Global solvability for viscous free surface flows of infinite depth in three and higher dimensions

Abstract

This paper is concerned with the global solvability for the Navier-Stokes equations describing viscous free surface flows of infinite depth in three and higher dimensions. We first prove time weighted estimates of solutions to a linearized system of the Navier-Stokes equations by time decay estimates of a $C_0$-analytic semigroup and maximal regularity estimates in an $L_p$-in-time and $L_q$-in-space setting with suitable $p$, $q$. The time weighted estimates then enable us to show the global solvability of the Navier-Stokes equations for small initial data by the contraction mapping principle.