Liouville-type theorems for steady Navier-Stokes system under helical symmetry or Navier boundary conditions

TL;DR

This paper establishes Liouville-type theorems for steady Navier-Stokes solutions under helical symmetry and Navier boundary conditions, showing conditions under which solutions are trivial or Poiseuille flows.

Contribution

It proves new Liouville-type theorems for steady Navier-Stokes equations with specific symmetries and boundary conditions, extending understanding of solution triviality.

Findings

Bounded helically symmetric solutions are constant.

Solutions with axisymmetric swirl or radial velocity decay are zero.

Non-large velocity solutions are Poiseuille flows.

Abstract

In this paper, the Liouville-type theorems for the steady Navier-Stokes system are investigated. First, we prove that any bounded smooth helically symmetric solution in $\mathbb{R}^3$ must be a constant vector. Second, for steady Navier-Stokes system in a slab supplemented with Navier boundary conditions, we prove that any bounded smooth solution must be zero if either the swirl or radial velocity is axisymmetric, or $ru^{r}$ decays to zero as $r$ tends to infinity. Finally, when the velocity is not big in $L^{\infty}$-space, the general three-dimensional steady Navier-Stokes flow in a slab with the Navier boundary conditions must be a Poiseuille type flow. The key idea of the proof is to establish Saint-Venant type estimates that characterize the growth of Dirichlet integral of nontrivial solutions.