On the local well-posedness of the 1D Green-Naghdi system on Sobolev spaces

Abstract

In this paper, we consider the local well-posedness of the 1D Green-Naghdi system. This system describes the evolution of water waves over an uneven bottom in the shallow water regime in terms of the water depth h and the horizontal velocity u. Using a Lagrangian formulation of this system on a Sobolev-type diffeomorphism group, we prove local well-posedness for (h,u) in the Sobolev space ([1-xi]+Hs(R))xHs+1(R),s>1/2, where xi : R -> R is the parameterization of the bottom and where we assume that the water surface has an equilibrium at height 1. This improves the present local well-posedness range by one degree.