The steady Navier-Stokes equations in a system of unbounded channels with sources and sinks

TL;DR

This paper proves the existence and uniqueness of solutions to the steady Navier-Stokes equations in complex unbounded channel systems with sources and sinks, extending classical results to more general, non-simply-connected domains.

Contribution

It generalizes classical existence results for Navier-Stokes in unbounded domains by allowing non-simply-connected geometries and non-small fluxes, and introduces a novel proof technique for key inequalities.

Findings

Existence of solutions with bounded Dirichlet integral in compact subsets.

Unique solvability and flow attainability for small data.

Development of a new proof method using contradiction and Morse-Sard-type theorems.

Abstract

The steady motion of a viscous incompressible fluid in a junction of unbounded channels with sources and sinks is modeled through the Navier-Stokes equations under inhomogeneous Dirichlet boundary conditions. In contrast to many previous works, the domain is not assumed to be simply-connected and the fluxes are not assumed to be small. In this very general setting, we prove the existence of a solution with a uniformly bounded Dirichlet integral in every compact subset. This is a generalization of the classical Ladyzhenskaya-Solonnikov result obtained under the additional assumption of zero boundary conditions. For small data of the problem we also prove the unique solvability and attainability of Couette-Poiseuille flows at infinity. The main novelty of our approach is the proof of the corresponding Leray-Hopf-type inequality by Leray's reductio ad absurdum argument (since the standard Hopf cutoff extension procedure does not work for general boundary data). For this contradiction approach, we use some fine properties of weak solutions to the Euler system based on Morse-Sard-type theorems in Sobolev spaces obtained by Bourgain, Korobkov & Kristensen.