We establish, in a rather general setting, an analogue of DiPerna-Lions theory on well-posedness of flows of ODE's associated to Sobolev vector fields. Key results are a well-posedness result for the continuity equation associated to suitably defined Sobolev vector fields, via a commutator estimate, and an abstract superposition principle in (possibly extended) metric measure spaces, via an embedding into $mathbb{R}^infty$. When specialized to the setting of Euclidean or infinite dimensional (e.g. Gaussian) spaces, large parts of previously known results are recovered at once. Moreover, the class of ${sf RCD}(K,infty)$ metric measure spaces object of extensive recent research fits into our framework. Therefore we provide, for the first time, well-posedness results for ODE's under low regularity assumptions on the velocity and in a nonsmooth context.

We establish, in a rather general setting, an analogue of DiPerna–Lions theory on well-posedness of flows of ODEs associated to Sobolev vector fields. Key results are a well-posedness result for the continuity equation associated to suitably defined Sobolev vector fields, via a commutator estimate, and an abstract superposition principle in (possibly extended) metric measure spaces, via an embedding into R∞. When specialized to the setting of Euclidean or infinite-dimensional (e.g., Gaussian) spaces, large parts of previously known results are recovered at once. Moreover, the class of RCD(K,∞) metric measure spaces, introduced by Ambrosio, Gigli and Savaré [Duke Math. J. 163:7 (2014) 1405–1490] and the object of extensive recent research, fits into our framework. Therefore we provide, for the first time, well-posedness results for ODEs under low regularity assumptions on the velocity and in a nonsmooth context.

Well-posedness of Lagrangian flows and continuity equations in metric measure spaces

AMBROSIO, Luigi;TREVISAN, DARIO
2014

Abstract

We establish, in a rather general setting, an analogue of DiPerna-Lions theory on well-posedness of flows of ODE's associated to Sobolev vector fields. Key results are a well-posedness result for the continuity equation associated to suitably defined Sobolev vector fields, via a commutator estimate, and an abstract superposition principle in (possibly extended) metric measure spaces, via an embedding into $mathbb{R}^infty$. When specialized to the setting of Euclidean or infinite dimensional (e.g. Gaussian) spaces, large parts of previously known results are recovered at once. Moreover, the class of ${sf RCD}(K,infty)$ metric measure spaces object of extensive recent research fits into our framework. Therefore we provide, for the first time, well-posedness results for ODE's under low regularity assumptions on the velocity and in a nonsmooth context.
Settore MAT/05 - Analisi Matematica
Mathematics - Functional Analysis; Mathematics - Functional Analysis; Mathematics - Analysis of PDEs; Mathematics - Metric Geometry; 35R05, 37C10, 30L99
File in questo prodotto:
File Dimensione Formato  
1402.4788.pdf

accesso aperto

Descrizione: Preprint dell'articolo poi pubblicato
Tipologia: Accepted version (post-print)
Licenza: Creative commons
Dimensione 575.53 kB
Formato Adobe PDF
575.53 kB Adobe PDF Visualizza/Apri
apde-v7-n5-p06-p.pdf

Accesso chiuso

Descrizione: journal article full text
Tipologia: Published version
Licenza: Non pubblico
Dimensione 1.44 MB
Formato Adobe PDF
1.44 MB Adobe PDF   Visualizza/Apri   Richiedi una copia

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: http://hdl.handle.net/11384/57121
 Attenzione

Attenzione! I dati visualizzati non sono stati sottoposti a validazione da parte dell'ateneo

Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus 50
  • ???jsp.display-item.citation.isi??? 48
social impact