On well-posedness and blowup criteria for the magnetohydrodynamics with the Hall and ion-slip effects

In this paper, we prove some regularity criteria and the local well-posedness of strong solutions to the magnetohydrodynamics with the Hall and ion-slip effects. We also establish global existence and time decay rate for small data.

[1]  H. Triebel.  Theory Of Function Spaces , 1983 .

[2]  Rainer Grauer,et al.  Axisymmetric Flows in Hall-MHD: A Tendency Towards Finite-Time Singularity Formation , 2005 .

[3]  Giovanni P. Galdi,et al.  A new approach to energy theory in the stability of fluid motion , 1990 .

[4]  Eric Dumas,et al.  On the Weak Solutions to the Maxwell–Landau–Lifshitz Equations and to the Hall–Magneto–Hydrodynamic Equations , 2013, Communications in Mathematical Physics.

[5]  Local Well-Posedness for the Hall-MHD Equations with Fractional Magnetic Diffusion , 2014, 1404.0486.

[6]  Dongho Chae,et al.  On the blow-up criterion and small data global existence for the Hall-magnetohydrodynamics , 2013, 1305.4681.

[7]  V. Solonnikov,et al.  On an initial boundary-value problem for the equation of magnetohydrodynamics with the hall and ion-slip effects , 1997 .

[8]  M. Maiellaro.  Uniqueness of M.H.D. thermodiffusive mixture flows with hall and ion-slip effects , 1977 .

[9]  Dongho Chae,et al.  Singularity formation for the incompressible Hall-MHD equations without resistivity , 2013, 1312.5519.

[10]  Dongho Chae,et al.  On the temporal decay for the Hall-magnetohydrodynamic equations , 2013, 1302.4601.

[11]  R. Temam,et al.  Navier-Stokes equations: theory and numerical analysis: R. Teman North-Holland, Amsterdam and New York. 1977. 454 pp. US $45.00 , 1978 .

[12]  Pierre Degond,et al.  Well-posedness for Hall-magnetohydrodynamics , 2012, 1212.3919.

[13]  Yong Zhou,et al.  Regularity criteria for the 3D MHD equations in terms of the pressure , 2006 .

[14]  E. Stein,et al.  Hp spaces of several variables , 1972 .

[15]  Gen Nakamura,et al.  Well-posedness for the axisymmetric incompressible viscous Hall-magnetohydrodynamic equations , 2013, Appl. Math. Lett..

[16]  Pierre Degond,et al.  Kinetic formulation and global existence for the Hall-Magneto-hydrodynamics system , 2011 .