Approximate dynamics of a class of stochastic wave equations with white noise

Guanggan Chen; Qin Li; Yunyun Wei

doi:10.3934/dcdsb.2021033

Article Contents

2022, Volume 27, Issue 1: 73-101. Doi: 10.3934/dcdsb.2021033

This issue Previous Article Asymptotic behaviour of the solutions for a weakly damped anisotropic sixth-order Schrödinger type equation in

$ \mathbb{R}^2 $

Next Article Modeling, approximation, and time optimal temperature control for binder removal from ceramics

Approximate dynamics of a class of stochastic wave equations with white noise

1.
School of Mathematical Science, and V.C. & V.R. Key Lab, Sichuan Normal University, Chengdu, 610068, China
2.
College of Management Science, Chengdu University of Technology, Chengdu, 610059, China

^* Corresponding author: Guanggan Chen
^* Corresponding author: Guanggan Chen

Received: August 2020

Revised: November 2020

Early access: February 2021

Published: January 2022

The first author is supported by the National Science Foundation of China (Grants No. 11571245).

Abstract / Introduction Full Text(HTML) Related Papers Cited by

Abstract

This work is concerned with a stochastic wave equation driven by a white noise. Borrowing from the invariant random cone and employing the backward solvability argument, this wave system is approximated by a finite dimensional wave equation with a white noise. Especially, the finite dimension is explicit, accurate and determined by the coefficient of this wave system; and further originating from an Ornstein-Uhlenbek process and applying Banach space norm estimation, this wave system is approximated by a finite dimensional wave equation with a smooth colored noise.

Keywords:

Mathematics Subject Classification: Primary: 37L55, 35L05, 37D10; Secondary: 58J65.

Citation:

Full Text(HTML)

