Navier and Stokes meet Poincaré and Dulac. (English) Zbl 1457.35024

This nice survey discuss asymptotic behavior of solutions of the Navier-Stokes system with potential forces either with space periodic conditions or in bounded domains, beginning with lower bounds on solutions and the limit of the ratio of the enstrophy over the energy, and culminating in the normal form of those equations in the sense generalizing Poincaré and Dulac theory of normal forms of (ordinary) differential equations. On the top of the announcements (and sometimes sketches or ideas of the proofs) of results, their clear geometric interpretations (invariant manifolds etc.) and historical miscellanea are given. Moreover, open problems are listed, and an extensive set of references is provided.


35Q30 Navier-Stokes equations
76D05 Navier-Stokes equations for incompressible viscous fluids
37G05 Normal forms for dynamical systems
37L10 Normal forms, center manifold theory, bifurcation theory for infinite-dimensional dissipative dynamical systems
35C20 Asymptotic expansions of solutions to PDEs
Full Text: DOI arXiv


[1] V. I. Arnold,Geometrical methods in the theory of ordinary differential equations, second edition, Grundlehren Math. Wiss. (Fundamental Principles of Mathematical Sciences), Springer-Verlag, 1988, 250.
[2] C. Bardos and L. Tartar,Sur l’unicit´e r´etrograde des ´equations paraboliques et quelques questions voisines, Arch. Rational Mech. Anal., 1973, 50, 10-25. · Zbl 0258.35039
[3] V. G. Bondarevsky,A method of finding large sets of data generating global solutions to nonlinear equations: applications to the Navier-Stokes equation, C. R. Acad. Sci. Paris S´er. I Math., 1996, 322(4), 333-338. · Zbl 0846.35099
[4] L. Brandolese,Asymptotic behavior of the energy and pointwise estimates for the solutions to the Navier-Stokes equations, Rev. Mat. Iberoamericana, 2004, 20(1), 223-256. · Zbl 1057.35026
[5] L. Brandolese,Space-time decay of Navier-Stokes flows invariant under rotations, Math.Ann., 2004, 329(4), 685-706. · Zbl 1080.35062
[6] A. D. Bruno,Normal forms of differential equations, Soviet Math. Dokl., 1964, 5, 1105-1108. · Zbl 0145.10202
[7] A. D. Bruno,Analytical form of differential equations (I, II), Trans. Moscow Math. Soc., 1971, 25, 131-288, 1972, 26, 199-239.
[8] A. D. Bruno,Local methods in nonlinear differential equations, Springer-Verlag: Berlin-Heidelberg-New York-London-Paris-Tokyo, 1989.
[9] A. D. Bruno,Power Geometry in Algebraic and Differential Equations, Elsevier Science (North-Holland), Amsterdam, 2000.
[10] X. Cabr´e, E. Fontich and R. D. L. Llave,The parametrization method for invariant manifolds III: overview and applications, J. Diff. Equations, 2005, 218, 444-515. · Zbl 1101.37019
[11] M. Cannone, Y. Meyer and F. Planchon,Solutions auto-similaires des ´equations de Navier-Stokes, S´eminaire “Equations aux D´eriv´ees Partielles” de l’Ecole polytechnique, Expos´e VIII, 1993, (2), 209-216.
