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Global well-posedness of a 3D MHD model in porous media
1. | Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK |
2. | Department of Mathematics, Texas A & M University College Station, 3368 TAMU, College Station, TX 77843-3368, USA |
3. | Department of Mathematics & Statistics, King Fahd University of Petroleum and Minerals, Dhahran 31261, KSA |
In this paper we show the global well-posedness of solutions to a three-dimensional magnetohydrodynamical (MHD) model in porous media. Compared to the classical MHD equations, our system involves a nonlinear damping term in the momentum equations due to the "Brinkman-Forcheimer-extended-Darcy" law of flow in porous media.
References:
[1] |
S. Agmon, Lectures on Elliptic Boundary Value Problems, AMS Chelsea Publishing, Providence, RI, 2010.
doi: 10.1090/chel/369. |
[2] |
S. N. Antontsev and H. B. de Oliveira,
The Navier-Stokes problem modified by an absorption term, Applicable Analysis, 89 (2010), 1805-1825.
doi: 10.1080/00036811.2010.495341. |
[3] |
J. W. Barrett and W. B. Liu,
Finite element approximation of the parabolic $p$-laplacian, SIAM Journal on Numerical Analysis, 31 (1994), 413-428.
doi: 10.1137/0731022. |
[4] |
H. Bessaih, S. Trabelsi and H. Zorgati,
Existence and uniqueness of global solutions for the modified anisotropic 3D Navier-Stokes equations, ESAIM: Mathematical Modelling and Numerical Analysis, 50 (2016), 1817-1823.
doi: 10.1051/m2an/2016008. |
[5] |
X. J. Cai and Q. S. Jiu,
Weak and strong solutions for the incompressible Navier-Stokes equations with damping, Journal of Mathematical Analysis and Applications, 343 (2008), 799-809.
doi: 10.1016/j.jmaa.2008.01.041. |
[6] |
C. S. Cao and J. H. Wu,
Two regularity criteria for the 3D MHD equations, Journal of Differential Equations, 248 (2010), 2263-2274.
doi: 10.1016/j.jde.2009.09.020. |
[7] |
A. O. Çelebi, V. Kalantarov and D. U$\widetilde {\rm{g}}$gurlu,
Continuous dependence for the convective Brinkman-Forchheimer equations, Applicable Analysis, 84 (2005), 877-888.
doi: 10.1080/00036810500148911. |
[8] |
A. O. Çelebi, V. K. Kalantarov and D. Uǧurlu,
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. |
[9] |
P. Constantin and C. Foias, Navier-Stokes Equations, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1988.
![]() |
[10] |
J. É. J. Dupuit, Études Théoriques et Pratiques sur le Mouvement des eaux dans les Canaux Découverts et à Travers les Terrains Perméables: avec des Considérations Relatives au Régime des Grandes eaux, au Débouché à leur Donner, et à la Marche des Alluvions dans les Rivières à Fond Mobile, Dunod, 1863. Google Scholar |
[11] |
G. Duvaut and J. L. Lions,
Inéquations en thermoélasticité et magnétohydrodynamique, Archive for Rational Mechanics and Analysis, 46 (1972), 241-279.
doi: 10.1007/BF00250512. |
[12] |
P. Forchheimer, Wasserbewegung durch boden, Zeitz. Ver. Duetch Ing., 45 (1901), 1782-1788. Google Scholar |
[13] |
M. Fourar, G. Radilla, R. Lenormand and C. Moyne, On the non-linear behavior of a laminar single-phase flow through two and three-dimensional porous media, Advances in Water Resources, 27 (2004), 669-677. Google Scholar |
[14] |
C. He and Z. P. Xin,
On the regularity of weak solutions to the magnetohydrodynamic equations, Journal of Differential Equations, 213 (2005), 235-254.
doi: 10.1016/j.jde.2004.07.002. |
[15] |
C. He and Z. P. Xin,
Partial regularity of suitable weak solutions to the incompressible magnetohydrodynamic equations, Journal of Functional Analysis, 227 (2005), 113-152.
doi: 10.1016/j.jfa.2005.06.009. |
[16] |
C. Hsu and P. Cheng, Thermal dispersion in a porous medium, International Journal of Heat and Mass Transfer, 33 (1990), 1587-1597. Google Scholar |
[17] |
V. Kalantarov and S. Zelik,
Smooth attractors for the Brinkman-Forchheimer equations with fast growing nonlinearities, Communications on Pure and Applied Analysis, 11 (2012), 2037-2054.
