Generalized regularity and solution concepts for differential equations

TL;DR

This thesis investigates generalized solution concepts for differential equations, focusing on regularity issues in Colombeau theory and microlocal analysis of pullbacks, comparing these methods to other approaches.

Contribution

It introduces a microlocal analysis framework for Colombeau solutions and develops the concept of a generalized graph, advancing understanding of regularity and solution existence.

Findings

Colombeau solutions may lack distributional shadows.

Microlocal analysis of pullbacks reveals regularity properties.

Comparison of Colombeau techniques with other solution concepts.

Abstract

As the title ``Generalized regularity and solution concepts for differential equations'' suggests, the main topic of my thesis is the investigation of generalized solution concepts for differential equations, in particular first order hyperbolic partial differential equations with real-valued, non-smooth coefficients and their characteristic system of ordinary differential equations. In Colombeau theory there have been developed existence results that yield solutions for ordinary and partial differential equations beyond the scope of classical approaches. Nevertheless this comes at the price of sacrificing regularity (in general a Colombeau solution may even lack a distributional shadow). It is prevailing in the Colombeau setting that the question of mere existence of solutions is much easier to answer than to determine their regularity properties (i.e. if a distributional shadow exists and how regular it is). In order order to address these regularity question and encouraged by the fact that the solution of a (homogeneous) first order partial differential equation can be written as a pullback of the initial condition by the characteristic backward flow, a main topic of my thesis deals with the microlocal analysis of pullbacks of c-bounded Colombeau generalized functions. Another topic is the comparsion of Colombeau techniques for solving ordinary and partial differential equations to other generalized solution concepts, which has led to a joint article with G\"unther H\"ormann. A useful tool for this purpose is the concept of a generalized graph, which has been developed in the thesis.