We prove the $L_p$-$L_q$ maximal regularity of solutions to the Neumann problem for the Stokes equations with non-homogeneous boundary condition and divergence condition in a bounded domain. And as an application, we consider a free boundary problem for the Navier-Stokes equation. We prove a locally in time unique existence of solutions to this problem for any initial data and a globally in time unique existence of solutions to this problem for some small initial data.
Yoshihiro Shibata. Senjo Shimizu. "$L_p$-$L_q$ maximal regularity and viscous incompressible flows with free surface." Proc. Japan Acad. Ser. A Math. Sci. 81 (9) 151 - 155, Nov. 2005. https://doi.org/10.3792/pjaa.81.151