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September  2020, 19(9): 4479-4506. doi: 10.3934/cpaa.2020204

Exponential mixing for the fractional Magneto-Hydrodynamic equations with degenerate stochastic forcing

Xuhui Peng 1, , Jianhua Huang 2,, and  Yan Zheng 2,

1. 

MOE-LCSM, School of Mathematics and Statistics, Hunan Normal University, Changsha, 410081, China, Key Laboratory of Applied Statistics and Data Science, Hunan Normal University, College of Hunan Province, Changsha, 410081, China
2. 

College of Liberal Arts and Sciences, National University of Defense Technology, Changsha, 410073, China

* Corresponding author

Received  December 2019 Revised  March 2020 Published  June 2020

Fund Project: The first author was supported by Hunan Provincial Natural Science Foundation of China (No. 2019JJ50377), NSFC (No.11871476) and the Construct Program of the Key Discipline in Hunan Province. The last two were supported by the NSF of China(No.11771449)

We establish the existence, uniqueness and exponential attraction properties of an invariant measure for the MHD equations with degenerate stochastic forcing acting only in the magnetic equation. The central challenge is to establish time asymptotic smoothing properties of the associated Markovian semigroup corresponding to this system. Towards this aim we take full advantage of the characteristics of the advective structure to discover a novel Hörmander-type condition which only allows for several noises in the magnetic direction.

Keywords: Exponential mixing, Malliavin calculus, ergodic, fractional Magneto-Hydrodynamic equations.
Mathematics Subject Classification: Primary:60H15, 60H07.
Citation: Xuhui Peng, Jianhua Huang, Yan Zheng. Exponential mixing for the fractional Magneto-Hydrodynamic equations with degenerate stochastic forcing. Communications on Pure and Applied Analysis, 2020, 19 (9) : 4479-4506. doi: 10.3934/cpaa.2020204
References:
[1]

S. Albeverio, A. Debussche and L. Xu, Exponential mixing of the 3D stochastic Navier-Stokes equations driven by mildly degenerate noises, Appl. Math. Optim., 66 (2012), 273-308.  doi: 10.1007/s00245-012-9172-2.

[2]

V. Barbu and G. Da Prato, Existence and ergodicity for the two-dimensional stochastic Magnetohydrodynamics equations, Appl. Math. Optim., 56 (2007), 145-168.  doi: 10.1007/s00245-007-0882-2.

[3] H. Cababbes, Theoretical Magnetofluiddynamics, Academic Press, New york, 1970. 
[4]

C. Cao, D. Regmi and J. Wu, The 2D MHD equations with horizontal dissipation and horizontal magnetic diffusion, J. Differ. Equ., 253 (2013), 2661-2681.  doi: 10.1016/j.jde.2013.01.002.

[5]

C. Cao and J. Wu, Global regularity for the 2D MHD equations with mixed partial dissipation and magnetic diffusion, Adv. Math., 226 (2011), 1803-1822.  doi: 10.1016/j.aim.2010.08.017.

[6]

G. Duvaut and J. L. Lions, Inequations en Thermolasticit et magntohydrodynamique, Arch. Ration. Mech. Anal., 46 (1972), 241-79.  doi: 10.1007/BF00250512.

[7]

W. E and J. C. Mattingly, Ergodicity for the Navier-Stokes Equation with Degenerate Random Forcing: Finite-Dimensional Approximation, Commun. Pure Appl. Math., 54 (2001), 1386-1402.  doi: 10.1002/cpa.10007.

[8]

J. Földes, N. Glatt-Holtz, G. Richards and E. Thomann, Ergodic and mixing properties of the Boussinesq equations with a degenerate random forcing, J. Funct. Anal., 269 (2015), 2427-2504.  doi: 10.1016/j.jfa.2015.05.014.

[9]

H. Martin and J. C. Mattingly, Ergodicity of the 2D Navier-Stokes equations with degenerate stochastic forcing, Ann. Math., 164 (2006), 993-1032.  doi: 10.4007/annals.2006.164.993.

[10]

H. Martin and J. C. Mattingly, Spectral gaps in Wasserstein distances and the 2D stochastic Navier-Stokes equations, Ann. Probab., 36 (2008), 2050-2091.  doi: 10.1214/08-AOP392.

[11]

H. Martin and J. C. Mattingly, A theory of hypoellipticity and unique ergodicity for semilinear stochastic PDEs, Electron. J. Probab., 16 (2011), 658-738.  doi: 10.1214/EJP.v16-875.

[12]

J. Huang and T. Shen, Well-posedness and dynamics of the stochastic fractional magneto-hydrodynamic equations, Nonlinear Anal., 133 (2016), 102-133.  doi: 10.1016/j.na.2015.12.001.

