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Exponential decay of Lebesgue numbers
The existence of uniform attractors for 3D Brinkman-Forchheimer equations
1. | Department of Applied Mathematics, Shanghai Normal University, Shanghai 200234, China |
2. | Department of Mathematics and Information Science, Wenzhou University, Wenzhou, Zhejiang 325035, China |
3. | Department of Mathematics, Zhejiang Normal University, Jinhua, 321004, China |
References:
[1] |
A. V. Babin and M. I. Vishik, "Attractors of Evolution Equations," Translated and revised from the 1989 Russian original by Babin, Studies in Mathematics and its Applications, 25, North-Holland Publishing Co., Amsterdam, 1992. |
[2] |
O. Çelebi, V. Kalantarov and D. Uğurlu, On continuous dependence on coefficients of the Brinkman-Forchheimer equations, Applied Mathematics Letters, 19 (2006), 801-807.
doi: 10.1016/j.aml.2005.11.002. |
[3] |
V. V. Chepyzhov and M. I. Vishik, "Attractors for Equations of Mathematical Physics," AMS Colloquium Publications, 49, AMS, Providence, RI, 2002. |
[4] |
M. Firdaouss, J.-L. Guermond and P. Le Quéré, Nonlinear corrections to Darcy's law at low Reynolds numbers, Journal of Fluid Mechanics, 343 (1997), 331-350.
doi: 10.1017/S0022112097005843. |
[5] |
F. Franchi and B. Straughan, Continuous dependence and decay for the Forchheimer equations, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 459 (2003), 3195-3202.
doi: 10.1098/rspa.2003.1169. |
[6] |
T. Giorgi, Derivation of the Forchheimer law via matched asymptotic expansions, Transport in Porous Media, 29 (1997), 191-206.
doi: 10.1023/A:1006533931383. |
[7] |
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. |
[8] |
V. K. Kalantarov and S. Zelik, Smooth attractor for the Brinkman-Forchheimer equations with fast growing nonlinearities, preprint, 2011, arXiv:1101.4070. |
[9] |
S. Lu, Attractors for nonautonomous 2D Navier-Stokes equations with less regular normal forces, J. Differential Equations, 230 (2006), 196-212.
doi: 10.1016/j.jde.2006.07.009. |
[10] |
S. Lu, H. Wu and C. Zhong, Attractors for nonautonomous 2D Navier-Stokes equations with normal external forces, Disc. Cont. Dyn. Syst., 13 (2005), 701-719.
doi: 10.3934/dcds.2005.13.701. |
[11] |
Y. Ouyang and L. Yan, A note on the existence of a global attractor for the Brinkman-Forchheimer equations, Nonlinear Analysis, 70 (2009), 2054-2059.
doi: 10.1016/j.na.2008.02.121. |
[12] |
L. E. Payne, J. C. Song and B. Straugham, Continuous dependence and convergence results for Brinkman and Forchheimer models with variable viscosity, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 455 (1999), 2173-2190.
doi: 10.1098/rspa.1999.0398. |
[13] |
L. E. Payne and B. Straugham, Convergence and continuous dependence for the Brinkman-Forchheimer equations, Studies in Applied Mathematics, 102 (1999), 419-439.
doi: 10.1111/1467-9590.00116. |
[14] |
R. Rosa, The global attractor for the 2D Navier-Stokes flow on some unbounded domains, Nonlinear Analysis, 32 (1998), 71-85.
doi: 10.1016/S0362-546X(97)00453-7. |
[15] |
G. R. Sell and Y. You, "Dynamics of Evolutionary Equations," Applied Mathematical Sciences, 143, Springer-Verlag, New York, 2002. |
[16] |
A. Shenoy, Non-Newtonian fluid heat transfer in porous media, Adv. Heat transfer, 24 (1994), 101-190.
doi: 10.1016/S0065-2717(08)70233-8. |
[17] |
B. Straughan, "Stability and Wave Motion in Porous Media," Applied Mathematical Sciences, 165, Springer, New York, 2008. |
[18] |
D. Ugurlu, On the existence of a global attractor for the Brinkman-Forchheimer equations, Nonlinear Analysis, 68 (2008), 1986-1992.
doi: 10.1016/j.na.2007.01.025. |
[19] |
S. Whitaker, The Forchheimer equation: A theoretical development, Transport in Porous Media, 25 (1996), 27-62.
doi: 10.1007/BF00141261. |
[20] |
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. |
[21] |
C. Zhao and S. Zhou, L$^2$-compact uniform attractors for a non-autonomous incompressible non-Newtonian fluid with locally uniformly integrable external forces in distribution space, J. Math. Phys., 48 (2007), 12 pp.
