The Liouville Theorem for the Steady-State Navier–Stokes Problem for Axially Symmetric 3D Solutions in Absence of Swirl

We study the Navier–Stokes equations of steady motion of a viscous incompressible fluid in $${\mathbb{R}^{3}}$$R3. We prove that there are no nontrivial solution of these equations defined in the whole space $${\mathbb{R}^{3}}$$R3 for axially symmetric case with no swirl (the Liouville theorem). Also we prove the conditional Liouville type theorem for axial symmetric solutions to the Euler system.