Stability of Vortex Solutions to an Extended Navier-Stokes System

TL;DR

This paper investigates the stability and long-term behavior of vortex solutions in an extended, less constrained Navier-Stokes system in two dimensions, demonstrating stability results similar to classical Navier-Stokes under certain conditions.

Contribution

It establishes the global stability of Oseen vortices in the extended system, including cases with non-zero mean divergence, and proves global well-posedness.

Findings

Oseen vortex is globally asymptotically stable when initial divergence has mean zero.

Existence and stability of vortex solutions when initial divergence is not mean zero.

Global well-posedness of the extended Navier-Stokes system.

Abstract

We study the long-time behavior an extended Navier-Stokes system in $\R^2$ where the incompressibility constraint is relaxed. This is one of several "reduced models" of Grubb and Solonnikov '89 and was revisited recently (Liu, Liu, Pego '07) in bounded domains in order to explain the fast convergence of certain numerical schemes (Johnston, Liu '04). Our first result shows that if the initial divergence of the fluid velocity is mean zero, then the Oseen vortex is globally asymptotically stable. This is the same as the Gallay Wayne '05 result for the standard Navier-Stokes equations. When the initial divergence is not mean zero, we show that the analogue of the Oseen vortex exists and is stable under small perturbations. For completeness, we also prove global well-posedness of the system we study.