On the partial regularity of weak solutions for the magnetohydrodynamics system in ${\mathbb{R}}^4$

Abstract

In this paper, the partial regularity of the weak solutions to the magnetohydrodynamics (MHD) system in ${\mathbb{R}}^4$ is studied. In order to tackle the lack of compactness arising in the spatially high-dimensional setting, inspired by Wu [Arch. Rational Mech. Anal. 239 (2021), 1771–1808], we use the defect measures and prove the existence of partially regular weak solutions (satisfying certain local energy inequality) to the 4-dimensional MHD system. As an application, we obtain that the 2-dimensional Hausdorff dimension of singular sets of these weak solutions is finite.