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September  2013, 12(5): 2119-2144. doi: 10.3934/cpaa.2013.12.2119

Approximation of the trajectory attractor of the 3D MHD System

Gabriel Deugoue 1,

1. 

Department of Mathematics and Computer Science, University of Dschang, Cameroon

Received  May 2012 Revised  October 2012 Published  January 2013

We study the connection between the long-time dynamics of the 3D magnetohydrodynamic-$\alpha$ model and the exact 3D magnetohydrodynamic system. We prove that the trajectory attractor $U_\alpha$ of the 3D magnetohydrodynamic-$\alpha$ model converges to the trajectory attractor $U_0$ of the 3D magnetohydrodynamic system (in an appropriate topology) when $\alpha$ approaches zero.
Keywords: Navier-Stokes equations, convergence., Magnetohydrodynamic-alpha model, trajectory attractor, magnetohydrodynamic equations.
Mathematics Subject Classification: Primary: 35B40, 35B41; Secondary: 35B4.
Citation: Gabriel Deugoue. Approximation of the trajectory attractor of the 3D MHD System. Communications on Pure and Applied Analysis, 2013, 12 (5) : 2119-2144. doi: 10.3934/cpaa.2013.12.2119
References:
[1]

J. P. Aubin, Un théorème de compacité, C.R. Acad. Sci. Paris, 256 (1963), 5042-5044.

[2]

J. M. Ball, Continuity properties of global attractors of generalized semiflows and the Navier-Stokes equations, J. Nonlinear Sci., 7 (1997), 475-502. doi: 10.1007/s003329900037.

[3]

T. Caraballo, J. Langa and J. Valero, Global attractors for multivalued random dynamical systems, Nonlinear Anal., 48 (2002), 805-829. doi: 10.1016/S0362-546X(00)00216-9.

[4]

S. Chen, C. Foias, D. D. Holm, E. Oslon, E. S. Titi and S. Wynne, The Camassa-Holm equations as a closure model for turbulent channel and pipe flow, Phys. Rev. Lett., 81 (1998), 5338-5341. doi: 10.1103/PhysRevLett.81.5338.

[5]

S. Chen, C. Foias, D. D. Holm, E. Oslon, E. S. Titi and S. Wynne, A connection between the Camassa-Holm equations and turbulent flows in pipes and channels, Phys. Fluids, 11 (1999), 2343-2353. doi: 10.1063/1.870096.

[6]

S. Chen, C. Foias, D. D. Holm, E. Oslon, E. S. Titi and S. Wynne, The Camassa- Holm equations and turbulence, Physica D, 133 (1999), 49-65. doi: 10.1016/S0167-2789(99)00098-6.

[7]

S. Chen, D. D. Holm, L. G. Margolin, and R. Zhang, Direct numerical simulations of the Navier-Stokes-alpha model, Physica D, 133 (1999), 66-83. doi: 10.1016/S0167-2789(99)00099-8.

[8]

V. V. Chepyzhov and M. I. Vishik, Trajectory attractors for evolution equations, C.R. Acad. Sci. Paris Series I, 10 (1995), 1309-1314 . doi: 10.1016/S0021-7824(97)89978-3.

[9]

V. V. Chepyzhov and M. I. Vishik, Evolution equations and their trajectory attractors, J. Math. Pures Appl., 10 (1997), 913-964 . doi: 10.1016/S0021-7824(97)89978-3.

[10]

V. V. Chepyzhov and M. I. Vishik, Trajectory and global attractors of three-dimensional Navier-Stokes systems, Math. Notes, 71 (2002), 177-193. doi: 10.1023/A:1014190629738.

[11]

V. V. Chepyzhov and M. I. Vishik, "Attractors for Equations of Mathematical Physics," AMS Colloquium Publications , 2002.

[12]

V. V. Chepyzhov, E. S. Titi and M. I. Vishik, On the convergence of trajectory attractors of 3D Navier-Stokes-$\alpha$ model as alpha approaches 0, Mat. Sb., 198 (2007), 3-36. doi: 10.1070/SM2007v198n12ABEH003902.

[13]

V. V. Chepyzhov, E. S. Titi and M. I. Vishik, On the convergence of solutions of the Leray-$\alpha$ model to the trajectory attractor of the 3D Navier-Stokes system, Discrete Contin. Dyn. Syst., 17 (2007), 33-52. doi: 10.3934/dcds.2007.17.481.

[14]

V. V. Chepyzhov, E. S. Titi and M. I. Vishik, Trajectory attractor approximation of the 3D Navier-Stokes by a Leray-$\alpha$ model, Doklady Mathematics, 71 (2005), 92-95.

[15]

G. Deugoue, P. A. Razafimandimby and M. Sango, On the 3D stochastic magnetohydrodynamic-$\alpha$ model, Stochastic Processes and their Applications, 122 (2012), 2211-2248. doi: 10.1016/j.spa.2012.03.002.

