Existence and Uniqueness of Maximal Regular Flows for Non-smooth Vector Fields

被引：25

作者：

Ambrosio, Luigi ^{[1
]}

Colombo, Maria ^{[1
]}

Figalli, Alessio ^{[2
]}

机构：

[1] Scuola Normale Super Pisa, Pisa, Italy

[2] Univ Texas Austin, Austin, TX 78712 USA

来源：

ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS | 2015年 / 218卷 / 02期

基金：

美国国家科学基金会;

关键词：

BV; EQUATION; DIFFERENTIABILITY;

D O I：

10.1007/s00205-015-0875-9

中图分类号：

O29 [应用数学];

学科分类号：

070104 ;

摘要：

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.

引用

页码：1043 / 1081

页数：39