Thorough analysis of the Oseen system in 2D exterior domains

TL;DR

This paper develops $L_p$-estimates for the inhomogeneous Oseen system in 2D exterior domains, revealing its parabolic nature and implications for Navier-Stokes solvability with small velocities at infinity.

Contribution

It provides new $L_p$-estimates for the Oseen system in exterior domains, especially focusing on the half-space model and boundary conditions, advancing understanding of fluid flow near obstacles.

Findings

Establishment of $L_p$-estimates for the Oseen system in 2D exterior domains.

Identification of the parabolic character and wake region behind obstacles.

Implication of solvability results for the Navier-Stokes system with small velocity at infinity.

Abstract

We construct $L_p$-estimates for the inhomogeneous Oseen system studied in a two dimensional exterior domain $\Omega$ with inhomogeneous slip boundary conditions. The kernel of the paper is a result for the half space $\mathbb{R}^2_+$. Analysis of this model system shows us a parabolic character of the studied problem, resulting as an appearance of the wake region behind the obstacle. Main tools are given by the Fourier analysis to obtain the maximal regularity estimates. The results imply the solvability for the Navier-Stokes system for small velocity at infinity.