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

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)$(h,u)$ in the Sobolev space ([1−ξ]+Hs(R))×Hs+1(R),s>1/2$([1-\xi ]+H^s(\mathbb {R})) \times H^{s+1}(\mathbb {R}),\; s {>} 1/2$, where ξ:R→R$\xi :\mathbb {R}\rightarrow \mathbb {R}$ 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.