Communications in Mathematical Sciences

Volume 14 (2016)

Number 7

Stability of vortex solutions to an extended Navier–Stokes system

Pages: 1773 – 1797



Gung-Min Gie (Department of Mathematics, University of Louisville, Kentucky, U.S.A.)

Christopher Henderson (Department of Mathematics, Stanford University, Stanford, California, U.S.A.)

Gautam Iyer (Mathematical Sciences, Carnegie Mellon University, Pittsburgh, Pennsylvania, U.S.A.)

Landon Kavlie (Department of Mathematics, Statistics, and Computer Science, University of Illinois, Chicago, Il., U.S.A.)

Jared P. Whitehead (Department of Mathematics, Brigham Young University, Provo, Utah, U.S.A.)


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 (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.


Navier–Stokes equation, infinite energy solutions, extended system, long-time behavior, Lyapunov function, asymptotic stability

2010 Mathematics Subject Classification

35Q30, 65M06, 76D05, 76M25

Published 14 September 2016