We study the long-time behavior an extended Navier-Stokes system in R2
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.Comment: 24 pages, 1 figure, updated to add authors' contact information and
to address referee's comment