Proc. Japan Acad. Ser. A Math. Sci. 81 (9), 151-155, (Nov. 2005) DOI: 10.3792/pjaa.81.151
Yoshihiro Shibata, Senjo Shimizu
KEYWORDS: Stokes equations, Neumann boundary condition, maximal regularity, Navier-Stokes equations, free boundary problem, 35Q30, 76D05
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.