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The aim of this paper is to show the global wellposedness of theNavier-Stokes equations, including surface tension and gravity, with a freesurface in an unbounded domain such as bottomless ocean. In addition, it isproved that the solution decays polynomially as time $t$ tends to infinity. Toshow these results, we first use the Hanzawa transformation in order to reducethe problem in a time-dependent domain $\Omega_t\subset\mathbf{R}^3$, $t>0$, toa problem in the lower half-space $\mathbf{R}_-^3$. We then establish sometime-weighted estimate of solutions, in an $L_p$-in-time and $L_q$-in-spacesetting, for the linearized problem around the trivial steady state with thehelp of $L_r\text{-}L_s$ time decay estimates of semigroup. Next, thetime-weighted estimate, combined with the contraction mapping principle, showsthat the transformed problem in $\mathbf{R}_-^3$ admits a global-in-timesolution in the $L_p\text{-}L_q$ setting and that the solution decayspolynomially 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 originalproblem 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 differentexponents $p$, $q$ of $L_p\text{-}L_q$ setting play an essential role in ourapproach.

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