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 $∥|x|^αu(t)∥_q ≤ C(1+t^{\frac{α}{2} +ε} )t^{-\frac{3}{2}(1-\frac{1}{q})} ,$$t>0,$ where $ 0<α≤1$ and $\frac{3}{2}0$ holds, where $α>0$ and $1