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

TL;DR

This paper develops anisotropically weighted $L^q$-$L^r$ estimates for the Oseen semigroup in exterior domains and applies them to analyze the stability and behavior of Navier-Stokes flow past a rigid body in three-dimensional space.

Contribution

It introduces new anisotropic weighted estimates for the Oseen semigroup and applies these to study the stability and asymptotic behavior of Navier-Stokes flows in exterior domains.

Findings

Derived anisotropically weighted $L^q$-$L^r$ estimates for the Oseen semigroup.

Analyzed the stability of Navier-Stokes flow in weighted $L^q$ spaces.

Characterized the spatial-temporal behavior of nonstationary solutions.

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.