A priori estimates for rough PDEs with application to rough conservation   laws
  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. 
Aur\'elien Deya, Massimiliano Gubinelli, Martina Hofmanov\'a and Samy   Tindel
math.AP math.PR