# American Institute of Mathematical Sciences

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

 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.
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 & Continuous Dynamical Systems - A, 2009, 23 (4) : 1191-1204. doi: 10.3934/dcds.2009.23.1191
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