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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 Ln (n is dimension). The proof is based on the contraction mapping principle with the aid of L-L 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-L estimates in an exterior domain.

On an existence theorem for the Navier-Stokes equations with free slip boundary condition in exterior domain