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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
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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 · numdam:RSMUP_1961__31__308_0 · eudml:107065
[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 · numdam:RSMUP_1967__39__1_0 · eudml:107242
[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 · numdam:SJL_1971____A4_0
[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 · www.ams.org
[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 · www.emis.de
[28] Kuratowski ( C. ) [1] Topologie , vol. I ( Panstw. Wyd. Nauk ), Warszawa , 1958 . MR 90795 | Zbl 0102.37602 · Zbl 0102.37602 · eudml:219314
[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 · eudml:235182
[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 · numdam:PMIHES_1964__19__5_0 · eudml:103841
[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 · minidml.mathdoc.fr
[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 · numdam:RSMUP_1960__30__1_0 · eudml:107030
[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 · numdam:RSMUP_1962__32__374_0 · eudml:107089
[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 · numdam:MSMF_1966__6__3_0 · eudml:94544
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