Global Dissipative Solutions of the Camassa–Holm Equation
Alberto Bressan and Adrian Constantin
Abstract: This paper is concerned with the global existence of dissipative solutions to the Camassa–Holm equation after wave breaking. By introducing a new set of independent and dependent variables, the evolution problem is rewritten as an O.D.E. in an L∞ space, containing a non-local source term which is discontinuous but has bounded directional variation along a suitable cone of directions. For a given initial condition, the Cauchy problem has a unique solution obtained as fixed point of a contractive integral transformation. Returning to the original variables, we obtain a semigroup of global dissipative solutions, defined for every initial data in H1, and continuously depending on the initial data. The new variables resolve all singularities due to possible wave breaking and ensure that energy loss occurs only through wave breaking.