Linear ill-posedness of three-dimensional MHD boundary layer equations
Abstract
We prove linear ill-posedness in tangential Sobolev spaces for the three-dimensional resistive MHD boundary layer equations. We assume that a suitable linear combination of the initial tangential velocity components has a non-degenerate critical point and that both initial tangential magnetic components have a double zero at the same point. Compared with the corresponding two-dimensional MHD result, our construction requires only this double-zero condition, rather than a higher-order degeneracy of the magnetic shear. The proof combines the three-dimensional Prandtl critical layer construction with a vector correction to the two-dimensional magnetic quotient. This correction cancels the leading terms in both induction equations while preserving the magnetic divergence constraint. The resulting approximate solutions grow like exp(σt/√ε), and their residuals have a prefactor of order O(ε−3/8), up to logarithmic factors. A Duhamel argument then yields linear ill-posedness for every tangential derivative loss μ<1/8. Thus the three-dimensional Prandtl instability persists when the stabilizing tangential magnetic field degenerates.
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BibTeX
@misc{zhang2026linear,
title = {{Linear ill-posedness of three-dimensional MHD boundary layer equations}},
author = {Mingxue Zhang and Zhonger Wu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.14532},
eprint = {2609.14532},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.14532},
}
RIS
TY - UNPB
TI - Linear ill-posedness of three-dimensional MHD boundary layer equations
AU - Mingxue Zhang
AU - Zhonger Wu
PY - 2026
DA - 2026/09/13
PB - arXiv
DO - 10.48550/arxiv.2609.14532
UR - https://arxiv.org/abs/2609.14532
AB - We prove linear ill-posedness in tangential Sobolev spaces for the three-dimensional resistive MHD boundary layer equations. We assume that a suitable linear combination of the initial tangential velocity components has a non-degenerate critical point and that both initial tangential magnetic components have a double zero at the same point. Compared with the corresponding two-dimensional MHD result, our construction requires only this double-zero condition, rather than a higher-order degeneracy of the magnetic shear. The proof combines the three-dimensional Prandtl critical layer construction with a vector correction to the two-dimensional magnetic quotient. This correction cancels the leading terms in both induction equations while preserving the magnetic divergence constraint. The resulting approximate solutions grow like exp(σt/√ε), and their residuals have a prefactor of order O(ε^−3/8), up to logarithmic factors. A Duhamel argument then yields linear ill-posedness for every tangential derivative loss μ<1/8. Thus the three-dimensional Prandtl instability persists when the stabilizing tangential magnetic field degenerates.
ER -