Wave breaking for nonlinear nonlocal shallow water equations

On the blow-up of solutions to the integrable modified Camassa–Holm equation

“…(1.4) does not change sign, the Cauchy problem admits global solution for certain initial values [3,5,6], whereas solutions may blow up if their initial momentum density m 0 changes sign [3,4,5,6]. The existence of global weak solutions was investigated in [25,26].…”

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

“…Again the functionh is rescaled, and the system restarted. This construction is performed for each ξ ∈ R. As the system of ordinary differential equations is discontinuous, the global existence proof requires careful estimates; see Lemmas 5,6,8,[10][11][12][13]16.…”

“…The pattern of blow-up is known [10]: the solution remains continuous while the derivative u x tends to minus infinity at the blow-up point. For this reason, the blow-up for the CH equation is often characterized as wave breaking, and we will use this term extensively in this paper.…”

Geometric Integrability of the Camassa–Holm Equation. II

“…We once again refer to [22,43] for details. These comments imply that the well-known fact that smooth solutions to the Camassa-Holm equation can develop singularities in finite time [13,15,36] is not due to the lack of local conserved quantities, as suspected for instance in [36]. Rather, the point is that the local conserved quantities do not guarantee the existence of a priori bounds for the first derivative of the solutions u(x, t) to the CH equation.…”

“…In other words, we change our viewpoint, from thinking of the CH equation (63) as a differential equation for u(x, t), to considering (63) as an integro-differential equation for m. This approach has proved to be crucial for the analytic study of the CH equation: it has been used to determine whether solutions to CH are global in time or represent breaking waves [13,15,36], and it appears prominently in Lenells' construction of conservation laws [33], in the study of (multi)peakon dynamics and weak solutions [5,10,14], in the scattering/inverse scattering approach to CH [4,5,17], and also in a rigorous proof that the "least action principie" holds for CH [16].…”