Asymptotic analysis of the full Navier–Stokes–Fourier system: From compressible to incompressible fluid flows

E. Feireisl

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 62 (2007) no. 3, pp. 511-533

Résumé

This is a survey of new results related to the study of the full Navier–Stokes–Fourier system for a general compressible, viscous, and heat conducting fluid, and its asymptotic behaviour as the Mach number approaches zero. The classical Navier–Stokes system for an incompressible fluid with lift, combined with the corresponding heat equation, is a limiting case.

Bibliographie

[1] R. Klein, N. Botta, T. Schneider, C. D. Munz, S. Roller, A. Meister, L. Hoffmann, T. Sonar, “Asymptotic adaptive methods for multi-scale problems in fluid mechanics”, J. Engrg. Math., 39:1 (2001), 261–343 | DOI | MR | Zbl

[2] R. Klein, “Asymptotic analyses for atmospheric flows and the construction of asymptotically adaptive numerical methods”, ZAMM Z. Angew. Math. Mech., 80:11–12 (2000), 765–777 | 3.0.CO;2-1 class='badge bg-secondary rounded-pill ref-badge extid-badge'>DOI | MR | Zbl

[3] A. Majda, Introduction to PDEs and waves for the atmosphere and ocean, Courant Lect. Notes Math., 9, Amer. Math. Soc., Providence, RI, 2003 | MR | Zbl

[4] G. L. Eyink, “Local $4/5$-law and energy dissipation anomaly in turbulence”, Nonlinearity, 16:1 (2003), 137–145 | DOI | MR | Zbl

[5] T. Nagasawa, “A new energy inequality and partial regularity for weak solutions of Navier–Stokes equations”, J. Math. Fluid Mech., 3:1 (2001), 40–56 | DOI | MR | Zbl

[6] G. P. Galdi, An introduction to the mathematical theory of the Navier–Stokes equations. Vol. 1. Linearized steady problems, Springer Tracts Nat. Philos., 38, Springer-Verlag, New York, 1994 | MR | Zbl

[7] S. Klainerman, A. Majda, “Singular limits of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids”, Comm. Pure Appl. Math., 34:4 (1981), 481–524 | DOI | MR | Zbl

[8] D. B. Ebin, “The motion of slightly compressible fluids viewed as a motion with strong constraining force”, Ann. of Math. (2), 105:1 (1977), 141–200 | DOI | MR | Zbl

[9] T. Alazard, “Low Mach number flows and combustion”, SIAM J. Math. Anal., 38:4 (2006), 1186–1213 | DOI | MR | Zbl

[10] T. Alazard, “Low Mach number limit of the full Navier–Stokes equations”, Arch. Ration. Mech. Anal., 180:1 (2006), 1–73 | DOI | MR | Zbl

[11] R. Danchin, “Low Mach number limit for viscous compressible flows”, M2AN Math. Model. Numer. Anal., 39:3 (2005), 459–475 | DOI | MR | Zbl

[12] G. Métivier, S. Schochet, “The incompressible limit of the non-isentropic Euler equations”, Arch. Ration. Mech. Anal., 158:1 (2001), 61–90 | DOI | MR | Zbl

[13] S. Schochet, “The mathematical theory of low Mach number flows”, M2AN Math. Model. Numer. Anal., 39:3 (2005), 441–458 | DOI | MR | Zbl

[14] P.-L. Lions, N. Masmoudi, “Incompressible limit for a viscous compressible fluid”, J. Math. Pures Appl. (9), 77:6 (1998), 585–627 | DOI | MR | Zbl

[15] P.-L. Lions, N. Masmoudi, “On a free boundary barotropic model”, Ann. Inst. H. Poincaré Anal. Non Linéaire, 16:3 (1999), 373–410 | DOI | MR | Zbl

[16] B. Desjardins, E. Grenier, P.-L. Lions, N. Masmoudi, “Incompressible limit for solutions of the isentropic Navier–Stokes equations with Dirichlet boundary conditions”, J. Math. Pures Appl. (9), 78:5 (1999), 461–471 | DOI | MR | Zbl

