We present a blow-up criterion for the periodic Camassa-Holm equation. The condition obtained for blow-up uses two of the conservation laws associated with the equation and improves upon some recent results.
Introduction. The nonlinear partial differential equation
is a bi-Hamiltonian system [12], and a model for shallow water waves, u(t, x) representing in nondimensional variables the water's free surface above a flat bottom [1] (see also [13]). The Camassa-Holm equation ( 1.1) is a re-expression of geodesic flow on the diffeomorphism group of the circle [7], [8], [17]. Moreover, (1.1) is an infinite-dimensional completely integrable Hamiltonian system [9]. Let us also point out that (1.1) has, for any speed c > 0, peaked traveling wave solutions called peakons [1], [10], [11], [14], [15]. Certain classical solutions of (1.1) develop singularities in finite time [2], [3], [6], [19] while others exist globally in time [4], [5]. The finite-time blow-up occurs only in the form of wave breaking [6], i.e. the solution remains bounded while its slope becomes unbounded in finite time. This feature has attracted a lot of attention in the research literature and several criteria for blow-up are available (see [19], [20], [21] and references therein). Some of these criteria use a conserved quantity for (1.1) in proving blow-up [4], [6], [18]. Recently, Zhou [20], [21] was able to use the interplay between two conserved quantities to derive a blow-up result. In this note we will improve Zhou's results by optimizing certain inequalities needed for this type of approach.