Global Conservative Solutions of the Camassa–Holm Equation

Abstract

Abstract. 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. -IntroductionThe nonlinear partial differential equationwas 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 1

Continuous and Discontinuous Piecewise Linear Solutions of the Linearly Forced Inviscid Burgers Equation

“…In all four cases mentioned above, the ODEs governing the soliton dynamics can be explicitly solved using inverse spectral methods [1,12,13,2,10,3]. In the forced Burgers case the ODEs are also easily solved directly by elementary methods, as we will see.…”

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“…[10,7,8] and the reference therein). In both cases, writing a balance law for the quantity u 2 x one obtains the a priori bound…”

A Continuous Interpolation Between Conservative and Dissipative Solutions for the Two-Component Camassa–holm System

“…Here, P is an integrated term which is defined below; see (4). When blow-up occurs, the energy density (u 2 + u 2 x ) d x becomes singular; that is, it becomes a measure containing a singular part.…”

“…Interpolation between conservative and dissipative solutions for the 2CH system 3 The proper way to continue the solution after blow-up is to rewrite the equation in terms of new variables, denoted Lagrangian variables, where the CH equation appears as a system of ordinary differential equations taking values in a Banach space in such a way that the blow-up in the original Eulerian variables (1) evaporates [4,5,27,29]. In the present literature, the analysis has been distinct for the two classes of solutions.…”

“…The next task is then to return to Eulerian coordinates, where the solution (u(t), µ(t)) for each positive time t satisfies u(t) ∈ H 1 (R), as well as being a weak global solution of (4), and µ(t) is a nonnegative Radon measure such that µ ac (t) = u 2 x (t, ·) d x. When u is a smooth solution, µ = µ ac , but, at a blow-up time t c , the singular part of µ, which we denote µ s , accounts for the singular part of the energy, as we have lim t↑t c ((u 2 (t, x) + u 2 x (t, x)) d x) = µ s (t c ) + (u 2 (t c , x) + u 2 x (t c , x)) d x.…”