Recent Titles in This Series 144 0. V. Troshkin, Nontraditional methods in mathematical hydrodynamics, 1995 143 V. A. Malyshev and R. A. Mintos, Linear infinite-particle operators, 1995 142 N. V. Krylov, Introduction to the theory of diffusion processes, 1995 141 A. A. Davydov, Qualitative theory of control systems, 1994 140 Aizik I. Volpert, Vitaly A. Volpert, and Vladimir A. Volpert, Traveling wave solutions of parabolic systems, 1994 139 I. V. Skrypnik, Methods for analysis of nonlinear elliptic boundary value problems, 1994 138 Yu. P. Razmyslov, Identities of algebras and their representations, 1994 137 F. I. Karpelevich and A. Ya. Kreinin, Heavy traffic limits for multiphase queues, 1994 136 Masayoshi Miyanishi, Algebraic geometry, 1994 135 Masaru Takeuchi, Modern spherical functions, 1994 134 V. V. Prasolov, Problems and theorems in linear algebra, 1994 133 P. I. Naumkin and I. A. Shishmarev, Nonlinear nonlocal equations in the theory of waves, 1994 132 Hajime Urakawa, Calculus of variations and harmonic maps, 1993 131 V. V. Sharko, Functions on manifolds: Algebraic a'nd topological aspects, 1993 130 V. V. Vershinin, Cobordisms and spectral sequences, 1993 129 Mitsuo Morimoto, An introduction to Sato's hyperfunctions, 1993 128 V. P. Orevkov, Complexity of proofs and their transformations in axiomatic theories, 1993 127 F. L. Zak, Tangents and secants of algebraic varieties, 1993 126 M. L. Agranovskii, Invariant function spaces on homogeneous manifolds of Lie groups and applications, 1993 125 Masayoshi Nagata, Theory of commutative fields, 1993 124 Masahisa Adachi, Embeddings and immersions, 1993 123 M.A. Akivis and B. A. Rosenfeld, Elie Cartan (1869-1951), 1993 122 Zhang Guan-Hou, Theory of entire and meromorphic functions: Deficient and asymptotic values and singular directions, 1993 121 I. B. Fesenko and S. V. Vostokov, Local fields and their extensions: A constructive approach, 1993 120 Takeyuki Hida and Masuyuki Hitsuda, Gaussian processes, 1993 119 M. V. Karasev and V. P. Maslov, Nonlinear Poisson brackets. Geometry and quantization, 1993 I 18 Kenkichi Iwasawa, Algebraic functions, 1993 117 Boris Zilber, Uncountably categorical theories, 1993 116 G. M. Fel'dman, Arithmetic of probability distributions, and characterization problems on abelian groups, 1993 I I 5 Nikolai V. Ivanov, Subgroups of Teichmilller modular groups, 1992 I 14 Seizo Ito, Diffusion equations, 1992 113 Michail Zhitomirskii, Typical singularities of differential I-forms and Pfaffian equations, 1992 112 S. A. Lomov, Introduction to the general theory of singular perturbations, 1992 111 Simon Gindikin, Tube domains and the Cauchy problem, 1992 110 B. V. Shabat, Introduction to complex analysis Part II. Functions of several variables, 1992 109 Isao Miyadera, Nonlinear semigroups, 1992 108 Takeo Yokonuma, Tensor spaces and exterior algebra, 1992 107 B. M. Makarov, M. G. Goluzina, A. A. Lodkin, and A. N. Podkorytov, Selected problems in real analysis, 1992 106 G.-C. Wen, Conformal mappings and boundary value problems, 1992 105 D.R. Yafaev, Mathematical scattering theory: General theory, 1992 104 R. L. Dobrushin, R. Kotecky, and S. Shlosman, Wulff construction: A global shape from local interaction, 1992 !03 A. K. Tsikh, Multidimensional residues and their applications, 1992 (See the AMS catalog for earlier titles) Nontraditional Methods in Mathematical Hydrodynamics Translations of MATHEMATICAL MONOGRAPHS Volume 144 Nontraditional Methods in Mathematical Hydrodynamics 0. V. Troshkin 0. B. Tpoumrn HETPA)J:111..(11 OHHbIE 3A)J:AqJ1 MATEMATJ1qECKOH rl1)J:PO)J:l1HAMl1Kl1 Translated by Peter Zhevandrov from an original Russian manuscript Translation edited by Simeon Ivanov 1991 Mathematics Subject Classification. Primary 76C05, 76D05, 76Fxx; Secondary 35Q30, 76Bxx, 76Cxx, 76Exx. ABSTRACT. The main goal of the book is to present a number of features of mathematical models for incompressible fluids. Three basic systems of hydrodynamical equations are considered, namely, the system of stationary Euler equations for flows of ideal (nonviscous) fluid, stationary Navier-Stokes equations for flows of a viscous fluid, and Reynolds equations for the mean velocity field, pressure, and pair one-point velocity correlations of turbulent flows. The analysis concerns algebraic or geometric properties of vector fields generated by these equations, such as the general arrangement of streamlines, the character and distribution of singular points, etc. The information provided by this analysis is used for the investigation of the conditions for unique solvability of a number of problems for these quasilinear systems. The