Stationary solutions to the compressible Navier–Stokes system with general boundary conditions
Annales de l'I.H.P. Analyse non linéaire, Tome 35 (2018) no. 6, pp. 1457-1475.

We consider the stationary compressible Navier–Stokes system supplemented with general inhomogeneous boundary conditions. Assuming the pressure to be given by the standard hard sphere EOS we show existence of weak solutions for arbitrarily large boundary data.

DOI : 10.1016/j.anihpc.2018.01.001
Mots clés : Compressible Navier–Stokes system, Stationary solution, Inhomogeneous boundary conditions
@article{AIHPC_2018__35_6_1457_0,
     author = {Feireisl, Eduard and Novotn\'y, Anton{\'\i}n},
     title = {Stationary solutions to the compressible {Navier{\textendash}Stokes} system with general boundary conditions},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {1457--1475},
     publisher = {Elsevier},
     volume = {35},
     number = {6},
     year = {2018},
     doi = {10.1016/j.anihpc.2018.01.001},
     mrnumber = {3846233},
     zbl = {1405.35138},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.anihpc.2018.01.001/}
}
TY  - JOUR
AU  - Feireisl, Eduard
AU  - Novotný, Antonín
TI  - Stationary solutions to the compressible Navier–Stokes system with general boundary conditions
JO  - Annales de l'I.H.P. Analyse non linéaire
PY  - 2018
SP  - 1457
EP  - 1475
VL  - 35
IS  - 6
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.anihpc.2018.01.001/
DO  - 10.1016/j.anihpc.2018.01.001
LA  - en
ID  - AIHPC_2018__35_6_1457_0
ER  - 
%0 Journal Article
%A Feireisl, Eduard
%A Novotný, Antonín
%T Stationary solutions to the compressible Navier–Stokes system with general boundary conditions
%J Annales de l'I.H.P. Analyse non linéaire
%D 2018
%P 1457-1475
%V 35
%N 6
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.anihpc.2018.01.001/
%R 10.1016/j.anihpc.2018.01.001
%G en
%F AIHPC_2018__35_6_1457_0
Feireisl, Eduard; Novotný, Antonín. Stationary solutions to the compressible Navier–Stokes system with general boundary conditions. Annales de l'I.H.P. Analyse non linéaire, Tome 35 (2018) no. 6, pp. 1457-1475. doi : 10.1016/j.anihpc.2018.01.001. http://www.numdam.org/articles/10.1016/j.anihpc.2018.01.001/

[1] Armitage, G.H.; Kuran, Ü. The convexity of a domain and the superharmonicity of the signed distance function, Proc. Am. Math. Soc., Volume 93 (1985), pp. 598–600 | DOI | MR | Zbl

[2] Březina, J.; Novotný, A. On weak solutions of steady Navier–Stokes equations for monatomic gas, Comment. Math. Univ. Carol., Volume 49 (2008), pp. 611–632 | MR | Zbl

[3] Carnahan, N.F.; Starling, K.E. Equation of state for nonattracting rigid spheres, J. Chem. Phys., Volume 51 (1980), pp. 635–638

[4] I.S. Ciuperca, E. Feireisl, M. Jai, A. Petrov, A rigorous derivation of Reynolds equations of lubrication for compressible fluids, preprint, 2016.

[5] DiPerna, R.J.; Lions, P.-L. Ordinary differential equations, transport theory and Sobolev spaces, Invent. Math., Volume 98 (1989), pp. 511–547 | DOI | MR | Zbl

[6] Feireisl, E.; Novotný, A. Singular Limits in Thermodynamics of Viscous Fluids, Birkhäuser-Verlag, Basel, 2009 | DOI | MR | Zbl

[7] Finn, R. On the steady-state solutions of the Navier–Stokes equations. III, Acta Math., Volume 105 (1961), pp. 197–244 | DOI | MR | Zbl

[8] Foote, R.L. Regularity of the distance function, Proc. Am. Math. Soc., Volume 92 (1984), pp. 153–155 | MR | Zbl

[9] Frehse, J.; Steinhauer, M.; Weigant, W. The Dirichlet problem for steady viscous compressible flow in 3-D, J. Math. Pures Appl. (9), Volume 97 (2012) no. 2, pp. 85–97 | DOI | MR | Zbl

[10] Galdi, G.P. An Introduction to the Mathematical Theory of the Navier–Stokes Equations, Springer-Verlag, New York, 2003

[11] Hopf, E. Ein allgemeiner Endlichkeitssatz der Hydrodynamik, Math. Ann., Volume 117 (1940/1941), pp. 764–775 | DOI | JFM | MR

[12] Jiang, S.; Zhou, C. Existence of weak solutions to the three dimensional steady compressible Navier–Stokes equations, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 28 (2011), pp. 485–498 | Numdam | MR | Zbl

[13] Lions, P.-L. Mathematical Topics in Fluid Dynamics, vol. 2, Compressible Models, Oxford Science Publication, Oxford, 1998 | MR | Zbl

[14] Mucha, P.B.; Piasecki, T. Compressible perturbation of Poiseuille type flow, J. Math. Pures Appl. (9), Volume 102 (2014) no. 2, pp. 338–363 | DOI | MR | Zbl

[15] Nittka, R. Quasilinear elliptic and parabolic Robin problems on Lipschitz domains, NoDEA Nonlinear Differ. Equ. Appl., Volume 20 (2013) no. 3, pp. 1125–1155 | DOI | MR | Zbl

[16] Novotný, A.; Straškraba, I. Introduction to the Mathematical Theory of Compressible Flow, Oxford University Press, Oxford, 2004 | DOI | MR | Zbl

[17] Piasecki, T. On an inhomogeneous slip-inflow boundary value problem for a steady flow of a viscous compressible fluid in a cylindrical domain, J. Differ. Equ., Volume 248 (2010) no. 8, pp. 2171–2198 | DOI | MR | Zbl

[18] Piasecki, T.; Pokorný, M. Strong solutions to the Navier–Stokes–Fourier system with slip-inflow boundary conditions, Z. Angew. Math. Mech., Volume 94 (2014) no. 12, pp. 1035–1057 | DOI | MR | Zbl

[19] Plotnikov, P.I.; Ruban, E.V.; Sokolowski, J. Inhomogeneous boundary value problems for compressible Navier–Stokes equations: well-posedness and sensitivity analysis, SIAM J. Math. Anal., Volume 40 (2008) no. 3, pp. 1152–1200 | DOI | MR | Zbl

[20] Plotnikov, P.I.; Ruban, E.V.; Sokolowski, J. Inhomogeneous boundary value problems for compressible Navier–Stokes and transport equations, J. Math. Pures Appl. (9), Volume 92 (2009) no. 2, pp. 113–162 | DOI | MR | Zbl

[21] Plotnikov, P.I.; Ruban, E.V.; Sokołowski, J. Shape sensitivity analysis for compressible Navier–Stokes equations, System Modeling and Optimization, IFIP Adv. Inf. Commun. Technol., vol. 312, Springer, Berlin, 2009, pp. 430–447 | MR | Zbl

[22] Plotnikov, P.I.; Sokolowski, J. Stationary solutions of Navier–Stokes equations for diatomic gases, Russ. Math. Surv., Volume 62 (2007), pp. 561–593 | DOI | MR | Zbl

[23] Tartar, L.; Knopps, L.J. Compensated compactness and applications to partial differential equations, Nonlinear Anal. and Mech., Heriot-Watt Sympos., Research Notes in Math., vol. 39, Pitman, Boston, 1975, pp. 136–211

Cité par Sources :