A continuous interpolation between conservative and dissipative solutions for the two-component Camassa-Holm system

Abstract

We introduce a novel solution concept, denoted $\alpha$-dissipative solutions, that provides a continuous interpolation between conservative and dissipative solutions of the Cauchy problem for the two-component Camassa-Holm system on the line with vanishing asymptotics. All the $\alpha$-dissipative solutions are global weak solutions of the same equation in Eulerian coordinates, yet they exhibit rather distinct behavior at wave breaking. The solutions are constructed after a transformation into Lagrangian variables, where the solution is carefully modified at wave breaking.