A good introduction to gradient flows

Question

I would like to learn more about gradient flows and their application to the theory of PDEs. I was wondering if anyone had any favourite texts that they thought provided a good and solid introduction to the topic?

There is a similar question here which was asked in 2012. They seemed to recommend the text Partial Differential Equations by L. Evans.

I ask the question again because it hasn't received a great deal of attention and also there may be more modern texts available now which give an introduction to gradient flows.

Thanks for any recommendations given.

EDIT: I realise there is the book by Gigli et al. which discusses gradient flows on metric spaces. I would prefer to work with something describing the theory on $\mathbb{R}^n$ if such a text was available. However it may be the case that this book simply is the best resource out there.

Answer 1

I suggest the Bible of Gradient Flows, the book "Gradient Flows: In Metric Spaces and in the Space of Probability Measures" by Luigi Ambrosio, Nicola Gigli, and Giuseppe Savare