Skip to main content
SpringerLink
Log in

Critical Function Spaces for the Well-Posedness of the Navier-Stokes Initial Value Problem

  • Reference work entry
  • First Online:
Handbook of Mathematical Analysis in Mechanics of Viscous Fluids

Abstract

In this paper the homogeneous, incompressible Navier-Stokes equations are considered, and a number of results are reviewed which are related to the scaling of the equations. More specifically the initial value problem is studied in scale-invariant function spaces, insisting on the special role of the “largest” scale-invariant function space; the specificity of two space dimensions is recalled, in terms of the velocity field and the vorticity. Some examples of arbitrarily large initial data giving rise to a global solution are also provided, as well as a study of the long-time behavior of global solutions and their behavior at blow-up time (supposing such a time exists).

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 1,799.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 2,499.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. P. Auscher, S. Dubois, P. Tchamitchian, On the stability of global solutions to Navier-Stokes equations in the space. J. Math. Pures Appl. (9) 83, 673–697 (2004)

    Google Scholar 

  2. H. Bae, A. Biswas, E. Tadmor, Analyticity and decay estimates of the Navier-Stokes equations in critical Besov spaces. Arch. Ration. Mech. Anal. 205, 963–991 (2012)

    Article  MathSciNet  Google Scholar 

  3. H. Bahouri, J.-Y. Chemin, R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations. Grundlehren der mathematischen Wissenschaften (Springer, Heidelberg/New York, 2011)

    Google Scholar 

  4. H. Bahouri, A. Cohen, G. Koch, A general wavelet-based profile decomposition in the critical embedding of function spaces. Confluentes Mathematici 3, 1–25 (2011)

    Article  MathSciNet  Google Scholar 

  5. H. Bahouri, P. Gérard, High frequency approximation of solutions to critical nonlinear wave equations. Am. J. Math. 121, 131–175 (1999)

    Article  MathSciNet  Google Scholar 

  6. H. Beirao da Veiga, On a family of results concerning direction of vorticity and regularity for the Navier-Stokes equations. Ann. Univ. Ferrara Sez. VII Sci. Mat. 60(1), 23–34 (2014)

    Article  MathSciNet  Google Scholar 

  7. J. Bourgain, N. Pavlović, Ill-posedness of the Navier-Stokes equations in a critical space in 3D. J. Funct. Anal., 255, 2233–2247 (2008)

    Article  MathSciNet  Google Scholar 

  8. L. Brandolese, M. Schonbek, Large time decay and growth for solutions of a viscous Boussinesq system. Trans. Am. Math. Soc. 364(10), 5057–5090 (2012)

    Article  MathSciNet  Google Scholar 

  9. H. Brézis, J.-M. Coron, Convergence of solutions of H-Systems or how to blow bubbles. Arch. Ration. Mech. Anal. 89, 21–86 (1985)

    Article  MathSciNet  Google Scholar 

  10. L. Caffarelli, R. Kohn, L. Nirenberg, Partial regularity of suitable weak solutions to the Navier-Stokes equations. Commun. Pure Appl. Math. 35, 771–831 (1982)

    Article  MathSciNet  Google Scholar 

  11. C. Calderón, Existence of weak solutions for the Navier-Stokes equations with initial data in Lp. Trans. Am. Math. Soc. 318, 179–200 (1990)

    MATH  Google Scholar 

  12. M. Cannone, Ondelettes, paraproduits et Navier–Stokes (Diderot éditeur, Arts et Sciences, 1995)

    MATH  Google Scholar 

  13. M. Cannone, A generalization of a theorem by Kato on Navier-Stokes equations. Rev. Mat. Iberoamericana 13(3), 515–541 (1997)

    Article  MathSciNet  Google Scholar 

  14. C. Cao, E.S. Titi, Global regularity criterion for the 3D Navier-Stokes equations involving one entry of the velocity gradient tensor. Arch. Ration. Mech. Anal. 202, 919–932 (2011)

    Article  MathSciNet  Google Scholar 

  15. J.-Y. Chemin, Le système de Navier-Stokes incompressible soixante-dix ans après J. Leray. Séminaires et Congrès 9, 99–123 (2004)

    MATH  Google Scholar 

  16. J.-Y. Chemin, About weak-strong uniqueness for the 3D incompressible Navier-Stokes system. Commun. Pure Appl. Math. 64(12), 1587–1598 (2011)

