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We consider the large time behavior of the solution to the $3$D Navier-Stokesequation with horizontal viscosity $\Delta_{\rm h} u=\partial_1^2u+\partial_2^2 u$ and show that the $L^p$ decay rate of the horizontalcomponents of the velocity field coinsides to that of the $2$D heat kernel,while the vertical component decays like the $3$D heat kernel. Moreover, weconsider the asymptotic expansion of the solution and find that a portion ofthe nonlinear term affect the leading term of the horizontal components of thevelocity field, whereas the leading term of the vertical component is given byonly the linear solution.

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