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


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 $\alpha$.


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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.


Published: January, 2016
First available in Project Euclid: 25 January 2016

zbMATH: 1343.35187
MathSciNet: MR3454561
Digital Object Identifier: 10.2969/jmsj/06810295

Primary: 35Q30

Keywords: decay , Oseen semigroup , plane exterior domain

Rights: Copyright © 2016 Mathematical Society of Japan


Vol.68 • No. 1 • January, 2016
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