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

摘要

Let v(x,t)=v~re_r+v~θeθ+v ~ze_z be a solution to the three-dimensional incompressible axially-symmetric Navier-Stokes equations. Denote by b=v~re_r+v~ze_z the radial-axial vector field. Under a general scaling invariant condition on b, we prove that the quantity Γ=rv~θ is H?lder 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.