Steady compressible Navier-Stokes-Fourier system in two space dimensions

Petra Pecharová; Milan Pokorný

Commentationes Mathematicae Universitatis Carolinae (2010)

  • Volume: 51, Issue: 4, page 653-679
  • ISSN: 0010-2628

Abstract

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We study steady flow of a compressible heat conducting viscous fluid in a bounded two-dimensional domain, described by the Navier-Stokes-Fourier system. We assume that the pressure is given by the constitutive equation p ( ρ , θ ) ∼ ρ γ + ρ θ , where ρ is the density and θ is the temperature. For γ > 2 , we prove existence of a weak solution to these equations without any assumption on the smallness of the data. The proof uses special approximation of the original problem, which guarantees the pointwise boundedness of the density. Thus we get a solution with density in L ∞ ( Ω ) and temperature and velocity in W 1 , q ( Ω ) for any q < ∞ .

How to cite

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Pecharová, Petra, and Pokorný, Milan. "Steady compressible Navier-Stokes-Fourier system in two space dimensions." Commentationes Mathematicae Universitatis Carolinae 51.4 (2010): 653-679. <http://eudml.org/doc/247028>.

@article{Pecharová2010,
abstract = {We study steady flow of a compressible heat conducting viscous fluid in a bounded two-dimensional domain, described by the Navier-Stokes-Fourier system. We assume that the pressure is given by the constitutive equation $p(\rho , \theta ) \sim \rho ^\gamma + \rho \theta $, where $\rho $ is the density and $\theta $ is the temperature. For $\gamma > 2$, we prove existence of a weak solution to these equations without any assumption on the smallness of the data. The proof uses special approximation of the original problem, which guarantees the pointwise boundedness of the density. Thus we get a solution with density in $L^\infty (\Omega )$ and temperature and velocity in $W^\{1,q\} (\Omega )$ for any $q < \infty $.},
author = {Pecharová, Petra, Pokorný, Milan},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {steady compressible Navier--Stokes--Fourier equations; slip boundary condition; weak solutions; large data; steady compressible Navier-Stokes-Fourier equations; slip boundary condition; weak solution; large data},
language = {eng},
number = {4},
pages = {653-679},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Steady compressible Navier-Stokes-Fourier system in two space dimensions},
url = {http://eudml.org/doc/247028},
volume = {51},
year = {2010},
}

TY - JOUR
AU - Pecharová, Petra
AU - Pokorný, Milan
TI - Steady compressible Navier-Stokes-Fourier system in two space dimensions
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2010
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 51
IS - 4
SP - 653
EP - 679
AB - We study steady flow of a compressible heat conducting viscous fluid in a bounded two-dimensional domain, described by the Navier-Stokes-Fourier system. We assume that the pressure is given by the constitutive equation $p(\rho , \theta ) \sim \rho ^\gamma + \rho \theta $, where $\rho $ is the density and $\theta $ is the temperature. For $\gamma > 2$, we prove existence of a weak solution to these equations without any assumption on the smallness of the data. The proof uses special approximation of the original problem, which guarantees the pointwise boundedness of the density. Thus we get a solution with density in $L^\infty (\Omega )$ and temperature and velocity in $W^{1,q} (\Omega )$ for any $q < \infty $.
LA - eng
KW - steady compressible Navier--Stokes--Fourier equations; slip boundary condition; weak solutions; large data; steady compressible Navier-Stokes-Fourier equations; slip boundary condition; weak solution; large data
UR - http://eudml.org/doc/247028
ER -

References

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  10. Novotný A., Straškraba I., Introduction to The Mathematical Theory of Compressible Flow, Oxford Lecture Series in Mathematics and its Applications, 27, Oxford University Press, Oxford, 2004. MR2084891
  11. Pecharová P., Steady compressible Navier–Stokes–Fourier equations in two space dimensions, Master Degree Thesis, MFF UK, Praha, 2009. 
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