Steady solutions to the Navier-Stokes-Fourier system for dense compressible fluid

Šimon Axman, Piotr Bogusław Mucha, Milan Pokorný



We establish existence of strong solutions to the stationary Navier-Stokes-Fourier system for compressible flows with density dependent viscosities in regime of heat conducting fluids with very high densities. In comparison to the known results considering the low Mach number case, we work in the $L^p$-setting combining the methods for the weak solutions with the method of decomposition. Moreover, the magnitude of gradient of the density as well as other data are not limited, our only assumption is the given total mass must be sufficiently large.


Steady compressible Navier-Stokes-Fourier system; low Mach number limit; strong solution; denisty dependent viscosities; large data; existence via Schauder typefixed point theorem

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S. Agmon, A. Douglis and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I, Comm. Pure Appl. Math. 12 (1959), 623–727.

T. Alazard, Low Mach number limit of the full Navier–Stokes equations, Arch. Ration. Mech. Anal. 180 (2006), no. 1, 1–73.

Š. Axmann, P.B. Mucha and M. Pokorný, Steady solutions to viscous shallow water equations. The case of heavy water, Commun. Math. Sci. 15 (2017), no. 5, 1385–1402.

H. Beirão Da Veiga, An Lp -theory for the n-dimensional, stationary, compressible Navier–Stokes equations, and the incompressible limit for compressible fluids. The equilibrium solutions, Comm. Math. Phys. 109 (1987), no. 2, 229–248.

D. Bresch and B. Desjardins, Existence of global weak solutions for a 2D viscous shallow water equations and convergence to the quasi-geostrophic model, Comm. Math. Phys. 238 (2003), 211–223.

D. Bresch, B. Desjardins and D. Gerard-Varet, On compressible Navier–Stokes equations with density dependent viscosities in bounded domains, J. Math. Pures Appl. (9) 87 (2007), 227–235.

H.J. Choe and B.J. Jin, Existence of solutions of stationary compressible Navier–Stokes equations with large force, J. Funct. Anal. 177 (2000), no. 1, 54–88.

Ch. Dou, F. Jiang, S. Jiang and Y.-F. Yang, Existence of strong solutions to the steady Navier–Stokes equations for a compressible heat-conductive fluid with large forces, J. Math. Pures Appl. 103, no. 5, 1163–1197.

R. Farwig, Stationary solutions of the Navier–Stokes equations for a compressible, viscous and heat-conductive fluid, SFB 256, University Bonn, 1988, preprint.

R. Farwig, Stationary solutions of compressible Navier–Stokes equations with slip boundary conditions, Comm. Partial Differential Equations 14 (1989), no. 11, 1579–1606.

E. Feireisl, Dynamics of Viscous Compressible Fluids, Oxford University Press, Oxford (2004).

E. Feireisl, P.B. Mucha, A. Novotný and M. Pokorný, Time-periodic solutions to the full Navier–Stokes–Fourier system, Arch. Ration. Mech. Anal. 204 (2012), 745–786.

P.B. Mucha, On cylindrical symmetric flows through pipe-like domains, J. Differential Equations 201 (2004), no. 2, 304–323.

P.B. Mucha and T. Piasecki, Compressible perturbation of Poiseuille type flow, J. Math. Pures Appl. (9) 102 (2014), no. 2, 338–363.

P.B. Mucha and M. Pokorný, The rot-div system in exterior domains, J. Math. Fluid Mech. 16 (2014), no. 4, 701–720.

P.B. Mucha and M. Pokorný, On the steady compressible Navier–Stokes–Fourier system, Comm. Math. Phys. 288 (2009), no. 1, 349–377.

P.B. Mucha and R. Rautmann, Convergence of Rothe’s scheme for the Navier–Stokes equations with slip conditions in 2D domains, ZAMM Z. Angew. Math. Mech. 86 (2006), no. 9, 691–701.

A. Novotný and M. Padula, Lp -approach to steady flows of viscous compressible fluids in exterior domains, Arch. Ration. Mech. Anal. 126 (1994), no. 3, 243–297.

M. Padula, Existence and continuous dependence for solutions to the equations of a onedimensional model in gas dynamics, Meccanica 17 (1981), 128–135.

M. Padula, Existence and uniqueness for viscous steady compressible motions, Proc. Sem. Fis. Mat., Dinamica dei Fluidi e dei gaz ionizzati, Trieste (1982).

M. Padula, Existence and uniqueness for viscous steady compressible motions, Arch. Ration. Mech. Anal. 97 (1987), no. 2, 89–102.

T. Piasecki and M. Pokorný, Strong solutions to the Navier–Stokes–Fourier system with slip–inflow boundary conditions, ZAMM Z. Angew. Math. Mech. 94 (2014), no. 12, 1035–1057.

V.A. Solonnikov, Overdetermined elliptic boundary-value problems, J. Sov. Math. 1 (1973), no. 4, 477–512.

A. Valli, Periodic and stationary solutions for compressible Navier–Stokes equations via a stability method, Ann. Sc. Norm. Sup. Pisa (1) 4 (1983), 607–646.

A. Valli, On the existence of stationary solutions to compressible Navier–Stokes equations, Ann. Inst. H. Poincaré Anal. Non Linéaire (1) 4 (1987), 99–113.


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