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

We prove global well-posedness of the two-dimensional averaged Euler, or Eu ler-alpha equations for initial potential vorticity of class L-2. This mode l generalizes the one-dimensional Fokas-Fuchssteiner-Camassa-Holm equation which describes the propagation of unidirectional waves on the surface of s hallow water. As such, it can be realized as a geodesic equation for the H- 1 metric on the Lie algebra of vector fields. Moreover, in two dimensions t he alpha-model obeys an advection equation for the so-called potential vort icity in close analogy to the vorticity form of the Euler equations. We con struct solutions to the weak form of the potential vorticity equation by ta king 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 L-2. (C) 2000 Elsevi er Science B.V. All rights reserved.