A note on the full non-linear stability of inviscid, planar flows with constant relative vorticity

The non-linear stability of inviscid, planar flows with constant relative vorticity is proved in the context of the quasi-geostrophic shallow-water theory, for simply connected fluid domains of arbitrary shape. First, the result is obtained relative to the enstrophy and kinetic energy norms and, then, it is extended to a “generalised energy” norm which is expressed through the former.