Regularity criterion of axisymmetric weak solutions to the 3D Navier–Stokes equations

We consider the regularity of axisymmetric weak solutions to the Navier–Stokes equations in R3. Let u be an axisymmetric weak solution in R3 ×(0,T ), w = curlu, and wθ be the azimuthal component of w in the cylindrical coordinates. Chae–Lee [D. Chae, J. Lee, On the regularity of axisymmetric solutions of the Navier–Stokes equations, Math. Z. 239 (2002) 645–671] proved the regularity of weak solutions under the condition wθ ∈ Lq (0,T ;Lr ), with 32 < r <∞, 2 q + 3 r 2. We deal with the marginal case r =∞which they excluded. It is proved that u becomes a regular solution if wθ ∈ L1(0,T ; ˙B 0∞ ,∞). © 2006 Elsevier Inc. All rights reserved