Ill-posedness of $2\frac12$D electron MHD

TL;DR

The paper investigates ill-posedness of the 2.5D electron MHD system $B_t+ \nabla\times ((\nabla\times B)\times B)=0$ with $B=\nabla\times (a\hat{e}_z)+b\hat{e}_z$, reformulated in $(x,y)$ through $a=a(x,y,t)$ and $b=b(x,y,t)$. By constructing a delicate initial data family in $H^β \times H^{β-1}$ for $1<β<4$, the authors build an approximating solution that drives norm inflation in $a$ while keeping the other component under control, and then perform a rigorous perturbation analysis around this approximant. They show that for arbitrarily small $δ>0$ there exists $(a_0,b_0)$ with $\|a_0\|_{H^β}+\|b_0\|_{H^{β-1}} \lesssim δ$ such that the true solution blows up in the homogeneous norms within time $T\in(0,δ]$, i.e., $\|a(T)\|_{\dot{H}^β}+\|b(T)\|_{\dot{H}^{β-1}} \gtrsim 1/δ$. The results hinge on exploiting the transport structure of the $a$-equation and constructing an oscillatory, radial-perturbed data in polar coordinates, providing a different ill-posedness mechanism from prior high-low frequency techniques in fluid models. This work clarifies the delicate borderline regularity behavior of 2.5D EMHD and highlights transport-induced norm inflation as a route to instability in a quasi-linear, supercritical setting.

Abstract

We consider the electron magnetohydrodynamics (MHD) in the context where the 3D magnetic field depends only on the two horizontal plane variables. In particular, the magnetic field takes the form $B=\nabla\times (a\vec e_z)+b\vec e_z$ with $a=a(x,y)$ and $b=b(x,y)$. Initial data $(a_0,b_0)$ is constructed in the Sobolev space $H^β\times H^{β-1}$ with $1<β<4$ such that the solution to this electron MHD system either escapes the space or develops norm inflation in $\dot H^β\times \dot H^{β-1}$.

Theorems & Definitions (8)

• Theorem 1.1
• Lemma 3.1
• Lemma 3.2
• Lemma 3.3
• Lemma 3.4
• Lemma 3.6
• Lemma 4.1
• Lemma 4.2