Applied Mathematical Sciences

(1)

Applied Mathematical Sciences

Volume 125

Series Editors

Anthony Bloch, Department of Mathematics, University of Michigan, Ann Arbor, MI, USA

C. L. Epstein, Department of Mathematics, University of Pennsylvania, Philadelphia, PA, USA

Alain Goriely, Department of Mathematics, University of Oxford, Oxford, UK Leslie Greengard, New York University, New York, NY, USA

Advisory Editors

J. Bell, Center for Computational Sciences and Engineering, Lawrence Berkeley National Laboratory, Berkeley, CA, USA

P. Constantin, Department of Mathematics, Princeton University, Princeton, NJ, USA

R. Durrett, Department of Mathematics, Duke University, Durham, CA, USA R. Kohn, Courant Institute of Mathematical Sciences, New York University, New York, NY, USA

R. Pego, Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA, USA

L. Ryzhik, Department of Mathematics, Stanford University, Stanford, CA, USA A. Singer, Department of Mathematics, Princeton University, Princeton, NY, USA A. Stevens, Department of Applied Mathematics, University of Münster, Münster, Germany

S. Wright, Computer Sciences Department, University of Wisconsin, Madison, WI, USA

Founding Editors

F. John, New York University, New York, NY, USA J. P. LaSalle, Brown University, Providence, RI, USA L. Sirovich, Brown University, Providence, RI, USA

(2)

The mathematization of all sciences, the fading of traditional scientiﬁc boundaries, the impact of computer technology, the growing importance of computer modeling and the necessity of scientiﬁc planning all create the need both in education and research for books that are introductory to and abreast of these developments. The purpose of this series is to provide such books, suitable for the user of mathematics, the mathematician interested in applications, and the student scientist. In particular, this series will provide an outlet for topics of immediate interest because of the novelty of its treatment of an application or of mathematics being applied or lying close to applications. These books should be accessible to readers versed in mathematics or science and engineering, and will feature a lively tutorial style, a focus on topics of current interest, and present clear exposition of broad appeal.

A compliment to the Applied Mathematical Sciences series is the Texts in Applied Mathematics series, which publishes textbooks suitable for advanced undergraduate and beginning graduate courses.

More information about this series athttp://www.springer.com/series/34

(3)

Vladimir I. Arnold

^•

Boris A. Khesin

Topological Methods

in Hydrodynamics

Second Edition

123

(4)

Vladimir I. Arnold

Russian Academy of Sciences Steklov Mathematical Institute Moscow, Russia

Boris A. Khesin

Department of Mathematics University of Toronto Toronto, ON, Canada

ISSN 0066-5452 ISSN 2196-968X (electronic) Applied Mathematical Sciences

ISBN 978-3-030-74277-5 ISBN 978-3-030-74278-2 (eBook) https://doi.org/10.1007/978-3-030-74278-2

This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, speciﬁcally the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microﬁlms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed.

The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a speciﬁc statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use.

The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional afﬁliations.

This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

(5)

In Lieu of a Preface to the Second Edition

Back in the mid-1980s Vladimir Igorevich Arnold once told us, his students, how diﬀerent the notion of “being young” (and in particular, being a young mathematician) is in diﬀerent societies. For instance, the Moscow Mathemat- ical Society awards an annual prize to a young mathematician under thirty years of age. The Fields Medal, as is well known, recognizes outstanding young mathematicians whose age does not exceed forty in the year of the Interna- tional Congress. Both of the above requirements are strictly enforced.

