An existence proof for the stationary compressible Stokes problem

A. Fettah; T. Gallouët; H. Lakehal

Annales de la faculté des sciences de Toulouse Mathématiques (2014)

  • Volume: 23, Issue: 4, page 847-875
  • ISSN: 0240-2963

Abstract

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In this paper, we prove the existence of a solution for a quite general stationary compressible Stokes problem including, in particular, gravity effects. The Equation Of State gives the pressure as an increasing superlinear function of the density. This existence result is obtained by passing to the limit on the solution of a viscous approximation of the continuity equation.

How to cite

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Fettah, A., Gallouët, T., and Lakehal, H.. "An existence proof for the stationary compressible Stokes problem." Annales de la faculté des sciences de Toulouse Mathématiques 23.4 (2014): 847-875. <http://eudml.org/doc/275296>.

@article{Fettah2014,
abstract = {In this paper, we prove the existence of a solution for a quite general stationary compressible Stokes problem including, in particular, gravity effects. The Equation Of State gives the pressure as an increasing superlinear function of the density. This existence result is obtained by passing to the limit on the solution of a viscous approximation of the continuity equation.},
author = {Fettah, A., Gallouët, T., Lakehal, H.},
journal = {Annales de la faculté des sciences de Toulouse Mathématiques},
keywords = {convection-diffusion; regularized problem; existence; compressible Stokes problem},
language = {eng},
number = {4},
pages = {847-875},
publisher = {Université Paul Sabatier, Toulouse},
title = {An existence proof for the stationary compressible Stokes problem},
url = {http://eudml.org/doc/275296},
volume = {23},
year = {2014},
}

TY - JOUR
AU - Fettah, A.
AU - Gallouët, T.
AU - Lakehal, H.
TI - An existence proof for the stationary compressible Stokes problem
JO - Annales de la faculté des sciences de Toulouse Mathématiques
PY - 2014
PB - Université Paul Sabatier, Toulouse
VL - 23
IS - 4
SP - 847
EP - 875
AB - In this paper, we prove the existence of a solution for a quite general stationary compressible Stokes problem including, in particular, gravity effects. The Equation Of State gives the pressure as an increasing superlinear function of the density. This existence result is obtained by passing to the limit on the solution of a viscous approximation of the continuity equation.
LA - eng
KW - convection-diffusion; regularized problem; existence; compressible Stokes problem
UR - http://eudml.org/doc/275296
ER -

References

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  1. Bijl (H.) and Wesseling (P.).— A unified method for computing incompressible and compressible ows in boundary-fitted coordinates. J. Comput. Phys., 141(2), p. 153-173 (1998). Zbl0918.76054MR1619651
  2. Bramble (J. H.).— A proof of the inf-sup condition for the Stokes equations on Lipschitz domains, Mathematical Models and Methods in Applied Sciences, 13, p. 361-371 (2003). Zbl1073.35184MR1977631
  3. Březina (J.), Novotný (A.).— On Weak Solutions of Steady Navier-Stokes Equations for Monatomic Gas, Comment. Math. Univ. Carolin. 49, p. 611-632 (2008). Zbl1212.35345MR2493941
  4. Droniou (J.), Vazquez (J. L.).— Noncoercive convection-diffusion elliptic problems with Neumann boundary conditions, Calc. Var. Partial Differential Equations 34, no. 4, p. 413-434 (2009). Zbl1167.35342MR2476418
  5. Eymard (R.), Gallouët (T.), Herbin (R.), Latché (J.-C.).— A convergent finite element-finite volume scheme for the compressible Stokes problem. Part II: the isentropic case. Math. Comp. 79, no. 270, p. 649-675 (2010). Zbl1197.35192MR2600538
  6. Feireisl (E.).— Dynamics of viscous compressible fluids. Oxford Lecture Series in Mathematics and its Applications, 26. Oxford University Press, Oxford (2004). Zbl1080.76001MR2040667
  7. Fettah (A.), Gallouët (T.).— Numerical approximation of the general compressible Stokes problem. IMA Journal of Numerical Analysis (2012). Zbl06190569MR3081489
  8. Frehse (J.), Steinhauer (M.), Weigant (W.).— The Dirichlet Problem for Viscous Compressible Isothermal Navier-Stokes Equations in Two-Dimensions, Archive Ration. Mech. Anal. 198, no. 1, p. 1-12 (2010). Zbl1229.35175MR2679367
  9. Frehse (J.), Steinhauer (M.), Weigant (W.).— The Dirichlet Problem for Steady Viscous Compressible Flow in 3-D, Journal de Mathématiques Pures et Appliquées 97, no. 2, p. 85-97 (2012). Zbl1233.35154MR2875292
  10. Gallouët (T.), Herbin (R.).— Mesure, Intégration, Probabilités. Ellipses (2013). Zbl1273.28001
  11. Gallouët (T.), Herbin (R.), Latché (J.-C.).— A convergent finite element-finite volume scheme for the compressible Stokes problem. I. The isothermal case. Math. Comp., 78(267), p. 1333-1352 (2009). Zbl1223.76041MR2501053
  12. Harlow (F.), Amsden (A.).— A numerical uid dynamics calculation method for all flow speeds. Journal of Computational Physics, 8, p. 197-213 (1971). Zbl0221.76011
  13. Jesslé (D.), Novotný (A.).— Existence of renormalized weak solutions to the steady equations describing compressible fluids in barotropic regimes, J. Math. Pures Appl. 99 no. 3, p. 280-296 (2013). Zbl06146387MR3017990
  14. Jiang (S.), Zhou (C.).— Existence of weak solutions to the three dimensional steady compressible Navier-Stokes equations, Annales IHP - Analyse Nonlinéaire 28, p. 485-498 (2011). Zbl1241.35149MR2823881
  15. Leray (J.).— Sur le mouvement d’un liquide visqueux emplissant l’espace. Acta Math., 63(1), p. 193-248 (1934). MR1555394
  16. Lions (P.-L.).— Mathematical topics in fluid mechanics -volume 2- compressible models. volume 10 of Oxford Lecture Series in Mathematics and its Applications. Oxford University Press (1998). Zbl0866.76002MR1637634
  17. Novotnỳ (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). Zbl1088.35051MR2359339
  18. 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, p. 531-550 (2002). Zbl1050.35074MR1967222
  19. Oran (E. S.), Boris (J. P.).— Numerical simulation of reactive flow. Cambridge University Press (2001). Zbl0980.76002
  20. Plotnikov (P.I.), Sokolowski (J.).— Stationary solutions of Navier-Stokes equations for diatomic gases, Russian Math. Surv. 62, p. 561-593 (2007). Zbl1139.76049MR2355421

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