Dispersion and regularity for scalar conservation laws in several space dimensions
The analysis of scalar conservation laws is harder in several space dimensions, than in one space dimension, because characteristic lines do not behave nicely. Functional analysis can be used because Kruzkhov's semi-group is $L^1$-contracting. Crandall raised the question of whether the continuous extension of the semi-group to $L^1$, or even to $L^1+L^\infty$, is meaningful: does it provide distributional solutions to the Cauchy problem ? With L. Silvestre (Chicago), we solved this question by establishing dispersive properties: in presence of non-linearity, the abstract solution becomes instaneously bounded in space, the fluxes make sense and the PDE, together with the entropy differential inequalities, are satisfied in the distributional sense. Next, we provide a non-trivial (in a sense that will be explained) estimate of Besov style, of solutions up to the time of the first shock formation.
These results are established by means of a functional analysis tool that we developped recently, called Compesnated Integrability.