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

2024/11/18 by Lai, Baishun, Zhang, Shihao
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2411.11419

Abstract

In this paper, we employ the localization technique in frequency space developed by Tao in \citeMR4337421 to investigate the quantitative estimates for the MHD equations. With the help of quantitative Carleman inequalities given by Tao in \citeMR4337421 and the pigeonhole principle, we establish the quantitative regularity for the critical L³ norm bounded solutions which enables us explicitly quantify the blow-up behavior in terms of L³ 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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