On some Boussinesq systems in two space dimensions: theory and numerical analysis

Abstract

A three-parameter family of Boussinesq type systems in two space dimensions is consid¬ ered. These systems approximate the three-dimensional Euler equations, and consist of three nonlinear dispersive wave equations that describe two-way propagation of long surface waves of small amplitude in ideal fluids over a horizontal bottom. For a subset of these systems it is proved that their Cauchy problem is locally well-posed in suitable Sobolev classes. Further, a class of these systems is discretized by Galerkin-fmite element methods, and error estimates are proved for the resulting continuous time semidiscretizations. Results of numerical experiments are also presented with the aim of studying properties of line solitary waves and expanding wave solutions of these systems. Abstract. A three-parameter family of Boussinesq type systems in two space dimensions is consid¬ ered. These systems approximate the three-dimensional Euler equations, and consist of three nonlinear dispersive wave equations that describe two-way propagation of long surface waves of small amplitude in ideal fluids over a horizontal bottom. For a subset of these systems it is proved that their Cauchy problem is locally well-posed in suitable Sobolev classes. Further, a class of these systems is discretized by Galerkin-fmite element methods, and error estimates are proved for the resulting continuous time semidiscretizations. Results of numerical experiments are also presented with the aim of studying properties of line solitary waves and expanding wave solutions of these systems.