This can be compared with the Bourbaki group, which comprises young French mathematicians and which, reportedly, has an age bar of ﬁfty. However, as Arnold elaborated the story, this limit is more ﬂexible: upon reaching this age, the Bourbaki member undergoes a “coconutization procedure.” The term is derived from a tradition of a certain barbaric tribe that allows its chief to carry out his duties until someone doubts his leadership abilities. Once the doubt arises, the chief is forced to climb to the top of a tall palm tree, and the whole tribe starts shaking it. If the chief is strong enough to hold on and survives the challenge, he is allowed to climb down and continue to lead the tribe until the next “reasonable doubt” in his leadership crosses someone’s mind. If his grip is weak and he falls down from the 20-meter-tall tree, he obviously needs to be replaced, and so the next chief is chosen. This tree is usually a coconut palm, which gave the name to the coconutization procedure.

As far as coconutization at the Bourbaki group is concerned, according to Arnold’s story, the unsuspecting member who reaches fifty is invited, as usual, to the next Bourbaki seminar. Somewhere in the middle of the talk, when most of the audience is already half asleep, the speaker, who is in on the game for that occasion, inserts some tedious half-page long definition. It is at this very moment that the scrutinized (“coconutized”) member is expected to interrupt the speaker by exclaiming something like, “But excuse me, only the empty set satisfies your definition!” If he does so, he has successfully passed the test and will remain a part of Bourbaki. If he missed this chance, nobody says a word, but he would probably not be invited to meetings any longer.

V

(6)

VI In Lieu of a Preface Arnold ﬁnished this story by quoting someone’s deﬁnition of youth in mathematics, which he liked best: “A mathematician is young as long as he reads works other than his own”!

Soon after this “storytelling” occasion, Arnold’s ﬁftieth birthday was cel- ebrated: in June 1987 his whole seminar went for a picnic in a suburb of Moscow. Among Arnold’s presents were a “Return to Arnold” stamp to mark the reprints he gave to his students to work on, an academic gown with a nicely decorated “swallowtail,” one of low-dimensional singularities, and such. But most importantly, he was presented with a poster containing a crossword puz- zle of various notions from his many research domains. Most of the questions were rather intricate, which predictably did not prevent Arnold from easily cracking virtually everything. But one question remained unresolved: “A simple alternative of life” for a ﬁve-letter word (in English translation). None of the ideas worked for quite some time. After a while, having made no progress on this question, Arnold pronounced sadly, “Now I myself have been coconutized....” But a second later, he perked up, a bright mischievous expression on his face: “This is a PURSE”! (In addition to the pirate’s alternative “your Purse or your Life”, the crossword authors meant the term “purse” in singularity theory standing for the description of the bifurcation diagram of the real simple singularity D₄⁺, also called the hyperbolic umbilic — hence the hint on “simple” alternative.)

Arnold’s interest to fluid dynamics can be traced back to his “younger years,” whatever definition one is using for that purpose. His 1966 paper in the Annales de l’Institut Fourier had the effect of a bombshell. Now, over fifty years later, virtually every paper related to the geometry of the hydrodynamic Euler equation or diffeomorphism groups cites this work of Arnold’s early on.

In the next four or ﬁve years, Arnold laid out the foundations for the study of hydrodynamic stability and for the use of Hamiltonian methods there, described the topology of steady ﬂows, etc. All these topics are described in the present book.

Apparently, Arnold’s interest in hydrodynamics is rooted in Kolmogorov’s turbulence study and began with the program outlined by Kolmogorov for his seminar in 1958–1959 (and presented in Section I.12 below). Kolmogorov conjectured stochastization in dynamical systems related to hydrodynamic PDEs as viscosity vanishes, which would imply the practical impossibility of long-term weather forecasts. Arnold’s take on hydrodynamics was, however, completely diﬀerent from Kolmogorov’s, in that it involved groups and topology.¹

Since the publication of the ﬁrst edition of the book Topological Methods in Hydrodynamics (over 20 years ago) and its Russian edition (over 10 years

1 See “On V.I. Arnold and Hydrodynamics” in the book ARNOLD: Swimming Against the Tide, AMS, Providence RI, 2014.

(7)

In Lieu of a Preface VII ago) there has appeared an enormous body of literature on topological ﬂuid mechanics. Many problems and open questions posed or discussed in the book have been solved or substantially advanced. It would be natural to revise the text to update the reader on recent developments in this vast area for the second edition of the book.

