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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

Christopher Henderson
Department of Mathematics
Stanford University
Stanford, CA 94305

Gautam Iyer
Department of Mathematical Sciences
Carnegie Mellon University
Pittsburgh, PA 15213

Landon Kavlie
Department of Mathematics, Statistics, and Computer Science
University of Illinois
Chicago, Chicago, IL 60607

Jared P. Whitehead
Mathematics Department
Brigham Young University
Provo, UT 84602

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.

Get the paper in its entirety as  14-CNA-028.pdf

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