Generalized solutions in PDE's and the Burgers' equation

TL;DR

This paper introduces Generalized Ultrafunction Solutions (GUS) for PDEs, including evolution problems like Burgers' equation, providing existence, uniqueness, and microscopic insights beyond classical solutions.

Contribution

It extends ultrafunction theory to evolution PDEs, establishing GUS existence and uniqueness, and offers microscopic descriptions of solutions such as Burgers' equation.

Findings

GUS exists and is unique for a broad class of PDEs.

GUS provides microscopic insights into Burgers' equation.

GUS generalizes classical and weak solutions.

Abstract

In many situations, the notion of function is not sufficient and it needs to be extended. A classical way to do this is to introduce the notion of weak solution; another approach is to use generalized functions. Ultrafunctions are a particular class of generalized functions that has been previously introduced and used to define generalized solutions of stationary problems in [4,7,9,11,12]. In this paper we generalize this notion in order to study also evolution problems. In particular, we introduce the notion of Generalized Ultrafunction Solution (GUS) for a large family of PDE's, and we confront it with classical strong and weak solutions. Moreover, we prove an existence and uniqueness result of GUS's for a large family of PDE's, including the nonlinear Schroedinger equation and the nonlinear wave equation. Finally, we study in detail GUS's of Burgers' equation, proving that (in a precise sense) the GUS's of this equation provide a description of the phenomenon at microscopic level.