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The regularized Boussinesq equations with partial dissipations in dimension two
The structure and stability of pullback attractors for 3D Brinkman-Forchheimer equation with delay
1. | College of Mathematics and Information Science, Henan Normal University, Xinxiang 453007, China |
2. | Laboratoire de Mathématiques et Applications, UMR CNRS 7348-SP2MI, Université de Poitiers, Boulevard Marie et Pierre Curie-Téléport 2, 86962, Chasseneuil Futuroscope Cedex, France |
3. | Department of Mathematics, China University of Mining and Technology, Xuzhou 221008, China |
4. | School of Mathematics and Statistics, Hubei University of Arts and Science, Xiangyang 441053, China |
This paper concerns the stability of pullback attractors for 3D Brinkman-Forchheimer equation with delays. By some regular estimates and the variable index to deal with the delay term, we get the sufficient conditions for asymptotic stability of trajectories inside the pullback attractors for a fluid flow model in porous medium by generalized Grashof numbers.
References:
[1] |
T. Caraballo and J. Real,
Navier-Stokes equations with delays, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 457 (2001), 2441-2453.
doi: 10.1098/rspa.2001.0807. |
[2] |
T. Caraballo and J. Real,
Attractors for 2D Navier-Stokes models with delays, J. Differential Equations, 205 (2004), 271-297.
doi: 10.1016/j.jde.2004.04.012. |
[3] |
A. N. Carvalho, J. A. Langa and J. C. Robinson, Attractors for Infinite-Dimensional Non-Autonomous Dynamical Systems, Springer, New York–Heidelberg–Dordrecht–London, 2013.
doi: 10.1007/978-1-4614-4581-4. |
[4] |
J. García-Luengo, P. Marín-Rubio and G. Planas,
Attractors for a double time-delayed 2D Navier-Stokes model, Disc. Contin. Dyn. Syst., 34 (2014), 4085-4105.
doi: 10.3934/dcds.2014.34.4085. |
[5] |
D. Li, Q. Liu and X. Ju, Uniform decay estimates for solutions of a class of retarded integral inequalities, J. Differential Equations, Accepted, 2020. Google Scholar |
[6] |
L. Li, X.-G. Yang, X. Li, X. Yan and Y. Lu,
Dynamics and stability of the 3D Brinkman-Forchheimer equation with variable delay (Ⅰ), Asymptot. Anal., 113 (2019), 167-194.
doi: 10.3233/ASY-181512. |
[7] |
J.-L. Lions, Quelques Méthodes de Résolution des Problémes aux Limites Non Linéaires, Dunod, Gauthier-Villars, Paris, 1969. |
[8] |
P. Marín-Rubio and J. Real,
Pullback attractors for 2D-Navier-Stokes equations with delays in continuous and sub-linear operators, Discrete Contin. Dyn. Syst., 26 (2010), 989-1006.
doi: 10.3934/dcds.2010.26.989. |
[9] |
D. A. Nield,
The limitations of the Brinkman-Forchheimer equation in modeling flow in a saturated porous medium and at an interface, Int. J. Heat Fluid Flow, 12 (1991), 269-272.
doi: 10.1016/0142-727X(91)90062-Z. |
[10] |
Y. Ouyang and L. Yang,
A note on the existence of a global attractor for the Brinkman-Forchheimer equations, Nonlinear Anal., 70 (2009), 2054-2059.
doi: 10.1016/j.na.2008.02.121. |
[11] |
Y. Qin, Integral and Discrete Inequalities and Their Applications, Vol. Ⅰ: Linear Inequalities and Vol. Ⅱ: Nonlinear Inequalities, Birkhäser, Basel/Boston/Berlin, 2016. |
[12] |
J. Simon,
Compact sets in the space $L^p(0, T;B)$, Ann. Mat. Pura Appl., 146 (1987), 65-96.
doi: 10.1007/BF01762360. |
[13] |
R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Second edition. Applied Mathematical Sciences, 68. Springer-Verlag, New York, 1997.
doi: 10.1007/978-1-4612-0645-3. |
[14] |
R. Temam and A. Miranville, Mathematical Modeling in Continuum Mechanics, Cambridge University Press, Cambridge, 2001.
![]() |
[15] |
D. Uǧurlu,
On the existence of a global attractor for the Brinkman-Forchheimer equations, Nonlinear Anal., 68 (2008), 1986-1992.
doi: 10.1016/j.na.2007.01.025. |
[16] |
B. Wang and S. Lin,
Existence of global attractors for the three-dimensional Brinkman-Forchheimer equation, Math. Meth. Appl. Sci., 31 (2008), 1479-1495.
doi: 10.1002/mma.985. |
[17] |
Y. Wang, X. Yang and Y. Lu, Remarks on nontrivial pullback attractors of the 2-D Navier-Stokes equations with delays, Math. Meth. Appl. Sci., 43 (2020), 1892-1900. Google Scholar |
[18] |
S. Whitaker,
The Forchheimer equation: A theoretical development, Transp. Porous Media, 25 (1996), 27-62.
doi: 10.1007/BF00141261. |
[19] |
S. Whitaker,
Flow in porous media Ⅰ: A theoretical derivation of Darcy's law, Transp. Porous Media, 1 (1986), 3-25.
