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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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. Google Scholar [17] G. Duvaut and J. L. Lions, Inéquations en thermoelasticité et magnétohydrodynamique,, Arch. Ration. Mech. Anal., 46 (1972), 241. 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. 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. 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. 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). doi: 10.1063/1.2360145. Google Scholar [22] J. L. Lions and E. Magenes, "Problèmes aux limites non homogènes et applications,", Vol.1 Dunod, (1968). Google Scholar [23] J. L. Lions, "Quelques méthodes de résolutions des problèmes aux limites non linéaires,", Dunod et Gauthier-Villars, (1969). Google Scholar [24] M. Sango, Magnetohydrodynamic turbulent flows: Existence results,, Physica D, 239 (2010), 912. 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. 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). 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. doi: 10.1002/cpa.3160360506. Google Scholar [28] R. Temam, "Navier-Stokes Equations, Theory and Numerical Analysis,", AMS-Chelsea Series, (2001). Google Scholar [29] R.Temam, "Infinite Dimensional Dynamical Systems in Mechanics and Physics,", 2nd ed., (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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. Google Scholar [17] G. Duvaut and J. L. Lions, Inéquations en thermoelasticité et magnétohydrodynamique,, Arch. Ration. Mech. Anal., 46 (1972), 241. 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. 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. 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. 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). doi: 10.1063/1.2360145. Google Scholar [22] J. L. Lions and E. Magenes, "Problèmes aux limites non homogènes et applications,", Vol.1 Dunod, (1968). Google Scholar [23] J. L. Lions, "Quelques méthodes de résolutions des problèmes aux limites non linéaires,", Dunod et Gauthier-Villars, (1969). Google Scholar [24] M. Sango, Magnetohydrodynamic turbulent flows: Existence results,, Physica D, 239 (2010), 912. 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. 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). 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. doi: 10.1002/cpa.3160360506. Google Scholar [28] R. Temam, "Navier-Stokes Equations, Theory and Numerical Analysis,", AMS-Chelsea Series, (2001). Google Scholar [29] R.Temam, "Infinite Dimensional Dynamical Systems in Mechanics and Physics,", 2nd ed., (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. 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. doi: 10.1016/j.jde.2006.07.005. Google Scholar
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