Magnetic norm inflation at the ℓ1 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 Rd, d≥2, in the endpoint space Ḃ−1∞,1(Rd)× Ḃ0∞,1(Rd). For every sufficiently small ε>0, we construct smooth divergence-free initial data (u0,b0), with b0 compactly supported and norm[Ḃ−1∞,1]u0 +norm[Ḃ0∞,1]b0<ε, such that the corresponding classical solution satisfies norm[Ḃ0∞,1]b(tε)>ε−1 for some 0<tε<ε. 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 ℓ1 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 L1 spaces and a spatial localization argument control the remainders and remove the auxiliary constant magnetic field.
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BibTeX
@misc{ban2026magnetic,
title = {{Magnetic norm inflation at the $\ell^1$ Besov endpoint for viscous non-resistive incompressible MHD}},
author = {Yingzhe Ban and Boling Guo and Maotuo Guo},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.17192},
eprint = {2609.17192},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.17192},
}
RIS
TY - UNPB
TI - Magnetic norm inflation at the ℓ^1 Besov endpoint for viscous non-resistive incompressible MHD
AU - Yingzhe Ban
AU - Boling Guo
AU - Maotuo Guo
PY - 2026
DA - 2026/09/15
PB - arXiv
DO - 10.48550/arxiv.2609.17192
UR - https://arxiv.org/abs/2609.17192
AB - We prove magnetic-field norm inflation at the origin for the viscous, non-resistive incompressible magnetohydrodynamic equations on R^d, d≥2, in the endpoint space Ḃ^−1_∞,1(R^d)× Ḃ^0_∞,1(R^d). For every sufficiently small ε>0, we construct smooth divergence-free initial data (u_0,b_0), with b_0 compactly supported and norm[Ḃ^−1_∞,1]u_0 +norm[Ḃ^0_∞,1]b_0<ε, such that the corresponding classical solution satisfies norm[Ḃ^0_∞,1]b(t_ε)>ε^−1 for some 0<t_ε<ε. 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 ℓ^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.
ER -