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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

 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.
Citation: Gabriel Deugoue. Approximation of the trajectory attractor of the 3D MHD System. Communications on Pure & 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [11] V. V. Chepyzhov and M. I. Vishik, "Attractors for Equations of Mathematical Physics," AMS Colloquium Publications , 2002.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [16] Y. A. Dubinskii, Weak convergence in nonlinear elliptic and parabolic equations, Mat. Sbornik, 4 (1965), 609-642.  Google Scholar [17] G. Duvaut and J. L. Lions, Inéquations en thermoelasticité et magnétohydrodynamique, Arch. Ration. Mech. Anal., 46 (1972), 241-279. doi: 10.1007/BF00250512.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [22] J. L. Lions and E. Magenes, "Problèmes aux limites non homogènes et applications," Vol.1 Dunod, Paris, 1968.  Google Scholar [23] J. L. Lions, "Quelques méthodes de résolutions des problèmes aux limites non linéaires," Dunod et Gauthier-Villars, Paris, 1969.  Google Scholar [24] M. Sango, Magnetohydrodynamic turbulent flows: Existence results, Physica D, 239 (2010), 912-923. doi: 10.1016/j.physd.2010.01.009.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [28] R. Temam, "Navier-Stokes Equations, Theory and Numerical Analysis," AMS-Chelsea Series, AMS, Providence, 2001.  Google Scholar [29] R.Temam, "Infinite Dimensional Dynamical Systems in Mechanics and Physics," 2nd ed., Springer, Berlin, 1997.  Google Scholar [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.  Google Scholar [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.  Google Scholar

show all references

##### References:
 [1] J. P. Aubin, Un théorème de compacité, C.R. Acad. Sci. Paris, 256 (1963), 5042-5044.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [11] V. V. Chepyzhov and M. I. Vishik, "Attractors for Equations of Mathematical Physics," AMS Colloquium Publications , 2002.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [16] Y. A. Dubinskii, Weak convergence in nonlinear elliptic and parabolic equations, Mat. Sbornik, 4 (1965), 609-642.  Google Scholar [17] G. Duvaut and J. L. Lions, Inéquations en thermoelasticité et magnétohydrodynamique, Arch. Ration. Mech. Anal., 46 (1972), 241-279. doi: 10.1007/BF00250512.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [22] J. L. Lions and E. Magenes, "Problèmes aux limites non homogènes et applications," Vol.1 Dunod, Paris, 1968.  Google Scholar [23] J. L. Lions, "Quelques méthodes de résolutions des problèmes aux limites non linéaires," Dunod et Gauthier-Villars, Paris, 1969.  Google Scholar [24] M. Sango, Magnetohydrodynamic turbulent flows: Existence results, Physica D, 239 (2010), 912-923. doi: 10.1016/j.physd.2010.01.009.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [28] R. Temam, "Navier-Stokes Equations, Theory and Numerical Analysis," AMS-Chelsea Series, AMS, Providence, 2001.  Google Scholar [29] R.Temam, "Infinite Dimensional Dynamical Systems in Mechanics and Physics," 2nd ed., Springer, Berlin, 1997.  Google Scholar [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.  Google Scholar [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.  Google Scholar
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