Hölder regularity of solutions and physical quantities for the ideal electron magnetohydrodynamic equations

Abstract

In this paper, we make the first attempt to figure out the differences on Hölder regularity in time of solutions and conserved physical quantities between the ideal electron magnetohydrodynamic equations concerning Hall term and the incompressible Euler equations involving convection term. It is shown that the regularity in time of magnetic field B B is C t α 2 C_{t}^{\frac {\alpha }2} provided it belongs to L t ∞ C x α L_{t}^{\infty } C_{x}^{\alpha } for any α > 0 \alpha >0 , its energy is C t 2 α 2 − α C_{t}^{\frac {2\alpha }{2-\alpha }} as long as B B belongs to L t ∞ B ˙ 3 , ∞ α L_{t}^{\infty } \dot {B}^{\alpha }_{3,\infty } for any 0 > α > 1 0>\alpha >1 and its magnetic helicity is C t 2 α + 1 2 − α C_{t}^{\frac {2\alpha +1}{2-\alpha }} supposing B B belongs to L t ∞ B ˙ 3 , ∞ α L_{t}^{\infty } \dot {B}^{\alpha }_{3,\infty } for any 0 > α > 1 2 0>\alpha >\frac 12 , which are quite different from the classical incompressible Euler equations.