Energy conservation for the non-resistive MHD equations with physical boundaries

Abstract

In this paper, we study the energy equality for weak solutions to the non-resistive MHD equations with physical boundaries. Although the equations of magnetic field $b$ are of hyperbolic type, and the boundary effects are considered, we still prove the global energy equality provided that $ u \in L^{q}_{loc}\left(0, T ; L^{p}(\Omega)\right) \text { for any } \frac{1}{q}+\frac{1}{p} \leq \frac{1}{2}, \text { with } p \geq 4,\text{ and } b \in L^{r}_{loc}\left(0, T ; L^{s}(\Omega)\right) \text { for any } \frac{1}{r}+\frac{1}{s} \leq \frac{1}{2}, \text { with } s \geq 4 $. In particular, compared with the existed results, we do not require any boundary layer assumptions and additional conditions on the pressure $P$. Our result requires the regularity of boundary $\partial\Omega$ is only Lipschitz which is the minimum requirement to make the boundary condition $b\cdot n$ sense. The proof is based on the important properties of weak solutions of the nonstationary Stokes system and the separate mollification of weak solutions from the boundary effect by considering a non-standard local energy equality and transform the boundary effects into the estimates of the gradient of cut-off functions.