Decay and Vanishing of some D-Solutions of the Navier–Stokes Equations.

An old problem since Leray (J Math Pure Appl (French) 9:1–82, 1933) 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) the 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, for 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 D-solution (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 (Carrillo et al. in Funct Anal, 2020. 10.1016/j.jfa.2020.108504) although the general idea is related. [ABSTRACT FROM AUTHOR]