Existence of permanent and breaking waves for a shallow water equation : a geometric approach

Adrian Constantin

Annales de l'institut Fourier (2000)

  • Volume: 50, Issue: 2, page 321-362
  • ISSN: 0373-0956

Abstract

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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. 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 to exhibit breakdown of the geodesic flow despite the existence of geodesics that can be continued indefinitely in time.

How to cite

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Constantin, Adrian. "Existence of permanent and breaking waves for a shallow water equation : a geometric approach." Annales de l'institut Fourier 50.2 (2000): 321-362. <http://eudml.org/doc/75421>.

@article{Constantin2000,
abstract = {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. 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 to exhibit breakdown of the geodesic flow despite the existence of geodesics that can be continued indefinitely in time.},
author = {Constantin, Adrian},
journal = {Annales de l'institut Fourier},
keywords = {nonlinear evolution equation; shallow water waves; global solutions; wave breaking; diffeomorphism group; Riemannian structure; geodesic flow},
language = {eng},
number = {2},
pages = {321-362},
publisher = {Association des Annales de l'Institut Fourier},
title = {Existence of permanent and breaking waves for a shallow water equation : a geometric approach},
url = {http://eudml.org/doc/75421},
volume = {50},
year = {2000},
}

TY - JOUR
AU - Constantin, Adrian
TI - Existence of permanent and breaking waves for a shallow water equation : a geometric approach
JO - Annales de l'institut Fourier
PY - 2000
PB - Association des Annales de l'Institut Fourier
VL - 50
IS - 2
SP - 321
EP - 362
AB - 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. 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 to exhibit breakdown of the geodesic flow despite the existence of geodesics that can be continued indefinitely in time.
LA - eng
KW - nonlinear evolution equation; shallow water waves; global solutions; wave breaking; diffeomorphism group; Riemannian structure; geodesic flow
UR - http://eudml.org/doc/75421
ER -

References

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