Estimates on Weighted L^q-Norms of the Nonstationary 3D Navier-Stokes Flow in an Exterior Domain

Abstract

This paper is devoted to estimates on weighted L^q- norms of the nonstationary 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 establish the weighted estimate $\left\||x|^{\alpha} u(t)\right\|_{q} \leq C\left(1+t^{\frac{\alpha}{2}+\varepsilon}\right) t^{-\frac{3}{2}\left(1-\frac{1}{q}\right)}$, t>0, where 0<α≤1 and $\frac{3}{2}<q<\infty$, or 1<α<2 and $\frac{3}{3-\alpha}<q<\infty$, 0<ε<1 is arbitrary, and $u_{0} \in L_{\sigma}^{3}(\Omega)$, $|x|^{\alpha} u_{0} \in L^{1}(\Omega)$ with $\left\|u_{0}\right\|_{3}$ sufficiently small. With the aid of the representation of the flow, we also prove that if in addition $u_{0} \in D_{a}^{1-1 / b, b}$ for some $\frac{6}{5} \leq a<\frac{3}{2}$ and 1<b<2 with $\frac{3}{a}+\frac{2}{b}=4$, then the optimal estimate $\left\||x|^{\alpha} u(t)\right\|_{q} \leq C\left(1+t^{\frac{\alpha}{2}}\right) t^{-\frac{3}{2}\left(1-\frac{1}{q}\right)}$, t>0 holds, where $\alpha>0$ and 1 < q < ∞. Compared with the literature, here no extra restriction is laid on the range of the exponents α and q.

Keywords

weighted estimates / navier-stokes flow / exterior domain / L¹-data