book contains a lot of examples of particular phenomena illustrating general ideas in the book. The book can be used by researchers and graduate students working in mathematical physics and hydrodynamics. Library of Congress Cataloging-in-Publication Data Troshkin, 0. V. [Hetraditsionnye zadachi matematicheskoi gidrodinamiki. English] Nontraditional methods in mathematical hydrodynamics / 0. V. Troshkin. p. cm. - (Translations of mathematical monographs, ISSN 0065-9282; v. 144) Includes bibliographical references p. - ). ISBN 0-8218-0285-2 (acid-free) I. Hydrodynamics. I. Title. II. Series. QA91 l.T7413 1995 532' .58-dc20 94-48581 CIP © Copyright 1995 by the American Mathematical Society. All rights reserved. Translation authorized by the All-Union Agency for Author's Rights, Moscow. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. § The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. 0 Printed on recycled paper. Information on Copying and Reprinting can be found in the back of this volume. This volume was typeset using AMS-TEX, the American Mathematical Society's TEX macro system. 10 9 8 7 6 5 4 3 2 I 00 99 98 97 96 95 Contents Introduction Chapter 1. Stationary Flows of an Ideal Fluid on the Plane 11 §1 . Pressure field in a neighborhood of a singular point 11 l. l. Nondegenerate singular points 11 1.2. The stream function 12 1.3. Euler fields 12 §2. Other properties of plane flows 14 2.1. Equivalent forms of Euler's equations 14 2.2. Reducible fields 15 2.3. Frozen-in vortices 16 2.4. Simple domains 18 2.5. Holomorphic flows 20 Chapter 2. Topology of Two-Dimensional Flows 23 §3. An extended version of some Morse constructions 23 3.1. Admissible stream functions 23 3.2. The basic integer relations 27 3.3. Graphs and chambers 30 3.4. The structure of admissible functions 33 §4. Portraits of potential flows 37 §5. Imposition of a vortex on a simple passing flow 46 §6. Imposition of a vortex on an inverse flow 49 Chapter 3. A Two-Dimensional Passing Flow Problem for Stationary Euler Equations 53 §7. Uniqueness of an analytic solution 53 7. 1. Formulation of the problem 53 7.2. The uniqueness theorem 56 §8. Analysis of the uniqueness of a smooth flow 58 8.1. The choice of the region of prescribed vorticity 58 8.2. Uniqueness of an open flow 58 8.3. Nonuniqueness of a developed flow 60 8.4. Problems on simple passing and inverse flows 62 §9. Existence theorems 65 9.1. Auxiliary boundary problem 65 9.2. The vortex catastrophe 72 §10. Regular functionals 74 10. l. The closure of a smooth operator 74 vii viii CONTENTS 10.2. The conditions for splitting 77 10.3. The existence of a critical point 80 10.4. An abstract version of the eigenvalue problem 83 Chapter 4. The Dissipative Top and the Navier-Stokes Equations 89 §1 1. Axioms of the dissipative top 89 11.1. Plane flows of a viscous incompressible fluid 89 11.2. Quasicompact Lie algebras 92 11.3. The equation of the top. A local uniqueness theorem 94 11.4. Types of dissipative tops 97 11.5. The existence and completeness of the system of principal rotations 101 11.6. The basic scalar relations 104 §1 2. Stability of a principal rotation 105 12.1. A nonlocal uniqueness theorem 105 12.2. A nonlocal theorem on stability 108 §1 3. Generalized solutions of the equation of the top 112 p.1. Spatial flows of a viscous incompressible fluid 112 13.2. An extension of the class of admissible right-hand sides 116 13.3. The existence of a generalized solution 117 §1 4. Examples of dissipative tops 123 14.1. Motions on the algebra .so(n, R) 123 14.2. Flows on two-dimensional manifolds 125 14.3. Flows on a three-dimensional torus 129 14.4. An example of violation of quasicompactness 130 Chapter 5. Specific Features of Turbulence Models 133 §15. Wave properties of turbulent flows 133 15.1. An "ideal" turbulent medium 133 15.2. Field of small perturbations 134 15.3. Characteristics 137 15.4. Effects of viscosity 140 15.5. A proposition on characteristics 146 §1 6. Passage to turbulent regime and bifurcation 151 16.1. The closure scheme 151 16.2. An illustrative example 154 16.3. Analysis of the bifurcation problem 154 16.4. The existence of a minimizing element 160 Appendix. Formal Constructions Connected with Euler Equations 163 §17. Continuous models of some Lagrangian systems 163 17.1. Notation 163 17 .2. Natural motions and the Euler equations 167 17.3. The stationary action principle 170 17.4. Translational movements and motions along geodesics 176 17. 5. Equations of relativistic hydrodynamics 177 17.6. The concept of the integrating factor 185 17.7. The invariant form of the Navier-Stokes equations 187 References 191