On the existence of global weak solutions to the Navier-Stokes equations for viscous compressible and heat conducting fluids. (English) Zbl 1122.35092
Summary: The purpose of this work is to investigate the problem of global in time existence of sequences of weak solutions to the Navier-Stokes equations for viscous compressible and heat conducting fluids. A class of density and temperature dependent viscosity and conductivity coefficients is considered. This result extends P.-L. Lions’ work in 1993 [C. R. Acad. Sci. Paris, Sér. I 317, 115–120 (1993; Zbl 0781.76072)] restricted to barotropic flows, and provides weak solutions “à la Leray” to the full compressible model that includes internal energy evolution equation with thermal conduction effects. A partial answer is therefore given to this currently widely open problem, described for instance in P.-L. Lions’ book [Mathematical Topics in Fluid Dynamics, vol. 2, Compressible Models, Oxford Science Publication, Oxford (1998; Zbl 0908.76004)]. The proof uses the generalization to the temperature dependent case, of a new mathematical entropy equality derived by the authors in [Some diffusive capillary models of Korteweg type, C. R. Acad. Sci., Paris, Sect. Méc. 332, No. 11, 881–886 (2004)]. The construction scheme of approximate solutions, using on additional regularizing effects such as capillarity, is provided by the authors in [J. Math. Pures Appl. (9) 86, No. 4, 362–368 (2006; Zbl 1121.35094)], and allows to use the stability arguments of this paper.
MSC:
| 35Q30 | Navier-Stokes equations |
| 76N10 | Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics |
| 35D05 | Existence of generalized solutions of PDE (MSC2000) |
References:
| [1] | Alazard, T., Incompressible limit of the non isentropic Euler equations with the solid wall boundary conditions, Adv. Differential Equations, 10, 19-44 (2005) · Zbl 1101.35050 |
| [2] | Alazard, T., Low Mach number limit of the full Navier-Stokes equations, Arch. Rational Mech. Anal., 180, 1, 1-73 (2006) · Zbl 1108.76061 |
| [3] | T. Alazard, D. Bresch, B. Desjardins, Low Mach number limit for weak solutions to the full Navier-Stokes equations, in preparation; T. Alazard, D. Bresch, B. Desjardins, Low Mach number limit for weak solutions to the full Navier-Stokes equations, in preparation |
| [4] | Amosov, A. A., The existence of global generalized solutions of the equations of one-dimensional motion of a real viscous gas with discontinuous data, Differential Equations, 4, 540-558 (2000) · Zbl 1010.76074 |
| [5] | Anderson, D. M.; McFadden, G. B.; Wheeler, A. A., Diffusive interface methods in fluid mechanics, Annu. Rev. Fluid Mech., 30, 139-165 (1998) · Zbl 1398.76051 |
| [6] | 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 |
| [7] | Bresch, D.; Desjardins, B., Some diffusive capillary models of Korteweg type, C. R. Acad. Sci., Paris, Section Mécanique, 332, 11, 881-886 (2004) · Zbl 1386.76070 |
| [8] | D. Bresch, B. Desjardins, Numerical approximation of compressible fluid models with density dependent viscosity, in preparation; D. Bresch, B. Desjardins, Numerical approximation of compressible fluid models with density dependent viscosity, in preparation · Zbl 1121.35094 |
| [9] | D. Bresch, B. Desjardins, D. Gérard-Varet, On compressible Navier-Stokes equations with density dependent viscosities, J. Math. Pures Appl. (2006), in press; D. Bresch, B. Desjardins, D. Gérard-Varet, On compressible Navier-Stokes equations with density dependent viscosities, J. Math. Pures Appl. (2006), in press · Zbl 1121.35093 |
| [10] | Bresch, D.; Desjardins, B.; Grenier, E.; Lin, C. K., Low Mach number limit of viscous polytropic flows: Formal asymptotics in the periodic case, Stud. Appl. Math., 109, 125-148 (2002) · Zbl 1114.76347 |
| [11] | Bresch, D.; Desjardins, B.; Lin, C. K., On some compressible fluid models: Korteweg, lubrication and shallow water systems, Comm. Partial Differential Equations, 28, 3-4, 1009-1037 (2003) |
| [12] | Bresch, D.; Desjardins, B., On the construction of approximate solutions for 2D viscous shallow water model and for compressible Navier-Stokes models, J. Math. Pures Appl., 86, 4, 362-368 (2006) · Zbl 1121.35094 |
| [13] | Bush, W. B., The hypersonic approximation for the shock structure of a perfect gas with the Sutherland viscosity law, J. Mécanique, 1, 321-336 (1962) |
| [14] | Caffarelli, L.; Kohn, R.; Nirenberg, L., Partial regularity of suitable weak solutions of the Navier-Stokes equations, Comm. Pure Appl. Math., 35, 771-831 (1982) · Zbl 0509.35067 |