doi: 10.3934/cpaa.2012.11.2037. |
[18] |
K. Kang and J. Lee,
Interior regularity criteria for suitable weak solutions of the magnetohydrodynamic equations, Journal of Differential Equations, 247 (2009), 2310-2330.
doi: 10.1016/j.jde.2009.07.016. |
[19] |
V. A. Liskevich and Y. A. Semenov, Some problems on markov semigroups, Schrödinger Operators, Markov Semigroups, Wavelet Analysis, Operator Algebras, Math. Top., Adv. Partial Differential Equations, Akademie Verlag, Berlin, 11 (1996), 163–217. |
[20] |
Y. Liu and C. H. Lin, Structural stability for Brinkman-Forchheimer equations, Electronic Journal of Differential Equations, 2007 (2007), 8 pp. |
[21] |
M. Louaked, N. Seloula, S. Y. Sun and S. Trabelsi,
A pseudocompressibility method for the incompressible Brinkman-Forchheimer equations, Differential and Integral Equations, 28 (2015), 361-382.
|
[22] |
M. Louaked, N. Seloula and S. Trabelsi,
Approximation of the unsteady Brinkman-Forchheimer equations by the pressure stabilization method, Numerical Methods for Partial Differential Equations, 33 (2017), 1949-1965.
doi: 10.1002/num.22173. |
[23] |
P. A. Markowich, E. S. Titi and S. Trabelsi,
Continuous data assimilation for the three-dimensional Brinkman-Forchheimer-extended Darcy model, Nonlinearity, 29 (2016), 1292-1328.
doi: 10.1088/0951-7715/29/4/1292. |
[24] |
J. E. McClure, W. G. Gray and C. T. Miller,
Beyond anisotropy: Examining non-darcy flow in asymmetric porous media, Transp. Porous Media, 84 (2010), 535-548.
doi: 10.1007/s11242-009-9518-7. |
[25] |
D. Nield, The limitations of the Brinkman-Forchheimer equation in modeling flow in a saturated porous medium and at an interface, International Journal of Heat and Fluid Flow, 12 (1991), 269-272. Google Scholar |
[26] |
Y. Ouyang and L.-e. Yang,
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. |
[27] |
M. Panfilov and M. Fourar, Physical splitting of nonlinear effects in high-velocity stable flow through porous media, Advances in Water Resources, 29 (2006), 30-41. Google Scholar |
[28] |
L. E. Payne and B. Straughan,
Convergence and continuous dependence for the Brinkman-Forchheimer equations, Studies in Applied Mathematics, 102 (1999), 419-439.
doi: 10.1111/1467-9590.00116. |
[29] |
M. Sermange and R. Temam,
Some mathematical questions related to the MHD equations, Communications on Pure and Applied Mathematics, 36 (1983), 635-664.
doi: 10.1002/cpa.3160360506. |
[30] |
B. Straughan, Stability and Wave Motion in Porous Media, Applied Mathematical Sciences, 165. Springer, New York, 2008. |
[31] |
R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Applied Mathematical Sciences, 68. Springer-Verlag, New York, 1988.
doi: 10.1007/978-1-4684-0313-8. |
[32] |
S. Trabelsi, Global well-posedness of a 3D Forchheimer-Bénard convection model in porous media, Submitted, (2018). Google Scholar |
[33] |
D. Uǧurlu,
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. |
[34] |
K. Vafai and R. Thiyagaraja, Analysis of flow and heat transfer at the interface region of a porous medium, International Journal of Heat and Mass Transfer, 30 (1987), 1391-1405. Google Scholar |
[35] |
B. X. Wang and S. Y. Lin,
Existence of global attractors for the three-dimensional Brinkman-Forchheimer equation, Mathematical Methods in the Applied Sciences, 31 (2008), 1479-1495.
doi: 10.1002/mma.985. |
[36] |
Y. C. You, C. D. Zhao and S. F. Zhou,
The existence of uniform attractors for 3D Brinkman-Forchheimer equations, Discrete & Continuous Dynamical Systems-A, 32 (2012), 3787-3800.
doi: 10.3934/dcds.2012.32.3787. |
[37] |
Z. J. Zhang, D. X. Zhong, S. J. Gao and S. L. Qiu,
Fundamental Serrin type regularity criteria for 3D MHD fluid passing through the porous medium, Applicable Analysis, 96 (2017), 2130-2139.
doi: 10.2298/FIL1705287Z. |
[38] |
Z. J. Zhang,
Refined regularity criteria for the MHD system involving only two components of the solution, Appl. Anal., 96 (2017), 2130-2139.
