On some decay properties of Stokes semigroup of compressible viscous fluid flow in a 2-dimensional exterior domain

In this paper, we proved the local energy decay and some Lp–Lq decay properties of solutions to the initial–boundary value problem for the Stokes equations of compressible viscous fluid flow in a 2-dimensional exterior domain. Kobayashi (1997) [19], and Kobayashi and Shibata (1999) [21], treated the same problem in a 3-dimensional exterior domain, and our results obtained in this paper are an extension of results in Kobayashi (1997) [19], and Kobayashi and Shibata (1999) [21] to the 2-dimensional case. The fundamental solution to the resolvent equations for the Stokes equations in R2 has a logarithmical singularity at λ=0, λ being a resolvent parameter, while it is continuous up to λ=0 in R3. This difference requires us a new idea to prove the local energy decay estimate. Once getting the local energy decay estimate, the required Lp–Lq decay estimates in the exterior domain are obtained by combining the local energy estimate and the Lp–Lq estimates in R2 by a cut-off technique.