Energy conservation of weak solutions and an Onsager-type criterion for the two-component Camassa–Holm system

In this paper, we investigate energy conservation for weak solutions to the periodic two-component Camassa–Holm system. We derive a local energy balance in the sense of distributions with an explicit defect term, including the mixed increments of the velocity and density. Under the basic energy bounds, we establish an Onsager-type energy-conservation criterion based on the integrability of uₓ and ρ. By identifying these solutions with the periodic conservative evolution, we also obtain a sufficient condition for uniqueness within the stated regularity class.