Wong-Zakai approximations and attractors for stochastic three-dimensional globally modified Navier-Stokes equations driven by nonlinear noise

Ho Thi Hang; Bui Kim My; Pham Tri Nguyen

doi:10.3934/dcdsb.2023124

Article Contents

2024, Volume 29, Issue 2: 1069-1104. Doi: 10.3934/dcdsb.2023124

This issue Previous Article A model for data transmission and its optimization Next Article Continuity in expectation of odd random attractors for stochastic Kuramoto-Sivashinsky equations

Wong-Zakai approximations and attractors for stochastic three-dimensional globally modified Navier-Stokes equations driven by nonlinear noise

1.
Department of Mathematics, Electric Power University, 235 Hoang Quoc Viet, Bac Tu Liem, Hanoi, Vietnam
2.
Faculty of Primary Education, Hanoi Pedagogical University 2, 32 Nguyen Van Linh, Xuan Hoa, Phuc Yen, Vinhphuc, Vietnam

^*Corresponding author: Bui Kim My
^*Corresponding author: Bui Kim My

Received: September 2022

Revised: March 2023

Early access: July 10, 2023

Published online: July 10, 2023

This research was funded by Hanoi Pedagogical University 2 under grant number HPU2.2022-UT-10.

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Abstract

We analyze the asymptotic behavior of the stochastic three dimensional globally modified Navier-Stokes equations with general Lipschitz nonlinear noise. By using the Wong-Zakai approximation, we first show that the approximate equation has a unique random attractor, and then when the stochastic equation is driven by a linear multiplicative noise or additive white noise, we show the convergence of solutions and attractors of Wong-Zakai approximations of the approximate random systems as the size of the approximation tends to zero.

Keywords:

Stochastic globally modified Navier-Stokes equations,
Random pullback attractor,
Wong-Zakai approximation,
Nonlinear noise.

Mathematics Subject Classification: Primary: 35Q35, 35B40, 35B41; Secondary: 60H15, 60H30.

