Local Well-posedness for the Magnetohydrodynamics in the different two liquids case

We prove the local well-posedness for a two phase problem of magnetohydrodynamics with a sharp interface. The solution is obtained in the maximal regularity space: $H_p^1((0,T),L_q)\cap L_p((0,T),H_q^2)$ with $1 < p, q < \infty$and2\mathrm{/}p+N\mathrm{/}q<1$, where $N$ is the space dimension