Global strong solutions to the inhomogeneous incompressible Navier-Stokes system in the exterior of a cylinder

TL;DR

This paper proves the existence of global strong axisymmetric solutions to the inhomogeneous incompressible Navier-Stokes equations outside a cylinder, allowing vacuum states and utilizing axisymmetry to establish key bounds.

Contribution

It establishes the first global strong solutions for the inhomogeneous Navier-Stokes system in exterior cylindrical domains with vacuum, leveraging axisymmetry for crucial estimates.

Findings

Existence of global strong solutions in exterior cylindrical domains.

Solutions can include vacuum states.

Key bounds on velocity field in specific function spaces.

Abstract

In this paper, the global strong axisymmetric solutions for the inhomogeneous incompressible Navier-Stokes system are established in the exterior of a cylinder subject to the Dirichlet boundary conditions. Moreover, the vacuum is allowed in these solutions. One of the key ingredients of the analysis is to obtain the ${L^{2}(s,T;L^{\infty}(\Omega))}$ bound for the velocity field, where the axisymmetry of the solutions plays an important role.