On Leray's Problem for Almost Periodic Flows

J. Math. Sci. Univ. Tokyo
Vol. 19 (2012), No. 1, Page 69--130.

Berselli, Luigi C.;Romito, Marco
On Leray's Problem for Almost Periodic Flows
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We prove existence and uniqueness for fully-developed (Poiseuille-type) flows in semi-infinite cylinders, in the setting of (time) almost-periodic functions. In the case of Stepanov almost-periodic functions the proof is based on a detailed variational analysis of a linear ``inverse'' problem, while in the Besicovitch setting the proof follows by a precise analysis in wave-numbers. Next, we use our results to construct a unique almost periodic solution to the so called ``Leray's problem'' concerning 3D fluid motion in two semi-infinite cylinders connected by a bounded reservoir. In the case of Stepanov functions we need a natural restriction on the size of the flux (with respect to the viscosity), while for Besicovitch solutions certain limitations on the generalised Fourier coefficients are requested.

Mathematics Subject Classification (2010): Primary 35Q30, Secondary 76D03, 35B15.
Received: 2011-11-29