| [15] | Chemin, J.-Y., Remarques sur l’existence globale pour le système de Navier-Stokes incompressible, SIAM J. Math. Anal., 23, 20-28 (1992) · Zbl 0762.35063 |
| [16] | Constantin, P., Remarks on the Navier-Stokes equations, Comm. Math. Phys., 20, 2, 279-318 (1994) · Zbl 0826.35093 |
| [17] | Constantin, P.; Fefferman, C., Direction of vorticity and the problem of global regularity for the Navier-Stokes equations, Indiana Univ. Math. J., 42, 775-789 (1993) · Zbl 0837.35113 |
| [18] | Danchin, R., Global existence in critical spaces for compressible viscous and heat conductive gases, Arch. Rational Mech. Anal., 160, 1-39 (2001) · Zbl 1018.76037 |
| [19] | Desjardins, B.; Esteban, M., On weak solutions for fluid-rigid structure interaction: Compressible and incompressible models, Comm. Partial Differential Equations, 25, 1399-1413 (2000) · Zbl 0953.35118 |
| [20] | Desjardins, B.; Grenier, E., Low Mach number limit of viscous compressible flows in the whole space, R. Soc. London Proc. Ser. A Math. Phys. Engrg. Sci., 455, 2271-2279 (1999) · Zbl 0934.76080 |
| [21] | Desjardins, B.; Grenier, E.; Lions, P.-L.; Masmoudi, N., Incompressible limit for solutions of the isentropic Navier-Stokes equations with Dirichlet boundary conditions, J. Math. Pures Appl., 78, 461-471 (1999) · Zbl 0992.35067 |
| [22] | Drew, G.; Passman, R. J., Theory of Multicomponent Fluids, Applied Mathematical Sciences, vol. 135 (1998), Springer-Verlag: Springer-Verlag Berlin/New York · Zbl 0919.76003 |
| [23] | E. Feireisl, A. Novotny, On the low Mach number limit for the full Navier-Stokes Fourier system, Arch. Rational Mech. Anal. (2006), in press; E. Feireisl, A. Novotny, On the low Mach number limit for the full Navier-Stokes Fourier system, Arch. Rational Mech. Anal. (2006), in press · Zbl 1147.76049 |
| [24] | Feireisl, E.; Novotny, A.; Petzeltova, H., On the existence of globally defined weak solutions to the Navier-Stokes equations of compressible isentropic fluids, J. Math. Fluid Dynam., 3, 358-392 (2001) · Zbl 0997.35043 |
| [25] | Feireisl, E., On the motion of a viscous, compressible, and heat conducting fluid, Indiana Univ. Math. J., 53, 1707-1740 (2004) |
| [26] | Feireisl, E., Dynamics of Viscous Compressible Fluids (2004), Oxford Science Publication: Oxford Science Publication Oxford · Zbl 1080.76001 |
| [27] | Foias, C.; Temam, R., Some analytic and geometric properties of the solutions of the evolution Navier-Stokes equations, J. Math. Pures Appl., 58, 339-368 (1979) · Zbl 0454.35073 |
| [28] | Fujita, H.; Kato, T., On the Navier-Stokes initial value problem I, Arch. Rational Mech. Anal., 16, 269-315 (1964) · Zbl 0126.42301 |
| [29] | Gerbeau, J.-F.; Perthame, B., Derivation of viscous Saint-Venant system for laminar shallow water; Numerical validation, Discrete Contin. Dynam. Systems—Series B, 1, 89-102 (2001) · Zbl 0997.76023 |
| [30] | O. Heuzé, Building of equations of state with numerous phase transitions—application to bismuth, in: 14th APS Topical Conference on Shock Compression of Condensed Matter, 2005; O. Heuzé, Building of equations of state with numerous phase transitions—application to bismuth, in: 14th APS Topical Conference on Shock Compression of Condensed Matter, 2005 |
| [31] | Hoff, D., Discontinuous solutions of the Navier-Stokes equations for multidimensional flows of heat conducting fluids, Arch. Rational Mech. Anal., 139, 303-354 (1997) · Zbl 0904.76074 |
| [32] | Hoff, D., Global solutions of the Navier-Stokes equations for multi-dimensional compressible flow with discontinuous initial data, J. Differential Equations, 120, 1, 215-254 (1995) · Zbl 0836.35120 |
| [33] | Hoff, D., 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, 1, 1-14 (1995) · Zbl 0836.76082 |
| [34] | Hoff, D.; Serre, D., The failure of continuous dependence on initial data for the Navier-Stokes equations of compressible flows, SIAM J. Appl. Math., 51, 887-898 (1991) · Zbl 0741.35057 |
| [35] | Ishii, M.; Zuber, N., Drag coefficient and relative velocity in bubbly, droplet or particulate flows, AIChE J., 25, 843-854 (1979) |
| [36] | Jiang, S., Global smooth solutions of the equations of a viscous, heat-conducting, one-dimensional gas with density-dependent viscosity, Math. Nachr., 190, 169-183 (1998) · Zbl 0927.35014 |
| [37] | S. Jiang, Z.P. Xin, P. Zhang, Global weak solutions to 1D compressible isentropic Navier-Stokes equations with density dependent viscosity, Methods Appl. Anal. (2005), in press; S. Jiang, Z.P. Xin, P. Zhang, Global weak solutions to 1D compressible isentropic Navier-Stokes equations with density dependent viscosity, Methods Appl. Anal. (2005), in press · Zbl 1110.35058 |