However, after the untimely departure of Vladimir Igorevich Arnold in 2010, I would not like to substantially change the text of this book, which was thoroughly discussed with him and veriﬁed on many examples. Instead, the second edition, in addition to editorial corrections, contains a specially prepared survey of recent developments in topological, geometric, and group- theoretic hydrodynamics with an independent bibliography.

I hope that this new edition will be useful to mathematicians, physicists, and students of various disciplines who are interested in various problems of applied fluid dynamics, as well as in purely theoretical questions arising in the topology and geometry of groups of diffeomorphisms and having a hydrodynamic flavor.

I am particularly indebted to A. Izosimov, O. Kozlovsky, D. Kramer, G. Misiolek, K. Modin, D. Peralta-Salas, and A. Shnirelman for fruitful discussions and to V. Shuvalov for his generous help with the book ﬁgures. This work on the second edition was partially supported by an NSERC research grant and a Simons Fellowship.

January 2021 Boris Khesin

(8)

Preface

“...ad alcuno, dico, di quelli, che troppo laconicamente vorrebbero vedere, nei più angusti spazii che possibil fusse, ristretti i filosofici insegnamenti, s´ı che sempre si usasse quella rigida e concisa maniera, spogliata di qualsivoglia vaghezza ed ornamento, che é propria dei puri geometri, li quali né pure una parola proferiscono che dalla assoluta necessitá non sia loro suggerita.

Ma io, all’incontro, non ascrivo a difetto in un trattato, ancorch´e indirizzato ad un solo scopo, interserire altre varie notizie, purch´e non siano totalmente separate e senza veruna coerenza annesse al principale instituto.”²

Galileo Galilei “Lettera al Principe Leopoldo di Toscana” (1623)

Hydrodynamics is one of those fundamental areas in mathematics where progress at any moment may be regarded as a standard to measure the real success of mathematical science. Many important achievements in this ﬁeld are based on profound theories rather than on experiments. In turn, those hydrodynamic theories stimulated developments in the domains of pure mathematics, such as complex analysis, topology, stability theory, bifurcation theory, and completely integrable dynamical systems. In spite of all this acknowl- edged success, hydrodynamics with its spectacular empirical laws remains a challenge for mathematicians. For instance, the phenomenon of turbulence

2 “... Some prefer to see the scientiﬁc teachings condensed too laconically into the smallest possible volume, so as always to use a rigid and concise manner that whatsoever lacks beauty and embellishment, and that is so common among pure geometers who do not pronounce a single word which is not of absolute necessity.

I, on the contrary, do not consider it a defect to insert in a treatise, albeit devoted to a single aim, other various remarks, as long as they are not out of place and without coherency with the main purpose,” see [Gal].

IX

(9)

X Preface has not yet acquired a rigorous mathematical theory. Furthermore, the existence problems for smooth solutions of hydrodynamic equations of a three- dimensional ﬂuid are still open.

The simplest but already very substantial mathematical model for fluid dynamics is hydrodynamics of an ideal (i.e., an incompressible and inviscid) homogeneous fluid. From the mathematical point of view, a theory of such a fluid filling a certain domain is nothing but a study of geodesics on the group of diffeomorphisms of the domain that preserve volume elements. The geodesics on this (infinite-dimensional) group are considered with respect to the right-invariant Riemannian metric given by the kinetic energy.

In 1765, L. Euler [Eul] published the equations of motion of a rigid body.

Eulerian motions are described as geodesics in the group of rotations of three- dimensional Euclidean space, where the group is provided with a left-invariant metric. In essence, the Euler theory of a rigid body is fully described by this invariance. The Euler equations can be extended in the same way to an arbitrary group. As a result, one obtains, for instance, the equations of a rigid body motion in a high-dimensional space and, especially interesting, the Euler equations of hydrodynamics of an ideal ﬂuid.

