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

Hirokazu Saito, Yoshihiro Shibata

Published 2019-12-20Version 1

The aim of this paper is to show the global existence of solutions to the Navier-Stokes equations, including surface tension and gravity, with a free surface on 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 on a time-dependent domain $\Omega_t$, $t>0$, to a problem on the lower half-space $\mathbf{R}_-^3=\{(x',x_3) \mid x'=(x_1,x_2)\in\mathbf{R}^2,x_3<0\}$. 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 $L_r\text{-}L_s$ decay estimates of semigroup associated with the linearized problem. Next, the time-weighted estimate, combined with the contraction mapping principle, shows that the transformed problem on $\mathbf{R}_-^3$ admits in the $L_p\text{-}L_q$ setting a unique solution global in time and that the solution decays polynomially as time $t$ tends to infinity under the assumption that the exponents $p$, $q$ 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 on $\mathbf{R}_-^3$ to prove our main results mentioned above for the original problem on $\Omega_t$. 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.