January  2011, 10(1): 309-326. doi: 10.3934/cpaa.2011.10.309

Regularity criteria for a magnetohydrodynamic-$\alpha$ model

1. 

Department of Mathematics, Zhejiang Normal University, Jinhua, Zhejiang 321004

2. 

Department of Applied Mathematics, Nanjing Forestry University, Nanjing, 210037

Received  January 2010 Revised  June 2010 Published  November 2010

We study the $n$-dimensional magnetohydrodynamic-$\alpha$ (MHD-$\alpha$) model in the whole space. Various regularity criteria are established. When $n=4$, uniqueness of weak solutions is also proved. As a corollary, the strong solution to this model exists globally, as $n \leq 4$.
Citation: Yong Zhou, Jishan Fan. Regularity criteria for a magnetohydrodynamic-$\alpha$ model. Communications on Pure & Applied Analysis, 2011, 10 (1) : 309-326. doi: 10.3934/cpaa.2011.10.309
References:
[1]

Q. Chen, C. Miao and Z. Zhang, On the regularity criterion of weak solution for the 3D viscous Magneto-Hydrodynamics equations,, Comm. Math. Phys., 284 (2008), 919. doi: doi:10.1007/s00220-008-0545-y. Google Scholar

[2]

Q. Chen, C. Miao and Z. Zhang, On the uniqueness of weak solutions for the 3D Navier-Stokes equations,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 2165. doi: doi:10.1016/j.anihpc.2009.01.008. Google Scholar

[3]

S. Chen, C. Foias, D. D. Holm, E. Olson, E. S. Titi and S. Wynne, Camassa-Holm equations as a closure model for turbulent channel and pipe flow,, Phys. Rev. Lett., 81 (1998), 5338. doi: doi:10.1103/PhysRevLett.81.5338. Google Scholar

[4]

S. Chen, C. Foias, D. D. Holm, E. Olson, E. S. Titi and S. Wynne, The Camassa-Holm equations and turbulence. Predictability: quantifying uncertainty in models of complex phenomena,, Phys. D, 133 (1999), 49. doi: doi:10.1016/S0167-2789(99)00098-6. Google Scholar

[5]

A. Cheskidov, D. D. Holm, E. Olson and E. S. Titi, On a Leray-$\alpha$ model of turbulence,, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 461 (2005), 629. Google Scholar

[6]

R. Coifman, P. L. Lions, Y. Meyer and S. Semmes, Compensated compactness and Hardy spaces,, J. Math. Pures Appl., 72 (1993), 247. Google Scholar

[7]

C. Foias, D. D. Holm and E. S. Titi, The three dimensional viscous Camassa-Holm equations, and their relation to the Navier-Stokes equations and turbulence theory.,, J. Dynam. Differential Equations, 14 (2002), 1. doi: doi:10.1023/A:1012984210582. Google Scholar

[8]

C. He and Z. Xin, On the regularity of weak solutions to the magnetohydrodynamic equations,, J. Differential Equations, 213 (2005), 235. doi: doi:10.1016/j.jde.2004.07.002. Google Scholar

[9]

D. D. Holm, Lagrangian averages, averaged Lagrangians, and the mean effects of fluctuations in fluid dynamics,, Chaos, 12 (2002), 518. doi: doi:10.1063/1.1460941. Google Scholar

[10]

D. D. Holm, J. E. Marsden and T. S. Ratiu, The Euler-Poincaré equations and semidirect products with applications to continuum theories.,, Adv. Math., 137 (1998), 1. doi: doi:10.1006/aima.1998.1721. Google Scholar

[11]

A. A. Ilyin, E. M. Lunasin and E. S. Titi, A Modified-Leray-$\alpha$ subgrid scale model of turbulence,, Nonlinearity, 19 (2006), 879. doi: doi:10.1088/0951-7715/19/4/006. Google Scholar

[12]

B. B. Kadomtsev, "Tokamak Plasma: a Complex Physical System,", Bristol: Institute of Physics, (1992). Google Scholar

[13]

T. Kato and G. Ponce, Commutator estimates and the Euler and Navier-Stokes equations,, Comm. Pure Appl. Math., 41 (1988), 891. doi: doi:10.1002/cpa.3160410704. Google Scholar

[14]

J. S. Linshiz and E. S. Titi, Analytical study of certain magnetohydrodynamic -$\alpha$ models,, J. Math. Phys., 48 (2007). Google Scholar

[15]

S. Machihara and T. Ozawa, Interpolation inequalities in Besov spaces,, Proc. Amer. Math. Soc., 131 (2003), 1553. doi: doi:10.1090/S0002-9939-02-06715-1. Google Scholar

[16]

J. E. Marsden and S. Shkoller, Global well-posedness for the Lagrangian averaged Navier-Stokes (LANS-$\alpha$) equations on bounded domains,, Topological methods in the physical sciences (London, 359 (2001), 1449. Google Scholar

[17]

Y. Meyer, Oscillating patterns in some nonlinear evolution equations,, pp. 101-187, (1871), 101. Google Scholar

[18]

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

[19]

J. Wu, Regularity criteria for the generalized MHD equations,, Comm. Partial Differential Equations, 33 (2008), 285. doi: doi:10.1080/03605300701382530. Google Scholar

[20]

Y. Zhou, Remarks on regularities for the 3D MHD equations,, Discrete Contin. Dyn. Syst., 12 (2005), 881. doi: doi:10.3934/dcds.2005.12.881. Google Scholar

[21]

Y. Zhou, Regularity criteria for the 3D MHD equations in terms of the pressure,, Internat. J. Non-Linear Mech., 41 (2006), 1174. doi: doi:10.1016/j.ijnonlinmec.2006.12.001. Google Scholar

[22]

Y. Zhou, Regularity criteria for the generalized viscous MHD equations,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 24 (2007), 491. doi: doi:10.1016/j.anihpc.2006.03.014. Google Scholar

[23]

Y. Zhou and J. Fan, Regularity criteria for the viscous Camassa-Holm equations,, Int. Math. Res. Not. IMRN, (2009), 2508. Google Scholar

[24]

Y. Zhou and J. Fan, On regularity criteria in terms of pressure for the 3D viscous MHD equations,, Preprint (2008)., (2008). Google Scholar

[25]

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. doi: doi:10.1007/s00033-009-0023-1. Google Scholar

show all references

References:
[1]

Q. Chen, C. Miao and Z. Zhang, On the regularity criterion of weak solution for the 3D viscous Magneto-Hydrodynamics equations,, Comm. Math. Phys., 284 (2008), 919. doi: doi:10.1007/s00220-008-0545-y. Google Scholar

[2]

Q. Chen, C. Miao and Z. Zhang, On the uniqueness of weak solutions for the 3D Navier-Stokes equations,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 2165. doi: doi:10.1016/j.anihpc.2009.01.008. Google Scholar

[3]

S. Chen, C. Foias, D. D. Holm, E. Olson, E. S. Titi and S. Wynne, Camassa-Holm equations as a closure model for turbulent channel and pipe flow,, Phys. Rev. Lett., 81 (1998), 5338. doi: doi:10.1103/PhysRevLett.81.5338. Google Scholar

[4]

S. Chen, C. Foias, D. D. Holm, E. Olson, E. S. Titi and S. Wynne, The Camassa-Holm equations and turbulence. Predictability: quantifying uncertainty in models of complex phenomena,, Phys. D, 133 (1999), 49. doi: doi:10.1016/S0167-2789(99)00098-6. Google Scholar

[5]

A. Cheskidov, D. D. Holm, E. Olson and E. S. Titi, On a Leray-$\alpha$ model of turbulence,, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 461 (2005), 629. Google Scholar

[6]

R. Coifman, P. L. Lions, Y. Meyer and S. Semmes, Compensated compactness and Hardy spaces,, J. Math. Pures Appl., 72 (1993), 247. Google Scholar

[7]

C. Foias, D. D. Holm and E. S. Titi, The three dimensional viscous Camassa-Holm equations, and their relation to the Navier-Stokes equations and turbulence theory.,, J. Dynam. Differential Equations, 14 (2002), 1. doi: doi:10.1023/A:1012984210582. Google Scholar

[8]

C. He and Z. Xin, On the regularity of weak solutions to the magnetohydrodynamic equations,, J. Differential Equations, 213 (2005), 235. doi: doi:10.1016/j.jde.2004.07.002. Google Scholar

[9]

D. D. Holm, Lagrangian averages, averaged Lagrangians, and the mean effects of fluctuations in fluid dynamics,, Chaos, 12 (2002), 518. doi: doi:10.1063/1.1460941. Google Scholar

[10]

D. D. Holm, J. E. Marsden and T. S. Ratiu, The Euler-Poincaré equations and semidirect products with applications to continuum theories.,, Adv. Math., 137 (1998), 1. doi: doi:10.1006/aima.1998.1721. Google Scholar

[11]

A. A. Ilyin, E. M. Lunasin and E. S. Titi, A Modified-Leray-$\alpha$ subgrid scale model of turbulence,, Nonlinearity, 19 (2006), 879. doi: doi:10.1088/0951-7715/19/4/006. Google Scholar

[12]

B. B. Kadomtsev, "Tokamak Plasma: a Complex Physical System,", Bristol: Institute of Physics, (1992). Google Scholar

[13]

T. Kato and G. Ponce, Commutator estimates and the Euler and Navier-Stokes equations,, Comm. Pure Appl. Math., 41 (1988), 891. doi: doi:10.1002/cpa.3160410704. Google Scholar

[14]

J. S. Linshiz and E. S. Titi, Analytical study of certain magnetohydrodynamic -$\alpha$ models,, J. Math. Phys., 48 (2007). Google Scholar

[15]

S. Machihara and T. Ozawa, Interpolation inequalities in Besov spaces,, Proc. Amer. Math. Soc., 131 (2003), 1553. doi: doi:10.1090/S0002-9939-02-06715-1. Google Scholar

[16]

J. E. Marsden and S. Shkoller, Global well-posedness for the Lagrangian averaged Navier-Stokes (LANS-$\alpha$) equations on bounded domains,, Topological methods in the physical sciences (London, 359 (2001), 1449. Google Scholar

[17]

Y. Meyer, Oscillating patterns in some nonlinear evolution equations,, pp. 101-187, (1871), 101. Google Scholar

[18]

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

[19]

J. Wu, Regularity criteria for the generalized MHD equations,, Comm. Partial Differential Equations, 33 (2008), 285. doi: doi:10.1080/03605300701382530. Google Scholar

[20]

Y. Zhou, Remarks on regularities for the 3D MHD equations,, Discrete Contin. Dyn. Syst., 12 (2005), 881. doi: doi:10.3934/dcds.2005.12.881. Google Scholar

[21]

Y. Zhou, Regularity criteria for the 3D MHD equations in terms of the pressure,, Internat. J. Non-Linear Mech., 41 (2006), 1174. doi: doi:10.1016/j.ijnonlinmec.2006.12.001. Google Scholar

[22]

Y. Zhou, Regularity criteria for the generalized viscous MHD equations,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 24 (2007), 491. doi: doi:10.1016/j.anihpc.2006.03.014. Google Scholar

[23]

Y. Zhou and J. Fan, Regularity criteria for the viscous Camassa-Holm equations,, Int. Math. Res. Not. IMRN, (2009), 2508. Google Scholar

[24]

Y. Zhou and J. Fan, On regularity criteria in terms of pressure for the 3D viscous MHD equations,, Preprint (2008)., (2008). Google Scholar

[25]

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. doi: doi:10.1007/s00033-009-0023-1. Google Scholar

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