Algebraic deformation for (S)PDEs

TL;DR

This paper introduces an algebraic deformation framework based on pre-Lie products, providing new insights into the algebraic structures underlying Regularity Structures and dispersive PDEs at low regularity.

Contribution

It develops a novel algebraic deformation approach for pre-Lie products, linking it to existing algebraic structures in PDE theory and Regularity Structures.

Findings

Deformation of pre-Lie products via Taylor deformation.

Construction of associative products using Guin-Oudom procedure.

Connection of algebraic structures to Hopf algebras in PDE analysis.

Abstract

We introduce a new algebraic framework based on the deformation of pre-Lie products. This allows us to provide a new construction of the algebraic objects at play in Regularity Structures in the work arXiv:1610.08468 and in arXiv:2005.01649 for deriving a general scheme for dispersive PDEs at low regularity. This construction also explains how the algebraic structure in arXiv:1610.08468 can be viewed as a deformation of the Butcher-Connes-Kreimer and the extraction-contraction Hopf algebras. We start by deforming various pre-Lie products via a Taylor deformation and then we apply the Guin-Oudom procedure which gives us an associative product whose adjoint can be compared with known coproducts. This work reveals that pre-Lie products and their deformation can be a central object in the study of (S)PDEs.