Liouville Type Theorems for the Stationary Navier–Stokes Equations in High-Dimension Without Vanishing Condition

Abstract

The Liouville theorem for smooth solutions with finite Dirichlet integrals and uniform vanishing conditions to high-dimension stationary Navier–Stokes equations was established as reported by Galdi (An introduction to the mathematical theory of the Navier–Stokes equations: Steady-state problems, Springer, New York, 2011). In this paper, we mainly concern with the Liouville type problem of weak solutions only with finite Dirichlet integral to the stationary Navier–Stokes equations on \(\mathbb {R}^d\) with \(d\ge 5\). We first establish a Liouville type theorem under some restrictions on the high-frequency part tending to infinity of velocity fields. Then, we show the uniqueness of weak solutions to the stationary fractional Navier–Stokes equations with finite critical Dirichlet integral by establishing another Liouville type theorem.