  We discuss the minimal integrability needed for the initial data, in order
that the Cauchy problem for a multi-dimensional conservation law admit an
entropy solution. In particular we allow unbounded initial data. We investigate
also the decay of the solution as time increases, in relation with the
nonlinearity. The main ingredient is our recent theory of divergence-free
positive symmetric tensor. We apply in particular the so-called compensated
integrability to a tensor which generalizes the one that L. Tartar used in one
space dimension. It allows us to establish a Strichartz-like inequality, in a
quasilinear context. This program is carried out in details for a
multi-dimensional version of the Burgers equation.
