On the global wellposedness for free boundary problem for the Navier-Stokes systems with surface tension

The aim of this paper is to show the global wellposedness of the Navier-Stokes equations, including surface tension and gravity, with a free surface in an unbounded domain such as bottomless ocean. In addition, it is proved that the solution decays polynomially as time $t$ tends to infinity. To show these results, we first use the Hanzawa transformation in order to reduce the problem in a time-dependent domain $\Omega_t\subset\mathbf{R}^3$, $t>0$, to a problem in the lower half-space $\mathbf{R}_-^3$. We then establish some time-weighted estimate of solutions, in an $L_p$-in-time and $L_q$-in-space setting, for the linearized problem around the trivial steady state with the help of $L_r\text{-}L_s$ time decay estimates of semigroup. Next, the time-weighted estimate, combined with the contraction mapping principle, shows that the transformed problem in $\mathbf{R}_-^3$ admits a global-in-time solution in the $L_p\text{-}L_q$ setting and that the solution decays polynomially as time $t$ tends to infinity under the assumption that $p$, $q$ satisfy the conditions: $2<p<\infty$, $3<q<16/5$, and $(2/p)+(3/q)<1$. Finally, we apply the inverse transformation of Hanzawa's one to the solution in $\mathbf{R}_-^3$ to prove our main results mentioned above for the original problem in $\Omega_t$. Here we want to emphasize that it is not allowed to take $p=q$ in the above assumption about $p$, $q$, which means that the different exponents $p$, $q$ of $L_p\text{-}L_q$ setting play an essential role in our approach

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

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