Sur la théorie globale des équations de Navier-Stokes compressible

Didier Bresch[1]; Benoît Desjardins[2]

  • [1] Laboratoire de Mathématiques, UMR 5127 CNRS, Université de Savoie, 73376 Le Bourget du Lac cedex, France
  • [2] Département de Mathématiques et Applications, E.N.S. Ulm, 45 rue d’Ulm, 75230 Paris cedex 05, France

Journées Équations aux dérivées partielles (2006)

  • Volume: 343, Issue: 3, page 1-26
  • ISSN: 0752-0360

Abstract

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Le but de cet article est de présenter quelques résultats mathématiques plus ou moins récents sur la théorie de l’existence globale en temps (solutions faibles et solutions fortes) pour les équations de Navier-Stokes compressibles en dimension supérieure ou égale à deux sans aucune hypothèse de symétrie sur le domaine et sans aucune hypothèse sur la taille des données initiales.

How to cite

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Bresch, Didier, and Desjardins, Benoît. "Sur la théorie globale des équations de Navier-Stokes compressible." Journées Équations aux dérivées partielles 343.3 (2006): 1-26. <http://eudml.org/doc/10621>.

@article{Bresch2006,
abstract = {Le but de cet article est de présenter quelques résultats mathématiques plus ou moins récents sur la théorie de l’existence globale en temps (solutions faibles et solutions fortes) pour les équations de Navier-Stokes compressibles en dimension supérieure ou égale à deux sans aucune hypothèse de symétrie sur le domaine et sans aucune hypothèse sur la taille des données initiales.},
affiliation = {Laboratoire de Mathématiques, UMR 5127 CNRS, Université de Savoie, 73376 Le Bourget du Lac cedex, France; Département de Mathématiques et Applications, E.N.S. Ulm, 45 rue d’Ulm, 75230 Paris cedex 05, France},
author = {Bresch, Didier, Desjardins, Benoît},
journal = {Journées Équations aux dérivées partielles},
keywords = {Équations de Navier-Stokes compressibles; existence globale; explosion; solutions faibles; solutions fortes; viscosités constantes; viscosités non constantes; fluides barotropes; fluides conducteurs de chaleur},
language = {fre},
month = {6},
number = {3},
pages = {1-26},
publisher = {Groupement de recherche 2434 du CNRS},
title = {Sur la théorie globale des équations de Navier-Stokes compressible},
url = {http://eudml.org/doc/10621},
volume = {343},
year = {2006},
}

TY - JOUR
AU - Bresch, Didier
AU - Desjardins, Benoît
TI - Sur la théorie globale des équations de Navier-Stokes compressible
JO - Journées Équations aux dérivées partielles
DA - 2006/6//
PB - Groupement de recherche 2434 du CNRS
VL - 343
IS - 3
SP - 1
EP - 26
AB - Le but de cet article est de présenter quelques résultats mathématiques plus ou moins récents sur la théorie de l’existence globale en temps (solutions faibles et solutions fortes) pour les équations de Navier-Stokes compressibles en dimension supérieure ou égale à deux sans aucune hypothèse de symétrie sur le domaine et sans aucune hypothèse sur la taille des données initiales.
LA - fre
KW - Équations de Navier-Stokes compressibles; existence globale; explosion; solutions faibles; solutions fortes; viscosités constantes; viscosités non constantes; fluides barotropes; fluides conducteurs de chaleur
UR - http://eudml.org/doc/10621
ER -

