Abstract
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.
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References
Batchelor G.K.: An introduction to fluid dynamics. Cambridge University Press, Cambridge (1967)
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)
Březina, J., Novotný, A.: On Weak Solutions of Steady Navier-Stokes Equations for Monatomic Gas. preprint, http://ncmm.karlin.mff.cuni.cz/research/Preprints
Ducomet B., Feireisl E.: On the dynamics of gaseous stars. Arch. Rat. Mech. Anal. 174(2), 221–266 (2004)
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
Feireisl, E.: Dynamics of viscous compressible fluids. Oxford Lecture Series in Mathematics and its Applications 26, Oxford: Oxford University Press, 2004
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)
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)
Lions, P.L.: Mathematical Topics in Fluid Mechanics, Vol. 2: Compressible Models. Oxford: Oxford Science Publications, 1998
Mucha P.B.: On cylindrical symmetric flows through pipe-like domains. J. Diff. Eq. 201(2), 304–323 (2004)
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)
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)
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)
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)
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)
Novotný A., Straškraba I.: Mathematical Theory of Compressible Flows. Oxford Science Publications, Oxford (2004)
Pokorný M., Mucha P.B.: 3D steady compressible Navier–Stokes equations. Cont. Discr. Dyn. Systems S1, 151–163 (2008)
Solonnikov V.A.: Overdetermined elliptic boundary value problems. Zap. Nauch. Sem. LOMI 21, 112–158 (1971)
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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Mucha, P.B., Pokorný, M. On the Steady Compressible Navier–Stokes–Fourier System. Commun. Math. Phys. 288, 349–377 (2009). https://doi.org/10.1007/s00220-009-0772-x
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DOI: https://doi.org/10.1007/s00220-009-0772-x