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

Abstract

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 \[ \dot B^{-1}_{\infty,1}(\R^d)\times \dot B^{0}_{\infty,1}(\R^d). \] For every sufficiently small $\epsilon >0$, we construct smooth divergence-free initial data $(u_0,b_0)$, with $b_0$ compactly supported and \[ \norm[\dot B^{-1}_{\infty,1}]{u_0} +\norm[\dot B^{0}_{\infty,1}]{b_0}<\varepsilon, \] such that the corresponding classical solution satisfies \[ \norm[\dot B^{0}_{\infty,1}]{b(t_\varepsilon)}>\varepsilon^{-1} \] 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 ${\ell }^1$ 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.