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:
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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.

show all references

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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