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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, Nonlinearity, 16 (2003), 1119-1149.  Google Scholar

[2]

R. Caflisch, I. Klapper and G. Steele, Remarks on singularities, dimension and energy dissipation for ideal hydrodynamics and MHD, Comm. Math. Phys., 184 (1997), 443-455. doi: 10.1007/s002200050067.  Google Scholar

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E. Casella, P. Secchi and P. Trebeschi, Global classical solutions for MHD system, J. Math. Fluid Mech., 5 (2003), 70-91. 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, Int. J. Non-Linear Mech., 32 (1997), 317-335. doi: 10.1016/S0020-7462(96)00056-X.  Google Scholar

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L.C. Evans, Partial Differential Equations, 2nd edition, Grad. Stud. Math, 19, Amer. Math. Soc., Providence, RI, 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), Ann. Scuola Norm. Sup. Pisa Cl. Sci., 5 (1978), 28-63.  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, J. Math. Fluid Mech., 12 (2010), 306-319. 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, Adv. Math., 137 (1998), 1-81. 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, Chaos, 12 (2002), 518-530. 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, Arch. Rational Mech. Anal., 128 (1994), 297-312. doi: 10.1007/BF00387710.  Google Scholar

[11]

D. Gilbarg and N.S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin-Heidelberg-New York, 1977.  Google Scholar

[12]

H. Kozono, Weak and classical solutions of the two-dimensional magnetohydrodynamic equations, Tohoku Math. J., 41 (1989), 471-488. 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, J. Math. Phys., 48 (2007), 065504, 28pp. 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, Physica D, 292-293 (2015), 51-61. doi: 10.1016/j.physd.2014.11.001.  Google Scholar

[16]

L. Nirenberg, On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa, 13 (1959), 115-162.  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, Appl. Math. Lett., 29 (2014), 26-29. 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, Nonlinearity, 16 (2003), 1119-1149.  Google Scholar

[2]

R. Caflisch, I. Klapper and G. Steele, Remarks on singularities, dimension and energy dissipation for ideal hydrodynamics and MHD, Comm. Math. Phys., 184 (1997), 443-455. doi: 10.1007/s002200050067.  Google Scholar

[3]

E. Casella, P. Secchi and P. Trebeschi, Global classical solutions for MHD system, J. Math. Fluid Mech., 5 (2003), 70-91. doi: 10.1007/s000210300003.  Google Scholar

[4]

D. Cioranescu and V. Girault, Weak and classical solutions of a family of second grade fluids, Int. J. Non-Linear Mech., 32 (1997), 317-335. doi: 10.1016/S0020-7462(96)00056-X.  Google Scholar

[5]

L.C. Evans, Partial Differential Equations, 2nd edition, Grad. Stud. Math, 19, Amer. Math. Soc., Providence, RI, 2010.  Google Scholar

[6]

C. Foias and R. Temam, Remarques sur les équations de Navier-Stokes stationnaires et les phénomènes successifs de bifurcation (French), Ann. Scuola Norm. Sup. Pisa Cl. Sci., 5 (1978), 28-63.  Google Scholar

[7]

J. Fan and T. Ozawa, Global Cauchy problem for the 2-D magnetohydrodynamic-$\alpha$ models with partial viscous terms, J. Math. Fluid Mech., 12 (2010), 306-319. 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, Adv. Math., 137 (1998), 1-81. 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, Chaos, 12 (2002), 518-530. 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, Arch. Rational Mech. Anal., 128 (1994), 297-312. doi: 10.1007/BF00387710.  Google Scholar

[11]

D. Gilbarg and N.S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin-Heidelberg-New York, 1977.  Google Scholar

[12]

H. Kozono, Weak and classical solutions of the two-dimensional magnetohydrodynamic equations, Tohoku Math. J., 41 (1989), 471-488. 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, J. Math. Phys., 48 (2007), 065504, 28pp. 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, Physica D, 292-293 (2015), 51-61. doi: 10.1016/j.physd.2014.11.001.  Google Scholar

[16]

L. Nirenberg, On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa, 13 (1959), 115-162.  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, Appl. Math. Lett., 29 (2014), 26-29. doi: 10.1016/j.aml.2013.10.009.  Google Scholar

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