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Abstract
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.
Mathematics Subject Classification: 34D, 35Q30, 76.
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