Stability of vortex solutions to an extended Navier-Stokes system

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 (1989) and was revisited recently [Liu, Liu,Pego 2007] in bounded domains in order to explain the fast convergence of certain numerical schemes [Johnston, Liu 2004]. 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 and Wayne 2005 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.