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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