    Article  MathSciNet  Google Scholar 

  17. J.-Y. Chemin, I. Gallagher, On the global wellposedness of the 3-D Navier-Stokes equations with large initial data. Annales de l’École Normale Supérieure 39, 679–698 (2006)

    Article  MathSciNet  Google Scholar 

  18. J.-Y. Chemin, I. Gallagher, Wellposedness and stability results for the Navier-Stokes equations in R3. Annales de l’Institut H. Poincaré, Analyse non linéaire 26(2), 599–624 (2009)

    Google Scholar 

  19. J.-Y. Chemin, I. Gallagher, Large, global solutions to the Navier-Stokes equations, slowly varying in one direction. Trans. Am. Math. Soc. 362(6), 2859–2873 (2010)

    Article  MathSciNet  Google Scholar 

  20. J.-Y. Chemin, I. Gallagher, M. Paicu, Global regularity for some classes of large solutions to the Navier-Stokes equations. Ann. Math. 173(2), 983–1012 (2011)

    Article  MathSciNet  Google Scholar 

  21. J.-Y. Chemin, I. Gallagher, P. Zhang, Sums of large global solutions to the incompressible Navier-Stokes equations. Journal für die reine und angewandte Mathematik 681, 65–82 (2013)

    MathSciNet  MATH  Google Scholar 

  22. J.-Y. Chemin, F. Planchon, Self-improving bounds for the Navier-Stokes equations. Bull. Soc. Math. Fr. 140(4), 583–597 (2012/2013)

    Article  MathSciNet  Google Scholar 

  23. J.-Y. Chemin, P. Zhang, On the critical one component regularity for 3-D Navier-Stokes system. Annales de l’École Normale Supérieure (to appear)

    Google Scholar 

  24. P. Constantin, C. Fefferman, Direction of vorticity and the problem of global regularity for the Navier-Stokes equations. Indiana Univ. Math. J. 42, 775–789 (1993)

    Article  MathSciNet  Google Scholar 

  25. P. Constantin, C. Foias, Navier-Stokes Equations (Chicago University Press, Chicago, 1988)

    MATH  Google Scholar 

  26. G.-H. Cottet, Equations de Navier-Stokes dans le plan avec tourbillon initial mesure. C. R. Acad. Sci. Paris Sér. I Math. 303, 105–108 (1986)

    MathSciNet  MATH  Google Scholar 

  27. L. Escauriaza, G.A. Seregin, V. Šverák, L3,∞-solutions of Navier-Stokes equations and backward uniqueness. Uspekhi Mat. Nauk 58(2(350)), 3–44 (2003)

    Google Scholar 

  28. R. Farwig, H. Sohr, Optimal initial value conditions for the existence of local strong solutions of the Navier-Stokes equations. Math. Ann. 345(3), 631–642 (2009)

    Article  MathSciNet  Google Scholar 

  29. R. Farwig, H. Sohr, W. Varnhorn, On optimal initial value conditions for local strong solutions of the Navier-Stokes equations. Ann. Univ. Ferrara Sez. VII Sci. Mat. 55(1), 89–110 (2009)

    Article  MathSciNet  Google Scholar 

  30. R. Farwig, H. Sohr, W. Varnhorn, Extensions of Serrin’s uniqueness and regularity conditions for the Navier-Stokes equations. J. Math. Fluid Mech. 14(3), 529–540 (2012)

    Article  MathSciNet  Google Scholar 

  31. H. Fujita, T. Kato, On the Navier-Stokes initial value problem I. Arch. Ration. Mech. Anal. 16, 269–315 (1964)

    Article  MathSciNet  Google Scholar 

  32. I. Gallagher, Profile decomposition for solutions of the Navier-Stokes equations. Bull. Soc. Math. Fr. 129(2), 285–316 (2001)

    Article  MathSciNet  Google Scholar 

  33. I. Gallagher, Th. Gallay, Uniqueness of solutions of the Navier-Stokes equation in R2 with measure-valued initial vorticity. Mathematische Annalen 332, 287–327 (2005)

    Article  MathSciNet  Google Scholar 

  34. I. Gallagher, Th. Gallay, P.-L. Lions, On the uniqueness of the solution of the two-dimensional Navier-Stokes equation with a Dirac mass as initial vorticity. Math. Nachrichten 278, 1665–1672 (2005)

