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.


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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