[12] D. Cao and L. Hoang,Long-time asymptotic expansions for Navier-Stokes equations with power-decaying forces,submitted, preprint. https://arxiv.org/abs/1803.05502. · Zbl 1439.35366
[13] L. Cattabriga,Su un problema al contorno relativo al sistema di equazioni di Stokes, Rendiconti del Seminario Matematico della Universit‘a di Padova, 1961, 31, 308-340. · Zbl 0116.18002
[14] J. Y. Chemin and I. Gallagher,On the global well-posedness of the 3-D NavierStokes equations with large initial data, Annales Scientifiques de l’Ecole Normale Sup´erieure de Paris, 2006, 39(4), 679-698. · Zbl 1124.35052
[15] J. Y. Chemin and I. Gallagher,Well-posedness and stability results for the Navier-Stokes equations inR3, Ann. Inst. H. Poincar´e Anal. Non Lin´eaire,
[16] Y. Chitour, D. Kateb and R. Long,Generic properties of the spectrum of the Stokes operator with Dirichlet boundary conditions inR3, Ann. Inst. H. Poincar´e Anal. Non Lin´eaire, 2016, 33, 119-167. · Zbl 1335.35163
[17] P. Constantin and C. Foias,Navier-Stokes equations, University of Chicago Press, 1988. · Zbl 0687.35071
[18] P. Constantin, C. Foias, I. Kukavica and A. J. Majda,Dirichlet quotients and 2D periodic Navier-Stokes equations, J. Math. Pures et Appl., 1997, 76, 125- 153. · Zbl 0874.35085
[19] O. Darrigol,Worlds of flow. A history of hydrodynamics from the Bernoullis to Prandtl, Oxford University Press, 2005. · Zbl 1094.76002
[20] H. Dulac,Solutions d’un syst‘eme d’´equations diff´erentielles dans le voisinage des valeurs singuli‘eres, Bull. Soc. Math. France, 1912, 40, 324-383.
[21] R. H. Dyer and D. E. Edmunds,Lower bounds for solutions of the Navier-Stokes equations, Proc. London Math. Soc., 1968, 18(3), 169-178. · Zbl 0157.57005
[22] C. Foias,Solutions statistiques des ´equations de Navier-Stokes, mimeographed notes, Cours au Coll‘ege de France, 1974.
[23] C. Foias, L. Hoang and B. Nicolaenko,On the helicity in 3D-periodic NavierStokes equations I. The non-statistical case, Proc. Lond. Math. Soc., 2007, 94(1), 53-90. · Zbl 1109.76015
[24] C. Foias, L. Hoang and B. Nicolaenko,On the helicity in 3D-periodic NavierStokes equations II. The statistical case, Comm. Math. Phys., 2009, 290(2), 679-717. · Zbl 1184.35239
[25] C. Foias, L. Hoang, E. Olson and M. Ziane,On the solutions to the normal form of the Navier-Stokes equations, Indiana Univ. Math. J., 2006, 55(2), 631-686. · Zbl 1246.76019
[26] C. Foias, L. Hoang, E. Olson and M. Ziane,The normal form of the NavierStokes equations in suitable normed spaces, Ann. Inst. H. Poincar´e Anal. Non Lin´eaire, 2009, 26(5), 1635-1673. · Zbl 1179.35212
[27] C. Foias, L. Hoang and J. C. Saut,Asymptotic integration of Navier-Stokes equations with potential forces. II. An explicit Poincar´e-Dulac normal form, J. Funct. Anal., 2011, 260(10), 3007-3035. · Zbl 1232.35115
[28] C. Foias, O. Manley, R. Rosa and R. Temam,Navier-Stokes equations and turbulence, Encyclopedia of Mathematics and its Applications, Cambridge University Press, 2001. · Zbl 0994.35002
[29] C. Foias, C. F. Mondaini, and E. S. Titi,A procedure of automatic reducing errors of measurements, in preparation.
