On an existence theorem for the Navier-Stokes equations with free slip boundary condition in exterior domain
This paper deals with a nonstationary problem for the Navier-Stokes equations with a free slip boundary condition in an exterior domain. We obtain a global in time unique solvability theorem and temporal asymptotic behavior of the global strong solution when the initial velocity is sufficiently small in the sense of $L^n$ ($n$ is dimension). The proof is based on the contraction mapping principle with the aid of $L^p$-$L^q$ estimates for the Stokes semigroup associated with a linearized problem, which is also discussed. In particular, we mainly discuss the local energy decay property of the semigroup which is a key estimate to prove the $L^p$-$L^q$ estimates in an exterior domain.