This paper is concerned with the asymptotic behavior of the stochastic three dimensional Brinkman-Forchheimer equations in some unbounded domains. We first define a continuous random dynamical system for the equations. Then by J. Ball's idea of energy equations, we obtain pullback asymptotic compactness of solutions and prove that the existence of a unique random attractor for the equations.
Citation: Shu Wang, Mengmeng Si, Rong Yang. Dynamics of stochastic 3D Brinkman-Forchheimer equations on unbounded domains[J]. Electronic Research Archive, 2023, 31(2): 904-927. doi: 10.3934/era.2023045
This paper is concerned with the asymptotic behavior of the stochastic three dimensional Brinkman-Forchheimer equations in some unbounded domains. We first define a continuous random dynamical system for the equations. Then by J. Ball's idea of energy equations, we obtain pullback asymptotic compactness of solutions and prove that the existence of a unique random attractor for the equations.
| [1] |
B. Wang, S. Lin, Existence of global attractors for the three-dimensional Brinkman-Forchheimer equation, Math. Methods Appl. Sci., 31 (2008), 1479–1495. https://doi.org/10.1002/mma.985 doi: 10.1002/mma.985
|
| [2] |
D. Ugurlu, On the existence of a global attractor for the Brinkman-Forchheimer equations, Nonlinear Anal. Theory Methods Appl., 68 (2008), 1986–1992. https://doi.org/10.1016/j.na.2007.01.025 doi: 10.1016/j.na.2007.01.025
|
| [3] |
X. G. Yang, L. Li, X. Yan, L. Ding, The structure and stability of pullback attractors for 3D Brinkman-Forchheimer equation with delay, Electron. Res. Arch., 28 (2020), 1395–1418. https://doi.org/10.3934/era.2020074 doi: 10.3934/era.2020074
|
| [4] |
J. R. Kang, J. Y. Park, Uniform attractors for non-autonomous Brinkman-Forchheimer equations with delay, Acta Math. Sinica, 29 (2013), 99–1006. https://doi.org/10.1007/s10114-013-1392-0 doi: 10.1007/s10114-013-1392-0
|
| [5] |
C. Zhao, L. Kong, G. Liu, M. Zhao, The trajectory attractor and its limiting behavior for the convective Brinkman-Forchheimer equations, Topological Methods Nonlinear Anal., 44 (2016), 413–433. https://doi.org/10.12775/tmna.2014.054 doi: 10.12775/tmna.2014.054
|
| [6] |
C. Zhao, Y. You, Approximation of the incompressible convective Brinkman-Forchheimer equations, J. Evol. Equations, 12 (2012), 767–788. https://doi.org/10.1007/s00028-012-0153-3 doi: 10.1007/s00028-012-0153-3
|
| [7] | L. Arnold, Random dynamical systems, in Dynamical Systems, (1995), 1–43. https://doi.org/10.1007/BFb0095238 |
| [8] |
P. W. Bates, H. Lisei, K. Lu, Attractors for stochastic lattice dynamical system, Stochastics Dyn., 6 (2006), 1–21. https://doi.org/10.1142/S0219493706001621 doi: 10.1142/S0219493706001621
|
| [9] |
P. W. Bates, K. Lu, B. Wang, Random attractors for stochastic reaction-diffusion equations on unbounded domains, J. Differ. Equations, 246 (2009), 845–869. https://doi.org/10.1016/j.jde.2008.05.017 doi: 10.1016/j.jde.2008.05.017
|
| [10] |
Z. Brzeźniak, T. Caraballo, J. A. Langa, Y. Li, G. Lukasiewiez, J. Real, Random attractors for stochastic 2D Navier-Stokes equations in some unbounded domains, J. Differ. Equations, 255 (2013), 3897–3919. https://doi.org/10.1016/j.jde.2013.07.043 doi: 10.1016/j.jde.2013.07.043
|
| [11] |
