Hyperbolic conservation laws: L-infinity data and granular matter

In my PhD Thesis I analyzed three problems. The first we studied n × n systems. We considered a particular case of hyperbolic non-linear systems, the straight-line systems and their perturbations. Both were analized in their most general form i.e. they are not necessarily conservative. We proved the existence of global solutions for the related Cauchy problems under the assumption that the perturbed system was sufficiently near to the straight-line one. The total variation of the initial data must be bounded, but not necessarily small. Also stability estimates hold. In an other work we consider a hyperbolic system of conservation laws with genuinely nonlinear characteristic fields. We extend the classical Glimm-Lax result proving the existence of solutions for L^infinity initial datum, relaxing the assumptions taken therein on the geometry of the shock–rarefaction curves. In the last part of the thesis we studied hyperbolic systems that represent the behaviour of granular matter and we have presented a new model It is a synthesis of the Hadeler–Kuttler and of the Savage–Hutter models. The result is a 3 × 3 system of balance laws able to describe the deposition–erosion dynamics, as in the former model, while being compliant with energy dissipation, as in the latter one.