The Ordinary Differential Equation with non-Lipschitz Vector Fields

Gianluca Crippa

Bollettino dell'Unione Matematica Italiana (2008)

  • Volume: 1, Issue: 2, page 333-348
  • ISSN: 0392-4041

Abstract

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In this note we survey some recent results on the well-posedness of the ordinary differential equation with non-Lipschitz vector fields. We introduce the notion of regular Lagrangian flow, which is the right concept of solution in this framework. We present two different approaches to the theory of regular Lagrangian flows. The first one is quite general and is based on the connection with the continuity equation, via the superposition principle. The second one exploits some quantitative a-priori estimates and provides stronger results in the case of Sobolev regularity of the vector field.

How to cite

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Crippa, Gianluca. "The Ordinary Differential Equation with non-Lipschitz Vector Fields." Bollettino dell'Unione Matematica Italiana 1.2 (2008): 333-348. <http://eudml.org/doc/290485>.

@article{Crippa2008,
abstract = {In this note we survey some recent results on the well-posedness of the ordinary differential equation with non-Lipschitz vector fields. We introduce the notion of regular Lagrangian flow, which is the right concept of solution in this framework. We present two different approaches to the theory of regular Lagrangian flows. The first one is quite general and is based on the connection with the continuity equation, via the superposition principle. The second one exploits some quantitative a-priori estimates and provides stronger results in the case of Sobolev regularity of the vector field.},
author = {Crippa, Gianluca},
journal = {Bollettino dell'Unione Matematica Italiana},
language = {eng},
month = {6},
number = {2},
pages = {333-348},
publisher = {Unione Matematica Italiana},
title = {The Ordinary Differential Equation with non-Lipschitz Vector Fields},
url = {http://eudml.org/doc/290485},
volume = {1},
year = {2008},
}

TY - JOUR
AU - Crippa, Gianluca
TI - The Ordinary Differential Equation with non-Lipschitz Vector Fields
JO - Bollettino dell'Unione Matematica Italiana
DA - 2008/6//
PB - Unione Matematica Italiana
VL - 1
IS - 2
SP - 333
EP - 348
AB - In this note we survey some recent results on the well-posedness of the ordinary differential equation with non-Lipschitz vector fields. We introduce the notion of regular Lagrangian flow, which is the right concept of solution in this framework. We present two different approaches to the theory of regular Lagrangian flows. The first one is quite general and is based on the connection with the continuity equation, via the superposition principle. The second one exploits some quantitative a-priori estimates and provides stronger results in the case of Sobolev regularity of the vector field.
LA - eng
UR - http://eudml.org/doc/290485
ER -

References

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