Global conservative solutions of the Camassa-Holm equation

This paper develops a new approach in the analysis of the Camassa-Holm equation. By introducing a new set of independent and dependent variables, the equation is transformed into a semilinear system, whose solutions are obtained as fixed points of a contractive transformation. These new variables resolve all singularities due to possible wave breaking. Returning to the original variables, we obtain a semigroup of global solutions, depending continuously on the initial data. Our solutions are conservative, in the sense that the total energy equals a constant, for almost every time. 0 Introduction The nonlinear partial differential equation ut − utxx + 3uux = 2uxuxx + uuxxx, t > 0, x ∈ IR, was derived by Camassa and Holm [CH] as a model for the propagation of shallow water waves, with u(t, x) representing the water’s free surface over a flat bed (see also the alternative derivation in [J]). The Camassa-Holm equation was actually obtained much earlier as an abstract bi-Hamiltonian partial differential equation with infinitely many conservation laws by Fokas and Fuchssteiner [FF] (see [L]). Nevertheles, Camassa and Holm put forward its derivation 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. the solitary waves retain their shape and speed after the nonlinear interaction with waves of the same type), features that prompted an ever increasing interest in the study of this equation. For a large class of initial data the Camassa-Holm equation is an integrable infinite dimensional Hamiltonian system. That is, by means of a Lax pair, it is possible to associate to each solution with initial data within this class some scattering data that evolve in time linearly at constant speed and from which the solution can be reconstructed in an explicit way (see [BSS2, CM1, C2]). In contrast to the Korteweg-de Vries equation, which is also an integrable model for shallow water waves, the Camassa-Holm equation possesses not only solutions that are global in time but models also wave breaking. Indeed, while some initial