Spatial Homogenization of Stochastic Wave Equation with Large Interaction

Jiang, Yongxin ; Wang, Wei ; Feng, Zhaosheng

Canadian mathematical bulletin, Tome 59 (2016) no. 3, pp. 542-552

Résumé

A dynamical approximation of a stochastic wave equation with large interaction is derived. A random invariant manifold is discussed. By a key linear transformation, the random invariant manifold is shown to be close to the random invariant manifold of a second-order stochastic ordinary differential equation.

Détail

DOI : 10.4153/CMB-2015-083-4

Mots-clés : 60F10, 60H15, 35Q55, stochasticwave equation, homogeneous system, approximation, random invariant manifold, Neumann boundary condition

Bibliographie

[1] [1] Arnold, L., Random dynamical systems. Springer Monographs in Mather Springer-Verlag, Berlin, 1998. Google Scholar | DOI

[2] [2] Carvalho, A. N. and Hale, J. K., Large diffusion with dispersion. Nonlineano. 12, 1139–1151. Google Scholar | DOI

[3] [3] Cerrai, S. and Freidlin, M., On the Smoluchowski-Kramers approximation infinite number of degrees of freedom.Probab. Theory Related Fields 135 363–394. Google Scholar | DOI

[4] [4] Cerrai, S. and Freidlin, M., Smoluchowski-Kramers approximation for a general class ofSPDEs. J. Evol. Equ. 6(2006), no. 4, 657–689. Google Scholar | DOI

[5] [5] Duan, J., Lu, K., and Schmalfuss, B., Invariant manifolds for stochastic partial differential equations. Ann. of Prob. 31(2003), no. 4, 2109–2135. Google Scholar | DOI

[6] [6] Duan, J., Lu, K., and Schmalfuss, B., Smooths stable and unstable manifolds for stochastic evolutionary equations. J. Dynam. Differential Equations 16(2004), no. 4, 949–972. Google Scholar | DOI

[7] [7] Liu, Z., Stochastic inertial manifolds for damped wave equations. Stoch.Dyn. 10(2010), no. 2, 211–230. Google Scholar | DOI

[8] [8] Lu, K. and Schmalfuss, B., Invariant manifolds for stochastic wave equations. J. Differential Equations 236(2007), no. 2, 460–492. Google Scholar | DOI

[9] [9] Lv, Y. and Roberts, A. J., Averaging approximation to singularly perturbed stochastic wave equation. J. Math. Phys. 53(2012), no. 6, 062702, 11. Google Scholar | DOI

[10] [10] Lv, Y. and Wang, W., Limit dynamics for stochastic wave equations. J. Differential Equations 244(2008), 1–23. Google Scholar | DOI

[11] [11] Lv, Y., Wang, W., and Roberts, A., Approximation of the random inertial manifold of singularly perturbed stochastic wave equations. Stoch.Dyn. 14(2014), no. 2,1350018, 21pp. Google Scholar | DOI

[12] [12] Mora, X., Finite-dimensional attracting invariant manifolds for damped semilinear wave equations. In: Contributions to nonlinear partial differential equations. II. (Paris,1985), Pitman Res. Notes Math. Ser., 155, Longman Sci. Tech., Harlow, 1987, pp. 172–183. Google Scholar

[13] [13] Pazy, A., Semigroups of linear operators and applications to partial differential equations. Applied Mathematical Sciences, 44, Springer-Verlag, New York, 1983. Google Scholar | DOI

[14] [14] Da Prato, G. and Zabczyk, J., Stochastic equations in infinite dimensions. Encyclopedia of Mathematics and its Applications, 44, Cambridge University Press, Cambridge, 1992. Google Scholar | DOI

[15] [15] Qin, W.-X., Spatial homogeneity and invariant manifolds for damped hyperbolic equations. Z. Angew.Math. Phys. 52(2001), 990–1016. Google Scholar | DOI

[16] [16] Roberts, A. J., Resolving the multitude of microscale interactions accurately models stochastic partial differential equations. LMS J. Comput. and Math. 9(2006), 193–221. Google Scholar | DOI

[17] [17] Roberts, A. J., Normal form transforms separate slow and fast modes in stochastic dynamical systems. 387(2008), no. 1,12-38. Google Scholar | DOI

[18] [18] Wang, W. and Lv, Y., Limit behavior of nonlinear stochastic wave equations with singular perturbation. Discrete Contin.Dyn.Syst. Ser. B 13(2010), no. 1,175-193. Google Scholar | DOI