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arXiv · 2411.11419

Quantitative regularity for the MHD equations via the localization technique in frequency space

Baishun Lai·Shihao Zhang
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Abstract

In this paper, we employ the localization technique in frequency space developed by Tao in \cite{MR4337421} to investigate the quantitative estimates for the MHD equations. With the help of quantitative Carleman inequalities given by Tao in \cite{MR4337421} and the pigeonhole principle, we establish the quantitative regularity for the critical $L^3$ norm bounded solutions which enables us explicitly quantify the blow-up behavior in terms of $L^3$ norm near a potential first-time singularity. Some technical innovations, such as introducing the corrector function, are required due to the fact that the scales are inconsistent between the magnetic field and the vorticity field.

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Baishun Lai, Shihao Zhang. 2024-11-18. Quantitative regularity for the MHD equations via the localization technique in frequency space. https://arxiv.org/abs/2411.11419

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