Related Papers

Cited by

References

[1]	P. Acquistapace and B. Terreni, An approach to Itô linear equations in Hilbert spaces by approximation of white noise with coloured noise, Stoch. Anal. Appl., 2 (1984), 131-186. doi: 10.1080/07362998408809031.
[2]	L. Arnold, Random Dynamical Systems, Springer-Verlag, Heidelberg, 1998. doi: 10.1007/978-3-662-12878-7.
[3]	S. Cerrai and M. I. Freidlin, On the Smoluchowski-Kramers approximation for a system with an infinite number of degrees of freedom, Probab. Theory Rel., 135 (2006), 363-394. doi: 10.1007/s00440-005-0465-0.
[4]	G. Chen, J. Duan and J. Zhang, Approximating dynamics of a singularly perturbed stochastic wave equation with a random dynamical boundary condition, SIAM. J. Math. Anal., 45 (2013), 2790-2814. doi: 10.1137/12088968X.
[5]	P. L. Chow, Asymptotics of solutions to semilinear stochastic wave equations, Ann. Appl. Probab., 16 (2006), 757-780. doi: 10.1214/105051606000000141.
[6]	P. L. Chow, Stochastic wave equations with polynomial nonlinearity, Ann. Appl. Probab., 12 (2002), 361-381. doi: 10.1214/aoap/1015961168.
[7]	G. da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge University Press, Cambridge, UK, 1992. doi: 10.1017/CBO9780511666223.
[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]	J. Duan, K. Lu and B. Schmalfuß, Smooth stable and unstable manifolds for stochastic evolutionary equations, J. Dyn. Differ. Equ., 16 (2004), 949-972. doi: 10.1007/s10884-004-7830-z.
[10]	X. Fan and Y. Wang, Fractal dimension of attractors for a stochastic wave equation with nonlinear damping and white noise, Stoch. Anal. Appl., 25 (2007), 381-396. doi: 10.1080/07362990601139602.
[11]	J. Garcia-Ojalvo and J. M. Sancho, Noise in Spatially Extended Systems, Springer-Verlag, Berlin, 1999. doi: 10.1007/978-1-4612-1536-3.
[12]	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.
[13]	M. Hairer and E. Pardoux, A Wong-Zakai theorem for stochastic PDEs, J. Math. Soc. Jpn., 67 (2015), 1551-1604. doi: 10.2969/jmsj/06741551.
[14]	J. Hale and G. Raugel, Upper semicontinuity of the attractor for a singularly perturbed hyperbolic equation, J. Differ. Equations, 73 (1988), 197-214. doi: 10.1016/0022-0396(88)90104-0.
[15]	W. Horsthemke and R. Lefever, Noise-induced Transitions: Theory and Applications in Physics, Chemistry, and Biology, Springer Series in Synergetics, Berlin, Springer, 1984. doi: 10.1007/3-540-36852-3.
[16]	N. Ikeda, S. Nakao and Y. Yamato, A class of approximations of Brownian motion, Publ. Res. I. Math. Sci., 13 (1977), 285-300. doi: 10.2977/prims/1195190109.
[17]	T. Jiang, X. Liu and J. Duan, A Wong-Zakai approximation for random invariant manifolds, J. Math. Phys., 58 (2017), 122701. doi: 10.1063/1.5017932.
[18]	D. Kelley and I. Melbourne, Smooth approximation of stochastic differential equations, Ann. Probab., 44 (2016), 479-520. doi: 10.1214/14-AOP979.
[19]	F. Konecny, On Wong-Zakai approximation of stochastic differential equations, J. Multivariate Anal., 13 (1983), 605-611. doi: 10.1016/0047-259X(83)90043-X.
[20]	T. Kurtz and P. Protter, Weak limit theorems for stochastic integrals and stochastic differential equations, Ann. Probab., 19 (1991), 1035-1070. doi: 10.1214/aop/1176990334.
[21]	K. Lu and B. Schmalfuß, Invariant manifolds for stochastic wave equations, J. Differ. Equations, 236 (2007), 460-492. doi: 10.1016/j.jde.2006.09.024.
[22]	Y. Lv and W. Wang, Limiting dynamics for stochastic wave equations, J. Differ. Equations, 244 (2008), 1-23. doi: 10.1016/j.jde.2007.10.009.
[23]	Y. Lv, W. Wang and A. J. Roberts, Approximation of the random inertial manifold of singularly perturbed stochastic wave equations, Stoch. Dynam., 14 (2014), 1350018. doi: 10.1142/S0219493713500184.
[24]	X. Mora, Finite-dimensional attracting invariant manifolds for damped semilinear wave equations, Contributions to Nonlinear Partial Differential Equations, 2 (1985), 172-183.
[25]	C. Mueller, Long time existence for the wave equation with a noise term, Ann. Probab., 25 (1997), 133-151. doi: 10.1214/aop/1024404282.
[26]	S. Nakao, On weak convergence of sequences of continuous local martingale, Ann. I. H. Poincare B, 22 (1986), 371-380.
[27]	E. Pardoux and A. Piatnitski, Homogenization of a singular random one-dimensional PDE with time-varying coefficients, Ann. Probab., 40 (2012), 1316-1356. doi: 10.1214/11-AOP650.
[28]	P. Protter, Approximations of solutions of stochastic differential equations driven by semimartingales, Ann. Probab., 13 (1985), 716-743. doi: 10.1214/aop/1176992905.
[29]	M. Reed and B. Simon, Methods of Modern Mathematical Physics Ⅱ, Academic Press, New York, 1975.
[30]	J. Shen and K. Lu, Wong-Zakai approximations and center manifolds of stochastic differential equations, J. Differ. Equations, 263 (2017), 4929-4977. doi: 10.1016/j.jde.2017.06.005.
[31]	G. Tessitore and J. Zabczyk, Wong-Zakai approximation of stochastic evolution equations, J. Evol. Equ., 6 (2006), 621-655. doi: 10.1007/s00028-006-0280-9.
[32]	X. Wang, K. Lu and B. Wang, Wong-Zakai approximations and attractors for stochastic reaction-diffusion equations on unbounded domains, J. Differ. Equations, 264 (2018), 378-424. doi: 10.1016/j.jde.2017.09.006.
[33]	W. Wang and J. Duan, A dynamical approximation for stochastic partial differential equations, J. Math. Phys., 48 (2007), 102701. doi: 10.1063/1.2800164.
[34]	G. Whitham, Linear and Nonlinear Waves, Wiley, New York, 1974.
[35]	E. Wong and M. Zakai, On the relation between ordinary and stochastic differential equations, Int. J. Eng. Sci., 3 (1965), 213-229. doi: 10.1016/0020-7225(65)90045-5.
[36]	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.
[37]	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.