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

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

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

Zeqi Zhu ^1, and Caidi Zhao ^2,,

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.

Keywords: Pullback attractor, invariant measures, regularized MHD equations.

Mathematics Subject Classification: 35B41, 35D99, 76F20.

Citation: Zeqi Zhu, Caidi Zhao. Pullback attractor and invariant measures for the three-dimensional regularized MHD equations. Discrete and Continuous Dynamical Systems, 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 and 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 and 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 and 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 and 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 and P. Secchi, Global existence for two regularized MHD models in three space-dimension, Port. Math., 68 (2011), 41-52.
[9]	M. Chekroun and 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 and R. Temam, Navier-Stokes Equations and Turbulence, Cambridge University Press, Cambridge, 2001.
[11]	J. García-Luengo, P. Marín-Rubio and 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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and B. Li, Analyticity of the global attractor for the 3D regularized MHD equations, E. J. Differential Equations, 2016 (2016), 1-20.
[33]	C. Zhao and 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 and 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 and 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 and 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 and 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 and 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 and 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 and P. Secchi, Global existence for two regularized MHD models in three space-dimension, Port. Math., 68 (2011), 41-52.
[9]	M. Chekroun and 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 and R. Temam, Navier-Stokes Equations and Turbulence, Cambridge University Press, Cambridge, 2001.
[11]	J. García-Luengo, P. Marín-Rubio and 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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and B. Li, Analyticity of the global attractor for the 3D regularized MHD equations, E. J. Differential Equations, 2016 (2016), 1-20.
[33]	C. Zhao and 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 and 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 and 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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