Local energy decay of solutions to the Oseen equation in the exterior domains

In this paper, we prove a local energy decay of the Oseen semigroup in the n-dimensional exterior domain (n greater than or equal to 3). In the three dimensional exterior domain case, Kobayashi and Shlbata [16] already proved the local energy decay. Our theorem is not only an extension of the results due to Kobayashi and Shlbata 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 L-p-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 L-n framework, and their properties of time decay.