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.
On Leray's problem for almost periodic flows / Berselli, LUIGI CARLO; Romito, Marco. - In: JOURNAL OF MATHEMATICAL SCIENCES UNIVERSITY OF TOKYO. - ISSN 1340-5705. - STAMPA. - 19:1(2012), pp. 69-130.