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On the $L^p-$ theory of Anisotropic singular perturbations of elliptic problems
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 |
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. |
[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. |
[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. |
[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. |
[5] |
L.C. Evans, Partial Differential Equations, 2nd edition, Grad. Stud. Math, 19, Amer. Math. Soc., Providence, RI, 2010. |
[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. |
[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. |
[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. |
[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. |
[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. |
[11] |
D. Gilbarg and N.S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin-Heidelberg-New York, 1977. |
[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. |
[13] |
O.A. Ladyzhenskaya, V.A. Solonnikov and N.U. Uraliceva, Linear and Quasi-linear Equations of Parabolic Type, American Mathematical Society, 1968. |
[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. |
[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. |
[16] |
L. Nirenberg, On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa, 13 (1959), 115-162. |
[17] |
K. Yamazaki, A remark on the two-dimensional Magnetohydrodynamic-alpha system,, preprint, ().
|
[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. |
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. |
[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. |
[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. |
[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. |
[5] |
L.C. Evans, Partial Differential Equations, 2nd edition, Grad. Stud. Math, 19, Amer. Math. Soc., Providence, RI, 2010. |
[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. |
[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. |
[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. |
[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. |
[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. |
[11] |
D. Gilbarg and N.S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin-Heidelberg-New York, 1977. |
[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. |
[13] |
O.A. Ladyzhenskaya, V.A. Solonnikov and N.U. Uraliceva, Linear and Quasi-linear Equations of Parabolic Type, American Mathematical Society, 1968. |
[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. |
[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. |
[16] |
L. Nirenberg, On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa, 13 (1959), 115-162. |
[17] |
K. Yamazaki, A remark on the two-dimensional Magnetohydrodynamic-alpha system,, preprint, ().
|
[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. |
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