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

TL;DR

This paper uses frequency space localization techniques to derive quantitative regularity estimates for solutions of the MHD equations, explicitly characterizing blow-up behavior near potential singularities.

Contribution

It introduces a novel application of Tao's frequency space localization and Carleman inequalities to quantify regularity and blow-up in MHD equations.

Findings

Established quantitative regularity for critical $L^3$ norm bounded solutions.

Explicitly quantified blow-up behavior near singularities.

Developed technical innovations like the corrector function for scale inconsistencies.

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.