On the foundations of nonlinear generalized functions I

TL;DR

This paper develops a diffeomorphism invariant algebra of generalized functions that includes distributions, unifies previous approaches, and applies to nonlinear differential equations with singularities.

Contribution

It constructs a canonical Colombeau-type algebra invariant under diffeomorphisms, unifying and completing earlier methods.

Findings

Unified previous approaches using differential calculus in infinite dimensions

Achieved classification results for the algebra

Applied to nonlinear differential equations with singularities

Abstract

We construct a diffeomorphism invariant (Colombeau-type) differential algebra canonically containing the space of distributions in the sense of L. Schwartz. Employing differential calculus in infinite dimensional (convenient) vector spaces, previous attempts in this direction are unified and completed. Several classification results are achieved and applications to nonlinear differential equations involving singularities are given.