<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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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 <i>n</i> ≥ 3 the theory was well developed by [<b>26</b>], [<b>10</b>] and [<b>11</b>], while 2D case has remained open because of difficulty arising from singularity like log √(λ+α<sup>2</sup>) 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 <i>L</i><sup><i>q</i></sup>-<i>L</i><sup><i>r</i></sup> 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.
- Journal of the Mathematical Society of Japan
Journal of the Mathematical Society of Japan 68(1), 295-346, 2016
The Mathematical Society of Japan