On Leray's problem for almost periodic flows

TL;DR

This paper establishes existence and uniqueness of almost periodic flows in semi-infinite cylinders, addressing Leray's problem for 3D fluid motion with different almost periodic function settings.

Contribution

It provides the first rigorous proof of Leray's problem for almost periodic flows, using variational and wave-number analysis for Stepanov and Besicovitch functions respectively.

Findings

Existence and uniqueness of almost periodic solutions in semi-infinite cylinders.

Construction of a unique almost periodic solution to Leray's problem.

Conditions on flux size and Fourier coefficients for solution existence.

Abstract

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, while for Besicovitch solutions certain limitations on the generalized Fourier coefficients are requested.