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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 $$\partial v/\partial t+v(\partial v/\partial x)+\partial p/\partial x=0$$ with “pressure” $$p(x)=(1/2)\int^\infty_{-\infty} e^{-|x-y|}(v^2+v'{}^2/2)dy$$.
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 to PDEs 76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction 37K40 Soliton theory, asymptotic behavior of solutions of infinite-dimensional Hamiltonian systems
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