Liouville type theorems on the steady Navier-Stokes equations in R3

Abstract

In this paper we study the Liouville type properties for solutions to the steady incompressible Navier-Stoks equations in ${\mathbf{R}}^3$. It is shown that any solution to the steady Navier-Stokes equations in ${\mathbf{R}}^3$ with finite Dirichlet integral and vanishing velocity field at far fields must be trivial. This solves an open problem. The key ingredients of the proof include a Hodge decomposition of the energy-flux and the observation that the square of the deformation matrix lies in the local Hardy space. As a by-product, we also obtain a Liouville type theorem for the steady density-dependent Navier-Stokes equations.

Our proof for Proposition 6 is incomplete. We need to check the validity of the estimate (14)

Cited by in corpus (3)

• On Liouville type theorems for the stationary MHD and the Hall-MHD systems in ${\mathbb{R}}^3$
• On Liouville-type theorems for the 2D stationary MHD equations
• A Review of results on axially symmetric Navier-Stokes equations, with addendum by X. Pan and Q. S. Zhang