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

Tran, C. V., Shepherd, T. G. ORCID: https://orcid.org/0000-0002-6631-9968 and Cho, H.-R. (2002) Stability of stationary solutions of the forced Navier-Stokes equations on the two-torus. Discrete and Continuous Dynamical Systems: Series B, 2 (4). pp. 483-494. ISSN 1531-3492 doi: 10.3934/dcdsb.2002.2.483

Abstract/Summary

We study the linear and nonlinear stability of stationary solutions of the forced two-dimensional Navier-Stokes equations on the domain [0,2π]x[0,2π/α], where α ϵ(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 χ2-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.