  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.
