Simplifying Some Dissipative-Dispersive Models

Abstract

We consider the effect of diffusive-dispersive small scale perturbations on nonlinear hyperbolic conservation laws addressing the study of zero diffusion-dispersion singular limits in order to understand “when can we use the hyperbolic (simplified) model instead of the perturbed ones?” Our arguments establish ‘integrity’ (convergence to the classical entropy solutions), but the major emphasis must be on ‘reliability’ (convergence to any physical solution) and ‘failure’ (divergence). The technique we use is based on energy methods and depends upon improved Tartar’s and DiPerna’s analytical settings on Young measures (for a concise review see, e.g., [2]). We aim for a realistic framework, with multi-space dimensional and physical dissipation-dispersion models.