[16]

Y. A. Dubinskii, Weak convergence in nonlinear elliptic and parabolic equations, Mat. Sbornik, 4 (1965), 609-642.

[17]

G. Duvaut and J. L. Lions, Inéquations en thermoelasticité et magnétohydrodynamique, Arch. Ration. Mech. Anal., 46 (1972), 241-279. doi: 10.1007/BF00250512.

[18]

C. Foias, D. D. Holm and E. S. Titi, The Navier-Stokes-alpha model of fluid turbulence, Physica D, 153 (2001), 505-519. doi: 10.1016/S0167-2789(01)00191-9.

[19]

C. Foias, D. D. Holm and E. S. Titi, The three dimensional viscous Camassa-Holm equations, and their relation to the Navier-Stokes equations and turbulence theory, Journal of Dynamics and Differential Equations, 14 (2002), 1-35. doi: 10.1023/A:1012984210582.

[20]

A. Kapustyan and J. Valero, Weak and strong attractors for the 3D Navier-Stokes system, J. Differential equations, 240 (2007), 249-278. doi: 10.1016/j.jde.2007.06.008.

[21]

J. S. Linshiz and E. S. Titi, Analytical study of certain magnetohydrodynamic-$\alpha$ models, J. Math. Phys., 48 (2007), 28pp. doi: 10.1063/1.2360145.

[22]

J. L. Lions and E. Magenes, "Problèmes aux limites non homogènes et applications," Vol.1 Dunod, Paris, 1968.

[23]

J. L. Lions, "Quelques méthodes de résolutions des problèmes aux limites non linéaires," Dunod et Gauthier-Villars, Paris, 1969.

[24]

M. Sango, Magnetohydrodynamic turbulent flows: Existence results, Physica D, 239 (2010), 912-923. doi: 10.1016/j.physd.2010.01.009.

[25]

V. Melnik and J. Valero, On attractors of multivalued semiflows and differential inclusions, Set-Valued Anal., 8 (2000), 375-403. doi: 10.1023/A:1008608431399.

[26]

P. D. Mininni, D. C. Montgomery and A. G. Pouquet, Numerical solutions of the three-dimensional magnetohydrodynamic alpha-model, Phys. Rev. E, 71 (2005), 046304. doi: 10.1103/PhysRevE.71.046304.

[27]

M. Sermange and R. Temam, Some mathematical questions related to the MHD equations, Comm. Pure Appl. Math., 36 (1983), 635-664. doi: 10.1002/cpa.3160360506.

[28]

R. Temam, "Navier-Stokes Equations, Theory and Numerical Analysis," AMS-Chelsea Series, AMS, Providence, 2001.

[29]

R.Temam, "Infinite Dimensional Dynamical Systems in Mechanics and Physics," 2nd ed., Springer, Berlin, 1997.

[30]

M. I. Vishik and V. V. Chepyzhov, Trajectory attractor and global attractors of three-dimensional Navier-Stokes systems, Mathematical Notes, 71 (2002), 177-193. doi: 10.1023/A:1014190629738.

[31]

Y. Wang and S. Zhou, Kernel sections and uniform attractors of multi-valued processes, J. Differential Equations, 232 (2007), 573-622. doi: 10.1016/j.jde.2006.07.005.

show all references

References:
[1]

J. P. Aubin, Un théorème de compacité, C.R. Acad. Sci. Paris, 256 (1963), 5042-5044.

[2]

J. M. Ball, Continuity properties of global attractors of generalized semiflows and the Navier-Stokes equations, J. Nonlinear Sci., 7 (1997), 475-502. doi: 10.1007/s003329900037.

[3]

T. Caraballo, J. Langa and J. Valero, Global attractors for multivalued random dynamical systems, Nonlinear Anal., 48 (2002), 805-829. doi: 10.1016/S0362-546X(00)00216-9.

[4]

S. Chen, C. Foias, D. D. Holm, E. Oslon, E. S. Titi and S. Wynne, The Camassa-Holm equations as a closure model for turbulent channel and pipe flow, Phys. Rev. Lett., 81 (1998), 5338-5341. doi: 10.1103/PhysRevLett.81.5338.

[5]

S. Chen, C. Foias, D. D. Holm, E. Oslon, E. S. Titi and S. Wynne, A connection between the Camassa-Holm equations and turbulent flows in pipes and channels, Phys. Fluids, 11 (1999), 2343-2353. doi: 10.1063/1.870096.

[6]

S. Chen, C. Foias, D. D. Holm, E. Oslon, E. S. Titi and S. Wynne, The Camassa- Holm equations and turbulence, Physica D, 133 (1999), 49-65. doi: 10.1016/S0167-2789(99)00098-6.

[7]

S. Chen, D. D. Holm, L. G. Margolin, and R. Zhang, Direct numerical simulations of the Navier-Stokes-alpha model, Physica D, 133 (1999), 66-83. doi: 10.1016/S0167-2789(99)00099-8.

[8]

V. V. Chepyzhov and M. I. Vishik, Trajectory attractors for evolution equations, C.R. Acad. Sci. Paris Series I, 10 (1995), 1309-1314 . doi: 10.1016/S0021-7824(97)89978-3.

