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Stability to the Incompressible Navier-Stokes Equations

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Abstract

This chapter is devoted to the presentation of a few basic incompressible fluid-dynamical equations in Eulerian form and some basic tools in analysis which will be used throughout this thesis. In the first section, we introduce three incompressible fluid-dynamical equations: the inhomogeneous Navier-Stokes equations, the homogeneous (classical) Navier-Stokes equations, and the anisotropic homogeneous Navier-Stokes equations. The second section is devoted to the presentation of some background material and main related results about these systems.

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References

  1. Lions, P.L.: Mathematical topics in fluid mechanics. Incompressible models. Oxford Lecture Series in Mathematics and its Applications, 3. Oxford Science Publications, vol. 1. Clarendon Press/Oxford University Press, New York (1996)

    Google Scholar 

  2. Pedlosky, J.: Geophysical Fluid Dynamics, 2nd edn, p. 710. Springer-Verlag, New York (1987)

    Book  MATH  Google Scholar 

  3. Leray, J.: Sur le mouvement d’un liquide visqueux remplissant l’espace. Acta Math. 63, 193–248 (1934)

    Article  MathSciNet  MATH  Google Scholar 

  4. Hopf, E.: über die Anfangwertaufgabe für die hydrodynamischen Grundgleichungen. Math. Nachr. 4, 213–231 (1951)

    Article  MathSciNet  MATH  Google Scholar 

  5. Serrin, J.: On the interior regularity of weak solutions of the Navier-Stokes equations. Arch. Rat. Mech. Anal. 9, 187–195 (1962)

    Article  MathSciNet  MATH  Google Scholar 

  6. Prodi, G.: Un teorema di unicità per le equazioni di Navier-Stokes. Ann. Mat. Pure. Appl. 48, 173–182 (1959)

    Article  MathSciNet  MATH  Google Scholar 

  7. Ohyama, T.: Interior regularity of weak solutions of the time-dependent Navier-Stokes equation. Proc. Japan Acad. 36, 273–277 (1960)

    Article  MathSciNet  MATH  Google Scholar 

  8. Giga, Y.: Solutions for semilinear Parabolic equations in \(L^p\) and regularity of weak solutions of the Navier-Stokes system. J. Differ. Equ. 62, 186–212 (1986)

    Article  MathSciNet  Google Scholar 

  9. Escauriaza, L., Seregin, G., Šverák, V.: On \(L^{3, \, \infty }\)-solutions to the Navier-Stokes equations and backward uniqueness. Usp. Mat. Nauk 58, 3–44 (2003)

    Google Scholar 

  10. Beirão da Veiga, H.: Concerning the regularity problem for the solutions of the Navier-Stokes equations. C.R. Acad. Sci. Paris Sér. I 321, 405–408 (1995)

    Google Scholar 

  11. Beirão da Veiga, H.: A new regularity class for the Navier-Stokes equations in \(\mathbb{R}^{n}\). Chin. Ann. Math. 16B, 407–412 (1995)

    Google Scholar 

  12. Tzvetkov, N.: Ill-posedness issues for nonlinear dispersive equations. In: Lectures on Nonlinear Dispersive Equations. GAKUTO International Series Mathematical Sciences and Applications, Gakkōtosho. vol. 27, pp. 63–103 (2006)

    Google Scholar 

  13. Kenig, C.E., Ponce, G., Vega, L.: On the ill-posedness of some canonical dispersive equations. Duke Math. J. 106, 617–633 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  14. Lemarié-Rieusset, P.G.: Recent developments in the Navier-Stokes problem. Chapman & Hall/CRC Research Notes in Mathematics, vol. 431. Chapman & Hall/CRC, Boca Raton (2002)

    Google Scholar 

  15. Chemin, J.-Y.: Localization in fourier space and Navier-Stokes system, In: Phase Space Analysis of Partial Differential Equations, Proceedings 2004, CRM series, pp. 53–136. Pisa (2004)

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  17. Kato, T.: Strong \(L^p\)-solutions of the Navier-Stokes equations in \(\mathbb{R}^{m}\) with applications to weak solutions. Math. Z. 187, 471–480 (1984)

    Google Scholar 

  18. Cannone, M., Meyer, Y., Planchon, F.: Solutions autosimilaires des équations de Navier-Stokes, Séminaire Équations aux Dérivées Partielles de l’École Polytechnique, 1993–1994

