# American Institue of Mathematical Sciences

2018, 38(3): 1461-1477. doi: 10.3934/dcds.2018060

## Pullback attractor and invariant measures for the three-dimensional regularized MHD equations

 1 State Key Laboratory of Geomechanics and Geotechnical Engineering, Institute of Rock and Soil Mechanics, Chinese Academy of Sciences, Wuhan 430071, China 2 Department of Mathematics and Information Sciences, Wenzhou University, Wenzhou 325035, China

* Corresponding author: Caidi Zhao

Received  May 2017 Published  December 2017

This article studies the three-dimensional regularized Magnetohydrodynamics (MHD) equations. Using the approach of energy equations, the authors prove that the associated process possesses a pullback attractor. Then they establish the unique existence of the family of invariant Borel probability measures which is supported by the pullback attractor.

Citation: Zeqi Zhu, Caidi Zhao. Pullback attractor and invariant measures for the three-dimensional regularized MHD equations. Discrete & Continuous Dynamical Systems - A, 2018, 38 (3) : 1461-1477. doi: 10.3934/dcds.2018060
##### References:
 [1] J. M. Ball, Global attractors for damped semilinear wave equations, Discrete Cont. Dyna. Syst.-A, 10 (2004), 31-52. [2] C. Cao, J. Wu, Two regularity criteria for the 3D MHD equations, J. Differential Equations, 248 (2010), 2263-2274. doi: 10.1016/j.jde.2009.09.020. [3] T. Caraballo, G. Lukaszewicz, J. Real, Pullback attractors for asymptotically compact non-autonomous dynamical system, Nonlinear Anal., 64 (2006), 484-498. doi: 10.1016/j.na.2005.03.111. [4] T. Caraballo, G. Lukaszewicz, J. Real, Invariant measures and statitical solutions of the globally modified Navier-Stokes equations, Discrete Cont. Dyna. Syst.-B, 10 (2008), 761-781. doi: 10.3934/dcdsb.2008.10.761. [5] A. N. Carvalho, J. A. Langa and J. C. Robinson, Attractors for Infinite-Dimensional NonAutonomous Dynamical Systems, Springer, New York, 2013. [6] D. Catania, P. Secchi, Global existence and finite dimensional global attractor for a 3D double viscous MHD-α model, Comm. Math. Sci., 8 (2010), 1021-1040. doi: 10.4310/CMS.2010.v8.n4.a12. [7] D. Catania, Global attractor and determining modes for a hyperbolic MHD turbulence Model, J. of Turbulence, 12 (2011), 1-20. [8] D. Catania, P. Secchi, Global existence for two regularized MHD models in three space-dimension, Port. Math., 68 (2011), 41-52. [9] M. Chekroun, N. E. Glatt-Holtz, Invariant measures for dissipative dynamical systems: Abstract results and applications, Comm. Math. Phys., 316 (2012), 723-761. doi: 10.1007/s00220-012-1515-y. [10] C. Foias, O. Manley, R. Rosa, R. Temam, Navier-Stokes Equations and Turbulence, Cambridge University Press, Cambridge, 2001. [11] J. García-Luengo, P. Marín-Rubio, J. Real, Pullback attractors for three-dimensional non-autonomous Navier-Stokes-Voigt equations, Nonlinearity, 25 (2012), 905-930. doi: 10.1088/0951-7715/25/4/905. [12] N. Ju, The H1-compact global attractor for the solutions to the Navier-Stokes equations in 2D unbounded domains, Nonlinearity, 13 (2000), 1227-1238. doi: 10.1088/0951-7715/13/4/313. [13] P. E. Kloeden, P. Marín-Rubio, J. Real, Equivalence of invariant measures and stationary statistical solutions for the autonomous globally modified Navier-Stokes equations, Comm. Pure Appl. Anal., 8 (2009), 785-802. doi: 10.3934/cpaa.2009.8.785. [14] A. Larios, E. S. Titi, On the higher-order global regularity of the inviscid Voigt-regularization of three-dimensional hydrodynamic models, Discrete Cont. Dyna. Syst.