References

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  1. I. Basov, V. Shelukhin. Generalized solutions to the equations of Bingham. compressible flows, Z. Angew. Math. Mech., 78, 1-8, (1998). Zbl0928.76009
  2. W. Borchers, H. Sohr. On the equation rot v = g and div u = f with zero boundary conditions. Hokkaido Math J., 19 : 67–87, (1990). Zbl0719.35014MR1039466
  3. D. Bresch, B. Desjardins. Stabilité de solutions faibles pour les équations de Navier-Stokes compressibles avec conductivité de chaleur. C.R. Acad. Sciences Paris, Section Mathématiques, vol. 343, Issue 3, 219–224, (2006). Zbl1217.76067MR2246342
  4. D. Bresch, B. Desjardins. Existence of global weak solutions for a 2D viscous shallow water equations and convergence to the quasi-geostrophic model. Commun. Math. Phys., 238, 1-2, (2003), p. 211–223. Zbl1037.76012MR1989675
  5. D. Bresch, B. Desjardins. Some diffusive capillary models of Korteweg type. C. R. Acad. Sciences, Paris, Section Mécanique. Vol 332 no 11 (2004), p 881–886. Zbl05953141
  6. D. Bresch, B. Desjardins. On the existence of global weak solutions to the Navier-Stokes equations for viscous compressible and heat conducting fluids. J. Maths. Pures et Appliquées, (2007)). Zbl1122.35092MR2297248
  7. D. Bresch, B. Desjardins On the construction of approximate solutions for the 2D viscous shallow water model and for compressible Navier-Stokes models. J. Maths. Pures et Appliquées, 86, 4, 362-368, (2006). Zbl1121.35094MR2257849
  8. D. Bresch, B. Desjardins, D. Gérard-Varet. Rotating fluids in a cylinder. numéro spécial « Some Evolution Equations and their Qualitative Properties », Série DCDS Série A, Vol. 11, Number 1, 47–82, (2004). Zbl1138.76446MR2073946
  9. D. Bresch, B. Desjardins, D. Gérard-Varet. On compressible Navier–Stokes equations with density dependent viscosities in bounded domains. J. Maths Pures et Appliquées, (2007). Zbl1121.35093MR2296806
  10. D. Bresch, B. Desjardins, J.–M. Ghidaglia, E. Grenier. Mathematical properties of the basic two fluid model. En préparation (2007). 
  11. D. Bresch, B. Desjardins, C.K. Lin. On some compressible fluid models : Korteweg, lubrication and shallow water systems. Comm. Part. Diff. Eqs. 28, 3–4, (2003), p. 1009–1037. Zbl1106.76436MR1978317
  12. Y. Cho, B.J. Jin. Blow up the viscous heat-conducting compressible flows. J. Math. Anal. 320, (2006), 819–826. Zbl1121.35110MR2225997
  13. R.R. Coifman, Y. Meyer. On commutators of singular integrals and bilinear singular integrals. Trans. Amer. Math. Soc., 212, (1975), 315–331. Zbl0324.44005MR380244
  14. R. Danchin. Global existence in critical spaces for compressible Navier-Stokes equations, Inventiones Mathematicae, 141, pages 579-614 (2000). Zbl0958.35100MR1779621
  15. R. Danchin. Global existence in critical spaces for compressible viscous and heat conductive gases, Arch. Rat. Mech. Anal., 160, 1-39 (2001). Zbl1018.76037MR1864120
  16. B. Desjardins. On weak solutions of the compressible isentropic Navier-Stokes equations. Appl. Math. Letters, 12, 107–111, (1999). Zbl0939.35142MR1750068
  17. R.J. DiPerna, P.–L. Lions. Ordinary differential equations, transport theory and Sobolev spaces. Invent. Math., 98, (1989), 511–547. Zbl0696.34049MR1022305
  18. P. Duhem. Recherches sur l’hydrodynamique. Ann. Toulouse 2 (1901-02). Zbl33.0770.01
  19. E. Feireisl. On the motion of a viscous, compressible and heat conducting fluid. Indiana Univ. Math. J. 53 (2004), no. 6, 1705–1738. Zbl1087.35078MR2106342
  20. E. Feireisl, A. Novotny, H., Petzeltova. On the existence of globally defined weak solutions to the Navier–Stokes equations of compressible isentropic fluids. J. Math. Fluid. Dynam., 3, (2001), p. 358–392. Zbl0997.35043MR1867887
  21. E. Feireisl. Dynamics of viscous compressible fluids. Oxford Science Publication, Oxford, (2004). Zbl1080.76001MR2040667
  22. E. Feireisl. Compressible Navier-Stokes equations with a non-monotone pressure law. J. Diff. Eqs, 184, 97-108, (2002). Zbl1012.76079MR1929148
  23. G.P. Galdi. An introduction to the mathematical theory of the Navier-Stokes equations, Volume I. Springer-Verlag, New York, (1994). Zbl0949.35005MR1284205
  24. M. Hillairet. Propagation of density-oscillations in solutions to barotropic compressible Navier-Stokes system. A paraître dans J.. Math. Fluid. Mech., (2007). Zbl1220.35118
  25. D. Hoff. Global solutions of the Navier-Stokes equations for multidimensional compressible flow with discontinuous initial data. J. Diff. Eqs, 120, (1995), 215–254. Zbl0836.35120MR1339675
  26. D. Hoff. Strong convergence to global solutions for multidimensional flows of compressible, viscous fluids with polytropic equations of state and discontinuous initial data. Arch. Rational Mech. Anal. 132 (1995), no. 1, 1–14. Zbl0836.76082MR1360077
