An existence proof for the stationary compressible Stokes problem
Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 23 (2014) no. 4, pp. 847-875.

Dans cet article, nous prouvons l’existence d’une solution pour le problème de Stokes compressible stationnaire en tenant compte, en particulier, des effets gravitaires. L’équation d’état donne la pression comme une fonction strictement croissante superlinéaire de la densité. L’existence de solution est obtenue en passant à la limite sur une approximation visqueuse de l’équation de continuité.

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.

@article{AFST_2014_6_23_4_847_0,
     author = {Fettah, A. and Gallou\"et, T. and Lakehal, H.},
     title = {An existence proof for the stationary compressible {Stokes} problem},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     pages = {847--875},
     publisher = {Universit\'e Paul Sabatier, Institut de math\'ematiques},
     address = {Toulouse},
     volume = {Ser. 6, 23},
     number = {4},
     year = {2014},
     doi = {10.5802/afst.1427},
     mrnumber = {3270426},
     zbl = {06374891},
     language = {en},
     url = {http://archive.numdam.org/articles/10.5802/afst.1427/}
}
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
DA  - 2014///
SP  - 847
EP  - 875
VL  - Ser. 6, 23
IS  - 4
PB  - Université Paul Sabatier, Institut de mathématiques
PP  - Toulouse
UR  - http://archive.numdam.org/articles/10.5802/afst.1427/
UR  - https://www.ams.org/mathscinet-getitem?mr=3270426
UR  - https://zbmath.org/?q=an%3A06374891
UR  - https://doi.org/10.5802/afst.1427
DO  - 10.5802/afst.1427
LA  - en
ID  - AFST_2014_6_23_4_847_0
ER  - 
Fettah, A.; Gallouët, T.; Lakehal, H. An existence proof for the stationary compressible Stokes problem. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 23 (2014) no. 4, pp. 847-875. doi : 10.5802/afst.1427. http://archive.numdam.org/articles/10.5802/afst.1427/

[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). | MR 1619651 | Zbl 0918.76054

[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). | MR 1977631 | Zbl 1073.35184

[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). | MR 2493941 | Zbl 1212.35345

[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). | MR 2476418 | Zbl 1167.35342

[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). | MR 2600538 | Zbl 1197.35192

[6] Feireisl (E.).— Dynamics of viscous compressible fluids. Oxford Lecture Series in Mathematics and its Applications, 26. Oxford University Press, Oxford (2004). | MR 2040667 | Zbl 1080.76001

[7] Fettah (A.), Gallouët (T.).— Numerical approximation of the general compressible Stokes problem. IMA Journal of Numerical Analysis (2012). | MR 3081489

[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). | MR 2679367 | Zbl 1229.35175

[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). | MR 2875292 | Zbl 1233.35154

[10] Gallouët (T.), Herbin (R.).— Mesure, Intégration, Probabilités. Ellipses (2013). | Zbl 1273.28001

[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). | MR 2501053 | Zbl 1223.76041

[12] Harlow (F.), Amsden (A.).— A numerical uid dynamics calculation method for all flow speeds. Journal of Computational Physics, 8, p. 197-213 (1971). | Zbl 0221.76011

[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). | MR 3017990

[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). | Numdam | MR 2823881 | Zbl 1241.35149

[15] Leray (J.).— Sur le mouvement d’un liquide visqueux emplissant l’espace. Acta Math., 63(1), p. 193-248 (1934). | MR 1555394

[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). | MR 1637634 | Zbl 0866.76002

[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). | MR 2359339 | Zbl 1088.35051

[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). | MR 1967222 | Zbl 1050.35074

[19] Oran (E. S.), Boris (J. P.).— Numerical simulation of reactive flow. Cambridge University Press (2001). | Zbl 0980.76002

[20] Plotnikov (P.I.), Sokolowski (J.).— Stationary solutions of Navier-Stokes equations for diatomic gases, Russian Math. Surv. 62, p. 561-593 (2007). | MR 2355421 | Zbl 1139.76049

Cité par Sources :