$L^q$-$L^r$ estimate of the Oseen flow in plane exterior domains

Abstract

We consider the initial value problem for the Oseen system in plane exterior domains and study the large time behavior of solutions. For the space dimension $n\geq 3$ the theory was well developed by [26], [10] and [11], while 2D case has remained open because of difficulty arising from singularity like $\log\sqrt{\lambda+\alpha^2}$ of the Oseen resolvent, where $\lambda$ is the resolvent parameter and $\alpha$ is the Oseen parameter. In this paper we derive the local energy decay of the Oseen semigroup and apply it to deduction of $L^q$-$L^r$ estimates. The dependence of estimates on the Oseen parameter $\alpha$ is also discussed. The proof relies on detailed analysis of asymptotic structure of the fundamental solution of the Oseen resolvent with respect to both $\lambda$ and $\alpha$.