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  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]

Alessandro Gondolo, Fernando Guevara Vasquez. Characterization and synthesis of Rayleigh damped elastodynamic networks. Networks & Heterogeneous Media, 2014, 9 (2) : 299-314. doi: 10.3934/nhm.2014.9.299
[2]

Charlene Kalle, Niels Langeveld, Marta Maggioni, Sara Munday. Matching for a family of infinite measure continued fraction transformations. Discrete & Continuous Dynamical Systems - A, 2020, 40 (11) : 6309-6330. doi: 10.3934/dcds.2020281
[3]

Y. Latushkin, B. Layton. The optimal gap condition for invariant manifolds. Discrete & Continuous Dynamical Systems - A, 1999, 5 (2) : 233-268. doi: 10.3934/dcds.1999.5.233
[4]

Christopher Bose, Rua Murray. Minimum 'energy' approximations of invariant measures for nonsingular transformations. Discrete & Continuous Dynamical Systems - A, 2006, 14 (3) : 597-615. doi: 10.3934/dcds.2006.14.597
[5]

Marian Gidea, Rafael de la Llave, Tere M. Seara. A general mechanism of instability in Hamiltonian systems: Skipping along a normally hyperbolic invariant manifold. Discrete & Continuous Dynamical Systems - A, 2020, 40 (12) : 6795-6813. doi: 10.3934/dcds.2020166
[6]

Manfred Einsiedler, Elon Lindenstrauss. On measures invariant under diagonalizable actions: the Rank-One case and the general Low-Entropy method. Journal of Modern Dynamics, 2008, 2 (1) : 83-128. doi: 10.3934/jmd.2008.2.83
[7]

Vladimir Georgiev, Sandra Lucente. Focusing nlkg equation with singular potential. Communications on Pure & Applied Analysis, 2018, 17 (4) : 1387-1406. doi: 10.3934/cpaa.2018068
[8]

Daoyin He, Ingo Witt, Huicheng Yin. On the strauss index of semilinear tricomi equation. Communications on Pure & Applied Analysis, 2020, 19 (10) : 4817-4838. doi: 10.3934/cpaa.2020213
[9]

Diana Keller. Optimal control of a linear stochastic Schrödinger equation. Conference Publications, 2013, 2013 (special) : 437-446. doi: 10.3934/proc.2013.2013.437
[10]

Simone Cacace, Maurizio Falcone. A dynamic domain decomposition for the eikonal-diffusion equation. Discrete & Continuous Dynamical Systems - S, 2016, 9 (1) : 109-123. doi: 10.3934/dcdss.2016.9.109
[11]

Naeem M. H. Alkoumi, Pedro J. Torres. Estimates on the number of limit cycles of a generalized Abel equation. Discrete & Continuous Dynamical Systems - A, 2011, 31 (1) : 25-34. doi: 10.3934/dcds.2011.31.25
[12]

Jumpei Inoue, Kousuke Kuto. On the unboundedness of the ratio of species and resources for the diffusive logistic equation. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2441-2450. doi: 10.3934/dcdsb.2020186
[13]

Wentao Huang, Jianlin Xiang. Soliton solutions for a quasilinear Schrödinger equation with critical exponent. Communications on Pure & Applied Analysis, 2016, 15 (4) : 1309-1333. doi: 10.3934/cpaa.2016.15.1309
[14]

Kin Ming Hui, Soojung Kim. Asymptotic large time behavior of singular solutions of the fast diffusion equation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (11) : 5943-5977. doi: 10.3934/dcds.2017258
[15]

Thierry Cazenave, Ivan Naumkin. Local smooth solutions of the nonlinear Klein-gordon equation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (5) : 1649-1672. doi: 10.3934/dcdss.2020448
[16]

Jiaquan Liu, Xiangqing Liu, Zhi-Qiang Wang. Sign-changing solutions for a parameter-dependent quasilinear equation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (5) : 1779-1799. doi: 10.3934/dcdss.2020454
[17]

Murat Uzunca, Ayşe Sarıaydın-Filibelioǧlu. Adaptive discontinuous galerkin finite elements for advective Allen-Cahn equation. Numerical Algebra, Control & Optimization, 2021, 11 (2) : 269-281. doi: 10.3934/naco.2020025
[18]

Nhu N. Nguyen, George Yin. Stochastic partial differential equation models for spatially dependent predator-prey equations. Discrete & Continuous Dynamical Systems - B, 2020, 25 (1) : 117-139. doi: 10.3934/dcdsb.2019175
[19]

Ademir Fernando Pazoto, Lionel Rosier. Uniform stabilization in weighted Sobolev spaces for the KdV equation posed on the half-line. Discrete & Continuous Dynamical Systems - B, 2010, 14 (4) : 1511-1535. doi: 10.3934/dcdsb.2010.14.1511
[20]

Abdulrazzaq T. Abed, Azzam S. Y. Aladool. Applying particle swarm optimization based on Padé approximant to solve ordinary differential equation. Numerical Algebra, Control & Optimization, 2021  doi: 10.3934/naco.2021008

2019 Impact Factor: 1.27

Tools

Metrics

  • PDF downloads (115)
  • HTML views (297)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences