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July  2016, 15(4): 1179-1191. doi: 10.3934/cpaa.2016.15.1179

On the initial boundary value problem for certain 2D MHD-$\alpha$ equations without velocity viscosity

Jitao Liu 1,

1. 

Instituto de Matemática, Universidade Federal do Rio de Janeiro, Cidade Universitária - Ilha do Fundão, Caixa Postal 68530, 21941-909 Rio de Janeiro, RJ, Brazil

Received  May 2015 Revised  February 2016 Published  April 2016

This paper is concerned with the initial boundary value problem of certain 2D MHD-$\alpha$ equations without velocity viscosity over a bounded domain with smooth boundary. We show that the equations have a unique global smooth solution $(u,b)$ for $W^{4,p}\times H^4$ initial data and physical boundary condition.
Keywords: initial-boundary problem, MHD-$\alpha$ equations, global regularity..
Mathematics Subject Classification: 35Q35, 76D0.
Citation: Jitao Liu. On the initial boundary value problem for certain 2D MHD-$\alpha$ equations without velocity viscosity. Communications on Pure & Applied Analysis, 2016, 15 (4) : 1179-1191. doi: 10.3934/cpaa.2016.15.1179
References:
[1]

A.V. Busuioc and T.S. Ratiu, The second grade fluid and averaged Euler equations with Navier-slip boundary conditions,, \emph{Nonlinearity}, 16 (2003), 1119.   Google Scholar

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R. Caflisch, I. Klapper and G. Steele, Remarks on singularities, dimension and energy dissipation for ideal hydrodynamics and MHD,, \emph{Comm. Math. Phys.}, 184 (1997), 443.  doi: 10.1007/s002200050067.  Google Scholar

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E. Casella, P. Secchi and P. Trebeschi, Global classical solutions for MHD system,, \emph{J. Math. Fluid Mech.}, 5 (2003), 70.  doi: 10.1007/s000210300003.  Google Scholar

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D. Cioranescu and V. Girault, Weak and classical solutions of a family of second grade fluids,, \emph{Int. J. Non-Linear Mech.}, 32 (1997), 317.  doi: 10.1016/S0020-7462(96)00056-X.  Google Scholar

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L.C. Evans, Partial Differential Equations,, 2$^{nd}$ edition, (2010).   Google Scholar

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C. Foias and R. Temam, Remarques sur les équations de Navier-Stokes stationnaires et les phénomènes successifs de bifurcation (French),, \emph{Ann. Scuola Norm. Sup. Pisa Cl. Sci.}, 5 (1978), 28.   Google Scholar

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J. Fan and T. Ozawa, Global Cauchy problem for the 2-D magnetohydrodynamic-$\alpha$ models with partial viscous terms,, \emph{J. Math. Fluid Mech.}, 12 (2010), 306.  doi: 10.1007/s00021-008-0289-7.  Google Scholar

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

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

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G.D. Galdi and A. Sequeira, A further existence results for classical solutions of the equations of a second-grade fluid,, \emph{Arch. Rational Mech. Anal.}, 128 (1994), 297.  doi: 10.1007/BF00387710.  Google Scholar

[11]

D. Gilbarg and N.S. Trudinger, Elliptic Partial Differential Equations of Second Order,, Springer-Verlag, (1977).   Google Scholar

[12]

H. Kozono, Weak and classical solutions of the two-dimensional magnetohydrodynamic equations,, \emph{Tohoku Math. J.}, 41 (1989), 471.  doi: 10.2748/tmj/1178227774.  Google Scholar

[13]

O.A. Ladyzhenskaya, V.A. Solonnikov and N.U. Uraliceva, Linear and Quasi-linear Equations of Parabolic Type,, American Mathematical Society, (1968).   Google Scholar

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J.S. Linshiz and E.S. Titi, Analytical study of certain magnetohydrodynamic-$\alpha$ models,, \emph{J. Math. Phys.}, 48 (2007).  doi: 10.1063/1.2360145.  Google Scholar

[15]

M.C. Lopes Filho, H.J. Nussenzveig Lopes, E.S. Titi and A. Zang, Convergence of the 2D Euler-$\alpha$ to Euler equations in the Dirichlet case: indifference to boundary layers,, \emph{Physica D}, 292-293 (2015), 292.  doi: 10.1016/j.physd.2014.11.001.  Google Scholar

