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Breakdown of a shallow water equation. (English) Zbl 0959.35140

The author discusses the following two questions respecting the shallow water equations:

1. When does the shallow water flow breakdown? 2. How does the shallow water equation breakdown? To answer these questions he studies the 1D equation $\partial v/\partial t+v\left(\partial v/\partial x\right)+\partial p/\partial x=0$ with “pressure” $p\left(x\right)=\left(1/2\right){\int }_{-\infty }^{\infty }{e}^{-|x-y|}\left({v}^{2}+{v}^{\text{'}}{}^{2}/2\right)dy$.

The introductory part of the paper is very useful for the interested reader, where he can find a historical sketch of the problem, the obtained results, and relations to other fields of mathematics.

##### MSC:
 35Q35 PDEs in connection with fluid mechanics 35B40 Asymptotic behavior of solutions of PDE 76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction 37K40 Soliton theory, asymptotic behavior of solutions