    Article  MathSciNet  Google Scholar 

  35. I. Gallagher, D. Iftimie, F. Planchon, Non explosion en temps grand et stabilite de solutions globales des equations de Navier–Stokes. Notes aux Comptes-Rendus de l’Academie des Sciences de Paris 334, Serie I, 289–292 (2002)

    Article  MathSciNet  Google Scholar 

  36. I. Gallagher, D. Iftimie, F. Planchon, Asymptotics and stability for global solutions to the Navier-Stokes equations. Annales de l’Institut Fourier 53, 1387–1424 (2003)

    Article  MathSciNet  Google Scholar 

  37. I. Gallagher, G.S. Koch, F. Planchon, A profile decomposition approach to the L t ∞(L x 3) Navier-Stokes regularity criterion. Math. Ann. 355(4), 1527–1559 (2013)

    Article  MathSciNet  Google Scholar 

  38. I. Gallagher, G.S. Koch, F. Planchon, Blow-up of critical Besov norms at a potential Navier-Stokes singularity. Comm. Math. Phys. (to accepted for publication)

    Google Scholar 

  39. I. Gallagher, M. Paicu, Remarks on the blow up of solutions to a toy model for the Navier-Stokes equations. Proc. Am. Math. Soc. 137(6), 2075–2083 (2009)

    Article  MathSciNet  Google Scholar 

  40. I. Gallagher, F. Planchon, On global infinite energy solutions to the Navier-Stokes equations in two dimensions. Arch. Ration. Mech. Anal. 161, 307–337 (2002)

    Article  MathSciNet  Google Scholar 

  41. Th. Gallay, C.E. Wayne, Global stability of vortex solutions of the two-dimensional Navier-Stokes equation. Commun. Math. Phys. 255(1), 97–129 (2005)

    Article  MathSciNet  Google Scholar 

  42. G.P. Galdi, An introduction to the Navier-Stokes initial-boundary value problem, in Fundamental Directions in Mathematical Fluid Mechanics, ed. by G.P. Galdi, J.G. Heywood, R. Rannacher. Advances in Mathematical Fluid Mechanics (Birkhäuser, Basel, 2000), pp. 1–70

    Chapter  Google Scholar 

  43. P. Gérard, Microlocal defect measures. Commun. Partial Differ. Equ. 16, 1761–1794 (1991)

    Article  MathSciNet  Google Scholar 

  44. P. Gérard, Description du défaut de compacité de l’injection de Sobolev. ESAIM Control Optim. Calc. Var. 3, 213–233 (1998)

    Article  MathSciNet  Google Scholar 

  45. P. Germain, The second iterate for the Navier-Stokes equation. J. Funct. Anal. 255(9), 2248–2264 (2008)

    Article  MathSciNet  Google Scholar 

  46. P. Germain, Équations de Navier-Stokes dans \(\mathbb{R}^{2}\): existence et comportement asymptotique de solutions d’énergie infinie. Bull. Sci. Math. 130(2), 123–151 (2006)

    Article  MathSciNet  Google Scholar 

  47. P. Germain, Multipliers, paramultipliers, and weak-strong uniqueness for the Navier-Stokes equations. J. Differ. Equ. 226(2), 373–428 (2006)

    Article  MathSciNet  Google Scholar 

  48. P. Germain, N. Pavlovic, G. Staffilani, Regularity of solutions to the Navier-Stokes equations evolving from small data in BMO1. Int. Math. Res. Not. 21, 35 (2007)

    MATH  Google Scholar 

  49. Y. Giga, Solutions for semilinear parabolic equations in Lp and regularity of weak solutions of the Navier-Stokes system. J. Differ. Equ. 62(2), 186–212 (1986)

    Article  Google Scholar 

  50. Y. Giga, T. Miyakawa, Solutions in Lr of the Navier-Stokes initial value problem. Arch. Ration. Mech. Anal. 89, 267–281 (1985)

    Article  Google Scholar 

  51. Y. Giga, T. Miyakawa, H. Osada, Two-dimensional Navier-Stokes flow with measures as initial vorticity. Arch. Ration. Mech. Anal. 104, 223–250 (1988)

    Article  MathSciNet  Google Scholar 

  52. Z. Grujić, R. Guberović, Localization of analytic regularity criteria on the vorticity and balance between the vorticity magnitude and coherence of the vorticity direction in the 3D NSE. Commun. Math. Phys. 298(2), 407–418 (2010)

