A priori bounds for quasi-linear SPDEs in the full sub-critical regime

TL;DR

This paper develops a framework for establishing local a priori bounds for quasi-linear stochastic partial differential equations driven by rough noise in the sub-critical regime, inspired by Hairer's regularity structures but using a more streamlined model.

Contribution

It introduces a simplified model indexed by multi-indices that captures symmetries, enabling the derivation of a priori estimates for quasi-linear SPDEs with algebraic renormalization.

Findings

Established local a priori bounds for solutions

Extended regularity structures approach to a broader class of equations

Demonstrated the effectiveness of the multi-index model in capturing symmetries

Abstract

This paper is concerned with quasi-linear parabolic equations driven by an additive forcing $\xi \in C^{\alpha-2}$, in the full sub-critical regime $\alpha \in (0,1)$. We are inspired by Hairer's regularity structures, however we work with a more parsimonious model indexed by multi-indices rather than trees. This allows us to capture additional symmetries which play a crucial role in our analysis. Assuming bounds on this model, which is modified in agreement with the concept of algebraic renormalization, we prove local a priori estimates on solutions to the quasi-linear equations modified by the corresponding counter terms.