Blowup for the forced electron MHD

Abstract

The electron magnetohydrodynamics (MHD) contains a highly nonlinear Hall term with an interesting structure. Exploring the Hall nonlinear structure, we investigate possible phenomena of finite time blow up for the electron MHD with a (non-rough) forcing. When the magnetic field has zero horizontal components, the vertical component equation has a mixing feature with the mixer being the current flow. By constructing a magnetic field profile whose current density is approximately a hyperbolic flow near the origin, we show blowup develops in finite time. In another setting when the magnetic field is a shear type, the Hall term vanishes, and finite time blowup can be obtained for the forced electron MHD as well.