[30] C. Foias and J. C. Saut,Limite du rapport de l’enstrophie sur l’´energie pour une solution des ´equations de Navier-Stokes, C. R. Acad. Sci. Paris S´er. I Math., 1981, 298, 241-244. · Zbl 0492.35063
[31] C. Foias and J. C. Saut,Asymptotic behavior ast→+∞of solutions of Navier-Stokes equations and nonlinear spectral manifolds, Indiana Univ. Math. J., 1984, 33(3), 459-477. · Zbl 0565.35087
[32] C. Foias and J. C. Saut,On the smoothness of nonlinear spectral manifolds of Navier-Stokes equations, Indiana Univ. Math. J., 1984, 33(6), 911-926. · Zbl 0572.35081
[33] C. Foias and J. C. Saut,Vari´et´es invariantes ‘a d´ecroissance lente pour les ´equations de Navier-Stokes avec forces potentielles, C. R. Acad. Sci. Paris S´er. I Math., 1986, 302, 563-566. · Zbl 0606.35064
[34] C. Foias and J. C. Saut,Linearization and normal form of the Navier-Stokes equations with potential forces, Ann. Inst. H. Poincar´e Anal. Non Lin´eaire, 1987, 4(1), 1-47. · Zbl 0635.35075
[35] C. Foias and J. C. Saut,Asymptotic integration of Navier-Stokes equations with potential forces. I., Indiana Univ. Math. J., 1991, 40(1), 305-320. · Zbl 0739.35066
[36] C. Foias and R. Temam,Gevrey class regularity for the solutions of the NavierStokes equations, J. Funct. Anal., 1989, 87(2), 359-369. · Zbl 0702.35203
[37] T. Gallay and S. Slijepcevic,Energy bounds for the two-dimensional NavierStokes equations in an infinite cylinder, Comm. in PDE, 2014, 39, 1741-1769. · Zbl 1304.35489
[38] T. Gallay and C. E. Wayne,Long-time asymptotics of the Navier-Stokes and vorticity equations onR3, R. Soc. Lond. Philos. Trans. Ser. A Math. Phys. · Zbl 1042.37058
[39] J. M. Ghidaglia,Long time behavior of solutions to abstract inequalities, application to thermohydraulic and MHD equations, J. Diff. Equations, 1986, 61(2), 268-294. · Zbl 0549.35102
[40] J. M. Ghidaglia and A. Marzocchi,Exact decay estimates for solutions to semilinear parabolic equations, Applicable analysis, 1991, 42, 69-81. · Zbl 0724.35015
[41] M. Ghisi, M. Gobbino and A. Haraux,A description of all possible decay rates for solutions of some semilinear parabolic equations, J. Math. Pures Appl., · Zbl 1403.35142
[42] H. Gispert,La France math´ematique. La Soci´et´e Math´ematique de France, Cahiers d’Histoire et de Philosophie des Sciences, 1991, 34, 1870-1914.
[43] C. Guillop´e,Remarques ‘a propos du comportement lorsquet→+∞des solutions des ´equations de Navier-Stokes associ´ees ‘a une force nulle, Bull. Soc. Math. France, 1983, 111(2), 151-180.
[44] P. Hartman,Ordinary differential equations, 2nd ed. Birka¨user, 1982. · Zbl 0476.34002
[45] L. T. Hoang and V. R. Martinez,Asymptotic expansion in Gevrey spaces for solutions of Navier-Stokes equations, Asymptot. Anal., 2017, 104(3-4), 167- 190. · Zbl 1375.35317
[46] L. T. Hoang and V. R. Martinez,Asymptotic expansion for solutions of the Navier-Stokes equations with non-potential body forces, J. Math. Anal. Appl., · Zbl 1394.35130
[47] I. Kukavica,Level sets of the vorticity and the stream function for the 2D periodic Navier-Stokes equations with potential forces, J. Diff. Equations, 1996, 126, 374-388. · Zbl 0847.35103
[48] I. Kukavica and E. Reis,Asymptotic expansion for solutions of the NavierStokes equations with potential forces, J. Diff. Equations, 2011, 250(1), 607- 622. · Zbl 1205.35200
[49] O. A. Ladyzhenskaya,The mathematical theory of viscous incompressible flow, Gordon and Breach Science Publishers, New York, 2nd English edn, revised and enlarged. Translated from the Russian by R. A. Silverman and J. Chu, Mathematics and Its Applications, 1969, 2. · Zbl 0184.52603
[50] J. Leray,Etude de diverses ´equations int´egrales non lin´eaires et de quelques probl‘emes que pose l’hydrodynamique, J. Math. Pures Appl., 1933, 12, 1-82. · Zbl 0006.16702
[51] J. Leray,Essai sur les mouvements plans d’un liquide visqueux que limitent des parois, J. Math. Pures Appl., 1934, 13, 331-418. · JFM 60.0727.01
[52] J. Leray,Essai sur le mouvement d’un liquide visqueux emplissant l’espace, Acta Math., 1934, 63, 193-248. · JFM 60.0726.05
[53] J. L. Lions,Quelques M´ethodes de Resolution des Probl‘emes aux Limites Non Lin´eaires, Dunod, Gouthier-Villars, Paris, 1969.
