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

We consider the spatial–temporal behavior of the Navier–Stokes flow past a rigid body in R3$\mathbb {R}^3$. This paper develops analysis in Lebesgue spaces with anisotropic weights (1+|x|)α(1+|x|−x1)β${(1+|x|)}^\alpha {(1+|x|-x_1)}^\beta$, which naturally arise in the asymptotic structure of fluid when the translational velocity of the body is parallel to the x1‐direction. We derive anisotropically weighted Lq$L^q$‐Lr$L^r$ 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 Lq$L^q$ spaces to get the spatial–temporal behavior of nonstationary solutions.