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

Zbl 0934.35153
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