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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 & 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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