×

Statistical study of Navier-Stokes equations. I. (English) Zbl 0283.76017


MSC:

76D05 Navier-Stokes equations for incompressible viscous fluids
35Q30 Navier-Stokes equations
65Z05 Applications to the sciences
76F05 Isotropic turbulence; homogeneous turbulence
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
60K35 Interacting random processes; statistical mechanics type models; percolation theory
PDFBibTeX XMLCite
Full Text: Numdam EuDML

References:

[1] Agmon ( S. ) [1] Lectures on Elliptic Boundary Value Problems ( Van Nostrand ), New York , 1965 . MR 178246 | Zbl 0142.37401 · Zbl 0142.37401
[2] Bass ( J. ) [1] Solutions turbulentes de certaines equations aux dérivées partielles , C. R. Acad. Sci. Paris , 249 ( 1959 ), 1456 - 1457 . MR 107450 | Zbl 0086.40702 · Zbl 0086.40702
[3] Sur l’existence des solutions turbulentes des équations de l’hydrodynamique , C. R. Acad. Sci. Paris , 252 ( 1961 ), 3392 - 3394 . MR 135371 | Zbl 0126.42803 · Zbl 0126.42803
[4] Batchelor ( G.K. ) [1] The Theory of homogeneous Turbulence , ( Cambridge Univ. Press ), Cambridge , 1967 . Zbl 0522.76051 · Zbl 0522.76051
[5] Bourbaki ( N. ) [1] Integration. Ch. 5. Intégration des mesures. Eléments de mathématiques . ( Hermann ), Paris , 1967 . MR 209424 | Zbl 0143.27101 · Zbl 0143.27101
[6] Cattabriga ( L. ) [1] Su un problema al contorno relativo al sistema di equazioni di Stokes . Rend. Sem. Mat. Univ. Padova , 31 ( 1961 ), 308 - 340 . Numdam | MR 138894 | Zbl 0116.18002 · Zbl 0116.18002
[7] Choquet ( G. ) [1] Lectures on Analysis. Vol. II. Representation Theory. ( W. A. Benjamin, Inc. ). New York , 1969 . MR 250012 | Zbl 0181.39602 · Zbl 0181.39602
[8] Dinculeanu ( N. ) [1] Vector Measures ( Pergamon Press ), London , 1967 . MR 206190
[9] Doob ( J.L. ) [1] Stochastic Processes . ( Wiley ), New York , 1953 . MR 58896 | Zbl 0053.26802 · Zbl 0053.26802
[10] Dubreil-Jacotin ( M.L. ) [1] Sur le passage des équations de Navier-Stokes aux équations de Reynolds , C. R. Acad. Sci. Paris , 244 ( 1957 ), 2887 - 2890 . MR 94053 | Zbl 0078.08402 · Zbl 0078.08402
[11] Dunford ( N. ) - Schwartz ( J.T. ) [1] Linear Operators - Part. I. General Theory ( Inters. Publ .), New York , 1958 . MR 1009162 | Zbl 0084.10402 · Zbl 0084.10402
[12] Foiaş ( C. ) [1] Ergodic problems in functional spaces related to Navier-Stokes equations . Proceedings Intern. Conf. Funct. Anal. and Rel. Topics. ( Tokyo , April 1969 ), 290 - 304 . MR 291885 | Zbl 0206.44701 · Zbl 0206.44701
[13] Solutions statistiques des equations d’évolution non linéaire . C.I.M.E. Varenna , 1970 . Zbl 0241.35063 · Zbl 0241.35063
[14] Foiaş ( C. ) - Prodi ( G. ) [1] Sur le comportement global des solutions non-stationnaires des équations de Navier-Stokes en dimension 2 . Ren. Sem. Mat. Univ. Padova, 39 ( 1967 ), 1 - 34 . Numdam | MR 223716 | Zbl 0176.54103 · Zbl 0176.54103
[15] Grisvard ( P. ) [1] Le comportement asymptotique des valeurs propres d’un opérateur. Séminaire sur les Equations aux dérivées partielles . ( Collège de France , 1971 ), IV , pp. 4.1-4. 12 . Numdam
[16] Hille ( E. ) - Phillips ( R.S. ) [1] Functional Analysis and Semi-groups ( Amer. Math. Soc. Coll. Publ .), New York , 1957 . MR 89373 | Zbl 0078.10004 · Zbl 0078.10004
[17] Hinze ( J.O. ) [1] Turbulence. An introduction and its Mechanism and Theory ( McGraw-Hill, Inc. ), New York , 1959 . MR 105962
[18] Hopf ( E. ) [1] A mathematical example displaying features of turbulence . Comm. Pure Appl. Math. , 1 ( 1948 ), 303 - 322 . MR 30113 | Zbl 0031.32901 · Zbl 0031.32901 · doi:10.1002/cpa.3160010401
[19] Uber die Anfangswertaufgabe für hydrodynamischen Grundgleichungen . Math. Nachrichten , 4 ( 1951 ), 2 - 13 - 251 . Zbl 0042.10604 · Zbl 0042.10604 · doi:10.1002/mana.3210040121
