Contractive relaxation systems and the scalar multidimensional conservation law

this article, entropy weak solutions of (1.1) are constructed as the singular limit of associated relaxation systems. The relaxation models can be loosely interpreted as (and their selection was motivated by) discrete velocity kinetic equations. The relaxation parameter plays the role of the mean free path and the system models the macroscopic conservation law at a "mesoscopic" level. In that sense they are a discrete velocity analogue of the kinetic equations introduced by Perthame and Tadmor [24], Lions, Perthame and Tadmor [19], which also describe entropy solutions of (1.1) as the mean free path tends to zero and provide a kinetic formulation for the scalar conservation law. Furthermore, we may go back to the "microscopic" level and build a stochastic interacting particle system, which: (a) in mesoscopic scales approximates the relaxation model and (b) in the fluid-dynamic limit converges to the entropy solution of a scalar conservation law. This issue was addressed by Perthame and Pulvirenti [23] for interacting particle systems corresponding to the kinetic formulation of conservation laws in [19], and is undertaken in a companion article [14] for interacting particle systems induced by the relaxation schemes proposed here. These results as well as part of the work in the present article are announced in the note [13]. The presence of relaxation mechanisms is widespread in both the continuum mechanics as well as the kinetic theory contexts. Relaxation is known to provide a subtle "dissipative" mechanism for discontinuities against the destabilizing effect of nonlinear response [20], as well as a damping effect on oscillations [3] - at least when assisted by nonlinear response. We refer to Chen, Levermore and Liu [3] for a discussion of a general mathematical framew...