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 of R-3, t > 0, to a problem in the lower half-space 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-L-s time decay estimates of semigroup. Next, the time-weighted estimate, combined with the contraction mapping principle, shows that the transformed problem in R--(3) admits a global-in-time solution in the L-p-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 < infinity, 3 < q < 16/5, and (2/p)+(3/q)<1. Finally, we apply the inverse transformation of Hanzawa's one to the solution in 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-L-q setting play an essential role in our approach.(c) 2023 Elsevier Inc. All rights reserved.