Euler’s theorems on the stability of rotations about the longest and short- est axes of the inertia ellipsoid have counterparts for an arbitrary group as well. In the case of hydrodynamics, these counterparts deliver nonlinear gen- eralizations of Rayleigh’s theorem on the stability of two-dimensional flows without inflection points of the velocity profile.

The description of ideal fluid flows by means of geodesics of the right- invariant metric allows one to apply methods of Riemannian geometry to the study of flows. It does not immediately imply that one has to start by con- structing a consistent theory of infinite-dimensional Riemannian manifolds.

The latter encounters serious analytical diﬃculties, related in particular to the absence of existence theorems for smooth solutions of the corresponding diﬀerential equations.

On the other hand, the strategy of applying geometric methods to the infinite-dimensional problems is as follows. Having established certain facts in the finite-dimensional situation (of geodesics for invariant metrics on finite- dimensional Lie groups), one uses the results to formulate the corresponding facts for the infinite-dimensional case of the diffeomorphism groups. These fi- nal results often can be proved directly, leaving aside the difficult questions of foundations for the intermediate steps (such as the existence of solutions on a given time interval). The results obtained in this way have an a priori charac- ter: the derived identities or inequalities take place for any reasonable meaning of “solutions,” provided that such solutions exist. The actual existence of the solutions remains an open question.

For example, we deduce the formulas for the Riemannian curvature of a group endowed with an invariant Riemannian metric. Applying these formulas to the case of the infinite-dimensional manifold whose geodesics are motions of an ideal fluid, we find that the curvature is negative in many di-

(10)

Preface XI rections. Negativity of the curvature implies instability of motion along the geodesics (which is well-known in Riemannian geometry of finite-dimensional manifolds). In the context of the (infinite-dimensional) case of the diffeomorphism group, we conclude that the ideal flow is unstable (in the sense that a small variation of the initial data implies large changes of the particle positions at a later time). Moreover, the curvature formulas allow one to estimate the increment of the exponential deviation of fluid particles with close initial positions and hence to predict the time period when the motion of fluid masses becomes essentially unpredictable.

For instance, in the simplest and utmost idealized model of the earth’s atmosphere (regarded as a two-dimensional ideal ﬂuid on the surface of a torus), the deviations grow by the factor of 10⁵in 2 months. This circumstance ensures that a dynamical weather forecast for such a period is practically impossible (however powerful the computers and however dense the grid of data used for this purpose).

The table of contents is essentially self explanatory. We have tried to make the chapters as independent of each other as possible. Cross references within the same chapter do not contain the chapter number.

For a ﬁrst acquaintance with the subject, we address the reader to the following sections in each chapter: Sections I.1-5 and I.12, Sections II.1 and II.3-4, Sections III.1-2 and III.4, Section IV.1, Sections V.1-2, Sections VI.1 and VI.4.

Some statements in this book may be new even for the experts. We mention the classification of local conservation laws in ideal hydrodynamics (Theorem I.9.9), M. Freedman’s solution of the A. Sakharov–Ya. Zeldovich problem on the energy minimization of an unknotted magnetic field (Theorem III.3.1), a discussion of the construction of manifold invariants from the energy bounds (Remark III.2.6), a discussion of a complex version of the Vassiliev knot invariants (in Section III.7.E), a nice remark of B. Zeldovich on the Lobachevsky triangle medians (Problem IV.1.4), the relation of the covariant derivative of a vector field and the inertia operator in hydrodynamics (Section IV.1.D), a digression on the Fokker–Planck equation (Section V.3.C), and the dynamo construction from the geodesic flow on surfaces of constant negative curvature (Section V.4.D).

