Paracontrolled calculus for quasilinear singular PDEs

TL;DR

This paper advances the high order paracontrolled calculus framework to analyze quasilinear singular PDEs, introducing infinite dimensional structures without relying on regularity structures or parametrized models.

Contribution

It develops a novel infinite dimensional paracontrolled calculus approach for quasilinear singular PDEs, differing from existing regularity structures methods.

Findings

Proves continuity results for key operators.

Introduces infinite dimensional paracontrolled structures.

Provides an alternative to regularity structures for singular PDEs.

Abstract

We develop further in this work the high order paracontrolled calculus setting to deal with the analytic part of the study of quasilinear singular PDEs. A number of continuity results for some operators are proved for that purpose. Unlike the regularity structures approach of the subject by Gerencser and Hairer, and Otto, Sauer, Smith and Weber, or Furlan and Gubinelli' study of the two dimensional quasilinear parabolic Anderson model equation, we do not use parametrised families of models or paraproducts to set the scene. We use instead infinite dimensional paracontrolled structures that we introduce here.