[54] T. Ma and S. Wang,Periodic structure of 2-D Navier-Stokes equations, J. Nonlinear Sci., 2005, 15, 133-158. · Zbl 1089.35044
[55] A. Majda, S. Y. Shim and X. Wang,Selective decay for geophysical flows, Methods and Applications of Analysis, 2000, 7(3), 551-554. · Zbl 1197.76140
[56] A. Majda and X. Wang,The selective decay principle for barotropic geophysical flows, Methods and Applications of Analysis, 2001, 8, 579-594. · Zbl 1021.76054
[57] A. Majda and X. Wang,Nonlinear Dynamics and Statistical Theories for Basic Geophysical Flows, Cambridge University Press, 2006. · Zbl 1141.86001
[58] T. Mikyakawa and M. E. Schonbek,On optimal decay rates for weak solutions to the Navier-Stokes equations, Math. Bohem., 2001, 126, 443-455. · Zbl 0981.35048
[59] G. Minea,Investigation of the Foias-Saut normalization in the finitedimensional case, J. Dynam. Differential Equations, 1998, 10(1), 189-207. · Zbl 0970.34045
[60] H. K. Moffatt,Some developments in theory of turbulence, J. Fluid Mech., 1981, 173, 303-356.
[61] H. K. Moffatt and A. Tsinober,Helicity in laminar and turbulent flow, Annual Rev. Fluid Mech., 1992, 24, 281-312. · Zbl 0751.76018
[62] J. J. Moreau,Constantes d’un ˆılot tourbillonnaire en fluide parfait barotrope, C. R. Acad. Sci. Paris, 1961, 252, 2810-2812. · Zbl 0151.41703
[63] M. Oliver and E. S. Titi,Remark on the rate of decay of higher derivatives of solutions to the Navier-Stokes equations inRn, J. Funct. Anal., 2000, 172, 1-18. · Zbl 0960.35081
[64] H. Poincar´e,Th‘ese Paris, 1879, reprinted in Oeuvres de Henri Poincar´e, Vol. I, Gauthier-Villars, Paris, 1928.
[65] H. Poincar´e,Les m´ethodes nouvelles de la M´ecanique C´eleste, Gauthier-Villars, Paris, 1889.
[66] G. Ponce, R. Racke, T. C. Sideris and E. S. Titi,Global stability of large solutions to the 3D Navier-Stokes equations, Comm. Math. Phys., 1994, 159, 329-341. · Zbl 0795.35082
[67] M. E. Schonbek,Large time behavior of solutions to the Navier-Stokes equations, Comm. in PDE, 1986, 11, 733-763. · Zbl 0607.35071
[68] J. Shatah,Normal forms and quadratic nonlinear Klein-Gordon equations, Comm. Pure Appl. Math., 1985, 38(5), 685-696. · Zbl 0597.35101
[69] Y. Shi,A Foias-Saut type of expansion for dissipative wave equations, Comm. in PDE, 2000, 25(11-12), 2287-2331. · Zbl 0963.35123
[70] R. Temam,Navier-Stokes Equations, AMS Chelsea Publishing, Providence, RI, 2001, Theory and numerical analysis, Reprint of the 1984 edition. · Zbl 0981.35001
[71] R. Temam,Navier-Stokes Equations and Nonlinear Functional Analysis, Second Edition, CBMS-NSF Regional Conference Series in Applied Mathematics, Society for Industrial and Applied Mathematics, 1995.
[72] M. Wiegner,Decay and stability inLpfor strong solutions of the Cauchy problem for the Navier-Stokes equations, in Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1990, 1431, 95-99.
[73] M. Q. Zhan,Selective decay principle for 2D magnetohydrodynamic flows, Asymptot. Anal., 2010, 67(34), 125-146. · Zbl 1200.35221
[74] M. Q. Zhan,Convergence of Dirichlet quotients and selective decay of 2D magnetohydrodynamic flows, J. Math. Anal. Appl., 2011, 380, 831-846. · Zbl 1216.35093
[75] Q.
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