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

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 $partialOmega$ is only Lipschitz which is the minimum requirement to make the boundary condition $bcdot n$ sense. To approach our result, we first separate the 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. Then, by establishing a sharp $L^2L^2$ estimate for pressure $P$, we use zero boundary conditions of $u$ to inhibit the boundary effect and obtain global energy equality by choosing suitable cut-off functions.