Publication 14-CNA-028
Stability of Vortex Solutions to an Extended
Navier-Stokes System
Gung-Min Gie
Department of Mathematics
University of Louisville
Louisville, KY 40292
gungmin.gie@louisville.edu
Christopher Henderson
Department of Mathematics
Stanford University
Stanford, CA 94305
chris@math.stanford.edu
Gautam Iyer
Department of Mathematical Sciences
Carnegie Mellon University
Pittsburgh, PA 15213
gautam@math.cmu.edu
Landon Kavlie
Department of Mathematics, Statistics, and Computer Science
University of Illinois
Chicago, Chicago, IL 60607
lkavli2@uic.edu
Jared P. Whitehead
Mathematics Department
Brigham Young University
Provo, UT 84602
whitehead@mathematics.byu.edu
Abstract: 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.
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