References needed for gradient flow

Question

Can anyone recommend lecture notes or (not too obscure) books that teaches me about gradient flow and what it has to do with PDEs? I did search but usually the material talks about dynamical systems which I don't want too much of. Something understandable by someone that knows about basic parabolic existence results would be good, and something that doesn't require knowledge of Riemannian geometry.

Thanks.

Answer 1

I would suggest

• Chapter 13 of Lectures on Optimal Transport by Luigi Ambrosio, Elia Brué, and Daniele Semola.

You could also consult

• Section 3.2 of the very new book An Invitation to Optimal Transport, Wasserstein Distances, and Gradient Flows by Alessio Figalli and Federico Glaudo,
• Sec. 8 of Optimal transport for applied mathematicians by Filippo Santambrogio
• Sec. 2.1 of $\{$Euclidean, metric, and Wasserstein$\}$ gradient flows: an overview by Filippo Santambrogio,
• Variational Modelling: Energies, gradient flows, and large deviations by Mark A. Peletier

as well as this talk by Katy Craig (from 31:28 onwards), this lecture by Brittany Hamfeldt.

Other standard (but more advanced) references are Optimal Transport - Old and New by Cedric Villani and Gradient flows: in metric spaces and in the space of probability measures by Luigi Ambrosio, Nicola Gigli, and Giuseppe Savarè.