# American Institute of Mathematical Sciences

October  2019, 39(10): 5543-5569. doi: 10.3934/dcds.2019244

## Propagation of long-crested water waves. Ⅱ. Bore propagation

 1 Department of Mathematics, Statistics and Computer Science, University of Illinois at Chicago, 851 S. Morgan Street, MC 249, Chicago, IL 60607, USA 2 Université de Bordeaux, Institut de mathématiques de Bordeaux, UMR CNRS 5251, 351 cours de la Libération, 33405 Talence, France 3 Université Paris–Est Créteil, Laboratoire d'analyse et de mathématiques appliquées, UMR CNRS 8050, 61 avenue du Général de Gaulle, 94010 Créteil Cedex, France

Received  February 2017 Published  July 2019

This essay is concerned with long-crested waves such as those arising in bore propagation. Such motions obtain on rivers when a surge of water invades an otherwise constantly flowing stretch and in the run-up of waves in the near-shore zone of large bodies of water. The dominating feature of the motion is that, in a standard $xyz-$coordinate system in which $z$ increases in the direction opposite to which gravity acts and $x$ increases in the principal direction of propagation, the depth of the fluid approaches a constant value $h_0>0$ as $x \to +\infty$ and another value $h_1>h_0$ as $x \to -\infty$. In an earlier work, the authors developed theory for an idealized model for such waves based on a Boussinesq system of equations. The local well-posedness theory developed in that article applies to the sort of initial data arising in modeling bore propagation. However, well-posedness on the longer, Boussinesq time scale was not dealt with in the case of bore propagation, though such results were established for motions where $h_1 = h_0$.

We argue that without a well-posedness theory at least on the Boussinesq time scale, such models for bore-propagation may not be of any practical use. The issue of well-posedness is complicated by the fact that the total energy of the idealized initial data is infinite.

The theory makes its way via the derivation of suitable approximations with which to compare the full solution. An interesting feature of the theory is the determination of dynamical boundary behavior that is not prescribed, but which the solution necessarily satisfies.

Citation: Jerry L. Bona, Thierry Colin, Colette Guillopé. Propagation of long-crested water waves. Ⅱ. Bore propagation. Discrete & Continuous Dynamical Systems - A, 2019, 39 (10) : 5543-5569. doi: 10.3934/dcds.2019244
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