Gradient flows in metric spaces and in the space of probability measures.

*(English)*Zbl 1090.35002
Lectures in Mathematics, ETH Zürich. Basel: Birkhäuser (ISBN 3-7643-2428-7/pbk). vii, 333 p. (2005).

The book under review deals with a systematic presentation of the theory of gradient flows and its application to some partial differential equations of evolution type. The volume is divided into two main parts.

The first part is devoted to the general theory of gradient flows in an arbitrary setting. The framework considered here is very general and needs only a metric structure in the ambient space. The key ingredients are the so-called “metric derivative” and the notion of curves of “maximal slope”, whose properties are studied in a very detailed way, also in connection to the De Giorgi’s concept of “minimizing movements”.

The second part is more concerned with the application of the results of the first part to the case when the ambient metric space is the space \({\mathcal P}_p(X)\) of all probabilities over a set \(X\), endowed with the \(p\)-Wasserstein metric \(W_p\). The connection to mass transportation problems is here emphasized, and the most important facts of transport theory are summarized, together with some basic tools of measure theory. The last two chapters deal with the development of a subdifferential calculus in \({\mathcal P}_p(X)\) and with some interesting applications to partial differential equations of evolution type as for instance:

– the linear transport equation with a potential;

– the nonlinear diffusion equation;

– the drift nonlocal diffusion equation;

– the Fokker-Planck equation in infinite dimension.

The first part is devoted to the general theory of gradient flows in an arbitrary setting. The framework considered here is very general and needs only a metric structure in the ambient space. The key ingredients are the so-called “metric derivative” and the notion of curves of “maximal slope”, whose properties are studied in a very detailed way, also in connection to the De Giorgi’s concept of “minimizing movements”.

The second part is more concerned with the application of the results of the first part to the case when the ambient metric space is the space \({\mathcal P}_p(X)\) of all probabilities over a set \(X\), endowed with the \(p\)-Wasserstein metric \(W_p\). The connection to mass transportation problems is here emphasized, and the most important facts of transport theory are summarized, together with some basic tools of measure theory. The last two chapters deal with the development of a subdifferential calculus in \({\mathcal P}_p(X)\) and with some interesting applications to partial differential equations of evolution type as for instance:

– the linear transport equation with a potential;

– the nonlinear diffusion equation;

– the drift nonlocal diffusion equation;

– the Fokker-Planck equation in infinite dimension.

Reviewer: Giuseppe Buttazzo (Pisa)

##### MSC:

35-02 | Research exposition (monographs, survey articles) pertaining to partial differential equations |

49-02 | Research exposition (monographs, survey articles) pertaining to calculus of variations and optimal control |

49J40 | Variational inequalities |

28A33 | Spaces of measures, convergence of measures |

35K55 | Nonlinear parabolic equations |

47H05 | Monotone operators and generalizations |

65M15 | Error bounds for initial value and initial-boundary value problems involving PDEs |