An introduction to the analysis of gradients systems

TL;DR

This paper provides an introductory overview of analytical methods for gradient-flow equations across various spaces, highlighting their commonalities and illustrating them with examples and counterexamples.

Contribution

It offers a comprehensive, accessible introduction to gradient-flow analysis in Hilbert, Banach, and metric spaces, including recent material added to enhance understanding.

Findings

Unified view of gradient-flow theories across different spaces

Illustrative examples and counterexamples provided

Extended notes from a lecture course

Abstract

The present notes provide an extended version of a small lecture course given at the Humboldt Universit\"at zu Berlin in the Winter Term 2022/23 (of 36 hours). The material starting in Section 5.4 was added afterwards. The aim of these notes to give an introductory overview on the analytical approaches for gradient-flow equations in Hilbert spaces, Banach spaces, and metric spaces and to show that on the first entry level these theories have a lot in common. The theories and their specific setups are illustrated by suitable examples and counterexamples.