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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 and 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.

[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.

[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.

[4]

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

[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.

[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.

[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.

[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.

[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.

[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.

[11]

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

[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.

[13]

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

[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.
[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.
[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.
[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.

[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.

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.

[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.

[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.

[4]

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

[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.

[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.

[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.

[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.

[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.

[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.

[11]

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

[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.

[13]

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

[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.
[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.
[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.
[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.

[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.

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