A priori estimates for rough PDEs with application to rough conservation laws

Aurélien Deya (1) , Massimiliano Gubinelli (2) , Martina Hofmanova (3) , Samy Tindel (4)

1 IECL - Institut Élie Cartan de Lorraine
2 Hausdorff Center for Mathematics and Institute for Numerical Simulation - University of Bonn
3 TU - Technical University of Berlin / Technische Universität Berlin
4 Department of mathematics Purdue University

We introduce a general framework to study PDEs driven by rough paths: we develop new a priori estimates based on a rough Gronwall lemma argument for weak solutions to rough PDEs. This will allow us to follow standard PDE strategies to obtain existence and uniqueness results. In particular our 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.

Domaines: Probabilités [math.PR]

Aurélien Deya, Massimiliano Gubinelli, Martina Hofmanova, Samy Tindel. A priori estimates for rough PDEs with application to rough conservation laws. Journal of Functional Analysis, 2019, 276 (12), pp.3577-3645. ⟨10.1016/j.jfa.2019.03.008⟩. ⟨hal-01308642⟩