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Consider an axis-symmetric suitable weak solution of 3D incompressible Navier-Stokes equation with nontrivial swirl, v = v r e r + v θ e θ + v z e z . Let z denote the axis of symmetry and r be the distance to the z-axis. If the solution satisfies a slightly supercritical assumption ( that is, |v| ≤ C (ln | ln r|) α r for α ∈ [0, 0.028] when r is small ), then we prove that v is regular. This extends the results in [5], [16], [18] where regularities under critical assumptions, such as |v| ≤ C r , were proven.Here and throughout the paper, we will use c and C to denote a generic constant. It may be different from line to line. Also we use A B to denote A ≤ CB.Remark 1.1 We note that Γ satisfies the equation (1.3) which enjoys the maximal principle. So the assumption sup x∈R 3 |Γ(·, −1)| < +∞ can assure that |Γ| ≤ C for all t ∈ [−1, 0) for some positive constant C.Readers can refer to [4] for the definition of suitable weak solutions.Recall the natural scaling of Navier-Stokes equations: If (v, p) is a solution of equations (1.1), then for any λ > 0, the following rescaled pair is also a solution:v λ (x, t) = λv(λx, λ 2 t), p λ (x, t) = λ 2 p(λx, λ 2 t).Denote b λ = v λ r e r + v λ z e z , we say our assumption (1.2) is supercritical which means, for a fixed point x 0 = (r 0 cos θ 0 , r 0 sin θ 0 , z 0 ), b λ (x 0 ) satisfies a bound

An old problem since Leray [Le] asks whether homogeneous D solutions of the 3 dimensional Navier-Stokes equation in R 3 or some noncompact domains are 0. In this paper, we give a positive solution to the problem in two cases: (1) full 3 dimensional slab case R 2 × [0, 1] with Dirichlet boundary condition (Theorem 1.1); (2) when the solution is axially symmetric and periodic in the vertical variable (Theorem 1.3).Also, in the slab case, we prove that even if the Dirichlet integral has some growth, axially symmetric solutions with Dirichlet boundary condition must be swirl free, namely u θ = 0, thus reducing the problem to essentially a "2 dimensional" problem. In addition, a general Dsolution (without the axial symmetry assumption) vanishes in R 3 if, in spherical coordinates, the positive radial component of the velocity decays at order -1 of the distance. The paper is self contained comparing with [CPZ] although the general idea is related.Since J ln r,

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