[13]

T. Komorowski and A. Walczuk, Central limit theorem for Markov processes with spectral gap in the Wasserstein metric, Stoch. Process. Their Appl., 122 2012, 2155-2184. doi: 10.1016/j.spa.2012.03.006.

[14]

S. Kuksin and A. Shirikyan, A Coupling Approach to Randomly Forced Nonlinear PDEs. I, Commun. Math. Phys., 221 (2001), 351-366.  doi: 10.1007/s002200100479.

[15]

S. Kuksin and A. Shirikyan, Coupling approach to white-forced nonlinear PDEs, J. Math. Pures Appl., 81 (2002), 567-602.  doi: 10.1016/S0021-7824(02)01259-X.

[16] S. Kuksin and A. Shirikyan, Mathematics of Two-Dimensional Turbulence, Cambridge University press, 2012.  doi: 10.1017/CBO9781139137119.
[17]

Z. Lei and Y. Zhou, BKM's criterion and Global weak solutions for magnetohydrodynamics with zero viscosity, Discrete Contin. Dyn. Syst., 25 (2009), 575-583.  doi: 10.3934/dcds.2009.25.575.

[18]

J. C. Mattingly, Exponential convergence for the stochastically forced Navier-Stokes equations and other partially dissipative dynamics, Commun. Math. Phys., 230 (2002), 421-462.  doi: 10.1007/s00220-002-0688-1.

[19] G. Da Parto and J. Zabcyzk, Ergodicity for Infinite Dimensional Systems, Cambridge University Press, Cambridge, 1996.  doi: 10.1017/CBO9780511662829.
[20]

M. Röckner and X. Zhang, Stochastic tamed 3D Navier-Stokes equations: existence, uniqueness and ergodicity, Probab. Theory Relat. Fields, 145 (2009), 211-267.  doi: 10.1007/s00440-008-0167-5.

[21]

M. Sermange and R. Temam, Some mathematical questions related to the MHD equations, Commun. Pure Appl. Math., 36 (1983), 635-664.  doi: 10.1002/cpa.3160360506.

[22]

T. Shen and J. Huang, Ergodicity of stochastic Magneto-Hydrodynamic equations driven by $\alpha$-stable noise, J. Math. Anal. Appl., 446 (2017), 746-769.  doi: 10.1016/j.jmaa.2016.08.050.

[23]

T. Shen, J. Huang and C. Zeng, Ergodicity of the 2D stochastic fractional Magneto-hydrodynamic equations driven by degenerate multiplicative noise, preprint.

[24]

J. Wu, Generalized MHD equations, J. Differ. Equ., 195 (2003), 284-312.  doi: 10.1016/j.jde.2003.07.007.

show all references

References:
[1]

S. Albeverio, A. Debussche and L. Xu, Exponential mixing of the 3D stochastic Navier-Stokes equations driven by mildly degenerate noises, Appl. Math. Optim., 66 (2012), 273-308.  doi: 10.1007/s00245-012-9172-2.

[2]

V. Barbu and G. Da Prato, Existence and ergodicity for the two-dimensional stochastic Magnetohydrodynamics equations, Appl. Math. Optim., 56 (2007), 145-168.  doi: 10.1007/s00245-007-0882-2.

[3] H. Cababbes, Theoretical Magnetofluiddynamics, Academic Press, New york, 1970. 
[4]

C. Cao, D. Regmi and J. Wu, The 2D MHD equations with horizontal dissipation and horizontal magnetic diffusion, J. Differ. Equ., 253 (2013), 2661-2681.  doi: 10.1016/j.jde.2013.01.002.

[5]

C. Cao and J. Wu, Global regularity for the 2D MHD equations with mixed partial dissipation and magnetic diffusion, Adv. Math., 226 (2011), 1803-1822.  doi: 10.1016/j.aim.2010.08.017.

[6]

G. Duvaut and J. L. Lions, Inequations en Thermolasticit et magntohydrodynamique, Arch. Ration. Mech. Anal., 46 (1972), 241-79.  doi: 10.1007/BF00250512.

[7]

W. E and J. C. Mattingly, Ergodicity for the Navier-Stokes Equation with Degenerate Random Forcing: Finite-Dimensional Approximation, Commun. Pure Appl. Math., 54 (2001), 1386-1402.  doi: 10.1002/cpa.10007.

[8]

J. Földes, N. Glatt-Holtz, G. Richards and E. Thomann, Ergodic and mixing properties of the Boussinesq equations with a degenerate random forcing, J. Funct. Anal., 269 (2015), 2427-2504.  doi: 10.1016/j.jfa.2015.05.014.

