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We consider the spatial-temporal behavior of the Navier-Stokes flow past arigid body in $\mathbb{R}^3$. The present paper develops analysis in Lebesguespaces with anisotropic weights $(1+|x|)^\alpha(1+|x|-x_1)^\beta$, whichnaturally arise in the asymptotic structure of fluid when the translationalvelocity of the body is parallel to the $x_1$-direction. We deriveanisotropically weighted $L^q$-$L^r$ estimates for the Oseen semigroup inexterior domains. As applications of those estimates, we study thestability/attainability of the Navier-Stokes flow in anisotropically weighted$L^q$ spaces to get the spatial-temporal behavior of nonstationary solutions.

Anisotropically weighted $L^q$-$L^r$ estimates of the Oseen semigroup in  exterior domains, with applications to the Navier-Stokes flow past a rigid  body