This work deals with the local rapid exponential stabilization for a Boussinesq system of KdVKdV type introduced by J. Bona, M. Chen and J.-C. Saut. This is a model for the motion of small amplitude long waves on the surface of an ideal fluid. Here, we will consider the Boussinesq system of KdV-KdV type posed on a finite domain, with homogeneous Dirichlet–Neumann boundary controls acting at the...

In this paper we examine a spatially two–dimensional forced, damped Boussinesq equation with quadratic nonlinearity. This equation describes small, nonlinear oscillations of a circular, elastic membrane in the presence of viscosity. The eigenfunction expansion method is used for the purpose of constructing solutions for small initial data analytically. The constructive method enables us to perf...

This paper focuses on the 2D incompressible Boussinesq equations with fractional dissipation, given by Λαu in the velocity equation and by Λβθ in the temperature equation, where Λ= √−Δ denotes the Zygmund operator. Due to the vortex stretching and the lack of sufficient dissipation, the global regularity problem for the supercritical regime α+β<1 remains an outstanding problem. This paper prese...

Three weakly nonlinear but fully dispersive Whitham–Boussinesq systems for uneven bathymetry are studied. The derivation and discretization of one system is presented. numerical solutions all three compared with wave gauge measurements from a series laboratory experiments conducted by Dingemans (Comparison computations Boussinesq-like models measurements. Delft Hydraulics memo H168412, 1994). r...

We consider the modulationally stable version of Kaup–Boussinesq system which models propagation nonlinear waves in various physical situations. It is shown that Whitham modulation equations for this model have a new type solutions describe trigonometric shock waves. In Gurevich–Pitaevskii problem evolution an initial discontinuity, these correspond to nonzero wave excitation on one sides disco...

The Boussinesq equation is a mathematics model of thermohydraulics, which consists of equations of fluid and temperature in the Boussinesq approximation.The deterministic case has been studied systematically by many authors (e.g., see [1– 3]). However, in many practical circumstances, small irregularity has to be taken into account.Thus, it is necessary to add to the equation a random force, wh...