Wong-Zakai approximations and attractors for stochastic wave equations driven by additive noise

Xiaohu Wang; Dingshi Li; Jun Shen

doi:10.3934/dcdsb.2020207

Article Contents

2021, Volume 26, Issue 5: 2829-2855. Doi: 10.3934/dcdsb.2020207

This issue Previous Article Global and exponential attractors for a nonlinear porous elastic system with delay term Next Article Periodic forcing on degenerate Hopf bifurcation

Wong-Zakai approximations and attractors for stochastic wave equations driven by additive noise

1.
Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, China
2.
School of Mathematics, Southwest Jiaotong University, Chengdu, Sichuan 610031, China

^* Corresponding author: Jun Shen, junshen85@163.com
^* Corresponding author: Jun Shen, junshen85@163.com

Received: September 30, 2019

Early access: July 2020

Published: May 2021

This work was supported by NSFC (11831012, 11971394 and 11871049), Fundamental Research Funds for the Central Universities (YJ201646, 2682019LK02), Sichuan Science and Technology Program (2019YJ0215) and International Visiting Program for Excellent Young Scholars of SCU

Abstract Full Text(HTML) Related Papers Cited by

Abstract

In this paper, we study the Wong-Zakai approximations given by a stationary process via Euler approximation of Brownian motion and the associated long term behavior of the stochastic wave equation driven by an additive white noise on unbounded domains. We first prove the existence and uniqueness of tempered pullback attractors for stochastic wave equation and its Wong-Zakai approximation. Then, we show that the attractor of the Wong-Zakai approximate equation converges to the one of the stochastic wave equation driven by additive noise as the correlation time of noise approaches zero.

Keywords:

Mathematics Subject Classification: Primary: 60H15; 35B40; 35B41.

Citation:

Full Text(HTML)

