Transfer of energy in Camassa–Holm and related models by use of nonunique characteristics

Abstract

We study the propagation of energy density in finite-energy weak solutions of the Camassa–Holm and related equations. Developing the methods based on generalized nonunique characteristics, we show that the parts of energy related to positive and negative slopes are one-sided weakly continuous and of bounded variation, which allows us to define certain measures of dissipation of both parts of energy. The result is a step towards the open problem of uniqueness of dissipative solutions of the Camassa–Holm equation.