  We introduce a general weak formulation for PDEs driven by rough paths, as
well as a new strategy to prove well-posedness. Our procedure is based on a
combination of fundamental a priori estimates with (rough) Gronwall-type
arguments. In particular this approach does not rely on any sort of
transformation formula (flow transformation, Feynman--Kac representation
formula etc.) and is therefore rather flexible. As an application, we study
conservation laws driven by rough paths establishing well--posedness for the
corresponding kinetic formulation.
