Decay rate for the incompressible flows in half spaces

Abstract

We show that the time decay rate of $L^2$ norm of weak solution for the Stokes equations and for the Navier–Stokes equations on the half spaces are $t^{-\frac{n}{2}(\frac{1}{r}-\frac{1}{2})-\frac{1}{2}}$ if the initial data $u_0\in L^2\cap L^{r}$ and $\int_{{\mathbb R}^n_+} |y_n u_0(y)|^r dy < \infty$ for $1 < r < 2$ . We also show that the decay rate is determined by the linear part of the weak solution. We use the heat kernel and Ukai's solution formula for the Stokes equations. It has been known up to now that the decay rate on the half space was $t^{-\frac{n}{2}(\frac{1}{r}-\frac{1}{2})}$ , which was obtained by Borchers and Miyakawa [1] and Ukai [9].