Stability of stationary solutions of the forced Navier-Stokes equations on the two-torus

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November 2002, 2(4): 483-494. doi: 10.3934/dcdsb.2002.2.483

Stability of stationary solutions of the forced Navier-Stokes equations on the two-torus

Chuong V. Tran ^1, , Theodore G. Shepherd ^1, and Han-Ru Cho ^1,

1.	Department of Physics, University of Toronto, 60 St. George Street, Toronto, ON, Canada M5S 1A7, Canada, Canada, Canada

Received January 2002 Revised June 2002 Published August 2002

Abstract
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We study the linear and nonlinear stability of stationary solutions of the forced two-dimensional Navier-Stokes equations on the domain $[0,2\pi]\times[0,2\pi/\alpha]$, where $\alpha\in(0,1]$, with doubly periodic boundary conditions. For the linear problem we employ the classical energy--enstrophy argument to derive some fundamental properties of unstable eigenmodes. From this it is shown that forces of pure $x_2$-modes having wavelengths greater than $2\pi$ do not give rise to linear instability of the corresponding primary stationary solutions. For the nonlinear problem, we prove the equivalence of nonlinear stability with respect to the energy and enstrophy norms. This equivalence is then applied to derive optimal conditions for nonlinear stability, including both the high- and low-Reynolds-number limits.

Keywords: Two-dimensional Navier–Stokes equations, linear stability, asymptotic (global) stability..

Mathematics Subject Classification: 34D, 35Q30, 7.

Citation: Chuong V. Tran, Theodore G. Shepherd, Han-Ru Cho. Stability of stationary solutions of the forced Navier-Stokes equations on the two-torus. Discrete & Continuous Dynamical Systems - B, 2002, 2 (4) : 483-494. doi: 10.3934/dcdsb.2002.2.483

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