Blowup for the forced electron MHD
Mimi Dai
TL;DR
The paper investigates finite-time blowup in forced electron MHD by exploiting the Hall-term nonlinear structure. It develops an explicit iterative construction based on hyperbolic current-flow profiles, assembling high-frequency perturbations that preserve $C^{2-\varepsilon}$ regularity while driving the current-density gradient to grow without bound, culminating in $\int_0^1 \|\nabla\times J(t)\|_{L^\infty}\,dt=\infty$. It also presents a shear-current profile alternative that achieves blowup under forcing, showing robustness of the mechanism beyond hyperbolic geometries. The results provide a Beale-Kato-Majda-type blowup framework for 3D forced electron MHD and offer insight into Hall-induced transport and reconnection dynamics under forcing.
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.