On the stationary Navier-Stokes equations in distorted pipes under energy-stable outflow boundary conditions

TL;DR

This paper proves the existence of weak solutions for stationary Navier-Stokes equations in distorted pipes with energy-stable outflow boundary conditions, using advanced mathematical techniques without restrictions on data.

Contribution

It introduces a novel existence proof for stationary Navier-Stokes solutions in complex pipe geometries with mixed boundary conditions, employing the Leray-Schauder principle and Bernoulli's law.

Findings

Existence of weak solutions is established without data restrictions.

Unique solvability is proven under small data assumptions.

The approach employs harmonic divergence-free vector fields and contradiction arguments.

Abstract

The steady motion of a viscous incompressible fluid in distorted pipes, of finite length, is modeled through the Navier-Stokes equations with mixed boundary conditions: the inflow is given by an arbitrary member of the Lions-Magenes class with positive influx, and the fluid motion is subject to a directional do-nothing boundary condition on the outlet, together with the standard no-slip assumption on the remaining walls of the domain. Existence of a weak solution to such Navier-Stokes system is proved without any restriction on the data (that is, inlet velocity and external force) by means of the Leray-Schauder Principle, in which the required a priori estimate is obtained by a contradiction argument that employs Bernoulli's law for solutions of the stationary Euler equations, as well as some properties of harmonic divergence-free vector fields. Under a suitable smallness assumption on the data, we also prove the unique solvability of the boundary-value problem.