doi: 10.1080/00036811.2016.1207245. |
[39] |
Z. J. Zhang, C. P. Wu and Z.-A. Yao,
Remarks on global regularity for the 3D MHD system with damping, Applied Mathematics and Computation, 333 (2018), 1-7.
doi: 10.1016/j.amc.2018.03.047. |
show all references
References:
[1] |
S. Agmon, Lectures on Elliptic Boundary Value Problems, AMS Chelsea Publishing, Providence, RI, 2010.
doi: 10.1090/chel/369. |
[2] |
S. N. Antontsev and H. B. de Oliveira,
The Navier-Stokes problem modified by an absorption term, Applicable Analysis, 89 (2010), 1805-1825.
doi: 10.1080/00036811.2010.495341. |
[3] |
J. W. Barrett and W. B. Liu,
Finite element approximation of the parabolic $p$-laplacian, SIAM Journal on Numerical Analysis, 31 (1994), 413-428.
doi: 10.1137/0731022. |
[4] |
H. Bessaih, S. Trabelsi and H. Zorgati,
Existence and uniqueness of global solutions for the modified anisotropic 3D Navier-Stokes equations, ESAIM: Mathematical Modelling and Numerical Analysis, 50 (2016), 1817-1823.
doi: 10.1051/m2an/2016008. |
[5] |
X. J. Cai and Q. S. Jiu,
Weak and strong solutions for the incompressible Navier-Stokes equations with damping, Journal of Mathematical Analysis and Applications, 343 (2008), 799-809.
doi: 10.1016/j.jmaa.2008.01.041. |
[6] |
C. S. Cao and J. H. Wu,
Two regularity criteria for the 3D MHD equations, Journal of Differential Equations, 248 (2010), 2263-2274.
doi: 10.1016/j.jde.2009.09.020. |
[7] |
A. O. Çelebi, V. Kalantarov and D. U$\widetilde {\rm{g}}$gurlu,
Continuous dependence for the convective Brinkman-Forchheimer equations, Applicable Analysis, 84 (2005), 877-888.
doi: 10.1080/00036810500148911. |
[8] |
A. O. Çelebi, V. K. Kalantarov and D. Uǧurlu,
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. |
[9] |
P. Constantin and C. Foias, Navier-Stokes Equations, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1988.
![]() |
[10] |
J. É. J. Dupuit, Études Théoriques et Pratiques sur le Mouvement des eaux dans les Canaux Découverts et à Travers les Terrains Perméables: avec des Considérations Relatives au Régime des Grandes eaux, au Débouché à leur Donner, et à la Marche des Alluvions dans les Rivières à Fond Mobile, Dunod, 1863. Google Scholar |
[11] |
G. Duvaut and J. L. Lions,
Inéquations en thermoélasticité et magnétohydrodynamique, Archive for Rational Mechanics and Analysis, 46 (1972), 241-279.
doi: 10.1007/BF00250512. |
[12] |
P. Forchheimer, Wasserbewegung durch boden, Zeitz. Ver. Duetch Ing., 45 (1901), 1782-1788. Google Scholar |
[13] |
M. Fourar, G. Radilla, R. Lenormand and C. Moyne, On the non-linear behavior of a laminar single-phase flow through two and three-dimensional porous media, Advances in Water Resources, 27 (2004), 669-677. Google Scholar |
[14] |
C. He and Z. P. Xin,
On the regularity of weak solutions to the magnetohydrodynamic equations, Journal of Differential Equations, 213 (2005), 235-254.
doi: 10.1016/j.jde.2004.07.002. |
[15] |
C. He and Z. P. Xin,
Partial regularity of suitable weak solutions to the incompressible magnetohydrodynamic equations, Journal of Functional Analysis, 227 (2005), 113-152.
doi: 10.1016/j.jfa.2005.06.009. |
[16] |
C. Hsu and P. Cheng, Thermal dispersion in a porous medium, International Journal of Heat and Mass Transfer, 33 (1990), 1587-1597. Google Scholar |
[17] |
V. Kalantarov and S. Zelik,
Smooth attractors for the Brinkman-Forchheimer equations with fast growing nonlinearities, Communications on Pure and Applied Analysis, 11 (2012), 2037-2054.
doi: 10.3934/cpaa.2012.11.2037. |
[18] |
K. Kang and J. Lee,
Interior regularity criteria for suitable weak solutions of the magnetohydrodynamic equations, Journal of Differential Equations, 247 (2009), 2310-2330.
