Large deviations for stochastic conservation laws

We consider a parabolic stochastic differential equation with a (nonlinear) transport term perturbed by a conservative noise and discuss the limit in which both the viscosity and the noise strength vanish. The lack of uniqueness for the formal limiting equation implies the existence of two different large deviations scales. The coarser one is analyzed in a Young measure setting and the rate functional vanishes on measure valued solutions to the conservation law. In the finer scale the rate functional is finite only on weak solutions and it is given by the entropy production corresponding to a specific entropy function determined by an Einstein relation.