Asymptotic behavior of random Navier-Stokes equations driven by Wong-Zakai approximations (original) (raw)

| | P. Acquistapace and B. Terreni , An approach to Itö linear equations in Hilbert spaces by approximation of white noise with coloured noise, Stochastic Anal. Appl., 2 (1984) , 131-186. doi: 10.1080/07362998408809031. | | ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ | | | L. Arnold, Random Dynamical Systems, Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-662-12878-7. | | | V. Bally , A. Millet and M. Sanz-Solé , Approximation and support theorem in Hölder norm for parabolic stochastic partial differential equations, Ann. Probab., 23 (1995) , 178-222. doi: 10.1214/aop/1176988383. | | | P. W. Bates , H. Lisei and K. Lu , Attractors for stochastic lattice dynamical systems, Stoch. Dyn., 6 (2006) , 1-21. doi: 10.1142/S0219493706001621. | | | 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. | | | Z. Brzeźniak , M. Capiński and F. Flandoli , A convergence result for stochastic partial differential equations, Stochastics, 24 (1988) , 423-445. doi: 10.1080/17442508808833526. | | | Z. Brzeźniak 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. | | | M. Capiński and N. J. Cutland , Existence of global stochastic flow and attractors for NavierStokes equations, Probab. Theory Relat. Fields, 115 (1999) , 121-151. doi: 10.1007/s004400050238. | | | T. Caraballo , M. Garrido-Atienza , B. Schmalfuss and J. Valero , Non-autonomous and random attractors for delay random semilinear equations without uniqueness, Discrete Contin. Dyn. Syst., 21 (2008) , 415-443. doi: 10.3934/dcds.2008.21.415. | | | T. Caraballo , J. Real and I. Chueshov , Pullback attractors for stochastic heat equations in materials with memory, Discrete Contin. Dyn. Syst. Ser. B, 9 (2008) , 525-539. doi: 10.3934/dcdsb.2008.9.525. | | | T. Caraballo and J. Langa , On the upper semicontinuity of cocycle attractors for non-autonomous and random dynamical systems, Dynamics of Continuous, Discrete and Impulsive Systems Series A: Mathematical Analysis, 10 (2003) , 491-513. | | | T. Caraballo , M. Garrido-Atienza , B. Schmalfuss and J. Valero , Asymptotic behaviour of a stochastic semilinear dissipative functional equation without uniqueness of solutions, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010) , 439-455. doi: 10.3934/dcdsb.2010.14.439. | | | T. Caraballo , M. Garrido-Atienza and T. Taniguchi , The existence and exponential behavior of solutions to stochastic delay evolution equations with a fractional Brownian motion, Nonlinear Anal., 74 (2011) , 3671-3684. doi: 10.1016/j.na.2011.02.047. | | | T. Caraballo , J. Langa , V. S. Melnik and J. Valero , Pullback attractors for nonautonomous and stochastic multivalued dynamical systems, Set-Valued Analysis, 11 (2003) , 153-201. doi: 10.1023/A:1022902802385. | | | I. Chueshov and M. Scheutzow , On the structure of attractors and invariant measures for a class of monotone random systems, Dynamical Systems, 19 (2004) , 127-144. doi: 10.1080/1468936042000207792. | | | I. Chueshov, Monotone Random Systems - Theory and Applications, Lecture Notes in Mathematics, 1779, Springer, Berlin, 2002. doi: 10.1007/b83277. | | | H. Crauel , A. Debussche and F. Flandoli , Random attractors, J. Dyn. Diff. Equat., 9 (1997) , 307-341. doi: 10.1007/BF02219225. | | | H. Crauel and F. Flandoli , Attractors for random dynamical systems, Probab. Th. Re. Fields, 100 (1994) , 365-393. doi: 10.1007/BF01193705. | | | A. Deya , M. Jolis and