Article
MSC:
35A01,
35D30,
35Q30,
35Q35,
76N10 | MR 2807430 | Zbl 1224.76108 | DOI: 10.1007/s10492-011-0013-4
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Keywords:
steady compressible Navier-Stokes-Fourier system; weak solution; entropy inequality; Orlicz spaces; compensated compactness; renormalized solution
steady compressible Navier-Stokes-Fourier system; weak solution; entropy inequality; Orlicz spaces; compensated compactness; renormalized solution
Summary:
We consider steady compressible Navier-Stokes-Fourier system in a bounded two-dimensional domain. We show the existence of a weak solution for arbitrarily large data for the pressure law $p(\varrho ,\vartheta ) \sim \varrho ^\gamma + \varrho \vartheta $ if $\gamma >1$ and $p(\varrho ,\vartheta ) \sim \varrho \ln ^\alpha (1+\varrho ) + \varrho \vartheta $ if $\gamma =1$, $\alpha >0$, depending on the model for the heat flux.
References:
[1] Erban, R.: On the existence of solutions to the Navier-Stokes equations of a two-dimensional compressible flow. Math. Methods Appl. Sci. 26 (2003), 489-517. DOI 10.1002/mma.362 | MR 1965718 | Zbl 1061.76073
[2] Feireisl, E., Novotný, A.: Singular Limits in Thermodynamics of Viscous Fluids. Advances in Mathematical Fluid Mechanics. Birkhäuser Basel (2009). MR 2499296
[3] Frehse, J., Steinhauer, M., Weigant, W.: The Dirichlet Problem for Steady Viscous Compressible Flow in 3-D. Preprint University of Bonn, SFB 611, No. 347 (2007), http://www.iam.uni-bonn.de/sfb611/
[4] Frehse, J., Steinhauer, M., Weigant, W.: The Dirichlet problem for viscous compressible isothermal Navier-Stokes equations in two-dimensions. Arch. Ration. Mech. Anal. 198 (2010), 1-12. DOI 10.1007/s00205-010-0338-2 | MR 2679367 | Zbl 1229.35175
[5] Kufner, A., John, O., Fučík, S.: Function spaces. Academia Praha (1977). MR 0482102
[6] Lions, P.-L.: Mathematical Topics in Fluid Dynamics, Vol. 2: Compressible Models. Oxford Lecture Series in Mathematics and Its Applications, Vol. 10. Clarendon Press Oxford (1998). MR 1637634
[7] Maligranda, L.: Orlicz Spaces and Interpolation. Univ. Estadual de Campinas Campinas (1989). MR 2264389 | Zbl 0874.46022
[8] Mucha, P. B., Pokorný, M.: On the steady compressible Navier-Stokes-Fourier system. Commun. Math. Phys. 288 (2009), 349-377. DOI 10.1007/s00220-009-0772-x | MR 2491627 | Zbl 1172.35467
[9] Mucha, P. B., Pokorný, M.: Weak solutions to equations of steady compressible heat conducting fluids. Math. Models Methods Appl. Sci. 20 (2010), 785-813. DOI 10.1142/S0218202510004441 | MR 2652619 | Zbl 1191.35207
[10] Novotný, A., Pokorný, M.: Steady compressible Navier-Stokes-Fourier system for monoatomic gas and its generalizations. J. Differ. Equations Accepted. See also Preprint Series of Nečas Center for Mathematical Modeling, http://www.karlin.mff.cuni.cz/ncmm/research/Preprints/servirPrintsYY.php?y=2010 Preprint No. 2010-021.
[11] Novotný, A., Pokorný, M.: Weak and variational solutions to steady equations for compressible heat conducting fluids. Submitted. See also Preprint Series of Nečas Center for Mathematical Modeling, http://www.karlin.mff.cuni.cz/ncmm/research/Preprints/servirPrintsYY.php?y=2010 Preprint No. 2010-023.
[12] Novotný, A., Straškraba, I.: Introduction to the Mathematical Theory of Compressible Flow. Oxford University Press Oxford (2004). MR 2084891 | Zbl 1088.35051
[13] Pecharová, P., Pokorný, M.: Steady compressible Navier-Stokes-Fourier system in two space dimensions. Commentat. Math. Univ. Carol (to appear). MR 2858268
[14] Plotnikov, P. I., Sokołowski, J.: On compactness, domain dependence and existence of steady state solutions to compressible isothermal Navier-Stokes equations. J. Math. Fluid Mech. 7 (2005), 529-573. DOI 10.1007/s00021-004-0134-6 | MR 2189674 | Zbl 1090.35140
[15] Plotnikov, P. I., Sokołowski, J.: Concentrations of stationary solutions to compressible Navier-Stokes equations. Commun. Math. Phys. 258 (2005), 567-608. DOI 10.1007/s00220-005-1358-x | MR 2172011
[16] Plotnikov, P. I., Sokołowski, J.: Stationary solutions of Navier-Stokes equations for diatomic gases. Russ. Math. Surv. 62 (2007), 561-593. DOI 10.1070/RM2007v062n03ABEH004414 | MR 2355421 | Zbl 1139.76049
[17] Vodák, R.: The problem $\div v=f$ and singular integrals on Orlicz spaces. Acta Univ. Palacki. Olomuc, Fac. Rerum Nat. Math. 41 (2002), 161-173. MR 1968228
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