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Strong convergence in averaging principle for stochastic hyperbolic-parabolic equations with two time-scales. (English) Zbl 1322.60111
Summary: This article deals with averaging principles for stochastic hyperbolic-parabolic equations with slow and fast time-scales. Under suitable conditions, the existence of an averaging equation eliminating the fast variable for this coupled system is proved. As a consequence, effective dynamics for the slow variable, which take the form of a stochastic wave equation, are derived. Also, the rate of strong convergence for the slow component towards the solution of the averaging equation is obtained as a byproduct.

MSC:
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
60F15 Strong limit theorems
70K70 Systems with slow and fast motions for nonlinear problems in mechanics
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[1] Bogoliubov, N. N.; Mitropolsky, Y. A., Asymptotic methods in the theory of non-linear oscillations, (1961), Gordon & Breach Science Publishers New York
[2] Bréhier, C. E., Strong and weak orders in averging for SPDEs, Stochastic Process. Appl., 122, 2553-2593, (2012) · Zbl 1266.60112
[3] Cardetti, F.; Choi, Y. S., A parabolic-hyperbolic system modelling a moving cell, Electron. J. Differential Equations, 95, 1-11, (2009) · Zbl 1178.35389
[4] Cerrai, S., A khasminkii type averaging principle for stochastic reaction-diffusion equations, Ann. Appl. Probab., 19, 3, 899-948, (2009) · Zbl 1191.60076
[5] Cerrai, S., Averaging principle for systems of reaction-diffusion equations with polynomial nonlinearities perturbed by multiplicative type noise, SIAM J. Math. Anal., 43, 6, 2482-2518, (2011) · Zbl 1239.60055
[6] Cerrai, S.; Freidlin, M. I., Averaging principle for a class of stochastic reaction-diffusion equations, Probab. Theory Related Fields, 144, 137-177, (2009) · Zbl 1176.60049
[7] Choi, Y. S.; Miller, Craig, Global existence of solutions to a coupled parabolic-hyperbolic system with moving boundary, Proc. Amer. Math. Soc., 139, 3257-3270, (2011) · Zbl 1232.35197
[8] Chow, P. L., Thermoelastic wave propagation in a random medium and some related problems, Internat. J. Engrg. Sci., 11, 253-971, (1973) · Zbl 0263.73007
[9] Chow, P. L., Stochastic partial differential equations, (2007), Chapman & Hall/CRC New York · Zbl 0365.76050
[10] Da Prato, G.; Zabczyk, J., Stochastic equations in infinite dimensions, (1992), Cambridge University Press Cambridge · Zbl 0761.60052
[11] Freidlin, M. I.; Wentzell, A. D., Random perturbation of dynamical systems, (1998), Springer-Verlag New York · Zbl 0922.60006
[12] Freidlin, M. I.; Wentzell, A. D., Long-time behavior of weakly coupled oscillators, J. Stat. Phys, 123, 1311-1337, (2006) · Zbl 1119.34044
[13] Fu, H.; Wan, L.; Wang, Y.; Liu, J., Strong convergence rate in averaging principle for stochastic fitzhug-Nagumo system with two time-scales, J. Math. Anal. Appl., 416, 609-628, (2014) · Zbl 1325.60107
[14] Givon, D.; Kevrekidis, I. G.; Kupferman, R., Strong convergence of projective integration schemes for singular perturbed stochastic differential systems, Commun. Math. Sci., 4, 707-729, (2006) · Zbl 1115.60036
[15] Khasminskii, R. Z., On the principle of averaging the Itô stochastic differential equations, Kibernetika, 4, 260-279, (1968), (in Russian) · Zbl 0231.60045
[16] Khasminskii, R. Z.; Yin, G., Limit behavior of two-time-scale diffusions revisited, J. Differential Equations, 212, 1, 85-113, (2005) · Zbl 1112.35014
[17] Kifer, Y., Some recent advance in averaging, (Modern Dynamical Systems and Applications, (2004), Cambridge University Press Cambridge, UK), 385-403 · Zbl 1147.37318
[18] Kifer, Y., Diffusion approximation for slow motion in fully coupled averaging, Probab. Theory Related Fields, 129, 157-181, (2004) · Zbl 1069.34070
[19] Kifer, Y., Another proof of the averaging principle for fully coupled dynamical systems with hyperbolic fast motions, Discrete Contin. Dyn. Syst., 13, 1187-1201, (2005) · Zbl 1103.37018
[20] Leung, A. W., Asymptotically stable invariant manifold for coupled nonlinear parabolic-hyperbolic partial differential equations, J. Differential Equations, 187, 184-200, (2003) · Zbl 1022.35006
[21] Leung, A. W., Stable invariant manifolds for coupled Navier-Stokes and second-order wave systems, Asymptot. Anal., 43, 339-357, (2005) · Zbl 1083.35010
[22] Liu, D., Strong convergence of principle of averaging for multiscale stochastic dynamical systems, Commun. Math. Sci., 8, 4, 999-1020, (2010) · Zbl 1208.60057
[23] Øksendal, B., Stochastic differential equations, (2003), Springer Berlin
[24] Rivera, J. E.M.; Racke, R., Smoothing properties, decay and global existence of solutions to nonlinear coupled systems of thermoelasticity type, SIAM J. Math. Anal., 26, 1547-1563, (1995) · Zbl 0842.35039
[25] Veretennikov, A. Y., On the averaging principle for systems of stochastic differential equations, Math. USSR-Sb., 69, 271-284, (1991) · Zbl 0724.60069
[26] Veretennikov, A. Yu., On large deviations in the averaging principle for SDEs with full dependence, Ann. Probab., 27, 284-296, (1999) · Zbl 0939.60012
[27] Wainrib, G., Double averaging principle for periodically forced slow-fast stochastic systems, Electron. Commun. Probab., 18, 51, 1-12, (2013) · Zbl 1297.70015
[28] Wu, S. H.; Chen, H.; Li, W. X., The local and global existence of the solutions of hyperbolic-parabolic system modeling biological phenomena, Acta Math. Sci., 28B, 1, 101-116, (2008) · Zbl 1150.35016
[29] Xu, Y.; Duan, J.; Xu, W., An averaging principle for stochastic dynamical systems with Lévy noise, Physica D, 240, 17, 1395-1401, (2011) · Zbl 1236.60060
[30] Xu, Y.; Guo, R.; Liu, D.; Zhang, H.; Duan, J., Stochastic averaging principle for dynamical systems with fractional Brownian motion, Discrete Contin. Dyn. Syst. Ser. B, 19, 4, 1197-1212, (2014) · Zbl 1314.60122
[31] Xu, Y.; Guo, R.; Xu, W., A limit theorem for the solutions of slow-fast systems with fractional Brownian motion, Theor. Appl. Mech. Lett., 4, (2014), ID 3-013003
[32] Xu, Y.; Pei, B.; Li, Y., An averaging principle for stochastic differential delay equations with fractional Brownian motion, Abstr. Appl. Anal., (2014), ID 479195
[33] Xu, Y.; Pei, B.; Li, Y., Approximation properties for solutions to non-Lipschitz stochastic differential equations with Lévy noise, Math. Methods Appl. Sci., 4, 1, (2014)
[34] Zhang, X.; Zuazua, E., Long-time behavior of a coupled heat-wave system arising in fluid-structure interaction, Arch. Ration. Mech. Anal., 184, 1, 49-120, (2007) · Zbl 1178.74075
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