doi: 10.1063/1.2709845. |
[22] |
C. Zhao and S. Zhou, Pullback attractors for a non-autonomous incompressible non-Newtonian fluid, J. Differential Equations, 238 (2007), 394-425.
doi: 10.1016/j.jde.2007.04.001. |
show all references
References:
[1] |
A. V. Babin and M. I. Vishik, "Attractors of Evolution Equations," Translated and revised from the 1989 Russian original by Babin, Studies in Mathematics and its Applications, 25, North-Holland Publishing Co., Amsterdam, 1992. |
[2] |
O. Çelebi, V. Kalantarov and D. Uğurlu, On continuous dependence on coefficients of the Brinkman-Forchheimer equations, Applied Mathematics Letters, 19 (2006), 801-807.
doi: 10.1016/j.aml.2005.11.002. |
[3] |
V. V. Chepyzhov and M. I. Vishik, "Attractors for Equations of Mathematical Physics," AMS Colloquium Publications, 49, AMS, Providence, RI, 2002. |
[4] |
M. Firdaouss, J.-L. Guermond and P. Le Quéré, Nonlinear corrections to Darcy's law at low Reynolds numbers, Journal of Fluid Mechanics, 343 (1997), 331-350.
doi: 10.1017/S0022112097005843. |
[5] |
F. Franchi and B. Straughan, Continuous dependence and decay for the Forchheimer equations, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 459 (2003), 3195-3202.
doi: 10.1098/rspa.2003.1169. |
[6] |
T. Giorgi, Derivation of the Forchheimer law via matched asymptotic expansions, Transport in Porous Media, 29 (1997), 191-206.
doi: 10.1023/A:1006533931383. |
[7] |
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. |
[8] |
V. K. Kalantarov and S. Zelik, Smooth attractor for the Brinkman-Forchheimer equations with fast growing nonlinearities, preprint, 2011, arXiv:1101.4070. |
[9] |
S. Lu, Attractors for nonautonomous 2D Navier-Stokes equations with less regular normal forces, J. Differential Equations, 230 (2006), 196-212.
doi: 10.1016/j.jde.2006.07.009. |
[10] |
S. Lu, H. Wu and C. Zhong, Attractors for nonautonomous 2D Navier-Stokes equations with normal external forces, Disc. Cont. Dyn. Syst., 13 (2005), 701-719.
doi: 10.3934/dcds.2005.13.701. |
[11] |
Y. Ouyang and L. Yan, A note on the existence of a global attractor for the Brinkman-Forchheimer equations, Nonlinear Analysis, 70 (2009), 2054-2059.
doi: 10.1016/j.na.2008.02.121. |
[12] |
L. E. Payne, J. C. Song and B. Straugham, Continuous dependence and convergence results for Brinkman and Forchheimer models with variable viscosity, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 455 (1999), 2173-2190.
doi: 10.1098/rspa.1999.0398. |
[13] |
L. E. Payne and B. Straugham, Convergence and continuous dependence for the Brinkman-Forchheimer equations, Studies in Applied Mathematics, 102 (1999), 419-439.
doi: 10.1111/1467-9590.00116. |
[14] |
R. Rosa, The global attractor for the 2D Navier-Stokes flow on some unbounded domains, Nonlinear Analysis, 32 (1998), 71-85.
doi: 10.1016/S0362-546X(97)00453-7. |
[15] |
G. R. Sell and Y. You, "Dynamics of Evolutionary Equations," Applied Mathematical Sciences, 143, Springer-Verlag, New York, 2002. |
[16] |
A. Shenoy, Non-Newtonian fluid heat transfer in porous media, Adv. Heat transfer, 24 (1994), 101-190.
doi: 10.1016/S0065-2717(08)70233-8. |
[17] |
B. Straughan, "Stability and Wave Motion in Porous Media," Applied Mathematical Sciences, 165, Springer, New York, 2008. |
[18] |
D. Ugurlu, On the existence of a global attractor for the Brinkman-Forchheimer equations, Nonlinear Analysis, 68 (2008), 1986-1992.
doi: 10.1016/j.na.2007.01.025. |
[19] |
S. Whitaker, The Forchheimer equation: A theoretical development, Transport in Porous Media, 25 (1996), 27-62.
doi: 10.1007/BF00141261. |
[20] |
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. |
[21] |
C. Zhao and S. Zhou, L$^2$-compact uniform attractors for a non-autonomous incompressible non-Newtonian fluid with locally uniformly integrable external forces in distribution space, J. Math. Phys., 48 (2007), 12 pp.
doi: 10.1063/1.2709845. |
[22] |
C. Zhao and S. Zhou, Pullback attractors for a non-autonomous incompressible non-Newtonian fluid, J. Differential Equations, 238 (2007), 394-425.
doi: 10.1016/j.jde.2007.04.001. |
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