Homogenization of nonlinear hyperbolic stochastic partial differential equations with nonlinear damping and forcing. (English) Zbl 1436.60065
Summary: In this paper we deal with the homogenization of stochastic nonlinear hyperbolic equations with periodically oscillating coefficients involving nonlinear damping and forcing driven by a multi-dimensional Wiener process. Using the two-scale convergence method and crucial probabilistic compactness results due to Prokhorov and Skorokhod, we show that the sequence of solutions of the original problem converges in suitable topologies to the solution of a homogenized problem, which is a nonlinear damped stochastic hyperbolic partial differential equation. More importantly, we also prove the convergence of the associated energies and establish a crucial corrector result.
MSC:
| 60H15 | Stochastic partial differential equations (aspects of stochastic analysis) |
| 80M35 | Asymptotic analysis for problems in thermodynamics and heat transfer |
| 80M40 | Homogenization for problems in thermodynamics and heat transfer |
| 35L70 | Second-order nonlinear hyperbolic equations |
Keywords:
homogenization; hyperbolic SPDEs; two-scale convergence; nonlinear damping and forcing; probabilistic compactness resultsReferences:
| [1] | A. Abdulle; M. J. Grote, Finite element heterogeneous multiscale method for the wave equation, SIAM, Multiscale Modeling and Simulation, 9, 766-792 (2011) · Zbl 1298.65145 · doi:10.1137/100800488 |
| [2] | A. Abdulle; W. E.; B. Engquist; E. Vanden-Eijnden, The heterogeneous multiscale method, Acta Numer., 21, 1-87 (2012) · Zbl 1255.65224 · doi:10.1017/S0962492912000025 |
| [3] | A. Abdulle; G. A. Pavliotis, Numerical methods for stochastic partial differential equations with multiple scales, J. Comput. Phys., 231, 2482-2497 (2012) · Zbl 1429.65013 · doi:10.1016/j.jcp.2011.11.039 |
| [4] | G. Allaire, Homogenization and two-scale convergence, SIAM J. Math. Anal., 23, 1482-1518 (1992) · Zbl 0770.35005 · doi:10.1137/0523084 |
| [5] | G. Allaire, Two-scale convergence: A new method in periodic homogenization. Nonlinear partial differential equations and their applicationsapplications, Collge de France Seminar, Vol. XII (Paris, 1991—1993), 1-14, Pitman Res. Notes Math. Ser., 302, Longman Sci. Tech., Harlow, 1994. · Zbl 0822.35011 |
| [6] | N. Bakhvalov and G. Panasenko, Homogenisation: Averaging Processes in Periodic Media. Mathematical Problems in the Mechanics of Composite Materials, Translated from the Russian by D. Lei(tes. Mathematics and its Applications (Soviet Series), 36. Kluwer Academic Publishers Group, Dordrecht, 1989. · Zbl 0692.73012 |
| [7] | A. Bensoussan, Some existence results for stochastic partial differential equations., Stochastic Partial Differential Equations and Applications (Trento, 1990), 37—53, Pitman Res. Notes Math. Ser., 268, Longman Sci. Tech., Harlow, 1992. · Zbl 0793.60067 |
| [8] | A. Bensoussan, Homogenization of a class of stochastic partial differential equations. Composite Media and Homogenization Theory (Trieste, 1990), 47-65, Progr. Nonlinear Differential Equations Appl., 5, Birkhuser Boston, Boston, MA, 1991. · Zbl 0737.60054 |
| [9] | A. Bensoussan, J.-L. Lions and G. Papanicolaou, Asymptotic Analysis for Periodic Structures, Corrected reprint of the 1978 original., AMS Chelsea Publishing, Providence, RI, 2011. · Zbl 1229.35001 |
| [10] | H. Bessaih; Y. Efendiev; F. Maris, Homogenization of Brinkman flows in heterogeneous dynamic media, Stoch. Partial Differ. Equ. Anal. Comput., 3, 479-505 (2015) · Zbl 1342.60103 · doi:10.1007/s40072-015-0058-6 |
| [11] | H. Bessaih; Y. Efendiev; F. Maris, Homogenization of the evolution Stokes equation in a perforated domain with a stochastic Fourier boundary condition, Netw. Heterog. Media, 10, 343-367 (2015) · Zbl 1336.60122 · doi:10.3934/nhm.2015.10.343 |
