Discrete and Continuous Dynamical Systems - Series A (DCDS-A)

Global regularity for the 3D axisymmetric MHD Equations with horizontal dissipation and vertical magnetic diffusion
Pages: 301 - 322, Issue 1, January 2015

doi:10.3934/dcds.2015.35.301      Abstract        References        Full text (419.2K)           Related Articles

Quansen Jiu - School of Mathematical Sciences & Beijing Center of Mathematics and Information Sciences, Capital Normal University, Beijing, 100048, China (email)
Jitao Liu - The Institute of Mathematical Sciences, The Chinese University of Hong Kong, China (email)

1 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.       
2 C. Cao and J. Wu, Global regularity for the 2D MHD equations with mixed partial dissipation and magnetic diffusion, Adv. Math., 226 (2011), 1803-1822.       
3 C. Cao and J. Wu, Two regularity criteria for the 3D MHD equations, J. Differential Equations, 248 (2010), 2263-2274.       
4 D. Chae and J. Lee, On the regularity of the axisymmetric solutions of the Navier-Stokes equations, Math. Z., 239 (2002), 645-671.       
5 J.-Y. Chemin, B. Desjardins, I. Gallagher and E. Grenier, Fluids with anisotropic viscosity, M2AN Math. Model. Numer. Anal., 34 (2000), 315-335.       
6 G. Duvaut and J. L. Lions, Inéquations en thermoélasticité et magnétohydrodynamique, Arch. Rational Mech. Anal., 46 (1972), 241-279.       
7 T. Hmidi and F. Rousset, Global well-posedness for the Euler-Boussinesq system with axisymmetric data, J. Funct. Anal., 260 (2011), 745-796.       
8 T. Hou, Z. Lei and C. Li, Global regularity of the 3D axi-symmetric Navier-Stokes equations with anisotropic data, Comm. Partial Differential Equations, 33 (2008), 1622-1637.       
9 O. A. Ladyzhenskaya, Unique global solvability of the three-dimensional Cauchy problem for the Navier-Stokes equations in the presence of axial symmetry,(Russian), Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 7 (1968), 155-177.       
10 Z. Lei, On axially symmetric incompressible Magnetohydrodynamics in three dimensions, preprint, arXiv:1212.5968v2.
11 S. Leonardi, J. Malek, J. Necas and M. Pokorny, On axially symmetric flows in $R^{3}$, Z. Anal. Anwendungen, 18 (1999), 639-649.       
12 F.-H. Lin, L. Xu and P. Zhang, Global small solutions to 2-D incompressible MHD system, preprint, arXiv:1302.5877v2.
13 C. Miao and X. Zheng, On the global well-posedness for the Boussinesq system with horizontal dissipation, Comm. Math. Phys., 321 (2013), 33-67.       
14 M. Sermange and R. Temam, Some mathematical questions related to the MHD equations, Comm. Pure Appl. Math., 36 (1983), 635-664.       
15 M. R. Ukhovskii and V. I. Yudovich, Axially symmetric flows of ideal and viscous fluids filling the whole space, J. Appl. Math. Mech., 32 (1968), 52-61.       
16 J. Wu, Viscous and inviscid magnetohydrodynamics equations, J. Anal. Math., 73 (1997), 251-265.       

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