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Zbl 0944.35062
Constantin, Adrian
Existence of permanent and breaking waves for a shallow water equation: a geometric approach. (English)
[J] Ann. Inst. Fourier 50, No.2, 321-362 (2000). ISSN 0373-0956; ISSN 1777-5310/e

Summary: The existence of global solutions and the phenomenon of blow-up of a solution in finite time for a recently derived shallow water equation are studied. We prove that the only way a classical solution could blow-up is as a breaking wave for which we determine the exact blow-up rate and, in some cases, the blow-up set. \par Using the correspondence between the shallow water equation and the geodesic flow on the manifold of diffeomorphisms of the line endowed with a weak Riemannian structure, we give sufficient conditions for the initial profile to develop into a global classical solution or into a breaking wave. With respect to the geometric properties of the diffeomorphism group, we prove that the metric spray is smooth infering that, locally, two points can be joined by a unique geodesic. The qualitative analysis of the shallow water equation is used to exhibit breakdown of the geodesic flow despite the existence of geodesics that can be continued indefinitely in time.

MSC 2000:: *35Q35 Other equations arising in fluid mechanics
37K25 Relations with differential geometry
58D05 Groups of diffeomorphisms and homeomorphisms as manifolds
76B15 Wave motions (fluid mechanics)

Keywords: nonlinear evolution equation; shallow water waves; global solutions; wave breaking; diffeomorphism group; Riemannian structure; geodesic flow

Cited in: Zbl 1195.35072 Zbl 1102.35021

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