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Decay and Vanishing of some D-Solutions of the Navier–Stokes Equations
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Speaker(s): Dr. Na Zhao
Time: Dec 1, 2020, 10:30-11:30
Venue: 201 Jinqizhigu Building, No.1 Tangling Road, Nanshan District, Shenzhen

An old problem since Leray 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; (2) when the solution is axially symmetric and periodic in the vertical variable. 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. This is a joint work with Dr. Bryan Carrillo, Prof. Xinghong Pan and Prof. Qi S. Zhang.