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

TL;DR

This paper develops a new weak formulation and proof strategy for rough PDEs, enabling well-posedness results for rough conservation laws without relying on transformation formulas.

Contribution

It introduces a general weak formulation for rough PDEs and a novel proof method based on a priori estimates and rough Gronwall arguments, enhancing flexibility.

Findings

Established well-posedness for rough conservation laws

Developed a transformation-free proof strategy

Provided a new framework for rough PDE analysis

Abstract

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.