Existence of weak solutions to the three-dimensional steady compressible Navier–Stokes equations

Jiang, Song; Zhou, Chunhui

Annales de l'I.H.P. Analyse non linéaire, Tome 28 (2011) no. 4, pp. 485-498.

We prove the existence of a spatially periodic weak solution to the steady compressible isentropic Navier–Stokes equations in ${ℝ}^3$ for any specific heat ratio $\gamma >1$. The proof is based on the weighted estimates of both pressure and kinetic energy for the approximate system which result in some higher integrability of the density, and the method of weak convergence.

Mots clés : Steady compressible Navier–Stokes equations, Existence for $ \gamma >1$, Potential estimate, Effective viscous flux

[1] M.E. Bogovskii, Solution of the first boundary value problem for the equation of continuity of an incompressible medium, Dokl. Akad. Nauk SSSR 248 no. 5 (1979), 1037-1040 | MR

[2] J. Březina, A. Novotný, On weak solutions of steady Navier–Stokes equations for monoatomic gas, Comment. Math. Univ. Carolinae 49 no. 4 (2008), 611-632 | EuDML | MR | Zbl

[3] E. Feireisl, Dynamics of Viscous Compressible Fluids, Oxford Science Publ., Clarendon Press, Oxford (2003) | MR

[4] E. Feireisl, On compactness of solutions to the compressible isentropic Navier–Stokes equations when the density is not integrable, Comment. Math. Univ. Carolinae 42 (2001), 83-98 | EuDML | MR | Zbl

[5] E. Feireisl, A. Novotný, H. Petzeltová, On the existence of globally defined weak solutions to the Navier–Stokes equations, J. Math. Fluid Mech. 3 (2001), 358-392 | MR | Zbl

[6] J. Frehse, S. Goj, M. Steinhauer, $L^p$-estimates for the Navier–Stokes equations for steady compressible flow, Manuscripta Math. 116 (2005), 265-275 | MR | Zbl

[7] J. Frehse, M. Steinhauer, W. Weigant, On stationary solutions for 2-D viscous compressible isothermal Navier–Stokes equations, J. Math. Fluid Mech. 13 (2010), 55-63 | MR | Zbl

[8] J. Frehse, M. Steinhauer, W. Weigant, The Dirichlet problem for steady viscous compressible flow in 3-D, J. Math. Pures Appl., doi:10.1016/j.matpur.2009.06.005. | MR

[9] D. Gilbarg, N.S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag (1983) | MR | Zbl

[10] S. Jiang, P. Zhang, Global spherically symmetry solutions of the compressible isentropic Navier–Stokes equations, Comm. Math. Phys. 215 (2001), 559-581 | MR | Zbl

[11] S. Jiang, P. Zhang, Axisymmetric solutions of the 3-D Navier–Stokes equations for compressible isentropic fluids, J. Math. Pure Appl. 82 (2003), 949-973 | MR | Zbl

[12] P.L. Lions, Mathematical Topics in Fluid Mechanics, vol. II, Compressible Models, Clarendon Press, Oxford (1998) | MR

[13] P.B. Mucha, M. Pokorný, Weak solutions to equations of steady compressible heat conducting fluids, Math. Models Meth. Appl. Sci. 20 (2010), 785-813 | MR | Zbl

[14] A. Novotný, I. Straškraba, Introduction to the Mathematical Theory of Compressible Flow, Oxford Univ. Press, Oxford (2004) | MR | Zbl

[15] P.I. Plotnikov, J. Sokolowski, Concentrations of solutions to time-discretized compressible Navier–Stokes equations, Comm. Math. Phys. 258 (2005), 567-608 | MR | Zbl

[16] E. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Univ. Press, Princeton, NJ (1970) | MR | Zbl