Abstract:
A nonlinear wave equation is developed, modeling the evolution in time of shallow water waves over a variable topography. As the usual assumptions of a perfect fluid and an irrotational flow are not made, the resulting model equation is dissipative due to the presence of a viscous boundary layer at the bottom of the flow region. The well-posedness of the Cauchy problem for classical solutions of this equation is addressed. In particular, it is established by means of various energy estimates and Soholev space embeddings that a long time classical (C2) solution to the Cauchy problem exists and is unique provided the initial data are small enough. An asymptotic result for the dependence of the lifespan of classical solutions upon the size of the initial data is given. Finally, a few results of an analytical nature (e.g., explicit computation of lifespans) are given for the model equation in one dimension with the hope of providing useful parameters for experimental validation of the nonlinear wave model. Numerical results display the propagation of the nonlinear wave in one and two spatial dimensions, and a comparison is made with the waves described by the familiar linear wave equation.