L. Quer-Sardanyons , The Stratonovich heat equation: A continuity result and weak approximations, Electron. J. Probab., 18 (2013) , 1-34. doi: 10.1214/EJP.v18-2004. | | | J. Duan and B. Schmalfuss , The 3D quasigeostrophic fluid dynamics under random forcing on boundary, Comm. Math. Sci., 1 (2003) , 133-151. doi: 10.4310/CMS.2003.v1.n1.a9. | | | 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. | | | F. Flandoli, Regularity Theory and Stochastic Flow for Parabolic SPDEs, Stochastics Monographs Vol. 9, Gordon and Breach Science Publishers SA, Singapore, 1995. | | | A. Ganguly, Wong-Zakai type convergence in infinite dimensions, Electron. J. Probab., 18 (2013), 34 pp. doi: 10.1214/EJP.v18-2650. | | | M. Garrido-Atienza and B. Schmalfuss , Ergodicity of the infinite dimensional fractional Brownian motion, J. Dynam. Differential Equations, 23 (2011) , 671-681. doi: 10.1007/s10884-011-9222-5. | | | M. Garrido-Atienza , A. Ogrowsky and B. Schmalfuss , Random differential equations with random delays, Stoch. Dyn., 11 (2011) , 369-388. doi: 10.1142/S0219493711003358. | | | M. Garrido-Atienza , B. Maslowski and B. Schmalfuss , Random attractors for stochastic equations driven by a fractional Brownian motion, International J. Bifurcation and Chaos, 20 (2010) , 2761-2782. doi: 10.1142/S0218127410027349. | | | B. Gess , W. Liu and M. Rockner , Random attractors for a class of stochastic partial differential equations driven by general additive noise, J. Differential Equations, 251 (2011) , 1225-1253. doi: 10.1016/j.jde.2011.02.013. | | | W. Grecksch and B. Schmalfuss , Approximation of the stochastic Navier-Stokes equation, Mat. Apl. Comput., 15 (1996) , 227-239. | | | I. Gyöngy and A. Shmatkov , Rate of convergence of Wong-Zakai approximations for stochastic partial differential equations, Applied Mathematics and Optimization, 54 (2006) , 315-341. doi: 10.1007/s00245-006-0873-2. | | | I. Gyöngy , On the approximation of stochastic partial differential equations, Ⅰ, Stochastics, 25 (1988) , 59-85. doi: 10.1080/17442508808833533. | | | I. Gyöngy , On the approximation of stochastic partial differential equations, Ⅱ, Stochastics, 26 (1989) , 129-164. doi: 10.1080/17442508908833554. | | | M. Hairer and E. Pardoux , A Wong-Zakai theorem for stochastic PDE, J. Math. Soc. Japan., 67 (2015) , 1551-1604. doi: 10.2969/jmsj/06741551. | | | J. Huang and W. Shen , Pullback attractors for nonautonomous and random parabolic equations on non-smooth domains, Discrete Contin. Dyn. Syst., 24 (2009) , 855-882. doi: 10.3934/dcds.2009.24.855. | | | N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, Kodansha, Ltd., Tokyo, 1981. | | | N. Ikeda , S. Nakao and Y. Yamato , A class of approximations of Brownian motion, Publ. RIMS, Kyoto Univ., 13 (1977) , 285-300. doi: 10.2977/prims/1195190109. | | | D. Kelley and I. Melbourne , Smooth approximation of stochastic differential equations, Ann. Probab., 44 (2016) , 479-520. doi: 10.1214/14-AOP979. | | | P. E. Kloeden and J. A. Langa , Flattening, squeezing and the existence of random attractors, Proc. Royal Soc. London Serie A., 463 (2007) , 163-181. doi: 10.1098/rspa.2006.1753. | | | 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. | | | 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. | | | T. Kurtz and P. Protter , Wong-Zakai corrections, random evolutions, and simulation schemes for sde. Stochastic Analysis: Liber Amicorum for Moshe Zakai, Academic Press, San Diego., (1991) , 331-346. | | | 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. | | | K. Lu