A Liouville theorem for the axially-symmetric Navier–Stokes equations

Abstract: Let be a solution to the three-dimensional incompressible axially-symmetric Navier–Stokes equations. Denote by the radial-axial vector field. Under a general scaling invariant condition on b, we prove that the quantity is Hölder continuous at , . As an application, we prove that the ancient weak solutions of axi-symmetric Navier–Stokes equations must be zero (which was raised by Koch, Nadirashvili, Seregin and Sverak (2009) in and Seregin and Sverak (2009) in as a conjecture) under the condition that . As another application, we prove that if , then v is regular. [Copyright &y& Elsevier]