[17] D. Bresch, B. Desjardins, E. Grenier, C.-K. Lin, “Low Mach number limit of viscous polytropic flows: Formal asymptotic in the periodic case”, Stud. Appl. Math., 109:2 (2002), 125–149 | DOI | MR | Zbl

[18] J. Leray, “Sur le mouvement d'un liquide visqueux emplissant l'espace”, Acta Math., 63:1 (1934), 193–248 | DOI | MR | Zbl

[19] E. Feireisl, Dynamics of viscous compressible fluids, Oxford Lecture Ser. Math. Appl., 26, Oxford Univ. Press, Oxford, 2004 | MR | Zbl

[20] E. Feireisl, “Mathematical theory of compressible, viscous, and heat conducting fluids”, Comput. Appl. Math. (to appear)

[21] E. Feireisl, “Stability of flows of real monoatomic gases”, Comm. Partial Differential Equations, 31:2 (2006), 325–348 | DOI | MR | Zbl

[22] R. Kh. Zeytounian, “Joseph Boussinesq and his approximation: a contemporary view”, C. R. Mécanique, 331:8 (2003), 575–586 | DOI

[23] E. Feireisl, A. Novotný, H. Petzeltová, “On the incompressible limit for the Navier–Stokes–Fourier system in domains with wavy bottoms”, Arch. Ration. Mech. Anal. (to appear)

[24] E. Feireisl, J. Málek, A. Novotný, I. Straškraba, “Anelastic approximation as a singular limit of the compressible Navier–Stokes system”, Comm. Partial Differential Equations (to appear)

[25] D. R. Durran, “Improving the anelastic approximation”, J. Atmospheric Sci., 46:11 (1989), 1453–1461 | 2.0.CO;2 class='badge bg-secondary rounded-pill ref-badge extid-badge'>DOI

[26] F. B. Lipps, R. S. Hemler, “A scale analysis of deep moist convection and some related numerical calculations”, J. Atmospheric Sci., 39:10 (1982), 2192–2210 | 2.0.CO;2 class='badge bg-secondary rounded-pill ref-badge extid-badge'>DOI

[27] O. A. Ladyzhenskaya, Matematicheskie voprosy dinamiki neszhimaemoi zhidkosti, 2-e izd., Nauka, M., 1970; O. A. Ladyzhenskaya, The mathematical theory of viscous incompressible flow, Gordon and Breach, New York–London–Paris, 1969 | MR | Zbl

[28] R. Temam, Navier–Stokes equations. Theory and numerical analysis, Stud. Math. Appl., 2, North-Holland, Amsterdam–New York–Oxford, 1977 | MR | Zbl

[29] L. Caffarelli, R. V. Kohn, L. Nirenberg, “Partial regularity of suitable weak solutions of the Navier–Stokes equations”, Comm. Pure Appl. Math., 35:6 (1982), 771–831 | DOI | MR | Zbl

[30] S. N. Antontsev, A. V. Kazhikhov, V. N. Monakhov, Kraevye zadachi mekhaniki neodnorodnykh zhidkostei, Nauka, Novosibirsk, 1983 ; S. N. Antontsev, A. V. Kazhikhov, V. N. Monakhov, Boundary value problems in mechanics of nonhomogeneous fluids, Stud. Math. Appl., 22, North-Holland, Amsterdam, 1990 | MR | Zbl | MR | Zbl

[31] P.-L. Lions, Mathematical topics in fluid mechanics. Vol. 1. Incompressible models, Oxford Lecture Ser. Math. Appl., 3, Clarendon Press, Oxford, 1996 | MR | Zbl

[32] B. Desjardins, “Regularity of weak solutions of the compressible isentropic Navier–Stokes equations”, Comm. Partial Differential Equations, 22:5–6 (1997), 977–1008 | DOI | MR | Zbl

