Large time behavior of solutions to the $3$D anisotropic Navier-Stokes equation

MIkihiro Fujii

Published 2021-08-26Version 1

We consider the large time behavior of the solution to the $3$D Navier-Stokes equation with horizontal viscosity $\Delta{\rm h} u=\partial_1^2 u+\partial_2^2 u$ and show that the $L^p$ decay rate of the horizontal components of the velocity fields coinside to that of the $2$D heat kernel, while the vertical component behaves like the $3$D heat kernel. Moreover, we find that the asymptotic profile of the horizontal components of the velocity field is different from that of the vertical component.