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

Abstract We consider the regularity of axisymmetric weak solutions to the Navier–Stokes equations in R 3 . Let u be an axisymmetric weak solution in R 3 × ( 0 , T ) , w = curl u , and w θ 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 w θ ∈ L q ( 0 , T ; L r ) , with 3 2 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 w θ ∈ L 1 ( 0 , T ; B ˙ ∞ , ∞ 0 ) .

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