Ill-posedness of 212 D electron MHD

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 = ∇× (a ez)+b ez with $ a = a(x, y) $ and $ b = b(x, y) $. For 3≤β<4, we construct smooth, compactly supported initial data (a₀, b₀) with arbitrarily small H^β× Hβ-1 norm and a prescribed arbitrarily short time tN>0 such that every solution continuous in this Sobolev space on [0, tN] develops norm inflation in H^β× Hβ-1 at tN. This complements the ill-posedness result of Jeong and Oh [17], which establishes unboundedness of the solution map at magnetic regularity Hʳ, r≥3, corresponding to (a, b)∈ Hʳ⁺¹× Hʳ. Our construction reaches the lower scalar regularity range 3≤β<4 through a transport-driven norm-inflation mechanism rather than degenerating wave packets.