H. Crauel, F. Flandoli, Attractors for random dynamical systems, Probab. Theory Relat. Fields, 100 (1994), 365–393. https://doi.org/10.1007/BF01193705 doi: 10.1007/BF01193705
|
| [12] |
H. Crauel, A. Debussche, F. Flandoli, Random attractors, J. Dyn. Differ. Equations, 9 (1997), 307–341. https://doi.org/10.1007/BF02219225 doi: 10.1007/BF02219225
|
| [13] |
R. Rosa, The global attractor for the 2D Navier-Stokes flow on some unbounded domains, Nonlinear Anal. Theory Methods Appl., 32 (1998), 71–85. https://doi.org/10.1016/S0362-546X(97)00453-7 doi: 10.1016/S0362-546X(97)00453-7
|
| [14] |
S. Wang, M. Si, R. Yang, Random attractors for non-autonomous stochastic Brinkman-Forchheimer equations on unbounded domains, Commun. Pure Appl. Anal., 21 (2022), 1621–1636. https://doi.org/10.3934/cpaa.2022034 doi: 10.3934/cpaa.2022034
|
| [15] |
M. Anguiano, Pullback attractors for a reaction-diffusion equation in a general nonempty open subset of $\mathbb{R}^{N}$ with non-autonomous forcing term in $H^{-1}$, Int. J. Bifurcation Chaos, 25 (2015), 1550164. https://doi.org/10.1142/S0218127415501643 doi: 10.1142/S0218127415501643
|
| [16] |
M. Anguiano, F. Morillas, J. Valero, On the Kneser property for reaction-diffusion equations in some unbounded domains with an $H^{-1}$-valued non-autonomous forcing term, Nonlinear Anal. Theory Methods Appl., 75 (2012), 2623–2636. https://doi.org/10.1016/j.na.2011.11.007 doi: 10.1016/j.na.2011.11.007
|
| [17] |
M. Anguiano, Pullback attractor for a non-autonomous reaction-diffusion equation in some unbounded domains, SeMA J., 51 (2010), 9–16. https://doi.org/10.1007/BF03322548 doi: 10.1007/BF03322548
|
| [18] | Z. Brzeźniak, Y. Li, Asymptotic compactness and absorbing sets for 2D stochastic Navier-Stokes equations on some unbounded domains, Trans. Amer. Math. Soc., 358 (2006), 5587–5629. Available from: https://www.ams.org/journals/tran/2006-358-12/S0002-9947-06-03923-7/S0002-9947-06-03923-7.pdf. |
| [19] | J. Simon, Équations de Navier-Stokes, Université Blaise Pascal, 2003. Available from: http://jsimon.vivrc.fr/maths/Simon-F11.pdf. |
| [20] |
J. M. Ball, Global attractor for damped semilinear wave equations, Discrete Contin. Dyn. Syst., 10 (2004), 31–52. https://doi.org/10.3934/dcds.2004.10.31 doi: 10.3934/dcds.2004.10.31
|
| [21] | R. Temam, Navier-Stokes Equations, North-Holland Publish Company, Amsterdam, 1979. |
| [22] |
M. J. Garrido-Atienza, P. Marín-Rubio, Navier-Stokes equations with delays on unbounded domains, Nonlinear Anal. Theory Methods Appl., 64 (2006), 1100–1118. https://doi.org/10.1016/j.na.2005.05.057 doi: 10.1016/j.na.2005.05.057
|
| [23] | J. L. Lions, Quelques Méthodes de Résolution des Problémes aux Limites Non Linéaires, Dunod, Gauthier-Villars, Paris, 1969. |
| [24] | K. Kinra, M. T. Mohan, ${H}^{1}$-random attractors for the 2D stochastic convective Brinkman-Forchheimer equations in some unbounded domains, preprint, arXiv: 2111.07841. |
| [25] | E. DiBenedetto, Degenerate Parabolic Equations, Springer-Verlag, New York, 1993. https://doi.org/10.1007/978-1-4612-0895-2 |
Shu Wang, Mengmeng Si, Rong Yang. Dynamics of stochastic 3D Brinkman-Forchheimer equations on unbounded domains[J]. Electronic Research Archive, 2023, 31(2): 904-927. doi: 10.3934/era.2023045
DownLoad: