In this paper, we obtain local energy decay estimates and - estimates of the solutions to the Stokes equations with Neumann boundary condition which is obtained as a linearized equation of the free boundary problem for the Navier-Stokes equations. Comparing with the non-slip boundary condition case, we have a better decay estimate for the gradient of the semigroup because of the null force at the boundary.
Yoshihiro SHIBATA. Senjo SHIMIZU. "Decay properties of the Stokes semigroup in exterior domains with Neumann boundary condition." J. Math. Soc. Japan 59 (1) 1 - 34, January, 2007. https://doi.org/10.2969/jmsj/1180135499