DISCRETE AND CONTINUOUS Website: http://AIMsciences.org DYNAMICAL SYSTEMS–SERIES B Volume 2, Number 4, November 2002 pp. 483–494 STABILITY OF STATIONARY SOLUTIONS OF THE FORCED NAVIER-STOKES EQUATIONS ON THE TWO-TORUS Chuong V. Tran, Theodore G. Shepherd, and Han-Ru Cho Department of Physics University of Toronto 60 St. George Street, Toronto, ON, Canada, M5S 1A7 (Communicated by Shouhong Wang) Abstract. We study the linear and nonlinear stability of stationary solutions of the forced two-dimensional Navier-Stokes equations on the domain [0, 2π] × [0, 2π/α], where α ∈ (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 2π 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. 1. Introduction. We consider 2D incompressible fluid flow in a doubly periodic rectangular domain T 2 = [0, 2π] × [0, 2π/α], where α ∈ (0, 1]. The fluid is assumed to be driven by a monoscale forcing and damped by various dissipation mechanisms including Ekman drag (a linear mechanical friction), hypoviscosity, molecular vis- cosity, and hyperviscosity. The 2D Navier-Stokes equations which govern the fluid motion are written in an abstract form in a function space H as du dt + B(u, u)+ A η u = f, u(t = 0) = u 0 . (1) A detailed description of the functional analysis setting for (1) is given in [1, 9, 10]. We recall that H is the L 2 -space of periodic, non-divergent functions representing the velocity u with vanishing average in T 2 . B(u, u)= P ((u ·∇)u) where P is the orthogonal projection in L 2 onto H , and A = −P Δ= −ΔP . The number η will be called the degree of viscosity. When η = 1 we have the usual molecular viscosity, while η = 0 corresponds to Ekman drag. The cases η> 1 and η< 1 correspond to hyperviscosity and hypoviscosity, respectively. Note that the generalized viscosity coefficient is taken to be unity. The eigenfunctions of A which form an orthonormal basis of H are given by (see [6, 7]) e k = √ α √ 2 π|k| k ′ cos k · x, (2) 1991 Mathematics Subject Classification. 34D, 35Q30, 76. Key words and phrases. Two-dimensional Navier–Stokes equations, linear stability, asymptotic (global) stability. 1