Energy equality of MHD system under a weaker condition on magnetic field

Abstract

We prove the energy equality of MHD system in the space founded by Cheskidov et al. (Nonlinearity 21:1233–1252, 2008) and Berselli and Chiodaroli (Nonlinear Anal 192:111704, 2020). It is clarified that the energy equality is established for a larger class of the magnetic field than that of velocity field. Most of the cases, we deal with the energy equality of MHD system in bounded domain. On the other hand, if the spacial integrability exponents of the weak solution are large, it is necessary to use the Besov space, which is suitable for us to handle freely derivatives of the nonlinear convection term. Only in this case we deal with the energy equality of MHD system in the whole space. Our result covers most of previous theorems on validity of the energy equality on the Navier–Stokes equations.