-B, 14 (2010), 603-627. doi: 10.3934/dcdsb.2010.14.603. [15] A. Larios, E. S. Titi, Higher-order global regularity of an inviscid Voigt regularization of the three-dimensional inviscid resistive magnetohydrodynamic equations, J. Math. Fluid Mech., 16 (2014), 59-76. doi: 10.1007/s00021-013-0136-3. [16] B. Levant, F. Ramos, E. S. Titi, On the statistical properties of the 3D incompressible Navier-Stokes-Voigt model, Comm. Math. Sci., 8 (2010), 277-293. doi: 10.4310/CMS.2010.v8.n1.a14. [17] G. Łukaszewicz, W. Sadowski, Uniform attractor for 2D magneto-micropolar fluid flow in some unbounded domains, Z. Angew. Math. Phy., 55 (2004), 247-257. doi: 10.1007/s00033-003-1127-7. [18] G. Lukaszewicz, Pullback attractors and statistical solutions for 2-D Navier-Stokes equations, Discrete Cont. Dyna. Syst.-B, 9 (2008), 643-659. doi: 10.3934/dcdsb.2008.9.643. [19] G. Lukaszewicz, J. Real, J. C. Robinson, Invariant measures for dissipative dynamical systems and generalised Banach limits, J. Dynam. Differential Equations, 23 (2011), 225-250. doi: 10.1007/s10884-011-9213-6. [20] G. Lukaszewicz, J. C. Robinson, Invariant measures for non-autonomous dissipative dynamical systems, Discrete Cont. Dyna. Syst.-A, 34 (2014), 4211-4222. doi: 10.3934/dcds.2014.34.4211. [21] I. Moise, R. Rosa, X. Wang, Attractors for non-compact semigroup via energy equations, Nonlinearity, 11 (1998), 1369-1393. doi: 10.1088/0951-7715/11/5/012. [22] I. Moise, R. Rosa, X. Wang, Attractors for non-compact nonautonomous systems via energy equations, Discrete Cont. Dyna. Syst.-A, 10 (2004), 473-496. [23] R. Rosa, The global attractor for the 2D Navier-Stokes flow on some unbounded domains, Nonlinear Anal., 32 (1998), 71-85. doi: 10.1016/S0362-546X(97)00453-7. [24] M. Sermange, R. Temam, Some mathematical questions related to the MHD equations, Comm. Pure Appl. Math., 36 (1983), 635-664. doi: 10.1002/cpa.3160360506. [25] X. Wang, An energy equation for the weakly damped driven nonlinear Schrödinger equations and its application to their attactors, Physica D, 88 (1995), 167-175. doi: 10.1016/0167-2789(95)00196-B. [26] X. Wang, Upper-semicontinuity of stationary statistical properties of dissipative systems, Discrete Cont. Dyna. Syst.-A, 23 (2009), 521-540. [27] J. Wu, Regularity results for weak solutions of the 3D MHD equations, Discrete Cont. Dyna. Syst.-A, 10 (2004), 543-556. [28] C. Zhao, S. Zhou, Pullback attractors for nonautonomous incompressible non-Newtonian fluid, J. Differential Equations, 238 (2007), 394-425. doi: 10.1016/j.jde.2007.04.001. [29] C. Zhao, Y. Li, S. Zhou, Regularity of trajectory attractor and upper semicontinuity of global attractor for a 2D non-Newtonian fluid, J. Differential Equations, 247 (2009), 2331-2363. doi: 10.1016/j.jde.2009.07.031. [30] C. Zhao, Pullback asymptotic behavior of solutions for a non-autonomous non-Newtonian fluid on two-dimensional unbounded domains, J. Math. Phys., 53 (2012), 122702, 22 pp. [31] C. Zhao, G. Liu, W. Wang, Smooth pullback attractors for a non-autonomous 2D non-Newtonian fluid and their tempered behaviors, J. Math. Fluid Mech., 16 (2014), 243-262. doi: 10.1007/s00021-013-0153-2. [32] C. Zhao, B. Li, Analyticity of the global attractor for the 3D regularized MHD equations, E. J. Differential Equations, 2016 (2016), 1-20. [33] C. Zhao, W. Sun, Global well-posedness and pullback attractors for a two-dimensional non-autonomous micropolar fluid flows with infinite delays, Comm. Math. Sci., 15 (2017), 97-121. doi: 10.4310/CMS.2017.v15.n1.a5. [34] C. Zhao, L. Yang, Pullback attractors and invariant measures for the non-autonomous globally modified Navier-Stokes equations, Comm. Math. Sci., 15 (2017), 1565-1580. doi: 10.4310/CMS.2017.v15.n6.a4. [35] Y. Zhou, Regularity criteria for the generalized viscous MHD equations, Ann. I. H. Poincaré-AN, 24 (2007), 491-505. doi: 10.1016/j.anihpc.2006.03.014. [36] Y. Zhou, S. Gala, Regularity criteria for the solutions to the 3D MHD equations in the multiplier space, Z. Angew. Math. Phys., 61 (2010), 193-199. doi: 10.1007/s00033-009-0023-1.

