IUMJ

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

Authors: Yuko Enomoto and Yoshihiro Shibata

Issue: Volume 53 (2004), Issue 5, 1291-1330

Abstract:

In this paper, we prove a local energy decay of the Oseen semigroup in the $n$-dimensional exterior domain ($n \ge 3$). In the three-dimensional exterior domain case, Kobayashi and Shibata (T. Kobayashi, Yoshihiro Shibata, 'On the Oseen equation in the three-dimensional exterior domains', Math. Ann. 310 (1998), 1-45) 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 $L_p$-$L_q$ 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.