    Google Scholar 

  19. Chemin, J.-Y.: Théorémes d’unicité pour le systéme de Navier-Stokes tridimensionnel. J. Anal. Math. 77, 27–50 (1999)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  21. Cannone, M.: Ondelettes, Paraproduits et Navier-Stokes, Diderot Éditeur. Arts et Sciences, Lyon (1995)

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  25. Chemin, J.-Y., Desjardins, B., Gallagher, I., Grenier, E.: Fluids with anisotropic viscosity. Modélisation Mathématique et Analyse Numérique 34, 315–335 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  26. Chemin, J.-Y., Zhang, P.: On the global wellposedness to the 3-D incompressible anisotropic Navier-Stokes equations. Comm. Math. Phys. 272, 529–566 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  27. Paicu, M.: Équation anisotrope de Navier-Stokes dans des espaces critiques. Revista Matematica Iberoamericana 21, 179–235 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  28. Zhang, P.: Global smooth solutions to the 2-D nonhomogeneous Navier-Stokes equations, Int. Math. Res. Not. IMRN, 2008, Art. ID rnn 098, 26 pp

    Google Scholar 

  29. Chemin, J.-Y., Desjardins, B., Gallagher, I., Grenier, E.: Mathematical geophysics. An introduction to rotating fluids and the Navier-Stokes equations. Oxford Lecture Series in Mathematics and its Applications, vol 32. Clarendon Press/Oxford University Press, Oxford (2006)

    Google Scholar 

  30. Iftimie, D.: A uniqueness result for the Navier-Stokes equations with vanishing vertical viscosity. SIAM J. Math. Anal. 33, 1483–1493 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  31. Kazhikov, A.V.: Solvability of the initial-boundary value problem for the equations of the motion of an inhomogeneous viscous incompressible fluid, (Russian). Dokl. Akad. Nauk SSSR 216, 1008–1010 (1974)

    MathSciNet  Google Scholar 

  32. Ladyženskaja, O.A., Solonnikov, V.A.: The unique solvability of an initial-boundary value problem for viscous incompressible inhomogeneous fluids. (Russian) Boundary value problems of mathematical physics, and related questions of the theory of functions, 8, Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 52, 52–109, 218–219 (1975)

    Google Scholar 

  33. Danchin, R.: Local and global well-posedness results for flows of inhomogeneous viscous fluids. Adv. Differ. Equ. 9, 353–386 (2004)

    MathSciNet  MATH  Google Scholar 

  34. DiPerna, R.J., Lions, P.L., Equations différentielles ordinaires et équations de transport avec des coefficients irréguliers. In: Séminaire EDP 1988–1989, Ecole Polytechnique, Palaiseau (1989)

    Google Scholar 

  35. Desjardins, B.: Regularity results for two-dimensional flows of multiphase viscous fluids. Arch. Ration. Mech. Anal. 137, 135–158 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  36. Danchin, R.: Density-dependent incompressible viscous fluids in critical spaces. Proc. Roy. Soc. Edinburgh Sect. A 133, 1311–1334 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  37. Abidi, H.: Équation de Navier-Stokes avec densité et viscosité variables dans l’espace critique, Rev. Mat. Iberoam 23(2), 537–586 (2007)

    Google Scholar 

  38. Abidi, H., Paicu, M.: Existence globale pour un fluide inhomogéne. Ann. Inst. Fourier (Grenoble) 57, 883–917 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  39. Chemin, J.-Y., Lerner, N.: Flot de champs de vecteurs non lipschitziens et équations de Navier-Stokes. J. Differ. Equ. 121, 314–328 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  40. Bony, J.M.: Calcul symbolique et propagation des singularités pour les quations aux drivées partielles non linéaires. Ann. Sci. École Norm. Sup. 14(4), 209–246 (1981)

    Google Scholar 

  41. Iftimie, D.: The resolution of the Navier-Stokes equations in anisotropic spaces. Revista Matematica Iberoamericana 15, 1–36 (1999)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Northwest University, Xi’an, 710069, Shaanxi, People’s Republic of China

    Guilong Gui

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  1. Guilong Gui
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Gui, G. (2013). Introduction. In: Stability to the Incompressible Navier-Stokes Equations. Springer Theses. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-36028-2_1

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