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

We study the linear and nonlinear stability of stationary solutions of the forced two-dimensional Navier-Stokes equations on the domain [0, 2pi] x [0, 2pi/alpha], where alpha is an element of (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 2pi 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.