Center for                           Nonlinear Analysis CNA Home People Seminars Publications Workshops and Conferences CNA Working Groups CNA Comments Form Summer Schools Summer Undergraduate Institute PIRE Cooperation Graduate Topics Courses SIAM Chapter Seminar Positions Contact Publication 14-CNA-028 Stability of Vortex Solutions to an Extended Navier-Stokes System Gung-Min GieDepartment of Mathematics University of Louisville Louisville, KY 40292gungmin.gie@louisville.edu Christopher HendersonDepartment of Mathematics Stanford University Stanford, CA 94305chris@math.stanford.edu Gautam IyerDepartment of Mathematical Sciences Carnegie Mellon University Pittsburgh, PA 15213gautam@math.cmu.edu Landon KavlieDepartment of Mathematics, Statistics, and Computer Science University of Illinois Chicago, Chicago, IL 60607lkavli2@uic.edu Jared P. WhiteheadMathematics Department Brigham Young University Provo, UT 84602whitehead@mathematics.byu.eduAbstract: We study the long-time behavior an extended Navier-Stokes system in $\mathbb{R^2}$ 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.Get the paper in its entirety as  14-CNA-028.pdf