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On the positive solutions for a perturbed negative exponent problem on $\mathbb{R}^3$
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 |
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.
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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