  We investigate large deviations for a family of conservative stochastic PDEs
(conservation laws) in the asymptotic of jointly vanishing noise and viscosity.
We obtain a first large deviations principle in a space of Young measures. The
associated rate functional vanishes on a wide set, the so-called set of
measure-valued solutions to the limiting conservation law. We therefore
investigate a second order large deviations principle, thus providing a
quantitative characterization of non-entropic solutions to the conservation
law.
