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

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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

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

Abstract
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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:

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[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
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[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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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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