Nonlinear hyperbolic conservation laws: diffusive-dispersive limits

Abstract

We are concerned with diffusive-dispersive perturbations of nonlinear hyperbolic conservation laws: if we consider such applications as optimal design and dynamic control in structures or martensitic phase transitions in materials, oscillations are fundamental issues. While diffusion has a dissipative effect, dispersion has an oscillatory one, but in the previous examples they must be differently balanced if we want to select the proper physical solution. So, the main issue we are interested in is the study of zero diffusion-dispersion limits in order to respond to the questions: "when can we replace the given balance laws by simpler hyperbolic models?" and, "what must be the proper additional entropy condition?" While our arguments establish ''integrity'' (convergence to classical discontinuous solutions), the major emphasis is on ''reliability'' (convergence to any physical solution) and ''failure'' (divergence). The techniques we use are energy based and depend upon the improved DiPerna’s analytical setting on Young measures. We aim for a realistic framework, with multi-space dimensional and physical diffusion-dispersion models.