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References

[1]	L. Arnold, Random Dynamical Systems, Springer, Berlin, 1998. doi: 10.1007/978-3-662-12878-7.
[2]	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.
[3]	M. Capinński and N. J. Cutland, Existence of global stochastic flow and attractors for Navier-Stokes equations, Probab. Theory Relat. Fields, 115 (1999), 121-151. doi: 10.1007/s004400050238.
[4]	T. Caraballo and P. E. Kloeden, The three-dimensional globally modified Navier-Stokes equations: Recent developments, in Recent Trends in Dynamical Systems, Springer Proc. Math. Stat., Springer, Basel, 35 (2013), 473-492. doi: 10.1007/978-3-0348-0451-6_18.
[5]	T. Caraballo, P. E. Kloeden and J. Real, Unique strong solutions and $V$-attractors of a three dimensional system of globally modified Navier-Stokes equations, Adv. Nonlinear Stud., 6 (2006), 411-436. doi: 10.1515/ans-2006-0304.
[6]	T. Caraballo, P. E. Kloeden and J. Real, Invariant measures and statistical solutions of the globally modified Navier-Stokes equations, Discrete Contin. Dyn. Syst. Ser. B, 10 (2008), 761-781. doi: 10.3934/dcdsb.2008.10.761.
[7]	H. Crauel and F. Flandoli, Attractors for random dynamical systems, Probab. Theory Related Fields., 100 (1994), 365-393. doi: 10.1007/BF01193705.
[8]	G. Deugoué and T. Tachim Medjo, The stochastic 3D globally modified Navier-Stokes equations: Existence, uniqueness and asymptotic behavior, Commun. Pure Appl. Anal., 17 (2018), 2593-2621. doi: 10.3934/cpaa.2018123.
[9]	G. Deugoué and T. Tachim Medjo, Convergence of the solution of the stochastic 3D globally modified Cahn-Hilliard-Navier-Stokes equations, J. Differential Equations, 265 (2018), 545-592. doi: 10.1016/j.jde.2018.03.002.
[10]	G. Deugoué and T. Tachim Medjo, The exponential behavior of a stochastic globally modified Cahn-Hilliard-Navier-Stokes model with multiplicative noise, J. Math. Anal. Appl., 460 (2018), 140-163. doi: 10.1016/j.jmaa.2017.11.050.
[11]	F. Flandoli, Regularity Theory and Stochastic Flow for Parabolic SPDEs, Stochastics Monographs Vol. 9, Gordon and Breach Science Publishers SA, Singapore, 1995.
[12]	F. Flandoli and B. Schmalf ß, Random attractors for the 3D stochastic Navier-Stokes equation with multiplicative noise, Stoch. Stoch. Rep., 59 (1996), 21-45. doi: 10.1080/17442509608834083.
[13]	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.
[14]	A. Gu, B. Guo and B. Wang, Long term behavior of random Navier-Stokes equations driven by colored noise, Discrete Contin. Dyn. Syst., 25 (2020), 2495-2532. doi: 10.3934/dcdsb.2020020.
[15]	A. Gu, K. Lu and B. Wang, Asymptotic behavior of random Navier-Stokes equations driven by Wong-Zakai approximations, Discrete Contin. Dyn. Syst., 39 (2019), 185-218. doi: 10.3934/dcds.2019008.
[16]	N. Ju, Existence of global attractor for the three-dimensional modified Navier-Stokes equations, Nonlinearity, 14 (2001), 777-786. doi: 10.1088/0951-7715/14/4/306.
[17]	P. E. Kloeden, J. A. Langa and J. Real, Pullback $V$-attractors of a three dimensional system of nonautonomous globally modified Navier-Stokes equations: Existence and finite fractal dimension, Commun. Pure Appl. Anal., 6 (2007), 937-955. doi: 10.3934/cpaa.2007.6.937.
[18]	P. E. Kloeden, P. Marín-Rubio and J. Real, Equivalence of invariant measures and stationary statistical solution for the autonomous globally modified Navier-Stokes equations, Commun. Pure Appl. Anal., 8 (2009), 785-802. doi: 10.3934/cpaa.2009.8.785.
[19]	A. Krause and B. Wang, Pullback attractors of non-autonomous stochastic degenerate parabolic equations on unbounded domains, J. Math. Anal. Appl., 417 (2014), 1018-1038. doi: 10.1016/j.jmaa.2014.03.037.
[20]	K. Lu and B. Wang, Wong-Zakai approximations and long term behavior of stochastic partial differential equations, J. Dynam. Differ. Equ., 31 (2019), 1341-1371. doi: 10.1007/s10884-017-9626-y.
[21]	P. Marín-Rubio, A. M. Márquez-Durán and J. Real, Three dimensional system of globally modified Navier-Stokes equations with infinite delays, Discrete Contin. Dyn. Syst. Ser. B., 14 (2010), 655-673. doi: 10.3934/dcdsb.2010.14.655.
[22]	P. Marín-Rubio, A. M. Márquez-Durán and J. Real, Pullback attractors for globally modified Navier-Stokes equations with infinite delays, Discrete Contin. Dyn. Syst., 31 (2011), 779-796. doi: 10.3934/dcds.2011.31.779.
[23]	P. Marín-Rubio, A. M. Márquez-Durán and J. Real, Asymptotic behavior of solutions for a three dimensional system of globally modified Navier-Stokes equations with a locally Lipschitz delay term, Nonlinear Anal., 79 (2013), 68-79. doi: 10.1016/j.na.2012.11.006.
[24]	P. Marín-Rubio, J. Real and A. M. Márquez-Durán, On the convergence of solutions of globally modified Navier-Stokes equations with delays to solutions of Navier-Stokes equations with delays, Adv. Nonlinear Stud., 11 (2011), 917-927. doi: 10.1515/ans-2011-0409.
[25]	F. M. Mosa, Q. Ma and M. Y. A. Bakhet, Existence of random attractors for stochastic non-autonomous reaction-diffusion equation with multiplicative noise on $\mathbb{R}^N$, Korean J. Math., 26 (2018), 583-599. doi: 10.11568/kjm.2018.26.4.583.
[26]	M. Romito, The uniqueness of weak solutions of the globally modified Navier-Stokes equations, Adv. Nonlinear Stud., 9 (2009), 425-427. doi: 10.1515/ans-2009-0209.
[27]	R. Temam, Navier-Stokes Equations: Theory and Numerical Analysis, Studies in Mathematics and its Applications, Vol. 2. North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977.
[28]	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.
[29]	B. Wang, Existence and upper semicontinuity of attractors for stochastic equations with deterministic non-autonomous terms, Stoch. Dyn., 14 (2014), 1450009, 31 pages. doi: 10.1142/S0219493714500099.
[30]	X. Wang, K. Lu and B. Wang, Wong-Zakai approximations and attractors for stochastic reaction-diffusion equations on unbounded domains, J. Differential Equations, 264 (2018), 378-424. doi: 10.1016/j.jde.2017.09.006.
[31]	X. Wang, J. Shen, K. Lu and B. Wang, Wong-Zakai approximations and random attractors for non-autonomous stochastic lattice systems, J. Differential Equations, 280 (2021), 477-516. doi: 10.1016/j.jde.2021.01.026.
[32]	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.
[33]	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.
[34]	C. Zhao and T. Caraballo, Asymptotic regularity of trajectory attractor and trajectory statistical solution for the 3D globally modified Navier-Stokes equations, J. Differential Equations, 266 (2019), 7205-7229. doi: 10.1016/j.jde.2018.11.032.
[35]	C. Zhao and L. Yang, Pullback attractors and invariant measures for the non-autonomous globally modified Navier-Stokes equations, Commun. Math. Sci., 15 (2017), 1565-1580. doi: 10.4310/CMS.2017.v15.n6.a4.