Global well-posedness for the averaged Euler equations in two dimensions

We prove global well-posedness of the two-dimensional averaged Euler, or Euler-α equations for initial potential vorticity of class L2. This model generalizes the one-dimensional Fokas–Fuchssteiner–Camassa–Holm equation which describes the propagation of unidirectional waves on the surface of shallow water. As such, it can be realized as a geodesic equation for the H1 metric on the Lie algebra of vector fields. Moreover, in two dimensions the α-model obeys an advection equation for the so-called potential vorticity in close analogy to the vorticity form of the Euler equations. We construct solutions to the weak form of the potential vorticity equation by taking the inviscid limit of solutions to a system regularized by artificial viscosity. Since the streamfunction–vorticity relation is of order four, we can show uniqueness even for potential vorticities in L2.