Well-posedness, blow up, and global existence for an integrable shallow water equation. (English) Zbl 1061.35123

Local well-posedness is established for the following equation derived recently in [H. R. Dullin, G. A. Gottwald and D. D. Holm, Phys. Rev. Lett. 87, 194501 (2001)]: \[ \begin{aligned} & u_t - \alpha ^2 u_{txx} + c_0 u_x + 3uu_x + \gamma u_{xxx} = \alpha ^2( 2u_x u_{xx} + uu_{xxx}),\quad t > 0,\;x \in \mathbb R,\\ &u( {0,x}) = u_0 ( x),\quad x \in\mathbb R.\end{aligned}\tag{1} \] Here the constants \(\alpha ^2 \) and \(\frac{\gamma } {{c_0 }}\) are squares of length scales, and \(c_0 > 0\) is the linear wave speed of the undisturbed water at rest at spatial infinity, \(u\left( {t,x} \right)\) stands for the fluid velocity. This new equation describes unidirectional propagation of surface waves on a shallow layer of water which is at rest at infinity and combines the linear dispersion of the KdV equation with the nonlinear/nonlocal dispersion of the Camassa-Holm equation. It is completely integrable and its travelling wave solutions contain both the KdV solitons and the CH peakons as limiting cases. The paper is organized as follows. In Section 2, the local well-posedness of the initial value problem associated with (1) is proved. In Section 3, the author investigates the global existence of a solution. In Section 4, he gets the precise blowup scenario and gives an explosion criterion for a strong solution to (1) with rather general initial data. The last section is devoted to a sharp estimate from below and lower semicontinuity of the existence time of strong solutions.


35Q53 KdV equations (Korteweg-de Vries equations)
35Q58 Other completely integrable PDE (MSC2000)
76B25 Solitary waves for incompressible inviscid fluids
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
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