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

TL;DR

This paper proves the global energy equality for weak solutions of non-resistive MHD equations with physical boundaries under minimal boundary regularity, without boundary layer assumptions or extra pressure conditions.

Contribution

It establishes energy equality for non-resistive MHD equations with Lipschitz boundaries, relaxing previous boundary regularity and boundary layer assumptions.

Findings

Energy equality holds under specified integrability conditions.

No boundary layer assumptions are needed.

Boundary regularity only requires Lipschitz continuity.

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.