| [12] | P. Billingsley, Convergence of Probability Measures, John Wiley & Sons, Inc., New York, 1999. · Zbl 0944.60003 |
| [13] | A. Bourgeat; A. Mikelić; S. Wright, Stochastic two-scale convergence in the mean and applications, J. Reine Angew. Math., 456, 19-51 (1994) · Zbl 0808.60056 |
| [14] | A. Bourgeat and A. L. Piatnitski, Averaging of a singular random source term in a diffusion convection equation, SIAM J. Math. Anal., 42 (2010), 2626—2651. · Zbl 1228.35029 |
| [15] | S. Cerrai, Averaging principle for systems of reaction-diffusion equations with polynomial nonlinearities perturbed by multiplicative noise, SIAM J. Math. Anal., 43, 2482-2518 (2011) · Zbl 1239.60055 · doi:10.1137/100806710 |
| [16] | S. Cerrai; M. Freidlin, Averaging principle for a class of stochastic reaction-diffusion equations, Probab. Theory Related Fields, 144, 137-177 (2009) · Zbl 1176.60049 · doi:10.1007/s00440-008-0144-z |
| [17] | D. Cioranescu and P. Donato, An Introduction to Homogenization. Oxford Lecture Series in Mathematics and its Applications, 17. The Clarendon Press, Oxford University Press, New York, 1999. · Zbl 0939.35001 |
| [18] | G. Dal Maso; L. Modica, Nonlinear stochastic homogenization and ergodic theory, J.Rei. Ang. Math. B., 368, 28-42 (1986) · Zbl 0582.60034 |
| [19] | M. A. Diop and E. Pardoux, Averaging of a parabolic partial differential equation with random evolution. Seminar on Stochastic Analysis, Random Fields and Applications IV, Progr. Probab., 58, Birkh user, Basel, (2004), 111-128. · Zbl 1063.35026 |
| [20] | Y. Gorb; F. Maris; B. Vernescu, Homogenization for rigid suspensions with random velocity-dependent interfacial forces, J. Math. Anal. Appl., 420, 632-668 (2014) · Zbl 1302.76192 · doi:10.1016/j.jmaa.2014.05.015 |
| [21] | N. Ichihara, Homogenization problem for stochastic partial differential equations of Zakai type, Stoch. Stoch. Rep., 76, 243-266 (2004) · Zbl 1056.60060 · doi:10.1080/10451120410001714107 |
| [22] | V. V. Jikov, S. M. Kozlov and O. A. Oleinik, Homogenization of Differential Operators and Integral Functionals, Translated from the Russian by G. A. Yosifian. Springer-Verlag, Berlin, 1994. · Zbl 0838.35001 |
| [23] | E. Y. Khruslov, Homogenized models of composite media, Composite Media and Homogenization Theory (Trieste, 1990), 159-182, Progr. Nonlinear Differential Equations Appl., 5, Birkhuser Boston, Boston, MA, 1991. · Zbl 0737.73009 |
| [24] | S. M. Kozlov, The averaging of random operators, Mat. Sb. (N.S.), 109(151) (1979), 188-202, 327. · Zbl 0415.60059 |
| [25] | S. V. Lototsky, Small perturbation of stochastic parabolic equations: A power series analysis, J. Funct. Anal., 193, 94-115 (2002) · Zbl 1009.60052 · doi:10.1006/jfan.2001.3923 |
| [26] | D. Lukkassen; G. Nguetseng; P. Wall, Two-scale convergence, Int. J. Pure Appl. Math., 2, 35-86 (2002) · Zbl 1061.35015 |
| [27] | M. Mohammed; M. Sango, Homogenization of linear hyperbolic stochastic partial differential equation with rapidly oscillating coefficients: The two scale convergence method, Asymptotic Analysis, 91, 341-371 (2015) · Zbl 1329.35048 |
| [28] | M. Mohammed; M. Sango, Homogenization of Neumann problem for hyperbolic stochastic partial differential equations in perforated domains, Asymptotic Analysis, 97, 301-327 (2016) · Zbl 1405.35267 · doi:10.3233/ASY-151355 |
| [29] | M. Mohammed and M. Sango, A Tartar approach to periodic homogenization of linear hyperbolic stochastic partial differential equation, Int. J. Mod. Phys. B, 30 (2016), 1640020, 9 pp. · Zbl 1351.35277 |
| [30] | M. Mohammed, Homogenization of nonlinear hyperbolic stochastic equation via Tartar’s method, J. Hyper. Differential Equations, 14, 323-340 (2017) · Zbl 1391.35039 · doi:10.1142/S0219891617500096 |
