 557-566. doi: 10.3934/dcds.2014.34.557
[11]

Steve Levandosky, Yue Liu. Stability and weak rotation limit of solitary waves of the Ostrovsky equation. Discrete & Continuous Dynamical Systems - B, 2007, 7 (4) : 793-806. doi: 10.3934/dcdsb.2007.7.793
[12]

Jonathan C. Mattingly, Etienne Pardoux. Invariant measure selection by noise. An example. Discrete & Continuous Dynamical Systems, 2014, 34 (10) : 4223-4257. doi: 10.3934/dcds.2014.34.4223
[13]

Luis Barreira and Jorg Schmeling. Invariant sets with zero measure and full Hausdorff dimension. Electronic Research Announcements, 1997, 3: 114-118.

[14]

V. Pata, Sergey Zelik. A remark on the damped wave equation. Communications on Pure & Applied Analysis, 2006, 5 (3) : 611-616. doi: 10.3934/cpaa.2006.5.611
[15]

Giuseppe Maria Coclite, Lorenzo di Ruvo. A note on the non-homogeneous initial boundary problem for an Ostrovsky-Hunter type equation. Discrete & Continuous Dynamical Systems - S, 2020, 13 (12) : 3357-3389. doi: 10.3934/dcdss.2020236
[16]

Simon Lloyd, Edson Vargas. Critical covering maps without absolutely continuous invariant probability measure. Discrete & Continuous Dynamical Systems, 2019, 39 (5) : 2393-2412. doi: 10.3934/dcds.2019101
[17]

Paola Mannucci, Claudio Marchi, Nicoletta Tchou. Asymptotic behaviour for operators of Grushin type: Invariant measure and singular perturbations. Discrete & Continuous Dynamical Systems - S, 2019, 12 (1) : 119-128. doi: 10.3934/dcdss.2019008
[18]

Boris Kalinin, Anatole Katok. Measure rigidity beyond uniform hyperbolicity: invariant measures for cartan actions on tori. Journal of Modern Dynamics, 2007, 1 (1) : 123-146. doi: 10.3934/jmd.2007.1.123
[19]

Boris Kalinin, Anatole Katok, Federico Rodriguez Hertz. Errata to "Measure rigidity beyond uniform hyperbolicity: Invariant measures for Cartan actions on tori" and "Uniqueness of large invariant measures for $\Zk$ actions with Cartan homotopy data". Journal of Modern Dynamics, 2010, 4 (1) : 207-209. doi: 10.3934/jmd.2010.4.207
[20]

Raffaele Folino, Corrado Lattanzio, Corrado Mascia. Motion of interfaces for a damped hyperbolic Allen–Cahn equation. Communications on Pure & Applied Analysis, 2020, 19 (9) : 4507-4543. doi: 10.3934/cpaa.2020205

2020 Impact Factor: 1.327

Tools

Metrics

  • PDF downloads (181)
  • HTML views (298)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences