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On the blow-up rate and the blow-up set of breaking waves for a shallow water equation. (English) Zbl 0954.35136
In a previous paper [Commun. Pure Appl. Math. 51, No. 5, 475-504 (1998; Zbl 0934.35153)], the authors proved the well-posedness of the Cauchy problem for the periodic Camassa-Holm equation with initial data \(u_0\) belonging to the Sobolev space \(H^3(\mathbb{S})\), \(\mathbb{S}\) the unit circle. Sufficient blow-up conditions where also given.
In the paper, which we review now, a detailed description of the blow-up phenomenon is provided. It is proved that if a solution \(u\) of the before mentioned problem blows-up in finite time \(T\), then necessarily the slope \(u_x\) becomes unbounded from below exactly at the moment \(T\) (that is, the blow-up occurs in the form of wave breaking) and the blow-up rate is \[ \lim_{t\to T} \Biggl((T- t)\min_{x\in\mathbb{S}} \{u_x(t, x)\}\Biggr)= -2. \] Using the relation which exists between the Cauchy problem for the periodic Camassa-Holm equation and the geodesic flow in the group of orientation preserving diffeomorphisms of the circle modeled on \(H^3(\mathbb{S})\), the blow-up set is exactly determined for a large subclass of initial data.

MSC:
35Q35 PDEs in connection with fluid mechanics
35B40 Asymptotic behavior of solutions to PDEs
35Q53 KdV equations (Korteweg-de Vries equations)
35L30 Initial value problems for higher-order hyperbolic equations
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