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

Let v(x,t)=vrer+vθeθ+vzez be a solution to the three-dimensional incompressible axially-symmetric Navier–Stokes equations. Denote by b=vrer+vzez the radial-axial vector field. Under a general scaling invariant condition on b, we prove that the quantity Γ=rvθ is Hölder continuous at r=0, t=0. 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 [15], and Seregin and Sverak (2009) in [26] as a conjecture) under the condition that b∈L∞([0,T],BMO−1). As another application, we prove that if b∈L∞([0,T],BMO−1), then v is regular.