A Liouville theorem for the helically symmetric steady Navier-Stokes system in $R^3$

In this paper, we prove that any bounded smooth helically symmetric solution to the steady Navier-Stokes system in $\mathbb{R}^3$ must be a constant vector. This Liouville-type theorem holds for any bounded three-dimensional solutions of the steady Navier-Stokes system in the whole space, when only the helically symmetric assumption rather than integrability or decay assumptions of the solutions is assumed. The key idea of the proof is based on the so-called helical identity and the periodicity of the flows along the axial direction for helically symmetric flows, which help to establish the Saint-Venant-type estimate.