Lipschitz Metrics for a Class of Nonlinear Wave Equations

The nonlinear wave equation utt-c(u)(c(u)ux)x=0 determines a flow of conservative solutions taking values in the space H¹(R). However, this flow is not continuous with respect to the natural H¹ distance. The aim of this paper is to construct a new metric which renders the flow uniformly Lipschitz continuous on bounded subsets of H¹(R). For this purpose, H¹ is given the structure of a Finsler manifold, where the norm of tangent vectors is defined in terms of an optimal transportation problem. For paths of piecewise smooth solutions, one can carefully estimate how the weighted length grows in time. By the generic regularity result proved in [7], these piecewise regular paths are dense and can be used to construct a geodesic distance with the desired Lipschitz property.