The main text of the book was also updated according to the 2007 Rus- sian edition and it discusses several new developments. The latter include the relation between the Eulerian and Lagrangian instability [Pre], Fredholmness of the geodesic exponential map on the diffeomorphism groups in the 2D set- ting [EbM, EbMP], universality of the Monge–Ampère equation in problems of fluid dynamics and optimal mass transport, a solution of the “short paths”

problem in the deﬁnition of the asymptotic Hopf invariant [Vog], as well as a Brownian interpretation of the higher-dimensional analogue of this invariant [Riv, Kh3]. The section written by A. Shnirelman was extended by adding a

(11)

XII Preface description of weak solutions of the 2D Euler equation in terms of continuum braids [Shn9]. Many open questions related to hydrodynamics were discussed in the book [Arn26].

(12)

Acknowledgments

We greatly benefited from the help of many people. We are sincerely grateful to all of them: F. Aicardi, J.-L. Brylinski, M.A. Berger, Yu.V. Chekanov, S. Childress, L.A. Dickey, D.G. Ebin, Ya. Eliashberg, L.D. Faddeev, V.V. Fock, M.H. Freedman, U. Frisch, A.D. Gilbert, V.L. Ginzburg, M.L. Gromov, M. Henon, M.-R. Herman, H. Hofer, Yu.S. Ilyashenko, K.M. Khanin, C. King, A.N. Kolmogorov, E.I. Korkina, V.V. Kozlov, O.A. Ladyzhen- skaya, P. Laurence, J. Leray, A.M. Lukatsky, M. Lyubich, S.V. Man- akov, J.E. Marsden, D. McDuff, A.S. Mishchenko, H.K. Moffatt, R. Mont- gomery, J.J. Moreau, J. Moser, N. Nekrasov, Yu.A. Neretin, S.P. Novikov, V.I. Oseledets, V.Yu. Ovsienko, D.A. Panov, L. Polterovich, M. Polyak, T.S. Ratiu, S. Resnick, C. Roger, A.A. Rosly, A.A. Ruzmaikin, A.D. Sakharov, L. Schwartz, D. Serre, B.Z. Shapiro, A.I. Shnirelman, M.A. Shubin, Ya.G. Sinai, S.L. Sobolev, D.D. Sokolov, S.L. Tabachnikov, A.N. Todorov, O.Ya. Viro, M.M. Vishik, V.A. Vladimirov, A. Weinstein, L.-S. Young, V.I. Yudovich, V.M. Zakalyukin, I.S. Zakharevich, E. Zehnder, V. Zeitlin, Ya.B. Zeldovich, A.V. Zorich, V.A. Zorich and many others.

Section IV.7 was written by A.I. Shnirelman, and an initial version of Section VI.5 was prepared by B.Z. Shapiro. Remark II.4.11 was written by J.E. Marsden. Special thanks go to O.S. Kozlovsky and G. Misiolek for the numerous discussions on diﬀerent topics of the book and for their many useful remarks. O.S. Kozlovsky has also provided us with his recent unpub- lished results for several sections in Chapter V (in particular, for Sections V.1.B,V.2.C,V.3.E).

Boris Khesin is deeply indebted to his wife Masha for her tireless moral support during the seemingly endless work on this book. We are grateful to A. Mekis and V. Shuvalov for their help with ﬁgures and to D. Kramer for his careful reading of the manuscript.

B.K. appreciates the kind hospitality of the Max-Planck Institut in Bonn, Institut des Hautes Etudes Scientifiques in Bures-sur-Yvette, Research Insti- tute for Mathematical Sciences in Kyoto, and Forschungsinstitut für Mathe- matik in Zürich during his work on this book. The preparation of this book

XIII

(13)

XIV Acknowledgments was partially supported by the Russian Basic Research Foundation, project 96-01-01104 (V.A.), by the Alfred P. Sloan Research Fellowship, and by the NSF and NSERC research grants DMS-9627782 and OGP-0194132 (B.K.).

(14)

Applied Mathematical Sciences