Sull'approssimazione follow-the-leader di leggi di conservazione scalari nonlineari e relativi modelli

The present PhD thesis deals with the approximation of nonlinear scalar conservation laws via follow-the-leader type schemes. On the one hand, a particular attention is devoted to the different concepts of entropy solution, and the convergence of the follow-the-leader scheme to a unique entropy solution satisfying a one-sided Lipschitz estimate similar to the Oleinik condition is proved. Its proof relies on the derivation of a discrete one-sided Lipschitz condition which is shown to hold, under reasonable assumptions, also in the limit. On the other hand, two different models are investigated, the first involving a nonlinear scalar conservation law with space dependent flux and the latter given by the Hughes model in the case of a linear running cost function. For each of these macroscopic models, various situations are analysed and, depending on the peculiar properties of the functions characterizing each of them, various microscopic models of the follow-the-leader type are introduced. The goal is to establish the convergence of the particle schemes to a possibly unique entropy solution (in the sense of Kruzkov) of the macroscopic model, and this result is achieved through a local maximum principle, a time continuity with respect to the 1-Wasserstein distance and, depending on the model, through a global BV in space compactness argument based on a uniform BV estimate or through a localized BV in space compactness argument based on a local BV bound. As far as the one-dimensional Hughes model is concerned, a byproduct of our convergence result is an existence result of entropy solutions with initial data yielding non-classical shocks which, to our knowledge, is currently not present in the literature.