[33] D. Hoff, “Discontinuous solutions of the Navier–Stokes equations for multidimensional flows of heat-conducting fluids”, Arch. Ration. Mech. Anal., 139:4 (1997), 303–354 | DOI | MR | Zbl

[34] D. Hoff, “Dynamics of singularity surfaces for compressible viscous flows in two space dimensions”, Comm. Pure Appl. Math., 55:11 (2002), 1365–1407 | DOI | MR | Zbl

[35] D. Hoff, D. Serre, “The failure of continuous dependence on initial data for the Navier–Stokes equations of compressible flow”, SIAM J. Appl. Math., 51:4 (1991), 887–898 | DOI | MR | Zbl

[36] V. A. Vaigant, “Primer nesuschestvovaniya v tselom po vremeni resheniya uravnenii Nave–Stoksa szhimaemoi vyazkoi barotropnoi zhidkosti”, Dokl. AN SSSR, 339:2 (1994), 155–156 ; V. A. Vaigant, “An example of the nonexistence with respect to time of the global solutions of Navier–Stokes equations for a compressible viscous barotropic fluid”, Russian Acad. Sci. Dokl. Math., 50:3 (1995), 397–399 | MR

[37] R. J. DiPerna, “Measure-valued solutions to conservation laws”, Arch. Ration. Mech. Anal., 88:3 (1985), 223–270 | DOI | MR | Zbl

[38] J. Málek, J. Nečas, M. Rokyta, M. Ružička, Weak and measure-valued solutions to evolutionary PDEs, Appl. Math. Math. Comput., 13, Chapman and Hall, London, 1996 | MR | Zbl

[39] P.-L. Lions, Mathematical topics in fluid mechanics. Vol. 2. Compressible models, Oxford Lecture Ser. Math. Appl., 10, Clarendon Press, Oxford, 1998 | MR | Zbl

[40] V. A. Vaigant, A. V. Kazhikhov, “O suschestvovanii globalnykh reshenii dvumernykh uravnenii Nave–Stoksa szhimaemoi vyazkoi zhidkosti”, Sib. matem. zhurn., 36:6 (1995), 1283–1316 | MR | Zbl

[41] D. Hoff, “Strong convergence to global solutions for multidimensional flows of compressible, viscous fluids with polytropic equations of state and discontinuous initial data”, Arch. Ration. Mech. Anal., 132:1 (1995), 1–14 | DOI | MR | Zbl

[42] D. Serre, “Variations de grande amplitude pour la densité d'un fluid viscueux compressible”, Phys. D, 48:1 (1991), 113–128 | DOI | MR | Zbl

[43] A. Novotný, I. Straškraba, Introduction to the mathematical theory of compressible flow, Oxford Lecture Ser. Math. Appl., 27, Oxford Univ. Press, Oxford, 2004 | MR | Zbl

[44] E. Feireisl, A. Novotný, “On a simple model of reacting compressible flows arising in astrophysics”, Proc. Roy. Soc. Edinburgh Sect. A, 135:6 (2005), 1169–1194 | DOI | MR | Zbl

[45] E. Becker, Gasdynamik, Teubner, Stuttgart, 1966 | MR | Zbl

[46] C. Buet, B. Després, “Asymptotic analysis of fluid models for the coupling of radiation and hydrodynamics”, J. Quant. Spectroscopy Radiative Transfer, 85:3–4 (2004), 385–418 | DOI

[47] J. Oxenius, Kinetic theory of particles and photons. Theoretical foundations of non-LTE plasma spectroscopy, Springer Ser. Electrophys., 20, Springer-Verlag, Berlin, 1986 | MR

[48] Ya. B. Zeldovich, Yu. P. Raizer, Fizika udarnykh voln i vysokotemperaturnykh gidrodinamicheskikh yavlenii, Fizmatgiz, M., 1963 ; Ya. B. Zel'dovich, Yu. P. Raizer, Physics of shock waves and high-temperature hydrodynamic phenomena, Academic Press, New York, 1966 | Zbl

