On the regularity of magneto-vorticity field and the global existence for the Hall magnetohydrodynamic equations

Abstract

In this paper, we investigate the incompressible viscous and resistive Hall magnetohydrodynamic equations (Hall MHD in short). We first study the regularity of the magneto-vorticity field $B+\omega$. In three dimensions, we derive some bounds of $B+\omega$ under a condition of the velocity field $u$. Moreover, if we consider the Hall MHD with 2D variables, the uniform-in-time bounds of $B+\omega$ come from the three dimensional case. The regularity of $B+\omega$ gives us a crucial clue of blow-up scenario and provides conditions of the existence of global-in-time solutions. In particular, we prove the global well-posedness of the Hall MHD (also the electron MHD) with 2D variables when the third component of the initial current density $J_0=\nabla \times B_0$ is sufficiently small. We also derive temporal decay rate of $B+\omega$.

arXiv comment21 pages

References in corpus (7)

• Linear Analysis of the Hall Effect in Protostellar Disks
• Kinetic formulation and global existence for the Hall-Magneto-hydrodynamics system
• Star Formation and the Hall Effect
• Local well-posedness for the Hall-MHD equations with fractional magnetic diffusion
• On partial regularity for the steady Hall magnetohydrodynamics system
• On some new global existence result of 3D Magnetohydrodynamic equations
• Regularity of the $3D$ stationary Hall magnetohydrodynamic equations on the plane