Abstract:

This paper studies weighted estimates for the 3D Navier-Stokes flow in an exterior domain. By multiplying the Navier-Stokes equation with a well selected vector field, an integral equation is derived, from which we prove that $\||x|^{\alpha}u(t)\|_{q}\leq C(1+t^{\frac{\alpha}{2}})t^{-\frac{3}{2}(\frac{1}{r}-\frac{1}{q})}$ for all $t>0$ under the initial condition $|x|^{\alpha}u_{0}\in L^{r}(\Omega)$ and $u_{0}\in L_{\sigma}^{3}(\Omega)$ with sufficiently small norm $\|u_{0}\|_{3}$, where $0<\alpha<3$, $1<r<3$ and $\max \left\{r, \frac{3}{2}\right\}＜q＜\infty$ meeting some other reasonable constraints. Compare with previous results, our initial condition in case $0<\alpha\leq2$ and restriction on $q$ is weaker.

Key words: exterior domain, Navier-Stokes flow, weighted estimate