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L. Nirenberg, On elliptic partial differential equations,, \emph{Ann. Scuola Norm. Sup. Pisa}, 13 (1959), 115.   Google Scholar

[17]

K. Yamazaki, A remark on the two-dimensional Magnetohydrodynamic-alpha system,, preprint, ().   Google Scholar

[18]

J. Zhao and M. Zhu, Global regularity for the incompressible MHD-$\alpha$ system with fractional diffusion,, \emph{Appl. Math. Lett.}, 29 (2014), 26.  doi: 10.1016/j.aml.2013.10.009.  Google Scholar

show all references

References:
[1]

A.V. Busuioc and T.S. Ratiu, The second grade fluid and averaged Euler equations with Navier-slip boundary conditions,, \emph{Nonlinearity}, 16 (2003), 1119.   Google Scholar

[2]

R. Caflisch, I. Klapper and G. Steele, Remarks on singularities, dimension and energy dissipation for ideal hydrodynamics and MHD,, \emph{Comm. Math. Phys.}, 184 (1997), 443.  doi: 10.1007/s002200050067.  Google Scholar

[3]

E. Casella, P. Secchi and P. Trebeschi, Global classical solutions for MHD system,, \emph{J. Math. Fluid Mech.}, 5 (2003), 70.  doi: 10.1007/s000210300003.  Google Scholar

[4]

D. Cioranescu and V. Girault, Weak and classical solutions of a family of second grade fluids,, \emph{Int. J. Non-Linear Mech.}, 32 (1997), 317.  doi: 10.1016/S0020-7462(96)00056-X.  Google Scholar

[5]

L.C. Evans, Partial Differential Equations,, 2$^{nd}$ edition, (2010).   Google Scholar

[6]

C. Foias and R. Temam, Remarques sur les équations de Navier-Stokes stationnaires et les phénomènes successifs de bifurcation (French),, \emph{Ann. Scuola Norm. Sup. Pisa Cl. Sci.}, 5 (1978), 28.   Google Scholar

[7]

J. Fan and T. Ozawa, Global Cauchy problem for the 2-D magnetohydrodynamic-$\alpha$ models with partial viscous terms,, \emph{J. Math. Fluid Mech.}, 12 (2010), 306.  doi: 10.1007/s00021-008-0289-7.  Google Scholar

[8]

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

[9]

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

[10]

G.D. Galdi and A. Sequeira, A further existence results for classical solutions of the equations of a second-grade fluid,, \emph{Arch. Rational Mech. Anal.}, 128 (1994), 297.  doi: 10.1007/BF00387710.  Google Scholar

[11]

D. Gilbarg and N.S. Trudinger, Elliptic Partial Differential Equations of Second Order,, Springer-Verlag, (1977).   Google Scholar

[12]

H. Kozono, Weak and classical solutions of the two-dimensional magnetohydrodynamic equations,, \emph{Tohoku Math. J.}, 41 (1989), 471.  doi: 10.2748/tmj/1178227774.  Google Scholar

[13]

O.A. Ladyzhenskaya, V.A. Solonnikov and N.U. Uraliceva, Linear and Quasi-linear Equations of Parabolic Type,, American Mathematical Society, (1968).   Google Scholar

[14]

J.S. Linshiz and E.S. Titi, Analytical study of certain magnetohydrodynamic-$\alpha$ models,, \emph{J. Math. Phys.}, 48 (2007).  doi: 10.1063/1.2360145.  Google Scholar

[15]

M.C. Lopes Filho, H.J. Nussenzveig Lopes, E.S. Titi and A. Zang, Convergence of the 2D Euler-$\alpha$ to Euler equations in the Dirichlet case: indifference to boundary layers,, \emph{Physica D}, 292-293 (2015), 292.  doi: 10.1016/j.physd.2014.11.001.  Google Scholar

[16]

L. Nirenberg, On elliptic partial differential equations,, \emph{Ann. Scuola Norm. Sup. Pisa}, 13 (1959), 115.   Google Scholar

[17]

K. Yamazaki, A remark on the two-dimensional Magnetohydrodynamic-alpha system,, preprint, ().   Google Scholar

[18]

J. Zhao and M. Zhu, Global regularity for the incompressible MHD-$\alpha$ system with fractional diffusion,, \emph{Appl. Math. Lett.}, 29 (2014), 26.  doi: 10.1016/j.aml.2013.10.009.  Google Scholar

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