Self-dual Partial Differential Systems and Their Variational Principles
Based on recent research by the author and his graduate students, this text describes novel variational formulations and resolutions of a large class of partial differential equations and evolutions, many of which are not amenable to the methods of the cl
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Nassif Ghoussoub
Self-dual Partial Differential Systems and Their Variational Principles
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Nassif Ghoussoub University of British Columbia Department of Mathematics Vancouver BC V6T 1Z2 Canada [email protected]
ISSN: 1439-7382 ISBN: 978-0-387-84896-9 DOI: 10.1007/978-0-387-84897-6
e-ISBN: 978-0-387-84897-6
Library of Congress Control Number: 2008938377 Mathematics Subject Classification (2000): 46-xx, 35-xx c Springer Science+Business Media, LLC 2009 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper springer.com
To Mireille.
Preface
How to solve partial differential systems by completing the square. This could well have been the title of this monograph as it grew into a project to develop a systematic approach for associating suitable nonnegative energy functionals to a large class of partial differential equations (PDEs) and evolutionary systems. The minima of these functionals are to be the solutions we seek, not because they are critical points (i.e., from the corresponding Euler-Lagrange equations) but from also being zeros of these functionals. The approach can be traced back to Bogomolnyi’s trick of “completing squares” in the basic equations of quantum field theory (e.g., Yang-Mills, Seiberg-Witten, Ginzburg-Landau, etc.,), which allows for the derivation of the so-called self (or antiself) dual version of these equations. In reality, the “self-dual Lagrangians” we consider here were inspired by a variational approach proposed – over 30 years ago – by Br´ezis and Ekeland for the heat equation and other gradient flows of convex energies. It is based on Fenchel-Legendre duality and can be used on any convex functional – not just quadratic ones – making them applicable in a wide range of problems. In retrospect, we realized that the “energy identities” satisfied by Leray’s solutions for the Navier-Stokes equations are also another manifestation of the concept of self-duality in the context of evolution equations. The book could have also been entitled How to solve nonlinear PDEs via convex analysis on phase space. Indeed, the self-dual vector fields we introduce and study here are natural extensions of gradients of convex energies – and hence of selfadjoint positive operators – which usually drive dissipative systems but also provide representations for the superposition of such gradients with skew-sy
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