Related Papers

Cited by

References

[1]	L. Arnold, Random Dynamical Systems, Springer, Berlin, 1998. doi: 10.1007/978-3-662-12878-7.
[2]	J. Ball, Global attractors for damped semilinear wave equations, Discrete Contin. Dynam. Syst., 10 (2004), 31-52. doi: 10.3934/dcds.2004.10.31.
[3]	P.W. Bates, K. Lu and B. Wang, Random attractors for stochastic reaction-diffusion equations on unbounded domains, J. Differential Equations, 246 (2009), 845-869. doi: 10.1016/j.jde.2008.05.017.
[4]	W.J. Beyn, B. Gess, P. Lescot and M. Röckner, The global random attractor for a class of stochastic porous media equations, Comm. Partial Differential Equations, 36 (2011), 446-469. doi: 10.1080/03605302.2010.523919.
[5]	Z. Brzezniak and F. Flandoli, Almost sure approximation of Wong-Zakai type for stochastic partial differential equations, Stochastic Process. Appl., 55 (1995), 329-358. doi: 10.1016/0304-4149(94)00037-T.
[6]	H. Crauel, A. Debussche and F. Flandoli, Random attractors, J. Dynam. Differential Equations, 9 (1997), 307-341. doi: 10.1007/BF02219225.
[7]	H. Crauel and F. Flandoli, Attractors for random dynamical systems, Probab. Theory Related Fields, 100 (1994), 365-393. doi: 10.1007/BF01193705.
[8]	J. Duan, K. Lu and B. Schmalfuss, Invariant manifolds for stochastic partial differential equations, Ann. Probab., 31 (2003), 2109-2135. doi: 10.1214/aop/1068646380.
[9]	F. Flandoli, Regularity Theory and Stochastic Flows for Parabolic SPDEs, Stochastics Monographs 9, Gordon and Breach, London, 1995.
[10]	F. Flandoli and B. Schmalfuss, Random attractors for the 3D stochastic Navier-Stokes equation with multiplicative noise, Stoch. Stoch. Rep., 59 (1996), 21-45. doi: 10.1080/17442509608834083.
[11]	H. Fu, X. Liu, J. Liu and X. Wang, On smooth approximation for random attractor of stochastic partial differential equations with multiplicative noise, Stoch. Dyn., 18 (2018), 1850040, 22pp. doi: 10.1142/s0219493718500405.
[12]	M. Garrido-Atienza, K. Lu and B. Schmalfuss, Random dynamical systems for stochastic evolution equations driven by multiplicative fractional Brownian noise with Hurst parameters $H\in(1/3, 1/2]$, SIAM J. Appl. Dyn. Syst., 15 (2016), 625-654. doi: 10.1137/15M1030303.
[13]	Z. Guo, X. Yan, W. Wang and X. Liu, Approximate the dynamical behavior for stochastic systems by Wong-Zakai approaching, J. Math. Anal. Appl., 457 (2018), 214-232. doi: 10.1016/j.jmaa.2017.08.004.
[14]	A. Gu, K. Lu and B. Wang, Asymptotic behavior of random Navier-Stokes equations driven by Wong-Zakai approximations, Discrete Contin. Dynam. Syst., 39 (2019), 185-218. doi: 10.3934/dcds.2019008.
[15]	A. Gu and B. Wang, Asymptotic behavior of random Fitzhugh-Nagumo systems driven by colored noise, Discrete Contin. Dynam. Syst. Ser. B, 23 (2018), 1689-1720. doi: 10.3934/dcdsb.2018072.
[16]	M. Hairer and E. Pardoux, A Wong-Zakai theorem for stochastic PDEs, J. Math. Soc. Japan, 67 (2015), 1551-1604. doi: 10.2969/jmsj/06741551.
[17]	T. Jiang, X. Liu and J. Duan, Approximation for random stable manifolds under multiplicative correlated noise, Discrete Contin. Dynam. Syst. Ser. B, 21 (2016), 3163-3174. doi: 10.3934/dcdsb.2016091.
[18]	D. Kelley and I. Melbourne, Smooth approximation of stochastic differential equations, Ann. Probab., 44 (2016), 479-520. doi: 10.1214/14-AOP979.
[19]	H. Li, Y. You and J. Tu, Random attractors and averaging for non-autonomous stochastic wave equations with nonlinear damping, J. Differential Equations, 258 (2015), 148-190. doi: 10.1016/j.jde.2014.09.007.
[20]	K. Lu and B. Wang, Wong-Zakai approximations and long term behavior of stochastic partial differential equations, J. Dynam. Differential Equations, 31 (2019), 1341-1371. doi: 10.1007/s10884-017-9626-y.
[21]	K. Lu and Q. Wang, Chaotic behavior in differential equations driven by a Brownian motion, J. Differential Equations, 251 (2011), 2853-2895. doi: 10.1016/j.jde.2011.05.032.
[22]	J. Shen and K. Lu, Wong-Zakai approximations and center manifolds of stochastic differential equations, J. Differential Equations, 263 (2017), 4929-4977. doi: 10.1016/j.jde.2017.06.005.
[23]	J. Shen, K. Lu and B. Wang, Convergence and center manifolds for differential equations driven by colored noise, Discrete Contin. Dynam. Syst., 39 (2019), 4797-4840. doi: 10.3934/dcds.2019196.
[24]	J. Shen, K. Lu and W. Zhang, Heteroclinic chaotic behavior driven by a Brownian motion, J. Differential Equations, 255 (2013), 4185-4225. doi: 10.1016/j.jde.2013.08.003.
[25]	B. Schmalfuss, Backward cocycle and attractors of stochastic differential equations, in International Seminar on Applied Mathematics-Nonlinear Dynamics: Attractor Approximation and Global Behavior (eds. V. Reitmann, T. Riedrich and N. Koksch), Technische Universität, Dresden, (1992), 185–192.
[26]	D. W. Stroock and S. R. S. Varadhan, On the support of diffusion processes with applications to the strong maximum principle, Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability, 3 (1972), 333-359.
[27]	H. J. Sussmann, An interpretation of stochastic differential equations as ordinary differential equations which depend on the sample point, Bull. Am. Math. Soc., 83 (1977), 296-298. doi: 10.1090/S0002-9904-1977-14312-7.
[28]	H. J. Sussmann, On the gap between deterministic and stochastic ordinary differential equations, Ann. Probab., 6 (1978), 19-41. doi: 10.1214/aop/1176995608.
[29]	R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4684-0313-8.
[30]	B. Wang, Asymptotic behavior of stochastic wave equations with critical exponents on $\mathbb{R}^3$, Trans. Amer. Math. Soc., 363 (2011), 3639-3663. doi: 10.1090/S0002-9947-2011-05247-5.
[31]	B. Wang, Sufficient and necessary criteria for existence of pullback attractors for non-compact random dynamical systems, J. Differential Equations, 253 (2012), 1544-1583. doi: 10.1016/j.jde.2012.05.015.
[32]	B. Wang, Existence and upper semicontinuity of attractors for stochastic equations with deterministic non-autonomous terms, Stoch. Dyn., 14 (2014), 1450009, 31pp.
[33]	B. Wang, Random attractors for non-autonomous stochastic wave equations with multiplicative noise, Discrete Contin. Dyn. Syst., 34 (2014), 269-300. doi: 10.3934/dcds.2014.34.269.
[34]	B. Wang, Multivalued non-autonomous random dynamical systems for wave equations without uniqueness, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 2011-2051. doi: 10.3934/dcdsb.2017119.
[35]	B. Wang and X. Gao, Random attractors for wave equations on unbounded domains, Discrete Contin. Dyn. Syst., (2009), 800–809. doi: 10.3934/proc.2009.2009.800.
[36]	X. Wang, K. Lu and B. Wang, Wong-Zakai approximations and attractors for stochastic reaction-diffusion equations on unbounded domains, J. Differential Equations, 264 (2018), 378-424. doi: 10.1016/j.jde.2017.09.006.
[37]	Z. Wang and S. Zhou, Random attractor for stochastic non-autonomous damped wave equation with critical exponent, Discrete Contin. Dyn. Syst., 37 (2017), 545-573. doi: 10.3934/dcds.2017022.
[38]	Z. Wang and S. Zhou, Random attractor and random exponential attractor for stochastic non-autonomous damped cubic wave equation with linear multiplicative white noise, Discrete Contin. Dyn. Syst., 38 (2018), 4767-4817. doi: 10.3934/dcds.2018210.
[39]	E. Wong and M. Zakai, On the relation between ordinary and stochastic diferential equations, Int. J. Eng. Sci., 3 (1965), 213-229. doi: 10.1016/0020-7225(65)90045-5.
[40]	E. Wong and M. Zakai, On the convergence of ordinary integrals to stochastic integrals, Ann. Math. Stat., 36 (1965), 1560-1564. doi: 10.1214/aoms/1177699916.
[41]	X. Yan, X. Liu and M. Yang, Random attractors of stochastic partial differential equations: A smooth approximation approach, Stoch. Anal. Appl., 35 (2017), 1007-1029. doi: 10.1080/07362994.2017.1345317.
[42]	S. Zhou, F. Yin and Z. Ou, Random attractor for damped nonlinear wave equations with white noise, SIAM J. Appl. Dyn. Syst., 4 (2005), 883-903. doi: 10.1137/050623097.