Orlicz-type Function Spaces and Generalized Gradient Flows with Degenerate Dissipation Potentials in Non-Reflexive Banach Spaces: Theory and Application

TL;DR

This thesis advances the mathematical understanding of nonlinear PDEs by establishing existence results for generalized gradient flows in non-reflexive Banach spaces and developing a comprehensive theory of Orlicz spaces with applications to complex differential equations.

Contribution

It introduces new existence results for solutions of gradient flows with degenerate dissipation and extends Orlicz space theory to handle infinity-valued integrands and arbitrary Banach-valued functions.

Findings

Proved existence of strong solutions for complex gradient flows.

Extended Orlicz space theory to infinity-valued integrands.

Demonstrated applicability to nonlinear PDEs like Allen-Cahn-Gurtin.

Abstract

This thesis explores two important areas in the mathematical analysis of nonlinear partial differential equations: Generalized gradient flows and vector valued Orlicz spaces. The first part deals with the existence of strong solutions for generalized gradient flows, overcoming challenges such as non-coercive and infinity-valued dissipation potentials and non-monotone subdifferential operators on non-reflexive Banach spaces. The second part focuses on the study of Banach-valued Orlicz spaces, a flexible class of Banach spaces for quantifying the growth of nonlinear functions. Besides improving many known results by imposing minimal assumptions, we extend the theory by handling infinity-valued Orlicz integrands and arbitrary Banach-values in the duality theory. The combination of these results offers a powerful tool for analyzing differential equations involving functions of arbitrary growth rates and leads to a significant improvement over previous results, demonstrated through the existence of weak solutions for a doubly nonlinear initial-boundary value problem of Allen-Cahn-Gurtin type.