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Low Regularity Norm Inflation of Axisymmetric Hall Magnetohydrodynamics in Sobolev Space
Abstract
In this paper, we study the axisymmetric Hall magnetohydrodynamics where u = urer+uzez and B = Bθeθ. We construct small, smooth, compactly supported, and axisymmetric initial data (u0,B0) and prove that there is a unique smooth axisymmetric solution with norm inflation in Hσ−1(ℝ3) × Hσ(ℝ3) for 1<σ<7/2. Moreover, we study the case with fractional dissipation (−Δ)βu and (−Δ)αB, modifying the initial data slightly and proving that the solution inflates in Hσ−1(ℝ3) × Hσ(ℝ3) for 1<σ<7/2−2max{α,β} and $ 0 \le α, β< 5/4$.
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BibTeX
@misc{zhu2026regularity,
title = {{Low Regularity Norm Inflation of Axisymmetric Hall Magnetohydrodynamics in Sobolev Space}},
author = {Haoming Zhu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.06730},
eprint = {2608.06730},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.06730},
}
RIS
TY - UNPB
TI - Low Regularity Norm Inflation of Axisymmetric Hall Magnetohydrodynamics in Sobolev Space
AU - Haoming Zhu
PY - 2026
DA - 2026/09/19
PB - arXiv
DO - 10.48550/arxiv.2608.06730
UR - https://arxiv.org/abs/2608.06730
AB - In this paper, we study the axisymmetric Hall magnetohydrodynamics where u = u^re_r+u^ze_z and B = B^θe_θ. We construct small, smooth, compactly supported, and axisymmetric initial data (u_0,B_0) and prove that there is a unique smooth axisymmetric solution with norm inflation in H^σ−1(ℝ^3) × H^σ(ℝ^3) for 1<σ<7/2. Moreover, we study the case with fractional dissipation (−Δ)^βu and (−Δ)^αB, modifying the initial data slightly and proving that the solution inflates in H^σ−1(ℝ^3) × H^σ(ℝ^3) for 1<σ<7/2−2max{α,β} and $ 0 \le α, β< 5/4$.
ER -