Sur la théorie globale des équations de Navier-Stokes compressible
Bresch, Didier ; Desjardins, Benoît
Journées équations aux dérivées partielles, (2006), p. 1-26 / Harvested from Numdam

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.

Publié le : 2006-01-01
DOI : https://doi.org/10.5802/jedp.30
Classification:  35Q30
Mots clés: Équations de Navier-Stokes compressibles, existence globale, explosion, solutions faibles, solutions fortes, viscosités constantes, viscosités non constantes, fluides barotropes, fluides conducteurs de chaleur.
@article{JEDP_2006____A3_0,
     author = {Bresch, Didier and Desjardins, Beno\^\i t},
     title = {Sur la th\'eorie globale des \'equations de Navier-Stokes compressible},
     journal = {Journ\'ees \'equations aux d\'eriv\'ees partielles},
     year = {2006},
     pages = {1-26},
     doi = {10.5802/jedp.30},
     language = {fr},
     url = {http://dml.mathdoc.fr/item/JEDP_2006____A3_0}
}
Bresch, Didier; Desjardins, Benoît. Sur la théorie globale des équations de Navier-Stokes compressible. Journées équations aux dérivées partielles,  (2006), pp. 1-26. doi : 10.5802/jedp.30. http://gdmltest.u-ga.fr/item/JEDP_2006____A3_0/

[1] I. Basov, V. Shelukhin. Generalized solutions to the equations of Bingham. compressible flows, Z. Angew. Math. Mech., 78, 1-8, (1998). | Zbl 0928.76009

[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). | MR 1039466 | Zbl 0719.35014

[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). | MR 2246342 | Zbl 05058417

[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. | MR 1989675 | Zbl 1037.76012

[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.

[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)). | MR 2297248 | Zbl 05134243

[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). | MR 2257849 | Zbl 05125007

[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). | MR 2073946 | Zbl 02113147

[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). | MR 2296806 | Zbl 05134240

[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. | MR 1978317 | Zbl 1106.76436

[12] Y. Cho, B.J. Jin. Blow up the viscous heat-conducting compressible flows. J. Math. Anal. 320, (2006), 819–826. | MR 2225997 | Zbl 05030912

[13] R.R. Coifman, Y. Meyer. On commutators of singular integrals and bilinear singular integrals. Trans. Amer. Math. Soc., 212, (1975), 315–331. | MR 380244 | Zbl 0324.44005

[14] R. Danchin. Global existence in critical spaces for compressible Navier-Stokes equations, Inventiones Mathematicae, 141, pages 579-614 (2000). | MR 1779621 | Zbl 0958.35100

[15] R. Danchin. Global existence in critical spaces for compressible viscous and heat conductive gases, Arch. Rat. Mech. Anal., 160, 1-39 (2001). | MR 1864120 | Zbl 1018.76037

[16] B. Desjardins. On weak solutions of the compressible isentropic Navier-Stokes equations. Appl. Math. Letters, 12, 107–111, (1999). | MR 1750068 | Zbl 0939.35142

[17] R.J. DiPerna, P.–L. Lions. Ordinary differential equations, transport theory and Sobolev spaces. Invent. Math., 98, (1989), 511–547. | MR 1022305 | Zbl 0696.34049

[18] P. Duhem. Recherches sur l’hydrodynamique. Ann. Toulouse 2 (1901-02). | Numdam

[19] E. Feireisl. On the motion of a viscous, compressible and heat conducting fluid. Indiana Univ. Math. J. 53 (2004), no. 6, 1705–1738. | MR 2106342 | Zbl 1087.35078

[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. | MR 1867887 | Zbl 0997.35043

[21] E. Feireisl. Dynamics of viscous compressible fluids. Oxford Science Publication, Oxford, (2004). | MR 2040667 | Zbl 1080.76001

[22] E. Feireisl. Compressible Navier-Stokes equations with a non-monotone pressure law. J. Diff. Eqs, 184, 97-108, (2002). | MR 1929148 | Zbl 1012.76079

[23] G.P. Galdi. An introduction to the mathematical theory of the Navier-Stokes equations, Volume I. Springer-Verlag, New York, (1994). | MR 1284205 | Zbl 0949.35005

[24] M. Hillairet. Propagation of density-oscillations in solutions to barotropic compressible Navier-Stokes system. A paraître dans J.. Math. Fluid. Mech., (2007).

[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. | MR 1339675 | Zbl 0836.35120

[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. | MR 1360077 | Zbl 0836.76082

[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. | MR 2213392 | Zbl 1100.76052

[28] D. Hoff. Discontinuous solutions of the Navier-Stokes equations for multidimensional heat-conducting flows. Arch. Rational Mech. Anal. 139, (1997), 303-354. | MR 1480244 | Zbl 0904.76074

[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. | MR 1117422 | Zbl 0741.35057

[30] A.V. Kazhikhov. The equations of potential flow of compressible viscous fluid at low Reynolds number. Acta Appl. Math. 37, (1994), 77–81. | MR 1308747 | Zbl 0815.35083

[31] P.–L. Lions. Mathematical topics in fluid mechanics : compressible models : vol. 2. Oxford University press, (1998). | MR 1637634 | Zbl 0908.76004

[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. | MR 1228976 | Zbl 0781.76072

[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). | MR 1409366 | Zbl 0851.35002

[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). | Zbl 0938.35121

[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). | MR 1709015 | Zbl 0928.35119

[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. | MR 564670 | Zbl 0429.76040

[37] A. Mellet, A. Vasseur. On the barotropic compressible Navier-Stokes equations. À paraître dans Comm. Partial Diff. Equations (2006). | MR 2304156 | Zbl 05150099

[38] S. Matusu-Necasova, M. Medvidova-Lukacova. Bipolar barotropic non-newtonian compressible fluids. Modél. math. anal. numér., vol. 34, no5, 923-934, (2000). | Numdam | MR 1837761 | Zbl 0992.76010

[39] F. Murat. Compacité par compensation. Annali della scuola normale superiore, 5, 487–507, (1978). | Numdam | MR 506997 | Zbl 0399.46022

[40] E. Nelson. Dynamical theories of brownian motion. Mathematical Notes, Princeton Univ. Press., (1967). | MR 214150 | Zbl 0165.58502

[41] A. Novotny, I. Straskraba. Introduction to the mathematical theory of compressible flow. Oxford lecture series in Mathematics and its applications, (2004). | Zbl 1088.35051

[42] M. Padula. Existence of global solutions for two-dimensional viscous compressible flows. J. Funct. Anal., (69), (1986), 1-20. | MR 864756 | Zbl 0633.76072

[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. | MR 867555 | Zbl 0597.76067

[44] D. Serre. Variations de grande amplitude pour la densité d’un fluide visqueux compressible. Phys. D 48 (1991), no. 1, 113–128. | Zbl 0739.35071

[45] J. Serrin. Mathematical principles of classical fluid mechanics, in Handbuch der Physik VIII, Springer-Verlage, 1959. | MR 108116

[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. | MR 481666 | Zbl 0338.35078

[47] A. Tani. On the first initial boundary value problem of compressible viscous fluid motion. Publ. Res. Inst. Math. Sci., 13, (1977), 193–253. | Zbl 0366.35070

[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. | MR 584398 | Zbl 0437.35004

[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. | MR 1316938 | Zbl 0877.35092

[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). | Zbl 0860.35098

[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. | MR 1488513 | Zbl 0937.35134

[52] V.I. Youdovitch. Two dimensional nonstationary problem on flow of ideal compressible fluid through the given domain. Mat. Sbornik, 64, (1994), 562–588.