[9]

H. Martin and J. C. Mattingly, Ergodicity of the 2D Navier-Stokes equations with degenerate stochastic forcing, Ann. Math., 164 (2006), 993-1032.  doi: 10.4007/annals.2006.164.993.

[10]

H. Martin and J. C. Mattingly, Spectral gaps in Wasserstein distances and the 2D stochastic Navier-Stokes equations, Ann. Probab., 36 (2008), 2050-2091.  doi: 10.1214/08-AOP392.

[11]

H. Martin and J. C. Mattingly, A theory of hypoellipticity and unique ergodicity for semilinear stochastic PDEs, Electron. J. Probab., 16 (2011), 658-738.  doi: 10.1214/EJP.v16-875.

[12]

J. Huang and T. Shen, Well-posedness and dynamics of the stochastic fractional magneto-hydrodynamic equations, Nonlinear Anal., 133 (2016), 102-133.  doi: 10.1016/j.na.2015.12.001.

[13]

T. Komorowski and A. Walczuk, Central limit theorem for Markov processes with spectral gap in the Wasserstein metric, Stoch. Process. Their Appl., 122 2012, 2155-2184. doi: 10.1016/j.spa.2012.03.006.

[14]

S. Kuksin and A. Shirikyan, A Coupling Approach to Randomly Forced Nonlinear PDEs. I, Commun. Math. Phys., 221 (2001), 351-366.  doi: 10.1007/s002200100479.

[15]

S. Kuksin and A. Shirikyan, Coupling approach to white-forced nonlinear PDEs, J. Math. Pures Appl., 81 (2002), 567-602.  doi: 10.1016/S0021-7824(02)01259-X.

[16] S. Kuksin and A. Shirikyan, Mathematics of Two-Dimensional Turbulence, Cambridge University press, 2012.  doi: 10.1017/CBO9781139137119.
[17]

Z. Lei and Y. Zhou, BKM's criterion and Global weak solutions for magnetohydrodynamics with zero viscosity, Discrete Contin. Dyn. Syst., 25 (2009), 575-583.  doi: 10.3934/dcds.2009.25.575.

[18]

J. C. Mattingly, Exponential convergence for the stochastically forced Navier-Stokes equations and other partially dissipative dynamics, Commun. Math. Phys., 230 (2002), 421-462.  doi: 10.1007/s00220-002-0688-1.

[19] G. Da Parto and J. Zabcyzk, Ergodicity for Infinite Dimensional Systems, Cambridge University Press, Cambridge, 1996.  doi: 10.1017/CBO9780511662829.
[20]

M. Röckner and X. Zhang, Stochastic tamed 3D Navier-Stokes equations: existence, uniqueness and ergodicity, Probab. Theory Relat. Fields, 145 (2009), 211-267.  doi: 10.1007/s00440-008-0167-5.

[21]

M. Sermange and R. Temam, Some mathematical questions related to the MHD equations, Commun. Pure Appl. Math., 36 (1983), 635-664.  doi: 10.1002/cpa.3160360506.

[22]

T. Shen and J. Huang, Ergodicity of stochastic Magneto-Hydrodynamic equations driven by $\alpha$-stable noise, J. Math. Anal. Appl., 446 (2017), 746-769.  doi: 10.1016/j.jmaa.2016.08.050.

[23]

T. Shen, J. Huang and C. Zeng, Ergodicity of the 2D stochastic fractional Magneto-hydrodynamic equations driven by degenerate multiplicative noise, preprint.

[24]

J. Wu, Generalized MHD equations, J. Differ. Equ., 195 (2003), 284-312.  doi: 10.1016/j.jde.2003.07.007.

Figure 1.  An illustration of how the new directions generated from the existing directions via the iterations of the chain of bracket computations. In this figure, $ m,m'\in \{0,1\}, \ell\in \mathcal{Z}_0. $ Solid arrows mean that the new function is generated from a Lie bracket, with the type of bracket indicated above the arrow. Dashed arrows with green color signify that the new element is generated as a linear combination of elements from the previous position. The dotted arrows with red color shows that the process is iterative. The doubled arrow with yellow color (→) shows that $ k\pm \ell $ is a element belongs to $ \mathcal{Z}_{2n+1} $ or $ \mathcal{Z}_{2n+2} $ actually.
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Figure 2.  An illustration of the structure of the lemmas that leads to the proof of Proposition 2. The solid arrows indicate that if one term is "small" then the other one "small" on a set of large measure(displayed up or left of the arrow), where the meaning of "smallness" is made precise in each lemma. The dashed arrows shows that the process is iterative. In this figure, $ m,m'\in \{0,1\}, \ell\in \mathcal{Z}_0. $ One may notice the close relationship between Figure 1 and Figure 1.
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