[9]

V. V. Chepyzhov and M. I. Vishik, Evolution equations and their trajectory attractors, J. Math. Pures Appl., 10 (1997), 913-964 . doi: 10.1016/S0021-7824(97)89978-3.

[10]

V. V. Chepyzhov and M. I. Vishik, Trajectory and global attractors of three-dimensional Navier-Stokes systems, Math. Notes, 71 (2002), 177-193. doi: 10.1023/A:1014190629738.

[11]

V. V. Chepyzhov and M. I. Vishik, "Attractors for Equations of Mathematical Physics," AMS Colloquium Publications , 2002.

[12]

V. V. Chepyzhov, E. S. Titi and M. I. Vishik, On the convergence of trajectory attractors of 3D Navier-Stokes-$\alpha$ model as alpha approaches 0, Mat. Sb., 198 (2007), 3-36. doi: 10.1070/SM2007v198n12ABEH003902.

[13]

V. V. Chepyzhov, E. S. Titi and M. I. Vishik, On the convergence of solutions of the Leray-$\alpha$ model to the trajectory attractor of the 3D Navier-Stokes system, Discrete Contin. Dyn. Syst., 17 (2007), 33-52. doi: 10.3934/dcds.2007.17.481.

[14]

V. V. Chepyzhov, E. S. Titi and M. I. Vishik, Trajectory attractor approximation of the 3D Navier-Stokes by a Leray-$\alpha$ model, Doklady Mathematics, 71 (2005), 92-95.

[15]

G. Deugoue, P. A. Razafimandimby and M. Sango, On the 3D stochastic magnetohydrodynamic-$\alpha$ model, Stochastic Processes and their Applications, 122 (2012), 2211-2248. doi: 10.1016/j.spa.2012.03.002.

[16]

Y. A. Dubinskii, Weak convergence in nonlinear elliptic and parabolic equations, Mat. Sbornik, 4 (1965), 609-642.

[17]

G. Duvaut and J. L. Lions, Inéquations en thermoelasticité et magnétohydrodynamique, Arch. Ration. Mech. Anal., 46 (1972), 241-279. doi: 10.1007/BF00250512.

[18]

C. Foias, D. D. Holm and E. S. Titi, The Navier-Stokes-alpha model of fluid turbulence, Physica D, 153 (2001), 505-519. doi: 10.1016/S0167-2789(01)00191-9.

[19]

C. Foias, D. D. Holm and E. S. Titi, The three dimensional viscous Camassa-Holm equations, and their relation to the Navier-Stokes equations and turbulence theory, Journal of Dynamics and Differential Equations, 14 (2002), 1-35. doi: 10.1023/A:1012984210582.

[20]

A. Kapustyan and J. Valero, Weak and strong attractors for the 3D Navier-Stokes system, J. Differential equations, 240 (2007), 249-278. doi: 10.1016/j.jde.2007.06.008.

[21]

J. S. Linshiz and E. S. Titi, Analytical study of certain magnetohydrodynamic-$\alpha$ models, J. Math. Phys., 48 (2007), 28pp. doi: 10.1063/1.2360145.

[22]

J. L. Lions and E. Magenes, "Problèmes aux limites non homogènes et applications," Vol.1 Dunod, Paris, 1968.

[23]

J. L. Lions, "Quelques méthodes de résolutions des problèmes aux limites non linéaires," Dunod et Gauthier-Villars, Paris, 1969.

[24]

M. Sango, Magnetohydrodynamic turbulent flows: Existence results, Physica D, 239 (2010), 912-923. doi: 10.1016/j.physd.2010.01.009.

[25]

V. Melnik and J. Valero, On attractors of multivalued semiflows and differential inclusions, Set-Valued Anal., 8 (2000), 375-403. doi: 10.1023/A:1008608431399.

[26]

P. D. Mininni, D. C. Montgomery and A. G. Pouquet, Numerical solutions of the three-dimensional magnetohydrodynamic alpha-model, Phys. Rev. E, 71 (2005), 046304. doi: 10.1103/PhysRevE.71.046304.

[27]

M. Sermange and R. Temam, Some mathematical questions related to the MHD equations, Comm. Pure Appl. Math., 36 (1983), 635-664. doi: 10.1002/cpa.3160360506.

[28]

R. Temam, "Navier-Stokes Equations, Theory and Numerical Analysis," AMS-Chelsea Series, AMS, Providence, 2001.

[29]

R.Temam, "Infinite Dimensional Dynamical Systems in Mechanics and Physics," 2nd ed., Springer, Berlin, 1997.

[30]

M. I. Vishik and V. V. Chepyzhov, Trajectory attractor and global attractors of three-dimensional Navier-Stokes systems, Mathematical Notes, 71 (2002), 177-193. doi: 10.1023/A:1014190629738.

[31]

Y. Wang and S. Zhou, Kernel sections and uniform attractors of multi-valued processes, J. Differential Equations, 232 (2007), 573-622. doi: 10.1016/j.jde.2006.07.005.

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