Abstract:

This paper deals with the weighted temporal-spatial estimates and strong solvability of the Navier-Stokes equation in ${\mathbb R}_{+}^{n}$. With the aid of Ukai's representation of the Stokes semigroup, and weighted inequalities for the fractional integral operators, $L^{r}$-$L^{q}$ estimates with mixed spatial weights are made for the Stokes flow. Then by means of Hardy's inequality, and interpolation method for the weak $L^{s}$ space, existence of the integral solution in $L^{b}(0, T;L^{q}({\mathbb R}_{+}^{n}))$ with temporal and spatial weights for the Navier-Stoke equation, where the initial velocity $u_{0}$ belongs to $L^{s}({\mathbb R}_{+}^{n})$ with the weight $w^{s-n}$ for some $n\leq s<\infty$ is established. This solution is proved to be the regular one provided $n=3$, $n\leq s\leq4$, and $u_{0}$ also lies in $L_{\sigma}^{2}({\mathbb R}_{+}^{n})$. Considering that $L_{w^{s-n}}^{s}({\mathbb R}_{+}^{n})$ does not coincide with $L^{s}({\mathbb R}_{+}^{n})$ whenever $s>n$, results obtained here can be viewed as useful supplements to the literatures.

Key words: Half space, Navier-Stokes equation, Weighted temporal-spatial estimate