Journal of Mathematical Fluid Mechanics

, Volume 19, Issue 4, pp 659–683 | Cite as

Dimension Reduction for the Full Navier–Stokes–Fourier system

  • Jan Březina
  • Ondřej Kreml
  • Václav MáchaEmail author


It is well known that the full Navier–Stokes–Fourier system does not possess a strong solution in three dimensions which causes problems in applications. However, when modeling the flow of a fluid in a thin long pipe, the influence of the cross section can be neglected and the flow is basically one-dimensional. This allows us to deal with strong solutions which are more convenient for numerical computations. The goal of this paper is to provide a rigorous justification of this approach. Namely, we prove that any suitable weak solution to the three-dimensional NSF system tends to a strong solution to the one-dimensional system as the thickness of the pipe tends to zero.


Navier–Stokes–Fourier system dimension reduction relative entropy 

Mathematics Subject Classification

35Q35 76N15 


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  1. 1.
    Antontsev, S.N., Kazhikhov, A.V., Monakhov, V.N.: Boundary Value Problems in Mechanics of Nonhomogeneous Fluids. Elsevier, New York (1990)zbMATHGoogle Scholar
  2. 2.
    Bella, P., Feireisl, E., Novotný, A.: Dimension reduction for compressible viscous fluid. Acta Appl. Math. 134, 111–121 (2014)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Carrillo, J., Jüngel, A., Markowich, P.A., Toscani, G., Unterreiter, A.: Entropy dissipation methods for degenerate parabolic problems and generalized Sobolev inequalities. Monatshefte Math. 133, 1–82 (2001)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Dafermos, C.M.: The second law of thermodynamics and stability. Arch. Ration. Mech. Anal. 70, 167–179 (1979)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Ducomet, B., Feireisl, E.: The equations of magnetohydrodynamics: on the interaction between matter and radiation in the evolution of gaseous stars. Commun. Math. Phys. 266, 595–629 (2006)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Ericksen, J.L.: Introduction to the Thermodynamics of Solids, revised ed. Applied Mathematical Sciences, vol. 131. Springer, New York (1998)CrossRefGoogle Scholar
  7. 7.
    Feireisl, E., Jin, B.J., Novotný, A.: Relative entropies, suitable weak solutions, and weak-strong uniqueness for the compressible Navier-Stoke system. J. Math. Fluid Mech. 14(4), 717–730 (2012)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Feireisl, E., Novotný, A.: Weak–strong uniqueness property for the full Navier–Stokes–Fourier system. Arch. Ration. Mech. Anal. 204, 683–706 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Feireisl, E., Novotný, A.: Singular Limits in Thermodynamics of Viscous Fluids. Birkhaüser, Berlin (2009)CrossRefzbMATHGoogle Scholar
  10. 10.
    Iftimie, D., Raugel, G., Sell, G.R.: Navier–Stokes equations in the 3D domains with Navier–Boundary conditions. Indiana Univ. Math. J. 56(3), 1083–1156 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Kawohl, B.: Global existence of large solutions to initial boundary value problems for a viscous heat-conducting, one-dimensional real gas. JDE 58, 76–103 (1985)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Ladyzhenskaya, O.A.: Solution “in the large” to the boundary value problem for the Naviesr-Stokes equations in two space variables. Sov. Phys. Dokl. 3: 1128–1131 (1958) Translation from. Dokl. Akad. Nauk SSSR 123, 427–429 (1958)Google Scholar
  13. 13.
    Maltese, D., Novotný, A.: Compressible Navier–Stokes equations on thin domains. J. Math. Fluid Mech. 16(3), 571–594 (2014)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Raugel, G., Sell, G.R.: Navier–Stokes equations in thin 3D domains III Existence of a Global Attractor. Turbulence in Fluid Flows, IMA Vol. Math. Appl., vol. 55, pp. 137–163. Springer, New York (1993)zbMATHGoogle Scholar
  15. 15.
    Raugel, G., Sell, G.R.: Navier–Stokes equations in thin 3D domains I. Global attractors and global regularity of solutions. J. Am. Math. Soc. 6(3), 503–568 (1993)zbMATHGoogle Scholar
  16. 16.
    Raugel, G., Sell, G. R.: Navier–Stokes equations in thin 3D domains II. Global regularity of spatially periodic solutions. Nonlinear Partial Differential Equations and Their Applications, Collège de France Seminar, vol. XI, Paris, 1989–1991, Pitman res. Notes Math. Ser. 299: 205–247, Longman Sci. Tech., Harlow (1994)Google Scholar
  17. 17.
    Saint-Raymond, L.: Hydrodynamic limits: some improvements of the relative entropy method. Ann. I. H. Poincaré Anal. 26, 705–744 (2009)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Valli, A.: An existence theorem for compressible viscous fluids. Ann. Mat. Pura Appl. 130(1), 197–213 (1982)CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    Vodák, R.: Asymptotic analysis of steady and nonsteady Navier–Stokes equations for barotropic compressible flow. Acta Appl. Math. 110(2), 991–1009 (2010)CrossRefzbMATHMathSciNetGoogle Scholar

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© Springer International Publishing 2016

Authors and Affiliations

  1. 1.Tokyo Institute of TechnologyTokyoJapan
  2. 2.Institute of Mathematics of the Czech Academy of SciencesPraha 1Czech Republic
  3. 3.Industry-University Research CenterYonsei UniversitySeoulRepublic of Korea

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