Existence of permanent and breaking waves for a shallow water equation: a geometric approach

The existence of global solutions and the phenomenon of blow-up of a solution in finite time for a recently derived shallow water equation are studied. We prove that the only way a classical solution could blow-up is as a breaking wave for which we determine the exact blow-up rate and, in some cases, the blow-up set. Using the correspondence between the shallow water equation and the geodesic flow on the manifold of diffeomorphisms of the line endowed with a weak Riemannian structure, we give sufficient conditions for the initial profile to develop into a global classical solution or into a breaking wave. With respect to the geometric properties of the diffeomorphism group, we prove that the metric spray is smooth infering that, locally, two points can be joined by a unique geodesic. The qualitative analysis of the shallow water equation is used to to exhibit breakdown of the geodesic flow despite the existence of geodesics that can be continued indefinitely in time.