The paper provides a comprehensive introduction to an extension of the theory of gradient flows to infinite-dimensional spaces which can cover the flows generated by some evolutionary PDEs. It is a slightly simplified version of the theory presented in the authors’ and N. Gigli’s book [Gradient flows in metric spaces and in the space of probability measures. Basel: Birkhäuser (2005; Zbl 1090.35002) and (2008; Zbl 1145.35001)]. The main object is the Wasserstein space, i.e., the space \(\mathcal P_2(\mathbb R^d)\) of probability measures on \(\mathbb R^d\) with finite quadratic moment, endowed with the Kantorovich-Rubinstein-Wasserstein distance. In concordance to the distance, in a suitable way notions of differential structure, tangent bundle, metric tensor, and geodesic curves can be defined on \(\mathcal P_2(\mathbb R^d)\), and the notion of gradient flow which provides the required link to PDEs can be introduced.
The exposition in the paper consists of six sections. Section 1 provides a review of the required notions from the measure theory. In Section 2, the definition of the Wasserstein space together with its basic properties is provided. Section 3 is devoted to the presentation of facts related to convexity of functionals on \(\mathcal P_2(\mathbb R^d)\). The subdifferential calculus on that space is the subject of Section 4. In Section 5, the gradient flows of geodesically convex functionals are considered. Finally, in Section 6, applications to evolution PDEs, including Kolmogorov-Fokker-Planck, nonlinear diffusion, and drift diffusion equations are given.
For the entire collection see [Zbl 1179.35003].