Skip to main content
Log in

Navier–Stokes–Fourier System with General Boundary Conditions

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

We consider the Navier–Stokes–Fourier system in a bounded domain \(\Omega \subset R^d\), \(d=2,3\), with physically realistic in/out flow boundary conditions. We develop a new concept of weak solutions satisfying a general form of relative energy inequality. The weak solutions exist globally in time for any finite energy initial data and comply with the weak–strong uniqueness principle.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Notes

  1. In what follows, we denote \({{\tilde{e}}}=e({{\tilde{\varrho }}},{{\tilde{\vartheta }}})\), \({{\tilde{p}}}=p({{\tilde{\varrho }}},{{\tilde{\vartheta }}})\) etc. if there is no danger of confusion.

  2. In what follows, we denote \(a{\mathop {\sim }\limits ^{<}}b\) if there exists \(c>0\) independent of n, \(\varepsilon \), \(\delta \) such that \(a\le c b\).

References

  1. Abbatiello, A., Feireisl, E., Novotný, A.: Generalized solutions to models of compressible viscous fluids. Discrete Contin. Dyn. Syst. 41, 1–28 (2021)

    Article  MathSciNet  Google Scholar 

  2. Ball, J.M.: A version of the fundamental theorem for Young measures. In: Rascle, et al. (eds.) PDE’s and Continuum Models of Phase Transitions. Lecture Notes in Physics, vol. 344, pp. 207–215. Springer, Berlin (1989)

  3. Ball, J.M., Murat, F.: Remarks on Chacons biting lemma. Proc. Am. Math. Soc. 107, 655–663 (1989)

    MathSciNet  MATH  Google Scholar 

  4. Bechtel, S.E., Rooney, F.J., Forest, M.G.: Connection between stability, convexity of internal energy, and the second law for compressible Newtonian fuids. J. Appl. Mech. 72, 299–300 (2005)

    Article  ADS  Google Scholar 

  5. Becker, E.: Gasdynamik. Teubner-Verlag, Stuttgart (1966)

    MATH  Google Scholar 

  6. Belgiorno, F.: Notes on the third law of thermodynamics, I. J. Phys. A 36, 8165–8193 (2003)

    Article  ADS  MathSciNet  Google Scholar 

  7. Belgiorno, F.: Notes on the third law of thermodynamics, II. J. Phys. A 36, 8195–8221 (2003)

    Article  ADS  MathSciNet  Google Scholar 

  8. Bresch, D., Desjardins, B.: Stabilité de solutions faibles globales pour les équations de Navier–Stokes compressibles avec température. C. R. Acad. Sci. Paris 343, 219–224 (2006)

    Article  MathSciNet  Google Scholar 

  9. Buckmaster, T., Vicol, V.: Convex integration and phenomenologies in turbulence. EMS Surv. Math. Sci. 6(1), 173–263 (2019)

    Article  MathSciNet  Google Scholar 

  10. Chang, T., Jin, B.J., Novotný, A.: Compressible Navier–Stokes system with general inflow-outflow boundary data. SIAM J. Math. Anal. 51(2), 1238–1278 (2019)

    Article  MathSciNet  Google Scholar 

  11. Chen, G.-Q., Torres, M., Ziemer, W.P.: Gauss-Green theorem for weakly differentiable vector fields, sets of finite perimeter, and balance laws. Commun. Pure Appl. Math. 62(2), 242–304 (2009)

    Article  MathSciNet  Google Scholar 

  12. Feireisl, E.: Relative entropies in thermodynamics of complete fluid systems. Discrete Contin. Dyn. Syst. Ser. A 32, 3059–3080 (2012)

    Article  MathSciNet  Google Scholar 

  13. Feireisl, E., Karper, T., Novotný, A.: A convergent numerical method for the Navier–Stokes–Fourier system. IMA J. Numer. Anal. 36(4), 1477–1535 (2016)

    Article  MathSciNet  Google Scholar 

  14. Feireisl, E., Lukáčová-Medvi\(\check{d}\)ová, M., Mizerová, H., She, B.: On the convergence of a finite volume method for the Navier–Stokes–Fourier (2019). Arxive Preprint Series, arxiv preprint No. 1903.08526

