Nonlinear partial differential equations on noncommutative Euclidean spaces

TL;DR

This paper explores the unique features of nonlinear partial differential equations on noncommutative Euclidean spaces, developing paradifferential calculus and revealing simplified analysis in the noncommutative setting.

Contribution

It introduces elementary paradifferential calculus for noncommutative Euclidean spaces and applies it to nonlinear evolution equations, highlighting novel properties and simplifications.

Findings

Paradifferential calculus adapted to noncommutative spaces

Simplified analysis of certain nonlinear equations in the noncommutative setting

Identification of peculiar features of PDEs on noncommutative Euclidean spaces

Abstract

Noncommutative Euclidean spaces -- otherwise known as Moyal spaces or quantum Euclidean spaces -- are a standard example of a non-compact noncommutative geometry. Recent progress in the harmonic analysis of these spaces gives us the opportunity to highlight some of their peculiar features. For example, the theory of nonlinear partial differential equations has unexpected properties in this noncommutative setting. We develop elementary aspects of paradifferential calculus for noncommutative Euclidean spaces and give some applications to nonlinear evolution equations. We demonstrate how the analysis of some equations radically simplifies in the strictly noncommutative setting.