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October  2009, 23(4): 1191-1204. doi: 10.3934/dcds.2009.23.1191

On initial-boundary value problems for a Boussinesq system of BBM-BBM type in a plane domain

V. A. Dougalis 1, , D. E. Mitsotakis 1, and  J.-C. Saut 2,

1. 

Department of Mathematics, University of Athens,15784 Zographou, and Institute of Applied and Computational Mathematics, FO.R.T.H., P.O. Box 1385, 71110 Heraklion, Greece, Greece
2. 

UMR de Mathématiques, Université de Paris-Sud, Bâtiment 425, P.O. Box 91405, Orsay, France

Received  May 2007 Revised  September 2007 Published  November 2008

We consider a Boussinesq system of BBM-BBM type in two space dimensions. This system approximates the three-dimensional Euler equations and consists of three coupled nonlinear dispersive wave equations that describe propagation of long surface waves of small amplitude in ideal fluids over a horizontal bottom. We show that the initial-boundary value problem for this system, posed on a bounded smooth plane domain with homogeneous Dirichlet or Neumann or reflective (mixed) boundary conditions, is locally well-posed in $H^1$. After making some remarks on the temporal interval of validity of these models, we discretize the system by a standard Galerkin-finite element method and present the results of some numerical experiments aimed at simulating two-dimensional surface wave flows in complex plane domains with a variety of initial and boundary conditions.
Keywords: BBM-BBM, initial-boundary value problems, Boussinesq systems.
Mathematics Subject Classification: 35Q53, 76B15, 65M60.
Citation: V. A. Dougalis, D. E. Mitsotakis, J.-C. Saut. On initial-boundary value problems for a Boussinesq system of BBM-BBM type in a plane domain. Discrete and Continuous Dynamical Systems, 2009, 23 (4) : 1191-1204. doi: 10.3934/dcds.2009.23.1191
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