Energy equality for weak solutions to the 3D magnetohydrodynamic equations in a bounded domain

Abstract

<p style='text-indent:20px;'>In this paper, we study the energy equality for weak solutions to the 3D homogeneous incompressible magnetohydrodynamic equations with viscosity and magnetic diffusion in a bounded domain. Two types of regularity conditions are imposed on weak solutions to ensure the energy equality. For the first type, some global integrability condition for the velocity <inline-formula><tex-math id="M1">\begin{document}$ \mathbf u $\end{document}</tex-math></inline-formula> is required, while for the magnetic field <inline-formula><tex-math id="M2">\begin{document}$ \mathbf b $\end{document}</tex-math></inline-formula> and the magnetic pressure <inline-formula><tex-math id="M3">\begin{document}$ \pi $\end{document}</tex-math></inline-formula>, some suitable integrability conditions near the boundary are sufficient. In contrast with the first type, the second type claims that if some additional interior integrability is imposed on <inline-formula><tex-math id="M4">\begin{document}$ \mathbf b $\end{document}</tex-math></inline-formula>, then the regularity on <inline-formula><tex-math id="M5">\begin{document}$ \mathbf u $\end{document}</tex-math></inline-formula> can be relaxed.</p>