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 $\alpha$.
Citation
Toshiaki HISHIDA. "$L^q$-$L^r$ estimate of the Oseen flow in plane exterior domains." J. Math. Soc. Japan 68 (1) 295 - 346, January, 2016. https://doi.org/10.2969/jmsj/06810295
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