2 research outputs found
Global regularity for the 2D MHD equations with partial hyperresistivity
This paper establishes the global existence and regularity for a system of
the two-dimensional (2D) magnetohydrodynamic (MHD) equations with only
directional hyperresistivity. More precisely, the equation of (the
horizontal component of the magnetic field) involves only vertical
hyperdiffusion (given by ) while the equation of
(the vertical component) has only horizontal hyperdiffusion (given by
), where and are directional
Fourier multiplier operators with the symbols being and ,
respectively. We prove that, for , this system always possesses a
unique global-in-time classical solution when the initial data is sufficiently
smooth. The model concerned here is rooted in the MHD equations with only
magnetic diffusion, which play a significant role in the study of magnetic
reconnection and magnetic turbulence. In certain physical regimes and under
suitable scaling, the magnetic diffusion becomes partial (given by part of the
Laplacian operator). There have been considerable recent developments on the
fundamental issue of whether classical solutions of these equations remain
smooth for all time. The papers of Cao-Wu-Yuan \cite{CaoWuYuan} and of Jiu-Zhao
\cite{JiuZhao2} obtained the global regularity when the magnetic diffusion is
given by the full fractional Laplacian with . The
main result presented in this paper requires only directional fractional
diffusion and yet we prove the regularization in all directions. The proof
makes use of a key observation on the structure of the nonlinearity in the MHD
equations and technical tools on Fourier multiplier operators such as the
H\"{o}rmander-Mikhlin multiplier theorem. The result presented here appears to
be the sharpest for the 2D MHD equations with partial magnetic diffusion.Comment: Accepted for publication in International Mathematics Research
Notice
Global regularity for the 2D MHD equations with horizontal dissipation and horizontal magnetic diffusion
This paper establishes the global regularity of classical solution to the 2D
MHD system with only horizontal dissipation and horizontal magnetic diffusion
in a strip domain when the initial data is
suitable small. To prove this, we combine the Littlewood-Paley decomposition
with anisotropic inequalities to establish a crucial commutator estimate. We
also analysis the asymptotic behavior of the solution. In addition, the global
existence and uniqueness of classical solution is obtained for the 2D
simplified tropical climate model with only horizontal dissipations.Comment: 41 page