    Article  MathSciNet  Google Scholar 

  53. T. Hmidi, S. Keraani, Blowup theory for the critical nonlinear Schrödinger equations revisited. Int. Math. Res. Not. 46, 2815–2828 (2005)

    Article  Google Scholar 

  54. D. Iftimie, G. Raugel, G.R. Sell, Navier-Stokes equations in thin 3D domains with Navier boundary conditions. Indiana Univ. Math. J. 56, 1083–1156 (2007)

    Article  MathSciNet  Google Scholar 

  55. S. Jaffard, Analysis of the lack of compactness in the critical Sobolev embeddings. J. Funct. Anal. 161, 384–396 (1999)

    Article  MathSciNet  Google Scholar 

  56. H. Jia, V. Şverák, Minimal L3-initial data for potential Navier-Stokes singularities. SIAM J. Math. Anal. 45, 1448–1459 (2013)

    Article  MathSciNet  Google Scholar 

  57. T. Kato, The Navier-Stokes equation for an incompressible fluid in \(\mathbb{R}^{2}\) with a measure as the initial vorticity. Differ. Integral Equ. 7, 949–966 (1994)

    MathSciNet  MATH  Google Scholar 

  58. C. Kenig, G. Koch, An alternative approach to regularity for the Navier-Stokes equations in critical spaces. Ann. Inst. H. Poincaré Anal. Non Linéaire 28(2), 159–187 (2011)

    Article  MathSciNet  Google Scholar 

  59. C.E. Kenig, F. Merle, Global well-posedness, scattering and blow up for the energy critical focusing non-linear wave equation. Acta Mathematica 201, 147–212 (2008)

    Article  MathSciNet  Google Scholar 

  60. S. Keraani, On the defect of compactness for the Strichartz estimates of the Schrödinger equation. J. Differ. Equ. 175, 353–392 (2001)

    Article  Google Scholar 

  61. G. Koch, Profile decompositions for critical Lebesgue and Besov space embeddings. Indiana Univ. Math. J. 59, 1801–1830 (2010)

    Article  MathSciNet  Google Scholar 

  62. H. Koch, D. Tataru, Well-posedness for the Navier-Stokes equations. Adv. Math. 157(1), 22–35 (2001)

    Article  MathSciNet  Google Scholar 

  63. H. Kozono, H. Sohr, Remark on uniqueness of weak solutions to the Navier-Stokes equations. Analysis 16(3), 255–271 (1996)

    Article  MathSciNet  Google Scholar 

  64. I. Kukavica, M. Ziane, One component regularity for the Navier-Stokes equations. Nonlinearity 19, 453–469 (2006)

    Article  MathSciNet  Google Scholar 

  65. I. Kukavica, M. Ziane, Navier-Stokes equations with regularity in one direction. J. Math. Phys. 48(6), 065203 (2007). 10pp

    Article  MathSciNet  Google Scholar 

  66. O. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow, 2nd English edn., revised and enlarged. Mathematics and Its Applications, vol. 2 (Gordon and Breach, New York/London/Paris 1969), xviii+224pp.

    Google Scholar 

  67. P.-G. Lemarié-Rieusset, Recent Developments in the Navier-Stokes Problem. Chapman and Hall/CRC Research Notes in Mathematics, vol. 43 (Chapman & Hall/CRC, Boca Raton, 2002)

    Book  Google Scholar 

  68. J. Leray, Essai sur le mouvement d’un liquide visqueux emplissant l’espace. Acta Mathematica 63, 193–248 (1933)

    Article  Google Scholar 

  69. J. Leray, Étude de diverses équations intégrales non linéaires et de quelques problèmes que pose l’hydrodynamique. J. Math. Pures. Appl. 12, 1–82 (1933)

    MATH  Google Scholar 

  70. D. Li, Ya. Sinai, Blow ups of complex solutions of the 3d-Navier-Stokes system and renormalization group method. J. Eur. Math. Soc. 10(2), 267–313 (2008)

    Google Scholar 

  71. F. Merle, L. Vega, Compactness at blow up time for L2 solutions of the critical nonlinear Schrödinger equation in 2D. Int. Math. Res. Not. 8, 399–425 (1998)

    Article  Google Scholar 

  72. P.-L. Lions, The concentration-compactness principle in the calculus of variations. The limit case I. Revista Matematica Iberoamericana 1(1), 145–201 (1985)

