<i>L</i><sup><i>q</i></sup>-<i>L</i><sup><i>r</i></sup> estimate of the Oseen flow in plane exterior domains

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  • L[q]-L[r] estimate of the Oseen flow in plane exterior domains

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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 ≥ 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 √(λ+α2) of the Oseen resolvent, where λ is the resolvent parameter and α is the Oseen parameter. In this paper we derive the local energy decay of the Oseen semigroup and apply it to deduction of Lq-Lr estimates. The dependence of estimates on the Oseen parameter α is also discussed. The proof relies on detailed analysis of asymptotic structure of the fundamental solution of the Oseen resolvent with respect to both λ and α.

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