Nonlinear Generalized Functions: their origin, some developments and recent advances

TL;DR

This paper discusses the development and applications of nonlinear generalized functions, highlighting their importance in modeling real-world phenomena and solving complex PDE systems where classical solutions do not exist.

Contribution

It introduces a nonlinear theory of generalized functions developed in the 1980s, enabling multiplication of distributions and addressing real-world modeling needs.

Findings

Existence and uniqueness results for PDE systems without classical solutions

Application to shock wave calculations in elasticity, cosmology, and fluid flows

Development of a natural and unavoidable mathematical framework for modeling the real world

Abstract

We expose some simple facts at the interplay between mathematics and the real world, putting in evidence mathematical objects " nonlinear generalized functions" that are needed to model the real world, which appear to have been generally neglected up to now by mathematicians. Then we describe how a "nonlinear theory of generalized functions" was obtained inside the Leopoldo Nachbin group of infinite dimensional holomorphy between 1980 and 1985. This new theory permits to multiply arbitrary distributions and contains the above mathematical objects, which shows that the features of this theory are natural and unavoidable for a mathematical description of the real world. Finally we present direct applications of the theory such as existence-uniqueness for systems of PDEs without classical solutions and calculations of shock waves for systems in non-divergence form done between 1985 and 1995, for which we give three examples of different nature: elasticity, cosmology, multifluid flows.