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

By A. Deya, Massimiliano Gubinelli, Martina Hofmanov'a, Samy Tindel. Journal of Functional Analysis, 2016

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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.

Subfields: Applied Mathematics, Computational Mechanics

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