  We investigate the density large deviation function for a multidimensional
conservation law in the vanishing viscosity limit, when the probability
concentrates on weak solutions of a hyperbolic conservation law conservation
law. When the conductivity and dif-fusivity matrices are proportional, i.e. an
Einstein-like relation is satisfied, the problem has been solved in [4]. When
this proportionality does not hold, we compute explicitly the large deviation
function for a step-like density profile, and we show that the associated
optimal current has a non trivial structure. We also derive a lower bound for
the large deviation function, valid for a general weak solution, and leave the
general large deviation function upper bound as a conjecture.