| [38] | Kazhikhov, A. V., The equation of potential flows of a compressible viscous fluid for small Reynolds numbers: existence, uniqueness and stabilization of solutions, Sibirsk. Mat. Zh., 34, 3, 70-80 (1993) · Zbl 0806.76077 |
| [39] | Kazhikhov, A.; Shelukhin, V., Unique global solution with respect to time of initial-boundary-value problems for one-dimensional equations of a viscous gas, J. Appl. Math. Mech., 41, 273-282 (1977) · Zbl 0393.76043 |
| [40] | Korteweg, D. J., Sur la forme que prennent les équations du mouvement des fluides si l’on tient compte des forces capillaires causées par des variations de densité considérables mais continues et sur la théorie de la capillarité dans l’hypothèse d’une variation continue de la densité, Archives Néerlandaises de Sciences Exactes et Naturelles, 1-24 (1901) · JFM 32.0756.02 |
| [41] | Ladyzhenskaya, O., The Mathematical Theory of Viscous Incompressible Flow (1969), Gordon and Breach: Gordon and Breach London · Zbl 0184.52603 |
| [42] | Leray, J., Sur le mouvement d’un liquide visqueux emplissant l’espace, Acta Math., 63, 193-248 (1934) · JFM 60.0726.05 |
| [43] | 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 |
| [44] | Lions, P.-L., Mathematical Topics in Fluid Dynamics, vol. 2, Compressible Models (1998), Oxford Science Publication: Oxford Science Publication Oxford · Zbl 0908.76004 |
| [45] | Liu, T.-P.; Xin, Z.-P.; Yang, T., Vacuum states for compressible flow, Discrete Contin. Dynam. Systems, 4, 1-32 (1998) · Zbl 0970.76084 |
| [46] | J.P. Vila, On the derivation of the Shallow Water model, 2006, in preparation; J.P. Vila, On the derivation of the Shallow Water model, 2006, in preparation |
| [47] | A. Matsumura, S. Yanagi, Uniform boundedness of the solutions for a one-dimensional isentropic model system of compressible viscous gas; A. Matsumura, S. Yanagi, Uniform boundedness of the solutions for a one-dimensional isentropic model system of compressible viscous gas · Zbl 0939.35153 |
| [48] | Matsumura, A.; Nishida, T., The initial value problem for the equations of motion of compressible and heat conductive fluids, Commun. Math. Phys., 89, 445-464 (1983) · Zbl 0543.76099 |
| [49] | A. Mellet, A. Vasseur, On the isentropic compressible Navier-Stokes equations, Comm. Partial Differential Equations (2006), submitted for publication; A. Mellet, A. Vasseur, On the isentropic compressible Navier-Stokes equations, Comm. Partial Differential Equations (2006), submitted for publication · Zbl 1149.35070 |
| [50] | Métivier, G.; Schochet, S., The incompressible limit of the non-isentropic Euler equations, Arch. Rational Mech. Anal., 158, 61-90 (2001) · Zbl 0974.76072 |
| [51] | S. Novo, Quelques problèmes aux limites pour des équations de Navier-Stokes compressibles et isentropiques, Thèse Université de Toulon et du Var, 2002; S. Novo, Quelques problèmes aux limites pour des équations de Navier-Stokes compressibles et isentropiques, Thèse Université de Toulon et du Var, 2002 |
| [52] | Novotny, A.; Straskraba, I., Introduction to the Mathematical Theory of Compressible Flow, Oxford Lecture Ser. Math. Appl. (2004), Oxford Univ. Press: Oxford Univ. Press Oxford · Zbl 1088.35051 |
| [53] | Solonnikov, V. A., Estimates for solutions of nonstationary Navier-Stokes systems, Zap. Nauchn. Sem. LOMI. Zap. Nauchn. Sem. LOMI, J. Soviet Math., 8, 467-529 (1977) · Zbl 0404.35081 |
| [54] | Tannehil, J. C.; Anderson, D. A.; Pletcher, R. H., Computational Fluid Mechanics and Heat Transfer (1997), Taylor and Francis: Taylor and Francis London, p. 259 |
| [55] | Valli, A.; Zajaczkowski, W., Navier-Stokes equations for compressible fluids: global existence and qualitative properties of the solutions in the general case, Commun. Math. Phys., 103, 2, 259-296 (1986) · Zbl 0611.76082 |
| [56] | Veigant, A.; Kazhikhov, A., On the existence of global solution to the two-dimensional Navier-Stokes equations for a compressible viscous flow, Siberian Math. J., 36, 1108-1141 (1995) · Zbl 0860.35098 |
| [57] | Wang, D., Global solution to the equations of viscous gas flows, Proc. Royal Soc. Edinburgh, 131, 437-449 (2001) · Zbl 0986.76076 |
| [58] | Yang, T., Some recent results on compressible flow with vacuum, Taiwanese J. Math., 4, 33-44 (2000) · Zbl 0958.35111 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
