IRIS

In this treatise we present some important results of the theory of the Wasserstein space of probability measures, with applications to diffusione equations and gradient flows.

Gradient Flows of Probability Measures

SAVARE', GIUSEPPE
2006-01-01

Abstract

In this treatise we present some important results of the theory of the Wasserstein space of probability measures, with applications to diffusione equations and gradient flows.
2006
Handbook of Differential Equations - Evolutionary Equations, Volume 3
DAFERMOS C.M.; FEIREISL E.
The Mathematics category includes resources dealing with mathematics, applied mathematics, statistics and probability.
Inglese
Internazionale
STAMPA
3
1
136
9780444528483
Elsevier
Amsterdam
PAESI BASSI
The material collected in this volume reflects the active present of this area of mathematics, ranging from the abstract theory of gradient flows to stochastic representations of non-linear parabolic PDE's. Articles will highlight the present as well as expected future directions of development of the field with particular emphasis on applications. The article by Ambrosio and Savare discusses the most recent development in the theory of gradient flow of probability measures. After an introduction reviewing the properties of the Wasserstein space and corresponding subdifferential calculus, applications are given to evolutionary partial differential equations. The contribution of Herrero provides a description of some mathematical approaches developed to account for quantitative as well as qualitative aspects of chemotaxis. Particular attention is paid to the limits of cell's capability to measure external cues on the one hand, and to provide an overall description of aggregation models for the slim mold Dictyostelium discoideum on the other. The chapter written by Masmoudi deals with a rather different topic - examples of singular limits in hydrodynamics. This is nowadays a well-studied issue given the amount of new results based on the development of the existence theory for rather general systems of equations in hydrodynamics. The paper by DeLellis addreses the most recent results for the transport equations with regard to possible applications in the theory of hyperbolic systems of conservation laws. Emphasis is put on the development of the theory in the case when the governing field is only a BV function. The chapter by Rein represents a comprehensive survey of results on the Poisson-Vlasov system in astrophysics. The question of global stability of steady states is addressed in detail. The contribution of Soner is devoted to different representations of non-linear parabolic equations in terms of Markov processes. After a brief introduction on the linear theory, a class of non-linear equations is investigated, with applications to stochastic control and differential games. The chapter written by Zuazua presents some of the recent progresses done on the problem of controllabilty of partial differential equations. The applications include the linear wave and heat equations,parabolic equations with coefficients of low regularity, and some fluid-structure interaction models.
Wasserstein spaces; Gradient flows; Optimal transport; Geodesic convexity; Measure valued solutions; Continuity equation
http://www.elsevier.com/wps/find/bookdescription.cws_home/709129/description
2 Contributo in Volume::2.1 Contributo in volume (Capitolo o Saggio)
2
268
none
Ambrosio, Luigi; Savare', Giuseppe
info:eu-repo/semantics/bookPart
File in questo prodotto:
Non ci sono file associati a questo prodotto.

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11571/132362
Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus ND
  • ???jsp.display-item.citation.isi??? ND
social impact