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##### References:
 [1] J. M. Ball, Global attractors for damped semilinear wave equations, Discrete Cont. Dyna. Syst.-A, 10 (2004), 31-52. [2] C. Cao, J. Wu, Two regularity criteria for the 3D MHD equations, J. Differential Equations, 248 (2010), 2263-2274. doi: 10.1016/j.jde.2009.09.020. [3] T. Caraballo, G. Lukaszewicz, J. Real, Pullback attractors for asymptotically compact non-autonomous dynamical system, Nonlinear Anal., 64 (2006), 484-498. doi: 10.1016/j.na.2005.03.111. [4] T. Caraballo, G. Lukaszewicz, J. Real, Invariant measures and statitical solutions of the globally modified Navier-Stokes equations, Discrete Cont. Dyna. Syst.-B, 10 (2008), 761-781. doi: 10.3934/dcdsb.2008.10.761. [5] A. N. Carvalho, J. A. Langa and J. C. Robinson, Attractors for Infinite-Dimensional NonAutonomous Dynamical Systems, Springer, New York, 2013. [6] D. Catania, P. Secchi, Global existence and finite dimensional global attractor for a 3D double viscous MHD-α model, Comm. Math. Sci., 8 (2010), 1021-1040. doi: 10.4310/CMS.2010.v8.n4.a12. [7] D. Catania, Global attractor and determining modes for a hyperbolic MHD turbulence Model, J. of Turbulence, 12 (2011), 1-20. [8] D. Catania, P. Secchi, Global existence for two regularized MHD models in three space-dimension, Port. Math., 68 (2011), 41-52. [9] M. Chekroun, N. E. Glatt-Holtz, Invariant measures for dissipative dynamical systems: Abstract results and applications, Comm. Math. Phys., 316 (2012), 723-761. doi: 10.1007/s00220-012-1515-y. [10] C. Foias, O. Manley, R. Rosa, R. Temam, Navier-Stokes Equations and Turbulence, Cambridge University Press, Cambridge, 2001. [11] J. García-Luengo, P. Marín-Rubio, J. Real, Pullback attractors for three-dimensional non-autonomous Navier-Stokes-Voigt equations, Nonlinearity, 25 (2012), 905-930. doi: 10.1088/0951-7715/25/4/905. [12] N. Ju, The H1-compact global attractor for the solutions to the Navier-Stokes equations in 2D unbounded domains, Nonlinearity, 13 (2000), 1227-1238. doi: 10.1088/0951-7715/13/4/313. [13] P. E. Kloeden, P. Marín-Rubio, J. Real, Equivalence of invariant measures and stationary statistical solutions for the autonomous globally modified Navier-Stokes equations, Comm. Pure Appl. Anal., 8 (2009), 785-802. doi: 10.3934/cpaa.2009.8.785. [14] A. Larios, E. S. Titi, On the higher-order global regularity of the inviscid Voigt-regularization of three-dimensional hydrodynamic models, Discrete Cont. Dyna. Syst.-B, 14 (2010), 603-627. doi: 10.3934/dcdsb.2010.14.603. [15] A. Larios, E. S. Titi, Higher-order global regularity of an inviscid Voigt regularization of the three-dimensional inviscid resistive magnetohydrodynamic equations, J. Math. Fluid Mech., 16 (2014), 59-76. doi: 10.1007/s00021-013-0136-3. [16] B. Levant, F. Ramos, E. S. Titi, On the statistical properties of the 3D incompressible Navier-Stokes-Voigt model, Comm. Math. Sci., 8 (2010), 277-293. doi: 10.4310/CMS.2010.v8.n1.a14. [17] G. Łukaszewicz, W. Sadowski, Uniform attractor for 2D magneto-micropolar fluid flow in some unbounded domains, Z. Angew. Math. Phy., 55 (2004), 247-257. doi: 10.1007/s00033-003-1127-7. [18] G. Lukaszewicz, Pullback attractors and statistical solutions for 2-D Navier-Stokes equations, Discrete Cont. Dyna. Syst.