Magnetic norm inflation below the velocity regularity threshold for viscous non-resistive MHD
Abstract
We study viscous, non-resistive magnetohydrodynamics on ℝd, d∈{2,3}. For every fixed magnetic exponent s>d/2 and every velocity exponent q<s−1, we prove norm inflation at the origin in Hq× Hs. Arbitrarily small divergence-free Schwartz data generate classical solutions whose magnetic Hs norm becomes arbitrarily large at arbitrarily small positive times, while the velocity remains arbitrarily small in Hq throughout the observation interval. Together with endpoint local well-posedness, this identifies q=s−1 as the sharp lower velocity threshold for fixed s>d/2. The construction uses a symmetric pair of high-frequency velocity packets and a fixed low-frequency magnetic profile. A Fourier component of their stretching interaction has a fixed sign and gains the power Ks−1−q on a short diffusive time scale. Subtracting an exact reference flow and using endpoint Besov estimates controls the full nonlinear remainder. Quantitative higher-order estimates ensure existence through the observation time and yield the uniform smallness of the velocity in the target norm.
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BibTeX
@misc{wang2026magnetic,
title = {{Magnetic norm inflation below the velocity regularity threshold for viscous non-resistive MHD}},
author = {Yinghui Wang},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.01124},
eprint = {2611.01124},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.01124},
}
RIS
TY - UNPB
TI - Magnetic norm inflation below the velocity regularity threshold for viscous non-resistive MHD
AU - Yinghui Wang
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.01124
UR - https://arxiv.org/abs/2611.01124
AB - We study viscous, non-resistive magnetohydrodynamics on ℝ^d, d∈{2,3}. For every fixed magnetic exponent s>d/2 and every velocity exponent q<s−1, we prove norm inflation at the origin in H^q× H^s. Arbitrarily small divergence-free Schwartz data generate classical solutions whose magnetic H^s norm becomes arbitrarily large at arbitrarily small positive times, while the velocity remains arbitrarily small in H^q throughout the observation interval. Together with endpoint local well-posedness, this identifies q=s−1 as the sharp lower velocity threshold for fixed s>d/2. The construction uses a symmetric pair of high-frequency velocity packets and a fixed low-frequency magnetic profile. A Fourier component of their stretching interaction has a fixed sign and gains the power K^s−1−q on a short diffusive time scale. Subtracting an exact reference flow and using endpoint Besov estimates controls the full nonlinear remainder. Quantitative higher-order estimates ensure existence through the observation time and yield the uniform smallness of the velocity in the target norm.
ER -