In this paper, we prove a local energy decay of the Oseen semigroup in the n-dimensional exterior domain (n ≥ 3). In the three dimensional exterior domain case, Kobayashi and Shibata  already proved the local energy decay. Our theorem is not only an extension of the results due to Kobayashi and Shibata to the n-dimensional case but also the complete study of the local energy decay theorem for the Oseen equation with optimal time decay rates. The local energy decay gives us a crucial step to obtain the Lp-L q estimates of the Oseen semigroup, which enable us to prove the unique existence of globally in time solutions to the Navier-Stokes equation in an exterior domain with small initial data in the Ln framework, and their properties of time decay.
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