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Continuity equations and ODE flows with non-smooth velocity*

Published online by Cambridge University Press:  01 December 2014

Luigi Ambrosio  and
Gianluca Crippa
Luigi Ambrosio
Affiliation:
Scuola Normale Superiore, Piazza Cavalieri 7, 56126 Pisa, Italy, ([email protected])
Gianluca Crippa
Affiliation:
Departement Mathematik und Informatik, Universität Basel, Mathematisches Institut, Rheinsprung 21, 4051 Basel, Switzerland, ([email protected])
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Abstract

In this paper we review many aspects of the well-posedness theory for the Cauchy problem for the continuity and transport equations and for the ordinary differential equation (ODE). In this framework, we deal with velocity fields that are not smooth, but enjoy suitable ‘weak differentiability’ assumptions. We first explore the connection between the partial differential equation (PDE) and the ODE in a very general non-smooth setting. Then we address the renormalization property for the PDE and prove that such a property holds for Sobolev velocity fields and for bounded variation velocity fields. Finally, we present an approach to the ODE theory based on quantitative estimates.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 144 , Issue 6 , December 2014 , pp. 1191 - 1244
Copyright
Copyright © Royal Society of Edinburgh 2014 

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Footnotes

*

This paper is a late addition to the papers surveying active areas in partial differential equations, published in issues 141.2 and 142.6, which were based on a series of mini-courses held in Edinburgh from 2010 to 2013.

References

* This paper is a late addition to the papers surveying active areas in partial differential equations, published in issues 141.2 and 142.6, which were based on a series of mini-courses held in Edinburgh from 2010 to 2013.