  27. D. Hoff. Uniqueness of Weak Solutions of the Navier–Stokes Equations of Multidimensional, Compressible Flow Siam J. Math. Anal. 37, 6, (2006), 1742–1760. Zbl1100.76052MR2213392
  28. D. Hoff. Discontinuous solutions of the Navier-Stokes equations for multidimensional heat-conducting flows. Arch. Rational Mech. Anal. 139, (1997), 303-354. Zbl0904.76074MR1480244
  29. D. Hoff, D. Serre. The failure of continuous dependence on initial data for the Navier-Stokes equations of compressible flow. SIAM J. Appl. Math., 51, (1991), 887–898. Zbl0741.35057MR1117422
  30. A.V. Kazhikhov. The equations of potential flow of compressible viscous fluid at low Reynolds number. Acta Appl. Math. 37, (1994), 77–81. Zbl0815.35083MR1308747
  31. P.–L. Lions. Mathematical topics in fluid mechanics : compressible models : vol. 2. Oxford University press, (1998). Zbl0908.76004MR1637634
  32. P.–L. Lions. Compacité des solutions des équations de Navier-Stokes compressibles isentropiques. C. R. Acad. Sciences, Paris, section Mathématique. 317, 1, (1993), 115–120. Zbl0781.76072MR1228976
  33. J. Málek, J. Necas, M. Rokyta, M. Ruzicka. Weak and measure-valued solutions to evolutionary PDEs. Applied Mathematics and Mathematical Computation 13, Chapman el Hall, (1996). Zbl0851.35002MR1409366
  34. A. Mamontov. Global solvability of the multidimensional Navier-Stokes equations of a compressible fluid with nonlinear viscosity, I. Siberian Math. J., 40, 352-362, (1999). Zbl0938.35121
  35. A. Mamontov. Global solvability of the multidimensional Navier-Stokes equations of a compressible fluid with nonlinear viscosity, II. Siberian Math. J., 40, 541-555, (1999). Zbl0928.35119MR1709015
  36. A. Matsumura, T. Nishida. The initial value problem for the equations of motion of viscous and heat-conductive gases. J. Math. Kyoto Univ., 20, (1980), 67–104. Zbl0429.76040MR564670
  37. A. Mellet, A. Vasseur. On the barotropic compressible Navier-Stokes equations. À paraître dans Comm. Partial Diff. Equations (2006). Zbl1149.35070MR2304156
  38. S. Matusu-Necasova, M. Medvidova-Lukacova. Bipolar barotropic non-newtonian compressible fluids. Modél. math. anal. numér., vol. 34, no5, 923-934, (2000). Zbl0992.76010MR1837761
  39. F. Murat. Compacité par compensation. Annali della scuola normale superiore, 5, 487–507, (1978). Zbl0399.46022MR506997
  40. E. Nelson. Dynamical theories of brownian motion. Mathematical Notes, Princeton Univ. Press., (1967). Zbl0165.58502MR214150
  41. A. Novotny, I. Straskraba. Introduction to the mathematical theory of compressible flow. Oxford lecture series in Mathematics and its applications, (2004). Zbl1088.35051
  42. M. Padula. Existence of global solutions for two-dimensional viscous compressible flows. J. Funct. Anal., (69), (1986), 1-20. Zbl0633.76072MR864756
  43. D. Serre. Solutions faibles globales des équations de Navier-Stokes pour un fluide compressible. C. R. Acad. Sci. Paris Sér. I Math. 303 (1986), no. 13, 639–642. Zbl0597.76067MR867555
  44. D. Serre. Variations de grande amplitude pour la densité d’un fluide visqueux compressible. Phys. D 48 (1991), no. 1, 113–128. Zbl0739.35071
  45. J. Serrin. Mathematical principles of classical fluid mechanics, in Handbuch der Physik VIII, Springer-Verlage, 1959. MR108116
  46. V.A. Solonnikov. Solvability of initial boundary value problem for the equation of motion of viscous compressible fluid. Steklov Inst. Seminars in Math., 56, (1976), 128–142. Zbl0338.35078MR481666
  47. A. Tani. On the first initial boundary value problem of compressible viscous fluid motion. Publ. Res. Inst. Math. Sci., 13, (1977), 193–253. Zbl0366.35070
  48. L. Tartar. Compensated compactness and applications to partial differential equation. In Nonlin. Anal. and Mech. (ed. L. Knopps), Res. Notes in Math. 39, Boston, 136–211, Heriot-Watt Sympos, Pitman. Zbl0437.35004MR584398
  49. V.A. Vaĭgant. An example of the nonexistence with respect to time of the global solution of Navier-Stokes equations for a compressible viscous barotropic fluid. Russian Acad. Sci. Dokl. Math. 50 (1995), no. 3, 397–399. Zbl0877.35092MR1316938
  50. V.A. Weigant, A.V. Kazhikhov. On the existence of global solutions to two-dimensional Navier-Stokes equations of compressible viscous fluids. Siberian Math. J., 36, (1995). Zbl0860.35098
  51. Z.P. Xin. Blowup of smooth solutions to the compressible Navier-Stokes equation with compact density. Comm. Pure Appl. Math. 51 (1998), no. 3, 229–240. (1995), 1108–1141. Zbl0937.35134MR1488513
  52. V.I. Youdovitch. Two dimensional nonstationary problem on flow of ideal compressible fluid through the given domain. Mat. Sbornik, 64, (1994), 562–588. 

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