Magnetic norm inflation at the ${\mathrm{\ell }}^1$ Besov endpoint for viscous non-resistive incompressible MHD

We prove magnetic-field norm inflation at the origin for the viscous, non-resistive incompressible magnetohydrodynamic equations on ${\mathbb{R}}^d$, $d\ge 2$, in the endpoint space $${\dot{B}}_{\mathrm{\infty },1}^{-1}({\mathbb{R}}^d)\times {\dot{B}}_{\mathrm{\infty },1}^0({\mathbb{R}}^d)\mathrm{.}$$ For every sufficiently small $\epsilon >0$, we construct smooth divergence-free initial data $(u_0,b_0)$, with $b_0$ compactly supported and $$\text{norm}[{\dot{B}}_{\mathrm{\infty },1}^{-1}]u_0+\text{norm}[{\dot{B}}_{\mathrm{\infty },1}^0]b_0<\epsilon ,$$ such that the corresponding classical solution satisfies $$\text{norm}[{\dot{B}}_{\mathrm{\infty },1}^0]b(t_{\epsilon })>{\epsilon }^{-1}$$ for some $0<t_{\epsilon }<\epsilon$. Consequently, no solution map agreeing with classical solutions can be continuous at the origin in the magnetic endpoint topology. The low–high paraproduct mechanism available when the dyadic summability exponent is greater than one gives no gain at the ${\mathrm{\ell }}^1$ endpoint. We instead combine a resonant four-wave velocity interaction with magnetic stretching to produce uniformly sized contributions on a growing family of dyadic shells. Frequency-localized test functionals extract the endpoint lower bound, while uniform Lagrangian estimates in weighted Fourier $L^1$ spaces and a spatial localization argument control the remainders and remove the auxiliary constant magnetic field.