doi: 10.1007/BF01036523. |
[20] |
X.-G. Yang, J. Zhang and S. Wang,
Stability and dynamics of a weak viscoelastic system with memory and nonlinear time-varying delay, Disc. Contin. Dyn. Syst., 40 (2020), 1493-1515.
doi: 10.3934/dcds.2020084. |
[21] |
Y. You, C. Zhao and S. Zhou,
The existence of uniform attractors for 3D Brinkman-Forchheimer equations, Disc. Contin. Dyn. Syst., 32 (2012), 3787-3800.
doi: 10.3934/dcds.2012.32.3787. |
show all references
References:
[1] |
T. Caraballo and J. Real,
Navier-Stokes equations with delays, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 457 (2001), 2441-2453.
doi: 10.1098/rspa.2001.0807. |
[2] |
T. Caraballo and J. Real,
Attractors for 2D Navier-Stokes models with delays, J. Differential Equations, 205 (2004), 271-297.
doi: 10.1016/j.jde.2004.04.012. |
[3] |
A. N. Carvalho, J. A. Langa and J. C. Robinson, Attractors for Infinite-Dimensional Non-Autonomous Dynamical Systems, Springer, New York–Heidelberg–Dordrecht–London, 2013.
doi: 10.1007/978-1-4614-4581-4. |
[4] |
J. García-Luengo, P. Marín-Rubio and G. Planas,
Attractors for a double time-delayed 2D Navier-Stokes model, Disc. Contin. Dyn. Syst., 34 (2014), 4085-4105.
doi: 10.3934/dcds.2014.34.4085. |
[5] |
D. Li, Q. Liu and X. Ju, Uniform decay estimates for solutions of a class of retarded integral inequalities, J. Differential Equations, Accepted, 2020. Google Scholar |
[6] |
L. Li, X.-G. Yang, X. Li, X. Yan and Y. Lu,
Dynamics and stability of the 3D Brinkman-Forchheimer equation with variable delay (Ⅰ), Asymptot. Anal., 113 (2019), 167-194.
doi: 10.3233/ASY-181512. |
[7] |
J.-L. Lions, Quelques Méthodes de Résolution des Problémes aux Limites Non Linéaires, Dunod, Gauthier-Villars, Paris, 1969. |
[8] |
P. Marín-Rubio and J. Real,
Pullback attractors for 2D-Navier-Stokes equations with delays in continuous and sub-linear operators, Discrete Contin. Dyn. Syst., 26 (2010), 989-1006.
doi: 10.3934/dcds.2010.26.989. |
[9] |
D. A. Nield,
The limitations of the Brinkman-Forchheimer equation in modeling flow in a saturated porous medium and at an interface, Int. J. Heat Fluid Flow, 12 (1991), 269-272.
doi: 10.1016/0142-727X(91)90062-Z. |
[10] |
Y. Ouyang and L. Yang,
A note on the existence of a global attractor for the Brinkman-Forchheimer equations, Nonlinear Anal., 70 (2009), 2054-2059.
doi: 10.1016/j.na.2008.02.121. |
[11] |
Y. Qin, Integral and Discrete Inequalities and Their Applications, Vol. Ⅰ: Linear Inequalities and Vol. Ⅱ: Nonlinear Inequalities, Birkhäser, Basel/Boston/Berlin, 2016. |
[12] |
J. Simon,
Compact sets in the space $L^p(0, T;B)$, Ann. Mat. Pura Appl., 146 (1987), 65-96.
doi: 10.1007/BF01762360. |
[13] |
R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Second edition. Applied Mathematical Sciences, 68. Springer-Verlag, New York, 1997.
doi: 10.1007/978-1-4612-0645-3. |
[14] |
R. Temam and A. Miranville, Mathematical Modeling in Continuum Mechanics, Cambridge University Press, Cambridge, 2001.
![]() |
[15] |
D. Uǧurlu,
On the existence of a global attractor for the Brinkman-Forchheimer equations, Nonlinear Anal., 68 (2008), 1986-1992.
doi: 10.1016/j.na.2007.01.025. |
[16] |
B. Wang and S. Lin,
Existence of global attractors for the three-dimensional Brinkman-Forchheimer equation, Math. Meth. Appl. Sci., 31 (2008), 1479-1495.
doi: 10.1002/mma.985. |
[17] |
Y. Wang, X. Yang and Y. Lu, Remarks on nontrivial pullback attractors of the 2-D Navier-Stokes equations with delays, Math. Meth. Appl. Sci., 43 (2020), 1892-1900. Google Scholar |
[18] |
S. Whitaker,
The Forchheimer equation: A theoretical development, Transp. Porous Media, 25 (1996), 27-62.
doi: 10.1007/BF00141261. |
[19] |
S. Whitaker,
Flow in porous media Ⅰ: A theoretical derivation of Darcy's law, Transp. Porous Media, 1 (1986), 3-25.
doi: 10.1007/BF01036523. |
[20] |
X.-G. Yang, J. Zhang and S. Wang,
Stability and dynamics of a weak viscoelastic system with memory and nonlinear time-varying delay, Disc. Contin. Dyn. Syst., 40 (2020), 1493-1515.
doi: 10.3934/dcds.2020084. |
[21] |
Y. You, C. Zhao and S. Zhou,
The existence of uniform attractors for 3D Brinkman-Forchheimer equations, Disc. Contin. Dyn. Syst., 32 (2012), 3787-3800.
doi: 10.3934/dcds.2012.32.3787. |
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