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

We consider the large time behavior of the solution to the 3D Navier–Stokes equation with horizontal viscosity Δhu=∂12u+∂22u and show that the Lp decay rate of the horizontal components of the velocity field coincides to that of the 2D heat kernel, while the vertical component decays like the 3D heat kernel. Moreover, we consider the asymptotic expansion of the solution and find that a portion of the nonlinear term affect the leading term of the horizontal components of the velocity field, whereas the leading term of the vertical component is given by only the linear solution.