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

Abstract

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

77 pages