Random attractors for non-autonomous stochastic Navier-Stokes-Voigt equations in some unbounded domains

Shu Wang; Mengmeng Si; Rong Yang

doi:10.3934/cpaa.2023062

Article Contents

2023, Volume 22, Issue 7: 2169-2185. Doi: 10.3934/cpaa.2023062

This issue Previous Article Existence of solutions for coupled hybrid systems of differential equations for microscopic dynamics and local concentrations Next Article The stochastic flocking model with far-field degenerate communication

Random attractors for non-autonomous stochastic Navier-Stokes-Voigt equations in some unbounded domains

Faculty of Sciences, Beijing University of Technology, P. R. China

^*Corresponding author: Mengmeng Si
^*Corresponding author: Mengmeng Si

Received: February 2023

Revised: April 2023

Early access: April 27, 2023

Published online: April 27, 2023

The work was supported by National Natural Science Foundation of China (Nos.11601021, 11831003, 11771031, and 12171111) and the Science and Technology Project of Beijing Municipal Education Commission (No.KZ202110005011).

Abstract / Introduction Full Text(HTML) Related Papers Cited by

Abstract

This paper is concerned with the asymptotic behavior for the three dimensional non-autonomous stochastic Navier-Stokes-Voigt equations on unbounded domains. A continuous non-autonomous random dynamical system for the equations is firstly established. We then obtain pullback asymptotic compactness of solutions and prove that the existence of tempered random attractors for the random dynamical system generated by the equations. Furthermore, we obtain that the tempered random attractors are periodic when the deterministic non-autonomous external term is periodic in time.

Keywords:

Stochastic Navier-Stokes-Voigt equations,
non-autonomous,
random attractors,
pullback asymptotic compactness.

Mathematics Subject Classification: Primary: 35Q30, 35Q35, 37H10, 35B41.

Citation:

Full Text(HTML)

Related Papers

Cited by

References

[1]	C. T. Anh and P. T. Trang, Pullback attractors for 3D Navier-Stokes-Voigt equations in some unbounded domains, P. Roy. Soc. Edinb. A, 143 (2013), 223-251. doi: 10.1017/S0308210511001491.
[2]	L. Arnold, Random Dynamical Systems, Springer-Verlag, 1998. doi: 10.1007/978-3-662-12878-7.
[3]	J. M. Ball, Global attractor for damped semilinear wave equations, Discrete Contin. Dyn. Syst., 6 (2004), 31-52. doi: 10.3934/dcds.2004.10.31.
[4]	P. W. Bates, H. Lisei and K. Lu, Attractors for stochastic lattice dynamical system, Stoch. Dyn., 6 (2006), 1-21. doi: 10.1142/S0219493706001621.
[5]	P. W. Bates, K. Lu and B. Wang, Random attractors for stochastic reaction-diffusion equations on unbounded domains, J. Differ. Equ., 246 (2009), 845-869. doi: 10.1016/j.jde.2008.05.017.
[6]	T. Q. Bao, Dynamics of stochastic three dimensional Navier-Stokes-Voigt equations on unbounded domains, J. Math. Anal. Appl., 419 (2014), 583-605. doi: 10.1016/j.jmaa.2014.05.003.
[7]	H. Crauel, A. Debussche and F. Flandoli, Random attractors, J. Dynam. Differ. Equ., 9 (1997), 307-341. doi: 10.1007/BF02219225.
[8]	H. Crauel and F. Flandoli, Attractors for random dynamical systems, Prob. Th. Rel. Fie., 100 (1994), 365-393. doi: 10.1007/BF01193705.
[9]	T. Caraballo, P. Kloeden and B. Schmalfuss, Exponentially stable stationary solutions for stochastic evolution equations and their perturbation, Appl. Math., 5 (2004), 183-207. doi: 10.1007/s00245-004-0802-1.
[10]	T. Caraballo, G. Lukaszewicz and J. Real, Pullback attractors for non-autonomous 2D-Navier-Stokes equations in some unbounded domains, C. R. Acad. Sci. Paris I, 342 (2006), 263-268. doi: 10.1016/j.crma.2005.12.015.
[11]	T. Caraballo, G. Lukaszewicz and J. Real, Pullback attractors for asymptotically compact non-autonomous dynamical systems, Nonlinear Anal., 64 (2006), 484-498. doi: 10.1016/j.na.2005.03.111.
[12]	H. Gao and C. Sun, Random dynamics of the 3D stochastic Navier-Stoke-Voight equations, Nonlinear Anal. RWA, 13 (2012), 1197-1205. doi: 10.1016/j.nonrwa.2011.09.013.
[13]	V. K. Kalantarov and E. S. Titi, Global attractor and determining modes for the 3D Navier-Stokes-Voight equations, Chin. Ann. Math. B, 30 (2009), 697-714. doi: 10.1007/s11401-009-0205-3.
[14]	V. K. Kalantarov and E. S. Titi, Gevrey regularity for the attractor of the 3D Navier-Stokes-Voight equations, J. Nonliner Sci., 19 (2009), 133-152. doi: 10.1007/s00332-008-9029-7.
[15]	A. P. Oskolkov, The uniqueness and solvability in the large of boundary value problems for the equations of motion of aqueous solutions of polymers, Zap. Nauchn. Semin. LOMI, 38 (1971), 98-136.
[16]	R. Rosa, The global attractor for the 2D Navier-Stokes flow on some unbounded domains, Nonlinear Anal., 32 (1998), 71-85. doi: 10.1016/S0362-546X(97)00453-7.
[17]	B. Schmalfuss, Backward cocycles and attractors of stochastic differential equations, International Seminar on Applied Mathematics-Nonlinear Dynamics: Attractor Approximation and Global Behavior, 1992,185-192.
[18]	J. Simon, Equations de Navier-Stokes, Université Blaise Pascal, Cours de DEA 2002-2003.
[19]	C. Sun and H. Gao, Hausdorff dimension of random attractor for stochastic Navier-Stokes-Voight equations and primitive equations, Dynam. Partial Differ. Equ., 7 (2010), 307-326. doi: 10.4310/DPDE.2010.v7.n4.a2.
[20]	R. Temam, Navier-Stokes Equations, North-Holland Publish Company, Amsterdam, 1979.
[21]	B. Wang, Periodic random attractors for stochastic Navier-Stokes equations on unbounded domains, Electron. J. Differ. Equ., 59 (2012), 1-18.
[22]	B. Wang, Sufficient and necessary criteria for existence of pullback attractors for non-compact random dynamical systems, J. Differ. Equ., 253 (2012), 1544-1583. doi: 10.1016/j.jde.2012.05.015.
[23]	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.
[24]	S. Wang, M. Si and R. Yang, Random attractors for non-autonomous stochastic Brinkman-Forchheimer equations on unbounded domains, Commun. Pure Appl. Anal., 21 (2022), 1621-1636. doi: 10.3934/cpaa.2022034.
[25]	S. Wang, M. Si and R. Yang, Dynamics of stochastic 3D Brinkman-Forchheimer equations on unbounded domains, Electron. Research Arch., 31 (2023), 904-927. doi: 10.3934/era.2023045.
[26]	G. Yue and C. K. Zhong, Attractors for autonomous and nonautonomous 3D Navier-Stokes-Voight equations, Discrete Cont. Dynam. Syst. Ser. B., 16 (2011), 985-1002. doi: 10.3934/dcdsb.2011.16.985.