caa a22 4500 605515956 CHVBK 20210128100711.0 cr unu---uuuuu 210128e20151101xx s 000 0 eng 10.1007/s00205-015-0875-9 doi (NATIONALLICENCE)springer-10.1007/s00205-015-0875-9 Existence and Uniqueness of Maximal Regular Flows for Non-smooth Vector Fields [Elektronische Daten] [Luigi Ambrosio, Maria Colombo, Alessio Figalli] In this paper we provide a complete analogy between the Cauchy-Lipschitz and the DiPerna-Lions theories for ODE's, by developing a local version of the DiPerna-Lions theory. More precisely, we prove the existence and uniqueness of a maximal regular flow for the DiPerna-Lions theory using only local regularity and summability assumptions on the vector field, in analogy with the classical theory, which uses only local regularity assumptions. We also study the behaviour of the ODE trajectories before the maximal existence time. Unlike the Cauchy-Lipschitz theory, this behaviour crucially depends on the nature of the bounds imposed on the spatial divergence of the vector field. In particular, a global assumption on the divergence is needed to obtain a proper blow-up of the trajectories. Springer-Verlag Berlin Heidelberg, 2015 Ambrosio Luigi Scuola Normale Superiore, Pisa, Italy aut Colombo Maria Scuola Normale Superiore, Pisa, Italy aut Figalli Alessio University of Texas at Austin, Austin, USA aut Archive for Rational Mechanics and Analysis Springer Berlin Heidelberg 218/2(2015-11-01), 1043-1081 0003-9527 218:2<1043 2015 218 205 https://doi.org/10.1007/s00205-015-0875-9 text/html Onlinezugriff via DOI BK010053 XK010053 XK010000 Metadata rights reserved Springer special CC-BY-NC licence nationallicence 1 research-article jats NATIONALLICENCE NATIONALLICENCE NL-springer NATIONALLICENCE 856 40 https://doi.org/10.1007/s00205-015-0875-9 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Ambrosio Luigi Scuola Normale Superiore, Pisa, Italy aut NATIONALLICENCE 700 1- Colombo Maria Scuola Normale Superiore, Pisa, Italy aut NATIONALLICENCE 700 1- Figalli Alessio University of Texas at Austin, Austin, USA aut NATIONALLICENCE 773 0- Archive for Rational Mechanics and Analysis Springer Berlin Heidelberg 218/2(2015-11-01), 1043-1081 0003-9527 218:2<1043 2015 218 205