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
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topBresch, 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
top- I. Basov, V. Shelukhin. Generalized solutions to the equations of Bingham. compressible flows, Z. Angew. Math. Mech., 78, 1-8, (1998). Zbl0928.76009
- W. Borchers, H. Sohr. On the equation and with zero boundary conditions. Hokkaido Math J., 19 : 67–87, (1990). Zbl0719.35014MR1039466
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- D. Bresch, B. Desjardins, J.–M. Ghidaglia, E. Grenier. Mathematical properties of the basic two fluid model. En préparation (2007).
- 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
- Y. Cho, B.J. Jin. Blow up the viscous heat-conducting compressible flows. J. Math. Anal. 320, (2006), 819–826. Zbl1121.35110MR2225997
- R.R. Coifman, Y. Meyer. On commutators of singular integrals and bilinear singular integrals. Trans. Amer. Math. Soc., 212, (1975), 315–331. Zbl0324.44005MR380244
- R. Danchin. Global existence in critical spaces for compressible Navier-Stokes equations, Inventiones Mathematicae, 141, pages 579-614 (2000). Zbl0958.35100MR1779621
- R. Danchin. Global existence in critical spaces for compressible viscous and heat conductive gases, Arch. Rat. Mech. Anal., 160, 1-39 (2001). Zbl1018.76037MR1864120
- B. Desjardins. On weak solutions of the compressible isentropic Navier-Stokes equations. Appl. Math. Letters, 12, 107–111, (1999). Zbl0939.35142MR1750068
- R.J. DiPerna, P.–L. Lions. Ordinary differential equations, transport theory and Sobolev spaces. Invent. Math., 98, (1989), 511–547. Zbl0696.34049MR1022305
- P. Duhem. Recherches sur l’hydrodynamique. Ann. Toulouse 2 (1901-02). Zbl33.0770.01
- 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
- 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
- E. Feireisl. Dynamics of viscous compressible fluids. Oxford Science Publication, Oxford, (2004). Zbl1080.76001MR2040667
- E. Feireisl. Compressible Navier-Stokes equations with a non-monotone pressure law. J. Diff. Eqs, 184, 97-108, (2002). Zbl1012.76079MR1929148
- G.P. Galdi. An introduction to the mathematical theory of the Navier-Stokes equations, Volume I. Springer-Verlag, New York, (1994). Zbl0949.35005MR1284205
- M. Hillairet. Propagation of density-oscillations in solutions to barotropic compressible Navier-Stokes system. A paraître dans J.. Math. Fluid. Mech., (2007). Zbl1220.35118
- 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
- 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
- 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
- D. Hoff. Discontinuous solutions of the Navier-Stokes equations for multidimensional heat-conducting flows. Arch. Rational Mech. Anal. 139, (1997), 303-354. Zbl0904.76074MR1480244
- 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
- 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
- P.–L. Lions. Mathematical topics in fluid mechanics : compressible models : vol. 2. Oxford University press, (1998). Zbl0908.76004MR1637634
- 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
- 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
- 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
- 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
- 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
- A. Mellet, A. Vasseur. On the barotropic compressible Navier-Stokes equations. À paraître dans Comm. Partial Diff. Equations (2006). Zbl1149.35070MR2304156
- S. Matusu-Necasova, M. Medvidova-Lukacova. Bipolar barotropic non-newtonian compressible fluids. Modél. math. anal. numér., vol. 34, no5, 923-934, (2000). Zbl0992.76010MR1837761
- F. Murat. Compacité par compensation. Annali della scuola normale superiore, 5, 487–507, (1978). Zbl0399.46022MR506997
- E. Nelson. Dynamical theories of brownian motion. Mathematical Notes, Princeton Univ. Press., (1967). Zbl0165.58502MR214150
- A. Novotny, I. Straskraba. Introduction to the mathematical theory of compressible flow. Oxford lecture series in Mathematics and its applications, (2004). Zbl1088.35051
- M. Padula. Existence of global solutions for two-dimensional viscous compressible flows. J. Funct. Anal., (69), (1986), 1-20. Zbl0633.76072MR864756
- 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
- D. Serre. Variations de grande amplitude pour la densité d’un fluide visqueux compressible. Phys. D 48 (1991), no. 1, 113–128. Zbl0739.35071
- J. Serrin. Mathematical principles of classical fluid mechanics, in Handbuch der Physik VIII, Springer-Verlage, 1959. MR108116
- 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
- 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
- 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
- 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
- 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
- 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
- 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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