December  2019, 11(4): 621-637. doi: 10.3934/jgm.2019031

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

* Corresponding author

This work is dedicated to Professor Darryl Holm on the occasion of his 70th birthday

Received  May 2018 Revised  March 2019 Published  November 2019

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.

Citation: Edriss S. Titi, Saber Trabelsi. Global well-posedness of a 3D MHD model in porous media. Journal of Geometric Mechanics, 2019, 11 (4) : 621-637. doi: 10.3934/jgm.2019031
References:
[1]

S. Agmon, Lectures on Elliptic Boundary Value Problems, AMS Chelsea Publishing, Providence, RI, 2010. doi: 10.1090/chel/369.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

H. BessaihS. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

A. O. ÇelebiV. 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.  Google Scholar

[8]

A. O. ÇelebiV. 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.  Google Scholar

[9] P. Constantin and C. Foias, Navier-Stokes Equations, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1988.   Google Scholar
[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.  Google Scholar

[12]

P. Forchheimer, Wasserbewegung durch boden, Zeitz. Ver. Duetch Ing., 45 (1901), 1782-1788.   Google Scholar

[13]

M. FourarG. RadillaR. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

Y. Liu and C. H. Lin, Structural stability for Brinkman-Forchheimer equations, Electronic Journal of Differential Equations, 2007 (2007), 8 pp.  Google Scholar

[21]

M. LouakedN. SeloulaS. Y. Sun and S. Trabelsi, A pseudocompressibility method for the incompressible Brinkman-Forchheimer equations, Differential and Integral Equations, 28 (2015), 361-382.   Google Scholar

[22]

M. LouakedN. 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.  Google Scholar

[23]

P. A. MarkowichE. 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.  Google Scholar

[24]

J. E. McClureW. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[30]

B. Straughan, Stability and Wave Motion in Porous Media, Applied Mathematical Sciences, 165. Springer, New York, 2008.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[36]

Y. C. YouC. 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.  Google Scholar

[37]

Z. J. ZhangD. X. ZhongS. 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.  Google Scholar

[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.  Google Scholar

[39]

Z. J. ZhangC. 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.  Google Scholar

show all references

References:
[1]

S. Agmon, Lectures on Elliptic Boundary Value Problems, AMS Chelsea Publishing, Providence, RI, 2010. doi: 10.1090/chel/369.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

H. BessaihS. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

A. O. ÇelebiV. 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.  Google Scholar

[8]

A. O. ÇelebiV. 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.  Google Scholar

[9] P. Constantin and C. Foias, Navier-Stokes Equations, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1988.   Google Scholar
[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.  Google Scholar

[12]

P. Forchheimer, Wasserbewegung durch boden, Zeitz. Ver. Duetch Ing., 45 (1901), 1782-1788.   Google Scholar

[13]

M. FourarG. RadillaR. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

Y. Liu and C. H. Lin, Structural stability for Brinkman-Forchheimer equations, Electronic Journal of Differential Equations, 2007 (2007), 8 pp.  Google Scholar

[21]

M. LouakedN. SeloulaS. Y. Sun and S. Trabelsi, A pseudocompressibility method for the incompressible Brinkman-Forchheimer equations, Differential and Integral Equations, 28 (2015), 361-382.   Google Scholar

[22]

M. LouakedN. 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.  Google Scholar

[23]

P. A. MarkowichE. 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.  Google Scholar

[24]

J. E. McClureW. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[30]

B. Straughan, Stability and Wave Motion in Porous Media, Applied Mathematical Sciences, 165. Springer, New York, 2008.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[36]

Y. C. YouC. 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.  Google Scholar

[37]

Z. J. ZhangD. X. ZhongS. 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.  Google Scholar

[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.  Google Scholar

[39]

Z. J. ZhangC. 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.  Google Scholar

[1]

Yuncheng You, Caidi Zhao, Shengfan Zhou. The existence of uniform attractors for 3D Brinkman-Forchheimer equations. Discrete & Continuous Dynamical Systems - A, 2012, 32 (10) : 3787-3800. doi: 10.3934/dcds.2012.32.3787

[2]

Chongsheng Cao. Sufficient conditions for the regularity to the 3D Navier-Stokes equations. Discrete & Continuous Dynamical Systems - A, 2010, 26 (4) : 1141-1151. doi: 10.3934/dcds.2010.26.1141

[3]

Xuanji Jia, Zaihong Jiang. An anisotropic regularity criterion for the 3D Navier-Stokes equations. Communications on Pure & Applied Analysis, 2013, 12 (3) : 1299-1306. doi: 10.3934/cpaa.2013.12.1299

[4]

Hui Chen, Daoyuan Fang, Ting Zhang. Regularity of 3D axisymmetric Navier-Stokes equations. Discrete & Continuous Dynamical Systems - A, 2017, 37 (4) : 1923-1939. doi: 10.3934/dcds.2017081