    Article  MathSciNet  Google Scholar 

  73. P.-L. Lions, The concentration-compactness principle in the calculus of variations. The limit case II. Revista Matematica Iberoamericana 1(2), 45–121 (1985)

    Article  MathSciNet  Google Scholar 

  74. A. Mahalov, B. Nicolaenko, Global solvability of three-dimensional Navier-Stokes equations with uniformly high initial vorticity (Russian. Russian summary). Uspekhi Mat. Nauk 58, 79–110 (2003); translation in Russ. Math. Surv. 58, 287–318 (2003)

    Google Scholar 

  75. A. Mahalov, E. Titi, S. Leibovich, Invariant helical subspaces for the Navier-Stokes equations. Arch. Ration. Mech. Anal. 112, 193–222 (1990)

    Article  MathSciNet  Google Scholar 

  76. J. Málek, J. Nečas, M. Pokorný, M. Schonbek, On possible singular solutions to the Navier-Stokes equations. Math. Nachr. 199 97–114 (1999)

    Article  MathSciNet  Google Scholar 

  77. K. Masuda, Weak solutions of the Navier-Stokes equation. Tôhoku Math. J. 36 623–646 (1984)

    Article  MathSciNet  Google Scholar 

  78. Y. Meyer, Wavelets, Paraproducts and Navier-Stokes Equations. Current Developments in Mathematics (International Press, Boston, 1997)

    Google Scholar 

  79. S. Montgomery-Smith, Finite-time blow up for a Navier-Stokes like equation. Proc. Am. Math. Soc. 129(10), 3025–3029 (2001)

    Article  MathSciNet  Google Scholar 

  80. J. Nečas, M. Ruzicka, V. Šverák, On Leray’s self-similar solutions of the Navier-Stokes equations. Acta Math. 176(2), 283–294 (1996)

    Article  MathSciNet  Google Scholar 

  81. J. Neustupa, A. Novotny, P. Penel, An interior regularity of a weak solution to the Navier-Stokes equations in dependence on one component of velocity. Topics in mathematical uid mechanics, 163–183, Quad. Mat., 10, Dept. Math., Seconda Univ. Napoli, Caserta (2002)

    Google Scholar 

  82. M. Paicu, Z. Zhang, Global well-posedness for 3D Navier-Stokes equations with ill-prepared initial data. J. Inst. Math. Jussieu 13(2), 395–411 (2014)

    Article  MathSciNet  Google Scholar 

  83. P. Penel, M. Pokorny, Some new regularity criteria for the Navier-Stokes equations containing gradient of the velocity. Appl. Math.49, 483–493 (2004)

    Article  MathSciNet  Google Scholar 

  84. N.C. Phuc, The Navier-Stokes equations in nonendpoint borderline Lorentz spaces (2014). arXiv:1407.5129

    Google Scholar 

  85. F. Planchon, Asymptotic behavior of global solutions to the Navier-Stokes equations in R3. Rev. Mat. Iberoam. 14(1), 71–93 (1998)

    Article  Google Scholar 

  86. G. Ponce, R. Racke, T. Sideris, E. Titi, Global stability of large solutions to the 3D Navier-Stokes equations. Commun. Math. Phys. 159, 329–341 (1994)

    Article  Google Scholar 

  87. E. Poulon, Behaviour of Navier-Stokes solutions with data in \(\dot{H}^{s}\) with \(1/2 < s < \frac{3} {2}\). Bulletin de la Société Mathématique de France (accepted)

    Google Scholar 

  88. G. Raugel, G.R. Sell, Navier-Stokes equations on thin 3D domains. I. Global attractors and global regularity of solutions. J. Am. Math. Soc. 6, 503–568 (1993)

    MATH  Google Scholar 

  89. J. Robinson, W. Sadowski, On the dimension of the singular set of solutions to the Navier-Stokes equations. Commun. Math. Phys. 309, 497–506 (2012)

    Article  MathSciNet  Google Scholar 

  90. I. Schindler, K. Tintarev, An abstract version of the concentration compactness principle. Revista Mathematica Complutense 15, 417–436 (2002)

    MathSciNet  MATH  Google Scholar 

  91. W. Rusin, V. Šverák, Minimal initial data for potential Navier-Stokes singularities. J. Funct. Anal. 260(3), 879–891 (2011)

    Article  MathSciNet  Google Scholar 

  92. M.E. Schonbek, L2 decay for weak solutions of the Navier-Stokes equation. Arch. Ration. Mech. Anal. 88, 209–222 (1985)

