Well − posedness of higher dimensional Camassa − Holm equations

Abstract

We formulate the n-dimensional Camassa-Holm (CH) equation on an arbitrary compact Riemannian manifold with a boundary and show that these equations are well-posed with respect to Dirichlet or Navier-slip boundary conditions. The method of proof consists in showing that the physically relevant H¹-like Riemannian metrics admit a smooth geodesic spray on the diffeomorphism groups associated with the above boundary conditions.