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On initial-boundary value problems for a Boussinesq system of BBM-BBM type in a plane domain. (English) Zbl 1155.35431

Summary: 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.

MSC:

35Q53 KdV equations (Korteweg-de Vries equations)
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
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