[49] S. Eliezer, A. Ghatak, H. Hora, An introduction to equations of states: theory and applications, Cambridge Univ. Press, Cambridge, 1986

[50] I. Müller, T. Ruggeri, Rational extended thermodynamics, Springer Tracts Nat. Philos., 37, Springer-Verlag, New York, 1998 | MR | Zbl

[51] G. Gallavotti, Statistical mechanics. A short treatise, Texts Monogr. Phys., Springer-Verlag, Berlin, 1999 | MR | Zbl

[52] S. E. Bechtel, F. J. Rooney, M. G. Forest, “Connection between stability, convexity of internal energy, and the second law for compressible Newtonian fuids”, J. Appl. Mech., 72:2 (2005), 299–300 | DOI | MR | Zbl

[53] R. J. DiPerna, P.-L. Lions, “Ordinary differential equations, transport theory and Sobolev spaces”, Invent. Math., 98:3 (1989), 511–547 | DOI | MR | Zbl

[54] B. Ducomet, E. Feireisl, “On the dynamics of gaseous stars”, Arch. Ration. Mech. Anal., 174:2 (2004), 221–266 | DOI | MR | Zbl

[55] K. R. Rajagopal, M. Ružička, A. R. Shrinivasa, “On the Oberbeck–Boussinesq approximation”, Math. Models Methods Appl. Sci., 6:8 (1996), 1157–1167 | DOI | MR | Zbl

[56] E. Feireisl, A. Novotný, “The Oberbeck–Boussinesq approximation as a singular limit of the full Navier–Stokes–Fourier system”, J. Math. Fluid Mech. (to appear)

[57] E. Feireisl, A. Novotný, “On the low Mach number limit for the full Navier–Stokes–Fourier system”, Arch. Ration. Mech. Anal. (to appear) | MR

[58] S. Schochet, “Fast singular limits of hyperbolic PDEs”, J. Differential Equations, 114:2 (1994), 476–512 | DOI | MR | Zbl

[59] N. Masmoudi, “Asymptotic problems and compressible and incompressible limits”, Advances in mathematical fluid mechanics (Paseky, Czech Republic, 1999), eds. J. Málek, J. Nečas, M. Rokyta, Springer, Berlin, 2000, 119–158 | MR | Zbl

[60] N. Garofalo, F. Segàla, “Another step toward the solution of the Pompeiu problem in the plane”, Comm. Partial Differential Equations, 18:3–4 (1993), 491–503 | DOI | MR | Zbl

[61] S. A. Williams, “Analyticity of the boundary for Lipschitz domains without Pompeiu property”, Indiana Univ. Math. J., 30:3 (1981), 357–369 | DOI | MR | Zbl

[62] R. Dalmasso, “A new result on the Pompeiu problem”, Trans. Amer. Math. Soc., 352:6 (2000), 2723–2736 | DOI | MR | Zbl

[63] M. I. Vishik, L. A. Lyusternik, “Regulyarnoe vyrozhdenie i pogranichnyi sloi dlya lineinykh differentsialnykh uravnenii s malym parametrom”, UMN, 12:5(77) (1957), 3–122 | MR | Zbl

[64] E. Feireisl, A. Novotný, H. Petzeltová, “On the existence of globally defined weak solutions to the Navier–Stokes equations”, J. Math. Fluid Mech., 3:4 (2001), 358–392 | DOI | MR | Zbl

[65] E. Feireisl, “Compressible Navier–Stokes equations with a non-monotone pressure law”, J. Differential Equations, 184:1 (2002), 97–108 | DOI | MR | Zbl

[66] A. Majda, “High Mach number combustion”, Reacting flows: combustion and chemical reactors, Part 1 (Ithaca, 1985), Lectures in Appl. Math., 24, Amer. Math. Soc., Providence, RI, 1986, 109–184 | MR