Discretely self-similar solutions for electron MHD

TL;DR

This work analyzes discretely self-similar solutions of the non-resistive electron MHD with Hall term, recasting the problem in a self-similar profile $H$ evolving in a logarithmic time variable. By deploying energy methods, radial cutoffs, time-periodicity, and precise decay/integrability conditions on $H$ and the current $J=\nabla\times H$, the authors establish a broad set of nonexistence results for nontrivial discretely self-similar blowups, covering both decaying and non-decaying regimes and a range of $\alpha$-dependent thresholds. The contributions include detailed proofs for multiple theorems, including cases with different $\alpha$ ranges and decay assumptions, as well as novel local energy inequalities and iterative arguments that force $H$ (and thus $B$) to vanish. Together, these results clarify when dispersive Hall dynamics preclude discretely self-similar singularities in the idealized EMHD model, providing rigorous constraints on possible singular behaviors and useful techniques for similar PDE blow-up analyses.

Abstract

We study discretely self-similar solutions for the electron magnetohydrodynamics (MHD) without resistivity. Under several different decay and non-decay conditions, we show the absence of non-trivial discretely self-similar blowup solutions.

Theorems & Definitions (9)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Lemma 7.1
• Proposition 7.2
• proof