[20] Statistical hydrodynamics and functional calculus . J. Rat. Mech. Anal. , 1 ( 1952 ), 87 - 123 . MR 59119 · Zbl 0049.41704
[21] Hörmander ( L. ) [1] Linear Partial differential Operators ( Springer Verl .), Berlin , 1963 . MR 404822
[22] Hunt ( J.N. ) [1] Incompressible Fluid mechanics ( Wiley ), New York , 1964 . MR 207282
[23] Ionescu Tulcea ( A. ) and (C. T.) [1] On the lifting property II. journ . Math. Mech. , 11 ( 1962 ), 777 - 795 . Zbl 0122.11701 · Zbl 0122.11701
[24] Jacobs ( K. ) [1] Ergodic Theory (Aarhus Univ. Mat. Inst. Lect.) , Aarhus , 1963 . MR 159922
[25] Kampé De Fériet ( J. ) [1] Statistical mechanics of continuous media . Proc. Symp. Appl. Math. , XIII ( 1962 ). MR 141260 | Zbl 0108.42802 · Zbl 0108.42802
[26] Kolmogorov ( A.N. ) - TIHOMIROV (V. M.) [1] \epsilon -entropy and \epsilon -capacity of sets in functional spaces, Uspehi Mat . Nauk , 14 ( 1959 ), 3 - 86 . Zbl 0133.06703 · Zbl 0133.06703
[27] Krlyov ( N. ) - BOGOLIUBOV (N. N.) [1] La théorie générale de la mesure dans son application a l’étude des systèmes dynamiques de la mécanique non linéaire . Ann. of Math. , 38 ( 1937 ), 65 - 113 . MR 1503326 | Zbl 0016.08604 | JFM 63.1002.01 · Zbl 0016.08604 · doi:10.2307/1968511
[28] Kuratowski ( C. ) [1] Topologie , vol. I ( Panstw. Wyd. Nauk ), Warszawa , 1958 . MR 90795 | Zbl 0102.37602 · Zbl 0102.37602
[29] Ladyzenskaya ( O.A. ) [1] Global solution of the boundary problem for Navier-Stokes equations in two spatial dimensions . Doklady Acad. Nauk SSSR , 123 ( 1958 ), 427 - 429 . Zbl 0090.41502 · Zbl 0090.41502
[30] The study of Navier-Stokes equations for stationary incompessible flows . Usp. Mat. Nauk. , 14 ( 1959 ), 75 - 97 .
[31] Mathematical Theory of viscous incompressible flows . ( Gordon & Breach Sci. Publ .), New York , 1963 . MR 155093
[32] Landau ( L. ) - Lifshitz ( E. ) [1] Mécanique des fluides, Physique Théorique , t. VI . (Ed. Mir ), Moscow , 1971 . · Zbl 0216.25801
[33] Leray ( J. ) [1] Etude de divers équations intégrales non-linéaire et de quelques problèmes que posent l’hydrodynamique . J. Math. Pures et Appl. , 9 e série, 12 ( 1933 ), 1 - 82 . Article | Zbl 0006.16702 · Zbl 0006.16702
[34] Essai sur le mouvement plan d’un liquide visqueux que limitent des parois . Journ. Math. Pures et Appl. , 9 e série, 13 ( 1934 ), 331 - 418 . JFM 60.0727.01 · JFM 60.0727.01
[35] Sur le mouvement d’un liquide visqueux emplissant l’espace . Acta Math. , 63 ( 1934 ), 193 - 248 . MR 1555394 | JFM 60.0726.05 · JFM 60.0726.05
[36] Lions ( J.L. ) [1] Quelques méthodes de résolution des problèmes aux limites non linéaires ( Dunod-Gauth. Vill .), Paris , 1969 . MR 259693 | Zbl 0189.40603 · Zbl 0189.40603
[37] Lions ( J.L. ) - Magenes ( E. ) [1] Problèmes aux limites non homogènes et applications , vol. I ( Dunod ), Paris , 1968 . Zbl 0165.10801 · Zbl 0165.10801
[38] Lions ( J.L. ) - PEETRE (J.) [1] Sur une classe d’espaces d’interpolation . Institut des Hautes Etudes Sc. Publ. Math. , 19 ( 1964 ), 5 - 68 . Numdam | MR 165343 | Zbl 0148.11403 · Zbl 0148.11403 · doi:10.1007/BF02684796
[39] Lions ( J.L. ) - PRODI (G.) [1] Un théorème d’existence et unicité dans les équations de Navier-Stokes en dimension 2 , C. R. Acad. Sci. Paris , 250 ( 1959 ), 3519 - 3521 . MR 108964 | Zbl 0091.42105 · Zbl 0091.42105
[40] Loève ( M. ) [1] Probability Theory . ( Van Nostrand Corp. Inc. ), Princeton , 1960 . MR 123342 | Zbl 0095.12201 · Zbl 0095.12201
[41] Loomis ( L.H. ) [1] An Introduction to Abstract Harmonic Analysis. ( Van Nostrand Comp., Inc. ), New York , 1953 . MR 54173 | Zbl 0052.11701 · Zbl 0052.11701
