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