-B, 9 (2008), 643-659. doi: 10.3934/dcdsb.2008.9.643. [19] G. Lukaszewicz, J. Real, J. C. Robinson, Invariant measures for dissipative dynamical systems and generalised Banach limits, J. Dynam. Differential Equations, 23 (2011), 225-250. doi: 10.1007/s10884-011-9213-6. [20] G. Lukaszewicz, J. C. Robinson, Invariant measures for non-autonomous dissipative dynamical systems, Discrete Cont. Dyna. Syst.-A, 34 (2014), 4211-4222. doi: 10.3934/dcds.2014.34.4211. [21] I. Moise, R. Rosa, X. Wang, Attractors for non-compact semigroup via energy equations, Nonlinearity, 11 (1998), 1369-1393. doi: 10.1088/0951-7715/11/5/012. [22] I. Moise, R. Rosa, X. Wang, Attractors for non-compact nonautonomous systems via energy equations, Discrete Cont. Dyna. Syst.-A, 10 (2004), 473-496. [23] R. Rosa, The global attractor for the 2D Navier-Stokes flow on some unbounded domains, Nonlinear Anal., 32 (1998), 71-85. doi: 10.1016/S0362-546X(97)00453-7. [24] M. Sermange, R. Temam, Some mathematical questions related to the MHD equations, Comm. Pure Appl. Math., 36 (1983), 635-664. doi: 10.1002/cpa.3160360506. [25] X. Wang, An energy equation for the weakly damped driven nonlinear Schrödinger equations and its application to their attactors, Physica D, 88 (1995), 167-175. doi: 10.1016/0167-2789(95)00196-B. [26] X. Wang, Upper-semicontinuity of stationary statistical properties of dissipative systems, Discrete Cont. Dyna. Syst.-A, 23 (2009), 521-540. [27] J. Wu, Regularity results for weak solutions of the 3D MHD equations, Discrete Cont. Dyna. Syst.-A, 10 (2004), 543-556. [28] C. Zhao, S. Zhou, Pullback attractors for nonautonomous incompressible non-Newtonian fluid, J. Differential Equations, 238 (2007), 394-425. doi: 10.1016/j.jde.2007.04.001. [29] C. Zhao, Y. Li, S. Zhou, Regularity of trajectory attractor and upper semicontinuity of global attractor for a 2D non-Newtonian fluid, J. Differential Equations, 247 (2009), 2331-2363. doi: 10.1016/j.jde.2009.07.031. [30] C. Zhao, Pullback asymptotic behavior of solutions for a non-autonomous non-Newtonian fluid on two-dimensional unbounded domains, J. Math. Phys., 53 (2012), 122702, 22 pp. [31] C. Zhao, G. Liu, W. Wang, Smooth pullback attractors for a non-autonomous 2D non-Newtonian fluid and their tempered behaviors, J. Math. Fluid Mech., 16 (2014), 243-262. doi: 10.1007/s00021-013-0153-2. [32] C. Zhao, B. Li, Analyticity of the global attractor for the 3D regularized MHD equations, E. J. Differential Equations, 2016 (2016), 1-20. [33] C. Zhao, W. Sun, Global well-posedness and pullback attractors for a two-dimensional non-autonomous micropolar fluid flows with infinite delays, Comm. Math. Sci., 15 (2017), 97-121. doi: 10.4310/CMS.2017.v15.n1.a5. [34] C. Zhao, L. Yang, Pullback attractors and invariant measures for the non-autonomous globally modified Navier-Stokes equations, Comm. Math. Sci., 15 (2017), 1565-1580. doi: 10.4310/CMS.2017.v15.n6.a4. [35] Y. Zhou, Regularity criteria for the generalized viscous MHD equations, Ann. I. H. Poincaré-AN, 24 (2007), 491-505. doi: 10.1016/j.anihpc.2006.03.014. [36] Y. Zhou, S. Gala, Regularity criteria for the solutions to the 3D MHD equations in the multiplier space, Z. Angew. Math. Phys., 61 (2010), 193-199. doi: 10.1007/s00033-009-0023-1.
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