[42] Malkus ( W.V.R. ) [1] Similarity arguments for fully developed turbulence . Suppl. at Nuovo Cimento , 22 ( 1961 ), serie X , 376 - 384 .
[43] Masuda ( K. ) [1] On the analyticity and the unique continuation theorem for solutions of the Navier-Stokes equation . Proc. Japan. Acad. , 43 ( 1967 ), 827 - 832 . Article | MR 247304 | Zbl 0204.26901 · Zbl 0204.26901 · doi:10.3792/pja/1195521421
[44] Monin ( A.S. ) - Yaglom ( A.M. ) [1] Statistical hydrodynamics. The Mechanics of turbulence. Part I ( Izd. Nauka ), Moscow , 1965 .
[45] Statistical hydrodynamics. The Mechanics of turbulence. Part II ( Izd. Nauka ), Moscow , 1967 .
[46] Nelson ( E. ) [1] Regular Probability measures on function space . Annals of Math. , 69 ( 1959 ), 630 - 643 . MR 105743 | Zbl 0087.13102 · Zbl 0087.13102 · doi:10.2307/1970027
[47] Nihoul ( J.C.J. ) [1] A kinematic theory of M.H.D. turbulent shear flow (A new approach to the Malkus theory of turbulence) . Revue Roum. Math . p. et appl., 12 ( 1967 ), 1503 - 1514 . Zbl 0148.45901 · Zbl 0148.45901
[48] Obukov ( A.M. ) [1] Statistical description of continuous fields . Proc. Geod. Inst. Akad. Nauk SSSR , 1954 , no. 24 (151), 3 - 42 . MR 67404
[49] Prodi ( G. ) [1] Un teorema di unicità per le equazioni di Navier-Stokes . Annali di Mat. pura e appl. ( IV ), 48 ( 1959 ), 173 - 182 . MR 126088 | Zbl 0148.08202 · Zbl 0148.08202 · doi:10.1007/BF02410664
[50] Qualche risultato riguardo alle equazioni di Navier-Stokes nel caso bidimensionale . Rend. Sem. Mat. Univ. Padova , 30 ( 1960 ), 16 - 23 . Numdam | MR 115017 | Zbl 0098.17204 · Zbl 0098.17204
[51] Teoremi ergodici per le equazioni della idrodinamica , C.I.M.E. , Roma , 1960 . Zbl 0117.10504 · Zbl 0117.10504
[52] On probability measures related to the Navier-Stokes equations in the 3-dimensional case (Air Force Res. Div. Contr. A.P.61(052)-414. Technical Note nr. 2 ( 1961 )), Trieste , 1961 .
[53] Résultats récents dans la théorie des équations de Navier Stokes. Les équations aux dérivées partielles ( Colloques Intern. CNRS ), Paris , 1962 ’. Zbl 0255.35076 · Zbl 0255.35076
[54] Teoremi di tipo locale per il sistema di Navier-Stokes e stabilità delle soluzioni stazionarie . Rend. Sem. Mat. Univ. Padova , 32 ( 1962 ), 374 - 397 . Numdam | MR 189354 | Zbl 0108.28602 · Zbl 0108.28602
[55] Reynolds ( O. ). [1] On the dynamical theory of incompressible viscous fluids and the determination of the criterion . Phil. Trans. Royal Soc. London , 186 ( 1894 ), 123 - 161 . JFM 26.0872.02 · JFM 26.0872.02
[56] Rosen ( G. ) [1] Functional Integration theory for incompressible fluid turbulence , The Phys. of Fluids , 10 ( 1967 ), 2614 - 2619 . Zbl 0204.28505 · Zbl 0204.28505 · doi:10.1063/1.1762083
[57] Riesz ( F. ) - Sz.-Nagy ( B. ) [1] Leçons d’analyse fonctionnelle ( Gauthier-Vill.-Akad.-Kiado ). Zbl 0064.35404 · Zbl 0064.35404
[58] Rudin ( W. ) [1] Fourier Analysis on Groups . ( Inters. Publ .), New York , 1962 . MR 152834 | Zbl 0107.09603 · Zbl 0107.09603
[59] Sansone ( G. ) [1] Equazioni differenziali nel campo reale. Parte I. ( Sc. Edizione ), Bologna , 1948 . JFM 67.0306.01 · JFM 67.0306.01
[60] Schlichting ( H. ) [1] Boundary Layer Theory ( 4 th edn. McGraw-Hill ), New York , 1960 . MR 76530 | Zbl 0096.20105 · Zbl 0096.20105
[61] Serrini ( J. ) [1] Mathematical Principle of Classical Fluid Mechanics (Handbuch der Physik , Band VIII / 1 , Stromungs Mechanik I ), Berlin , 1959 . MR 108116
[62] Vo-Khan ( K. ) [1] Etude des fonctions quasi-stationnaires et de leurs applications aux équations différentielles opérationnelles . Bull. Soc. Math. France , suppl. au no. de Juin 1966 , mémoire 6. Numdam | MR 198302 | Zbl 0165.49504 · Zbl 0165.49504
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.