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

Abstract

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 Ωt⊂R3, t>0, to a problem in the lower half-space R−3. We then establish some time-weighted estimate of solutions, in an Lp-in-time and Lq-in-space setting, for the linearized problem around the trivial steady state with the help of Lr-Ls 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 Lp-Lq setting and that the solution decays polynomially as time t tends to infinity under the assumption that p, q satisfy the conditions: 2<p<∞, 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 Ωt. 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 Lp-Lq setting play an essential role in our approach.