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

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