doi: 10.1016/j.jde.2009.07.016. |
[19] |
V. A. Liskevich and Y. A. Semenov, Some problems on markov semigroups, Schrödinger Operators, Markov Semigroups, Wavelet Analysis, Operator Algebras, Math. Top., Adv. Partial Differential Equations, Akademie Verlag, Berlin, 11 (1996), 163–217. |
[20] |
Y. Liu and C. H. Lin, Structural stability for Brinkman-Forchheimer equations, Electronic Journal of Differential Equations, 2007 (2007), 8 pp. |
[21] |
M. Louaked, N. Seloula, S. Y. Sun and S. Trabelsi,
A pseudocompressibility method for the incompressible Brinkman-Forchheimer equations, Differential and Integral Equations, 28 (2015), 361-382.
|
[22] |
M. Louaked, N. Seloula and S. Trabelsi,
Approximation of the unsteady Brinkman-Forchheimer equations by the pressure stabilization method, Numerical Methods for Partial Differential Equations, 33 (2017), 1949-1965.
doi: 10.1002/num.22173. |
[23] |
P. A. Markowich, E. S. Titi and S. Trabelsi,
Continuous data assimilation for the three-dimensional Brinkman-Forchheimer-extended Darcy model, Nonlinearity, 29 (2016), 1292-1328.
doi: 10.1088/0951-7715/29/4/1292. |
[24] |
J. E. McClure, W. G. Gray and C. T. Miller,
Beyond anisotropy: Examining non-darcy flow in asymmetric porous media, Transp. Porous Media, 84 (2010), 535-548.
doi: 10.1007/s11242-009-9518-7. |
[25] |
D. Nield, The limitations of the Brinkman-Forchheimer equation in modeling flow in a saturated porous medium and at an interface, International Journal of Heat and Fluid Flow, 12 (1991), 269-272. Google Scholar |
[26] |
Y. Ouyang and L.-e. Yang,
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. |
[27] |
M. Panfilov and M. Fourar, Physical splitting of nonlinear effects in high-velocity stable flow through porous media, Advances in Water Resources, 29 (2006), 30-41. Google Scholar |
[28] |
L. E. Payne and B. Straughan,
Convergence and continuous dependence for the Brinkman-Forchheimer equations, Studies in Applied Mathematics, 102 (1999), 419-439.
doi: 10.1111/1467-9590.00116. |
[29] |
M. Sermange and R. Temam,
Some mathematical questions related to the MHD equations, Communications on Pure and Applied Mathematics, 36 (1983), 635-664.
doi: 10.1002/cpa.3160360506. |
[30] |
B. Straughan, Stability and Wave Motion in Porous Media, Applied Mathematical Sciences, 165. Springer, New York, 2008. |
[31] |
R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Applied Mathematical Sciences, 68. Springer-Verlag, New York, 1988.
doi: 10.1007/978-1-4684-0313-8. |
[32] |
S. Trabelsi, Global well-posedness of a 3D Forchheimer-Bénard convection model in porous media, Submitted, (2018). Google Scholar |
[33] |
D. Uǧurlu,
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. |
[34] |
K. Vafai and R. Thiyagaraja, Analysis of flow and heat transfer at the interface region of a porous medium, International Journal of Heat and Mass Transfer, 30 (1987), 1391-1405. Google Scholar |
[35] |
B. X. Wang and S. Y. Lin,
Existence of global attractors for the three-dimensional Brinkman-Forchheimer equation, Mathematical Methods in the Applied Sciences, 31 (2008), 1479-1495.
doi: 10.1002/mma.985. |
[36] |
Y. C. You, C. D. Zhao and S. F. Zhou,
The existence of uniform attractors for 3D Brinkman-Forchheimer equations, Discrete & Continuous Dynamical Systems-A, 32 (2012), 3787-3800.
doi: 10.3934/dcds.2012.32.3787. |
[37] |
Z. J. Zhang, D. X. Zhong, S. J. Gao and S. L. Qiu,
Fundamental Serrin type regularity criteria for 3D MHD fluid passing through the porous medium, Applicable Analysis, 96 (2017), 2130-2139.
doi: 10.2298/FIL1705287Z. |
[38] |
Z. J. Zhang,
Refined regularity criteria for the MHD system involving only two components of the solution, Appl. Anal., 96 (2017), 2130-2139.
doi: 10.1080/00036811.2016.1207245. |
[39] |
Z. J. Zhang, C. P. Wu and Z.-A. Yao,
Remarks on global regularity for the 3D MHD system with damping, Applied Mathematics and Computation, 333 (2018), 1-7.
doi: 10.1016/j.amc.2018.03.047. |
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