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arXiv · 1808.10386

Decay and vanishing of some D-solutions of the Navier-Stokes equations

Bryan Carrillo·Xinghong Pan·Qi S. Zhang·Na Zhao
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Abstract

An old problem since Leray \cite{Le:1} asks whether homogeneous D solutions of the 3 dimensional Navier-Stokes equation in $\mathbb{R}^3$ or some noncompact domains are 0. In this paper, we give a positive solution to the problem in two special cases: (1) when the solution is axially symmetric and periodic in the vertical variable; (2) full 3 dimensional slab case $\mathbb{R}^2 \times [0, 1]$ with Dirichlet boundary condition. Other partial results are also presented. The paper is self contained comparing with the first part \cite{CPZ:1} although the general idea is related.

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Bryan Carrillo, Xinghong Pan, Qi S. Zhang, Na Zhao. 2018-08-30. Decay and vanishing of some D-solutions of the Navier-Stokes equations. https://doi.org/10.1007/s00205-020-01533-3

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