[5]

Vladimir V. Chepyzhov, E. S. Titi, Mark I. Vishik. On the convergence of solutions of the Leray-$\alpha $ model to the trajectory attractor of the 3D Navier-Stokes system. Discrete & Continuous Dynamical Systems - A, 2007, 17 (3) : 481-500. doi: 10.3934/dcds.2007.17.481

[6]

Thomas Y. Hou, Ruo Li. Nonexistence of locally self-similar blow-up for the 3D incompressible Navier-Stokes equations. Discrete & Continuous Dynamical Systems - A, 2007, 18 (4) : 637-642. doi: 10.3934/dcds.2007.18.637

[7]

Daoyuan Fang, Chenyin Qian. Regularity criterion for 3D Navier-Stokes equations in Besov spaces. Communications on Pure & Applied Analysis, 2014, 13 (2) : 585-603. doi: 10.3934/cpaa.2014.13.585

[8]

Yong Yang, Bingsheng Zhang. On the Kolmogorov entropy of the weak global attractor of 3D Navier-Stokes equations:Ⅰ. Discrete & Continuous Dynamical Systems - B, 2017, 22 (6) : 2339-2350. doi: 10.3934/dcdsb.2017101

[9]

Zoran Grujić. Regularity of forward-in-time self-similar solutions to the 3D Navier-Stokes equations. Discrete & Continuous Dynamical Systems - A, 2006, 14 (4) : 837-843. doi: 10.3934/dcds.2006.14.837

[10]

G. Deugoué, T. Tachim Medjo. The Stochastic 3D globally modified Navier-Stokes equations: Existence, uniqueness and asymptotic behavior. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2593-2621. doi: 10.3934/cpaa.2018123

[11]

Milan Pokorný, Piotr B. Mucha. 3D steady compressible Navier--Stokes equations. Discrete & Continuous Dynamical Systems - S, 2008, 1 (1) : 151-163. doi: 10.3934/dcdss.2008.1.151

[12]

Andrei Fursikov. Local existence theorems with unbounded set of input data and unboundedness of stable invariant manifolds for 3D Navier-Stokes equations. Discrete & Continuous Dynamical Systems - S, 2010, 3 (2) : 269-289. doi: 10.3934/dcdss.2010.3.269

[13]

Huicheng Yin, Lin Zhang. The global existence and large time behavior of smooth compressible fluid in an infinitely expanding ball, Ⅱ: 3D Navier-Stokes equations. Discrete & Continuous Dynamical Systems - A, 2018, 38 (3) : 1063-1102. doi: 10.3934/dcds.2018045

[14]

A. V. Fursikov. Stabilization for the 3D Navier-Stokes system by feedback boundary control. Discrete & Continuous Dynamical Systems - A, 2004, 10 (1&2) : 289-314. doi: 10.3934/dcds.2004.10.289

[15]

Kuanysh A. Bekmaganbetov, Gregory A. Chechkin, Vladimir V. Chepyzhov, Andrey Yu. Goritsky. Homogenization of trajectory attractors of 3D Navier-Stokes system with randomly oscillating force. Discrete & Continuous Dynamical Systems - A, 2017, 37 (5) : 2375-2393. doi: 10.3934/dcds.2017103

[16]

Jingrui Wang, Keyan Wang. Almost sure existence of global weak solutions to the 3D incompressible Navier-Stokes equation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (9) : 5003-5019. doi: 10.3934/dcds.2017215

[17]

Luigi C. Berselli. An elementary approach to the 3D Navier-Stokes equations with Navier boundary conditions: Existence and uniqueness of various classes of solutions in the flat boundary case.. Discrete & Continuous Dynamical Systems - S, 2010, 3 (2) : 199-219. doi: 10.3934/dcdss.2010.3.199

[18]

Chao Deng, Xiaohua Yao. Well-posedness and ill-posedness for the 3D generalized Navier-Stokes equations in $\dot{F}^{-\alpha,r}_{\frac{3}{\alpha-1}}$. Discrete & Continuous Dynamical Systems - A, 2014, 34 (2) : 437-459. doi: 10.3934/dcds.2014.34.437

[19]

Fabio Ramos, Edriss S. Titi. Invariant measures for the $3$D Navier-Stokes-Voigt equations and their Navier-Stokes limit. Discrete & Continuous Dynamical Systems - A, 2010, 28 (1) : 375-403. doi: 10.3934/dcds.2010.28.375

[20]

T. Tachim Medjo. Averaging of a 3D Lagrangian averaged Navier-Stokes-$\alpha$ model with oscillating external forces. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1281-1305. doi: 10.3934/cpaa.2011.10.1281

2018 Impact Factor: 0.525

Article outline

[Back to Top]