Global Dissipative Solutions of the Camassa-Holm Equation

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 ū ∈ H(IR), 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. 0 Introduction The Camassa-Holm equation ut − utxx + 3uux = 2uxuxx + uuxxx t > 0 , x ∈ IR , (0.1) models the propagation of water waves in the shallow water regime, when the wavelength is considerably larger than the average water depth. Here u(t, x) represents the water’s free surface over a flat bed [CH]. The equation (0.1) was first considered by Fokas and Fuchssteiner [FF] as an abstract bi-Hamiltonian P.D.E. with infinitely many conservation laws. For a detailed discussion of the conservation laws we refer to [I] and [L]. This equation attracted a lot of attention after Camassa and Holm [CH] derived it as a model for shallow water waves and discovered that it is formally integrable, in the sense that there is an associated Lax pair, and that its solitary waves are solitons, i.e. they retain their shape and speed after the interaction with waves of the same type. An alternative derivation of the equation as a model for shallow water waves was subsequently given by Johnson [J]. The Camassa-Holm equation has a very rich structure. For a large class of initial data the equation is an integrable infinite dimensional Hamiltonian system: to each solution with initial data in this class, one can associate some scattering data that evolve in time linearly at constant