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 L _p-L _q 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 ℝ ² has a logarithmical singularity at λ=0, λ being a resolvent parameter, while it is continuous up to λ=0 in ℝ ³. This difference requires us a new idea to prove the local energy decay estimate. Once getting the local energy decay estimate, the required L _p-L _q decay estimates in the exterior domain are obtained by combining the local energy estimate and the L _p-L _q estimates in ℝ ² by a cut-off technique.