On the maximal $L_p$-$L_q$ regularity theorem for the linearized electro-magnetic field equations with interface conditions

This paper deals with the maximal $L_p$-$L_q$ regularity theorem for the linearized electro-magnetic field equations with interface conditions and perfect wall condition. This problem is motivated by linearization of the coupled magnetohydrodynamics system, which generates two separate problems. The first problem is associated with the well studied Stokes system. Another problem related to the magnetic field is studied in this paper. The maximal $L_p$-$L_q$ regularity theorem for the Stokes equations with interface and non-slip boundary conditions has been proved by Pruess and Simonett [15], Maryani and Saito [12]. Combination of these results and the result obtained in this paper yields local well-posedness for MHD problem in the case of two incompressible liquids separated by a closed interface. We plan to prove it in a forthcoming paper.
The main part of the paper is devoted to proving the existence of $\mathcal{R}$ bounded solution operators associated with the generalized resolvent problem. The maximal $L_p$-$L_q$ regularity is established by applying the Weis operator valued Fourier multiplier theorem.