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On the steady compressible Navier-Stokes-Fourier system. (English) Zbl 1172.35467

Summary: We study the motion of the steady compressible heat conducting viscous fluid in a bounded three dimensional domain governed by the compressible Navier-Stokes-Fourier system. Our main result is the existence of a weak solution to these equations for arbitrarily large data. A key element of the proof is a special approximation of the original system guaranteeing pointwise uniform boundedness of the density as well as the positiveness of the temperature. Therefore the passage to the limit omits tedious technical tricks required by the standard theory. Basic estimates on the solutions are possible to obtain by a suitable choice of physically reasonable boundary conditions.

MSC:

35Q35 PDEs in connection with fluid mechanics
76N15 Gas dynamics (general theory)
76N10 Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics
35A35 Theoretical approximation in context of PDEs
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[1] Batchelor G.K.: An introduction to fluid dynamics. Cambridge University Press, Cambridge (1967) · Zbl 0152.44402
[2] Bause M., Heywood J.G., Novotný A., Padula M.: On some approximation schemes for steady compressible viscous flow. J. Math. Fluid Mech. 5(3), 201–230 (2003) · Zbl 1036.76054
[3] Březina, J., Novotný, A.: On Weak Solutions of Steady Navier-Stokes Equations for Monatomic Gas. preprint, http://ncmm.karlin.mff.cuni.cz/research/Preprints
[4] Ducomet B., Feireisl E.: On the dynamics of gaseous stars. Arch. Rat. Mech. Anal. 174(2), 221–266 (2004) · Zbl 1085.76061 · doi:10.1007/s00205-004-0326-5
[5] Frehse, J., Steinhauer, M., Weigant, V.: On Stationary Solutions for 2 - D Viscous Compressible Isothermal Navier-Stokes Equations. preprint, http://ncmm.karlin.mff.cuni.cz/research/Preprints · Zbl 1270.35341
[6] Feireisl, E.: Dynamics of viscous compressible fluids. Oxford Lecture Series in Mathematics and its Applications 26, Oxford: Oxford University Press, 2004 · Zbl 1080.76001
[7] Feireisl E., Novotný A., Petzeltová H.: On a class of physically admissible variational solutions to the Navier-Stokes-Fourier system. Z. Anal. Anwendungen 24(1), 75–101 (2005) · Zbl 1081.35075 · doi:10.4171/ZAA/1230
[8] Feireisl E., Novotný A.: Large time behaviour of flows of compressible, viscous, and heat conducting fluids. Math. Methods Appl. Sci. 29(11), 1237–1260 (2006) · Zbl 1105.35075 · doi:10.1002/mma.722
[9] Lions, P.L.: Mathematical Topics in Fluid Mechanics, Vol. 2: Compressible Models. Oxford: Oxford Science Publications, 1998 · Zbl 0908.76004
[10] Mucha P.B.: On cylindrical symmetric flows through pipe-like domains. J. Diff. Eq. 201(2), 304–323 (2004) · Zbl 1057.35032 · doi:10.1016/j.jde.2004.03.007
[11] Mucha P.B., Pokorný M.: On a new approach to the issue of existence and regularity for the steady compressible Navier–Stokes equations. Nonlinearity 19(8), 1747–1768 (2006) · Zbl 1189.35262 · doi:10.1088/0951-7715/19/8/003
[12] Mucha P.B., Rautmann R.: Convergence of Rothe’s scheme for the Navier-Stokes equations with slip conditions in 2D domains. ZAMM Z. Angew. Math. Mech. 86(9), 691–701 (2006) · Zbl 1108.76021 · doi:10.1002/zamm.200510274
[13] Novo S., Novotný A.: On the existence of weak solutions to the steady compressible Navier-Stokes equations when the density is not square integrable. J. Math. Kyoto Univ. 42(3), 531–550 (2002) · Zbl 1050.35074
[14] Novo S., Novotný A., Pokorný M.: Steady compressible Navier-Stokes equations in domains with non-compact boundaries. Math. Methods Appl. Sci. 28(12), 1445–1479 (2005) · Zbl 1075.35040 · doi:10.1002/mma.623
[15] Novotný A., Padula M.: L p -approach to steady flows of viscous compressible fluids in exterior domains. Arch. Rat. Mech. Anal. 126(3), 243–297 (1994) · Zbl 0809.76080 · doi:10.1007/BF00375644
[16] Novotný A., Straškraba I.: Mathematical Theory of Compressible Flows. Oxford Science Publications, Oxford (2004)
[17] Pokorný M., Mucha P.B.: 3D steady compressible Navier–Stokes equations. Cont. Discr. Dyn. Systems S1, 151–163 (2008) · Zbl 1140.76034
[18] Solonnikov V.A.: Overdetermined elliptic boundary value problems. Zap. Nauch. Sem. LOMI 21, 112–158 (1971) · Zbl 0231.35029
[19] Zajączkowski, W.: Existence and regularity of some elliptic systems in domains with edges. Dissertationes Math. (Rozprawy Mat.) 274 (1989) 95 pp
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