  15. Feireisl, E., Novotný, A.: Singular Limits in Thermodynamics of Viscous Fluids. Birkhäuser-Verlag, Basel (2009)

    Book  Google Scholar 

  16. Feireisl, E., Novotný, A.: Weak–strong uniqueness property for the full Navier–Stokes–Fourier system. Arch. Ration. Mech. Anal. 204, 683–706 (2012)

    Article  MathSciNet  Google Scholar 

  17. Feireisl, E., Novotný, A.: Singular Limits in Thermodynamics of Viscous Fluids. Advances in Mathematical Fluid Mechanics, 2nd edn. Birkhäuser/Springer, Cham (2017)

    Book  Google Scholar 

  18. Feireisl, E., Novotný, A.: On a simple model of reacting compressible flows arising in astrophysics. Proc. R. Sect. Soc. Edinb. Sect. A 135, 1169–1194 (2005)

    Article  MathSciNet  Google Scholar 

  19. Feireisl, E., Novotný, A., Petzeltová, H.: On the existence of globally defined weak solutions to the Navier–Stokes equations of compressible isentropic fluids. J. Math. Fluid Mech. 3, 358–392 (2001)

    Article  ADS  MathSciNet  Google Scholar 

  20. Feireisl, E., Novotný, A., Sun, Y.: Suitable weak solutions to the Navier–Stokes equations of compressible viscous fluids. Indiana Univ. Math. J. 60, 611–631 (2011)

    Article  MathSciNet  Google Scholar 

  21. Feireisl, E., Petzeltová, H.: On the long time behaviour of solutions to the Navier–Stokes–Fourier system with a time dependent driving force. J. Dyn. Differ. Equ. 19, 685–707 (2007)

    Article  MathSciNet  Google Scholar 

  22. Girinon, V.: Navier–Stokes equations with nonhomogeneous boundary conditions in a bounded three-dimensional domain. J. Math. Fluid Mech. 13, 309–339 (2011)

    Article  ADS  MathSciNet  Google Scholar 

  23. Kwon, Y.S., Novotný, A.: Dissipative solutions to compressible Navier–Stokes equations with general inflow-outflow data: existence, stability and weak-strong uniqueness. J. Math. Fluid Mech. 23(4), 1–27 (2021)

    MathSciNet  MATH  Google Scholar 

  24. Ladyzhenskaya, O.A., Solonnikov, V.A., Uraltseva, N.N.: Linear and Quasilinear Equations of Parabolic Type. Translations of Mathematical Monographs, vol. 23. American Mathematical Society, Providence (1968)

    Book  Google Scholar 

  25. Leray, J.: Sur le mouvement d’un liquide visqueux emplissant l’espace. Acta Math. 63, 193–248 (1934)

    Article  MathSciNet  Google Scholar 

  26. Lions, P.-L.: Mathematical Topics in Fluid Dynamics. Compressible Models, vol. 2. Oxford Science Publication, Oxford (1998)

    MATH  Google Scholar 

  27. Norman, D.E.: Chemically reacting fluid flows: weak solutions and global attractors. J. Differ. Equ. 152(1), 75–135 (1999)

    Article  ADS  MathSciNet  Google Scholar 

  28. Sprung, B.: Upper and lower bounds for the Bregman divergence. J. Inequal. Appl. Paper No. 4, 12 (2019)

  29. Yakhot, V., Orszag, S.A.: Renormalization group analysis of turbulence. I. Basic theory. J. Sci. Comput. 1(1), 3–51 (1986)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Eduard Feireisl.

Additional information

Communicated by A. Ionescu.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

The work of E.F. was supported by the Czech Sciences Foundation (GAČR), Grant Agreement 18-12719S.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Feireisl, E., Novotný, A. Navier–Stokes–Fourier System with General Boundary Conditions. Commun. Math. Phys. 386, 975–1010 (2021). https://doi.org/10.1007/s00220-021-04091-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-021-04091-1

Navigation