and B. Wang, Wong-Zakai approximations and long term behavior of stochastic partial differential equations, J. Dyn. Diff. Equat. (2017), https://doi.org/10.1007/s10884-017-9626-y. | | | R. Manthey , Weak convergence of solutions of the heat equation with Gaussian noise, Math. Nachr., 123 (1985) , 157-168. doi: 10.1002/mana.19851230115. | | | E. J. McShane , Stochastic differential equations and models of random processes, Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability (Univ. California, Berkeley, Calif., 1970/1971), 3 (1972) , 263-294. | | | S. Nakao , On weak convergence of sequences of continuous local martingale, Annales De L'I. H. P., Section B, 22 (1986) , 371-380. | | | S. Nakao and Y. Yamato, Approximation theorem on stochastic differential equations, Proc. International Symp. S.D.E., Kyoto. 1976,283-296. | | | A. Nowak , A Wong-Zakai type theorem for stochastic systems of Burgers equations, Panamer. Math. J., 16 (2006) , 1-25. | | | P. Protter , Approximations of solutions of stochastic differential equations driven by semimartingales, Ann. Probab., 13 (1985) , 716-743. doi: 10.1214/aop/1176992905. | | | B. Schmalfuss, Backward cocycles and attractors of stochastic differential equations, International Seminar on Applied Mathematics-Nonlinear Dynamics: Attractor Approximation and Global Behavior, 185-192, Dresden, 1992. | | | 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. | | | Z. Shen , S. Zhou and W. Shen , One-dimensional random attractor and rotation number of the stochastic damped sine-Gordon equation, J. Differential Equations, 248 (2010) , 1432-1457. doi: 10.1016/j.jde.2009.10.007. | | | D. W. Stook and S. R. S. Varadhan , On the support of diffusion processes with applications to the strong maximum principle, Proc. 6-th Berkeley Symp. on Math. Stat. and Prob., 3 (1972) , 333-359. | | | H. J. Sussmann , An interpretation of stochastic differential equations as ordinary differential equations which depend on the sample point, Bull. Amer. Math. Soc., 83 (1977) , 296-298. doi: 10.1090/S0002-9904-1977-14312-7. | | | H. J. Sussmann , On the gap between deterministic and stochastic ordinary differential equations, Ann. Probab., 6 (1978) , 19-41. doi: 10.1214/aop/1176995608. | | | R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3. | | | G. Tessitore and J. Zabczyk , Wong-Zakai approximations of stochastic evolution equations, J. Evol. Equ., 6 (2006) , 621-655. doi: 10.1007/s00028-006-0280-9. | | | B. Wang , Asymptotic behavior of stochastic wave equations with critical exponents on mathbbR3\mathbb{R}^3mathbbR3, Trans. Amer. Math. Soc., 363 (2011) , 3639-3663. doi: 10.1090/S0002-9947-2011-05247-5. | | | 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. | | | B. Wang, Existence and upper semicontinuity of attractors for stochastic equations with deterministic non-autonomous terms, Stoch. Dyn., 14 (2014), 1450009, 31pp. doi: 10.1142/S0219493714500099. | | | B. Wang , Random attractors for non-autonomous stochastic wave equations with multiplicative noise, Discrete Contin. Dyn. Syst. Ser. A, 34 (2014) , 269-300. doi: 10.3934/dcds.2014.34.269. | | | E. Wong and M. Zakai , On the convergence of ordinary integrals to stochastic integrals, Ann. Math. Statist., 36 (1965) , 1560-1564. doi: 10.1214/aoms/1177699916. | | | E. Wong and M. Zakai , On the relation between ordinary and stochastic differential equations, Internat. J. Engrg. Sci., 3 (1965) , 213-229. doi: 10.1016/0020-7225(65)90045-5. |