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Global weak solutions to a generic two-fluid model. (English) Zbl 1193.35146

Summary: This paper deals with mathematical properties of a generic two-fluid flow model commonly used in industrial applications. More precisely, we address the question of whether available mathematical results in the case of a single-fluid governed by the compressible barotropic Navier-Stokes equations may be extended to such a two-phase model. We focus on existence of global weak solutions, linear theory and determination of eigenvalues and invariant regions.

MSC:

35Q35 PDEs in connection with fluid mechanics
76T10 Liquid-gas two-phase flows, bubbly flows
76N10 Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics
35B45 A priori estimates in context of PDEs
35A35 Theoretical approximation in context of PDEs
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[1] Alazard T.: Low Mach number limit of the full Navier–Stokes equations. Arch. Ration. Mech. Anal. 180(1), 1–73 (2006) · Zbl 1108.76061 · doi:10.1007/s00205-005-0393-2
[2] Bresch D., Desjardins B.: On the existence of global weak solutions to the Navier–Stokes equations for viscous compressible and heat conducting fluids. J. Math. Pure Appl. 1(87), 57–90 (2007) · Zbl 1122.35092
[3] Bresch D., Desjardins B.: Stabilité de solutions faibles pour les équations de Navier–Stokes compressibles avec conductivité de chaleur. C. R. Acad. Sci. Paris Sect. Math. 343(3), 219–224 (2006) · Zbl 1217.76067
[4] Bresch D., Desjardins B.: 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), 211–223 (2003) · Zbl 1037.76012
[5] Bresch D., Desjardins B.: Some diffusive capillary models of Korteweg type. C. R. Acad. Sci. Paris Sect. Mec. 332(11), 881–886 (2004) · Zbl 1386.76070
[6] Bresch D., Desjardins B.: On the construction of approximate solutions for the 2d viscous shallow water model and for compressible Navier–Stokes models. J. Math. Pure Appl. 4(86), 262–268 (2006) · Zbl 1121.35094
[7] Bresch D., Desjardins B., Gérard-Varet D.: On compressible Navier–Stokes equations with density dependent viscosities in bounded domains. J. Math. Pure Appl. 2(87), 227–235 (2007) · Zbl 1121.35093
[8] Bresch D., Desjardins B., Lin C.K.: On some compressible fluid models: Korteweg, lubrication and shallow water systems. Comm. Part. Differ. Eqs. 28(3–4), 1009–1037 (2003) · Zbl 1106.76436
[9] Bresch, D., Desjardins, B., Ghidaglia, J.-M., Grenier, E.: The low Mach number limit and the generic two-phase model (2007, in preparation)
[10] Feireisl E.: Dynamics of Viscous Compressible Fluids. Oxford Science, Oxford (2004) · Zbl 1080.76001
[11] Ghidaglia J.-M., Kumbero A., Le Coq G.: On the numerical solution to the fluid models via a cell centered finite volume method. Eur. J. Mech. B. Fluids 20(6), 841–867 (2001) · Zbl 1059.76041 · doi:10.1016/S0997-7546(01)01150-5
[12] Haspot, B.: Étude d’équations liées à la mécanique des fluides compressibles. Thèse Université Paris 12 (2007)
[13] Ishii M.: Thermo-Fluid Dynamic Theory of Two-Phase Flow. Eyrolles, Paris (1975) · Zbl 0325.76135
[14] Lions P.-L.: Compacité des solutions des équations de Navier–Stokes compressibles isentropiques. C. R. Acad. Sci. Paris Sér. I. 317, 115–120 (1993) · Zbl 0781.76072
[15] Lions P.-L.: Mathematical Topics in Fluid Dynamics, Vol. 2, Compressible Models. Oxford Science, Oxford (1998)
[16] Ramos, D.: Quelques résultats mathématiques et simulations numériques d’écoulements régis par des modèles bifluides. Thèse de l’ENS Cachan (2002)
[17] Raviart P.A., Godlewski E.: Hyperbolic Systems of Conservation Laws. Ellipses, Paris (1991) · Zbl 0768.35059
[18] Rovarc’h, J.-M.: Solveurs tridimensionnels pour les écoulements de fluides diphasiques avec transferts d’énergie. Thèse de l’ENS Cachan (2006)
[19] Smoller J.: Shock Waves and Reaction Diffusion Equations. Springer, Berlin (1983) · Zbl 0508.35002
[20] Xin, Z.P.: Blowup of smooth solutions to the compressible Navier–Stokes equation with compact density. Comm. Pure Appl. Math. 51(3), 229–240 (1998); 1108–1141 (1995) · Zbl 0937.35134
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