Overview
- Self-contained introduction and an up to date survey on many aspects of the theory of transport equations and ordinary differential equations with non-smooth velocity fields
- Presentation of the literature on this topic by balancing heuristic motivations with rigorous proofs
- Two parts: the first one devoted to the PDE analysis, and the second one to ODE theory, relying both on the connection with the PDE and on some direct estimates
Part of the book series: Publications of the Scuola Normale Superiore (PSNS, volume 12)
Part of the book sub series: Theses (Scuola Normale Superiore) (TSNS)
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Keywords
- partial differential equations
About this book
The aim of this book is to provide a self-contained introduction and an up-to-date survey on many aspects of the theory of transport equations and ordinary differential equations with non-smooth velocity fields. The interest in this topic is motivated by important issues in nonlinear PDEs, in particular conservation laws and fluid mechanics. A fascinating feature of this research area, which is currently of concern in mathematics, is the interplay between PDE techniques and geometric measure theory techniques.
Several masterpieces appear in the related literature, balancing completely rigorous proofs with more heuristic arguments. A consistent part of the book is based on results obtained by the author in collaboration with other mathematicians. After a short introduction to the classical smooth theory, the book is divided into two parts. The first part focuses on the PDE aspect of the problem, presenting some general tools of this analysis, many well-posedness results, an abstract characterization of the well-posedness, and some examples showing the sharpness of the assumptions made. The second part, instead, deals with the ODE aspect of the problem, respectively by an abstract connection with the PDE, and by some direct and simple (but powerful) a priori estimates.
Reviews
From the reviews:
“The monograph is very well written and I recommend it both to people working in the field and to everyone who would like to learn the basics of the theory.” (Alessio Figalli, Mathematical Reviews, Issue 2010 k)Authors and Affiliations
Bibliographic Information
Book Title: The Flow Associated to Weakly Differentiable Vector Fields
Authors: Gianluca Crippa
Series Title: Publications of the Scuola Normale Superiore
Publisher: Edizioni della Normale Pisa
Copyright Information: Edizioni della Normale 2009
Softcover ISBN: 978-88-7642-340-6Published: 27 March 2009
Series ISSN: 2239-1460
Series E-ISSN: 2532-1668
Edition Number: 1
Number of Pages: XV, 167