On three-dimensional Hall-magnetohydrodynamic equations with partial dissipation

Baoying Du

doi:10.1186/s13661-022-01587-0

Abstract

In this paper, we address the Hall-MHD equations with partial dissipation. Applying some important inequalities (such as the logarithmic Sobolev inequality using BMO space, bilinear estimates in BMO space, Young’s inequality, cancellation property, interpolation inequality) and delicate energy estimates, we establish an improved blow-up criterion for the strong solution. Moreover, we also obtain the existence of the strong solution for small initial data, the smallness conditions of which are given by the suitable Sobolev norms.

Highlights

The incompressible Hall-magnetohydrodynamic equations with full dissipation in three dimensions read as: ⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨duitv+u(=u · ∇)u 0, + ∇(p π) = κ1ux1x1 κ2ux2x2 κ3ux3x3
We investigate the Hall-magnetohydrodynamic system with full viscosity and partial dissipation
Inspired by [8, 13, 19, 33], we find a new blow-up criterion for strong solution, which imposes the condition is (u, ∇B) ∈ L2(0, T; BMO)

Summary

Introduction

The incompressible Hall-magnetohydrodynamic equations with full dissipation in three dimensions read as:. Compared to usual MHD system and the Boussinesq equations, Hall-MHD equations involve ∇ × ((∇ × B) × B), it is Hall term and plays a crucial position in magnetic reconnection due to Ohm’s law. Hall-MHD is very important for such problems as magnetic reconnection in neutron stars, geo-dynamo, space plasmas, and star formation. The paper [7] got the local existence and global small solutions for the Hall-magnetohydrodynamics. Some results on the Boussinesq and MHD equations with partial viscosity were obtained in [5, 6, 15, 24]. Inspired by [8, 13, 19, 33], we find a new blow-up criterion for strong solution, which imposes the condition is (u, ∇B) ∈ L2(0, T; BMO).

Noticing the fact that

One can use the Hölder inequality to deduce that