    Article  Google Scholar 

  93. G. Seregin, A certain necessary condition of potential blow up for Navier-Stokes equations. Commun. Math. Phys. 312(3), 833–845 (2012)

    Article  MathSciNet  Google Scholar 

  94. Z. Skalák, P. Kučera, A note on coupling of velocity components in the Navier-Stokes equations. Zeitschrift für Angewandte Mathematik und Mechanik 84, 124–127 (2004)

    Article  MathSciNet  Google Scholar 

  95. H. Sohr, The Navier-Stokes Equations. An Elementary Functional Analytic Approach. Modern Birkhäuser Classics (Birkhäuser/Springer Basel AG, Basel, 2001), x+367 pp.

    Google Scholar 

  96. E. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals. Princeton Mathematical Series, vol. 43. Monographs in Harmonic Analysis, III (Princeton University Press, Princeton, 1993)

    Google Scholar 

  97. M. Struwe, A global compactness result for boundary value problems involving limiting nonlinearities. Mathematische Zeitschrift 187, 511–517 (1984)

    Article  MathSciNet  Google Scholar 

  98. L. Tartar, H-measures, a new approach for studying homogenisation, oscillations and concentration effects in partial differential equations. Proc. R. Soc. Edinb. 115, 193–230 (1990)

    Article  MathSciNet  Google Scholar 

  99. T. Tao, Finite time blowup for an averaged three-dimensional Navier-Stokes equation. arXiv:1402.0290

    Google Scholar 

  100. M. Ukhovskii, V. Iudovich, Axially symmetric flows of ideal and viscous fluids filling the whole space. Prikl. Mat. Meh. 32, 59–69 (Russian); translated as J. Appl. Math. Mech. 32, 52–61 (1968)

    Article  MathSciNet  Google Scholar 

  101. W. von Wahl, The Equations of Navier-Stokes and Abstract Parabolic Equations (Friedr. Vieweg & Sohn, Braunschweig, 1985)

    Book  Google Scholar 

  102. B. Wang, Ill-posedness for the Navier-Stokes equations in critical Besov spaces \(\dot{B}_{\infty,q}^{-1}\). Adv. Math. 268, 350–372 (2015)

    Article  MathSciNet  Google Scholar 

  103. F. Weissler, The Navier-Stokes initial value problem in Lp. Arch. Ration. Mech. Anal. 74, 219–230 (1980)

    Article  Google Scholar 

  104. M. Wiegner, Decay results for weak solutions of the Navier-Stokes equations on \(\mathbb{R}^{n}\). J. Lond. Math. Soc. 2, 303–313 (1987)

    Article  MathSciNet  Google Scholar 

  105. T. Yoneda, Ill-posedness of the 3D-Navier-Stokes equations in a generalized Besov space near BMO−1. J. Funct. Anal. 258, 3376–3387 (2010)

    Article  MathSciNet  Google Scholar 

  106. Y. Zhou, A new regularity criterion for the Navier-Stokes equations in terms of the direction of vorticity. Monatsh. Math. 144(3), 251–257 (2005)

    Article  MathSciNet  Google Scholar 

  107. Y. Zhou, M. Pokorny, On the regularity of the solutions of the Navier-Stokes equations via one velocity component. Nonlinearity 23, 1097–1107 (2010)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Paris-Diderot University, 75205 Cedex 13, Paris, France

    Isabelle Gallagher

Authors
  1. Isabelle Gallagher
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Isabelle Gallagher .

Editor information

Editors and Affiliations

  1. Graduate School of Mathematical Sciences, University of Tokyo, Meguro-ku, Tokyo, Japan

    Yoshikazu Giga

  2. Université de Toulon IMATH, Toulon, France

    Antonín Novotný

Rights and permissions

Reprints and permissions

Copyright information

© 2018 Springer International Publishing AG, part of Springer Nature

About this entry

Check for updates. Verify currency and authenticity via CrossMark

Cite this entry

Gallagher, I. (2018). Critical Function Spaces for the Well-Posedness of the Navier-Stokes Initial Value Problem. In: Giga, Y., Novotný, A. (eds) Handbook of Mathematical Analysis in Mechanics of Viscous Fluids. Springer, Cham. https://doi.org/10.1007/978-3-319-13344-7_12

Download citation

Publish with us

Policies and ethics

Search

Navigation