Plane Couette - symmetries

Symmetries of plane Couette flow

The Navier-Stokes equations for the plane Couette flow are equivariant under "shift-reflect" and "shift-rotate" transformations:

[action of isotropy group S]

By “equivariant” we mean that if u(x,t) is a solution, so are s₁u(x,t), s₂u(x,t), and s₁s₂u(x,t); the four solutions are physically equivalent. ( to see examples of such symmetry-related solutions). We refer to the space of velocity fields left invariant under the symmetry group

as the S-invariant subspace. Most of the exact invariant solutions (equilibria, periodic orbits) shown in what follows belong to this subspace; stable/unstable manifolds, travelling waves do not.