| [31] | F. Murat; L. Tartar, H-convergence in Topics in the mathematical Modelling of composite Materials. ed. A. Cherkaev and Kohn, Birkhauser. Boston, 31, 21-43 (1997) · Zbl 0920.35019 |
| [32] | G. Nguetseng, A general convergence result for a functional related to the theory of homogenization, SIAM J. Math. Anal., 20 (1989), 608-623. · Zbl 0688.35007 |
| [33] | G. Nguetseng; M. Sango; J. L. Woukeng, Reiterated ergodic algebras and applications, Comm. Math. Phys., 300, 835-876 (2010) · Zbl 1228.46049 · doi:10.1007/s00220-010-1127-3 |
| [34] | O. A. Oleinik, A. S. Shamaev and G. A. Yosifian, Mathematical Problems in elasticity and Homogenization, Studies in Mathematics and its Applications, 26. North-Holland Publishing Co., Amsterdam, 1992. · Zbl 0768.73003 |
| [35] | M. Ondreját, Uniqueness for stochastic evolution equations in Banach spaces, Dissertationes Mathematicae, 426, 1-63 (2004) · Zbl 1053.60071 · doi:10.4064/dm426-0-1 |
| [36] | A. Pankov, G-Convergence and Homogenization of Nonlinear Partial Differential Operators, Mathematics and its Applications, Kluwer Academic Publishers, Dordrecht, 1997. · Zbl 0883.35001 |
| [37] | G. Papanicolaou and S. R. S. Varadhan, Boundary value problems with rapidly oscillating random coefficients, Random Fields, Vol. I, II (Esztergom, 1979), 835-873, Colloq. Math. Soc. Janos Bolyai, 27, North-Holland, Amsterdam-New York, 1981. · Zbl 0499.60059 |
| [38] | E. Pardoux, Équations aux dérivées Partielles Stochastiques Non Linéaires Monotones, Thèse, Université Paris XI, 1975. · Zbl 0363.60041 |
| [39] | B. L. Rozovskiĭ, Stochastic Evolution Systems. Linear Theory and Applications to Nonlinear Filtering, Translated from the Russian by A. Yarkho. Mathematics and its Applications (Soviet Series), 35. Kluwer Academic Publishers Group, Dordrecht, 1990. xviii+315 pp · Zbl 0724.60070 |
| [40] | E. Sanchez-Palencia, Non-Homogeneous Media and Vibration Theory (Lecture Notes in Physics), Springer, 1980. · Zbl 0432.70002 |
| [41] | E. Sanchez-Palencia and A. Zaoui, Homogenization Techniques for Composite Media, Lecture Notes in Physics, 272. Springer-Verlag, Berlin, 1987. · Zbl 0619.00027 |
| [42] | M. Sango, Splitting-up scheme for nonlinear stochastic hyperbolic equations, Forum Math., 25, 931-965 (2013) · Zbl 1305.60054 · doi:10.1515/form.2011.138 |
| [43] | M. Sango, Homogenization of stochastic semilinear parabolic equations with non-Lipschitz forcings in domains with fine grained boundaries, Commun. Math. Sci., 12, 345-382 (2014) · Zbl 1308.60080 · doi:10.4310/CMS.2014.v12.n2.a7 |
| [44] | M. Sango, Asymptotic behavior of a stochastic evolution problem in a varying domain, Stochastic Anal. Appl., 20, 1331-1358 (2002) · Zbl 1069.60055 · doi:10.1081/SAP-120015835 |
| [45] | J. Simon, Compact sets in the space L_p(0, T; B), Ann. Mat. Pura Appl., IV. Ser., 146 (1987), 65-96. · Zbl 0629.46031 |
| [46] | I. V. Skrypnik, Methods for Analysis of Nonlinear Elliptic Boundary Value Problems, Nauka, Moscow, 1990. English translation in: Translations of Mathematical Monographs, AMS, Providence, 1994. · Zbl 0743.35026 |
| [47] | E. P. Souganidis, Stochastic homogenization of Hamilton-Jacobi equations and some applications, Asymptotic Analysis. IOS Press, 20, 1-11 (1999) · Zbl 0935.35008 |
| [48] | N. Svanstedt, Multiscale stochastic homogenization of monotone operators, Netw. Heterog. Media, 2, 181-192 (2007) · Zbl 1140.35341 · doi:10.3934/nhm.2007.2.181 |
| [49] | L. Tartar, Quelques remarques sur l’homogénésation, in Functional Analysis and Numerical Analysis, Proc. Japan-France Siminar 1976, ed. H. Fujitaa, Japanese Society for the Promotion of Science, (1977), 468-486. |
| [50] | L. Tartar, The General Theory of Homogenization, A personalized introduction. Lecture Notes of the Unione Matematica Italiana, 7. Springer-Verlag, Berlin; UMI, Bologna, 2009. · Zbl 1188.35004 |
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