On a Lagrangian formulation of the 1D Green-Naghdi system

In this paper we consider the 1D Green-Naghdi system. This system describes the evolution of water waves over a flat bottom in the shallow water regime in terms of the surface height h and the horizontal velocity u. We give a Lagrangian formulation of the 1D GreenNaghdi system on a Sobolev type diffeomorphism group. As an application of this formulation we prove local well-posedness for (h, u) in the Sobolev space (1 + Hs(R)) × Hs+1(R), s > 1/2. This improves the local well-posedness range for the 1D Green-Naghdi system.