Liouville-type theorems for Axisymmetric solutions to steady Navier-Stokes system in a layer domain

TL;DR

This paper establishes Liouville-type theorems for axisymmetric steady Navier-Stokes solutions in a layered domain, showing solutions are trivial under certain growth conditions of the outlet width.

Contribution

It provides new Liouville-type results for steady Navier-Stokes solutions in layered domains with boundary conditions, based on growth rate constraints.

Findings

Bounded solutions are trivial if outlet width grows slower than R^{1/2}.

D-solutions are trivial if outlet width grows slower than R^{4/5}.

A Saint-Venant type estimate is key to the proofs.

Abstract

In this paper, we investigate the Liouville-type theorems for axisymmetric solutions to steady Navier-Stokes system in a layer domain. The both cases for the flows supplemented with no-slip boundary and Navier boundary conditions are studied. If the width of the outlet grows at a rate less than $R^{\frac{1}{2}}$, any bounded solution is proved to be trivial. Meanwhile, if the width of the outlet grows at a rate less than $R^{\frac{4}{5}}$, every D-solution is proved to be trivial. The key idea of the proof is to establish a Saint-Venant type estimate that characterizes the growth of Dirichlet integral of nontrivial solutions.