Skip to main content
SpringerLink
Log in

Dissipative measure-valued solutions to the compressible Navier–Stokes system

  • Published:
Calculus of Variations and Partial Differential Equations Aims and scope Submit manuscript

Abstract

We introduce a new concept of dissipative measure-valued solution to the compressible Navier–Stokes system satisfying, in addition, a relevant form of the total energy balance. Then we show that a dissipative measure-valued and a standard smooth classical solution originating from the same initial data coincide (weak-strong uniqueness principle) as long as the latter exists. Such a result facilitates considerably the proof of convergence of solutions to various approximations including certain numerical schemes that are known to generate a measure-valued solution. As a byproduct we show that any measure-valued solution with bounded density component that starts from smooth initial data is necessarily a classical one.

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

Access this article

Log in via an institution

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

References

  1. Alibert, J.-J., Bouchitté, G.: Non-uniform integrability and generalized young measure. J. Convex Anal. 4, 129–148 (1997)

    MathSciNet  MATH  Google Scholar 

  2. Brenier, Y., De Lellis, C., Székelyhidi Jr., L.: Weak-strong uniqueness for measure-valued solutions. Comm. Math. Phys. 305(2), 351–361 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  3. Brenner, H.: Navier–Stokes revisited. Phys. A 349(1–2), 60–132 (2005)

    Article  MathSciNet  Google Scholar 

  4. Cho, Y., Choe, H.J., Kim, H.: Unique solvability of the initial boundary value problems for compressible viscous fluids. J. Math. Pures. Appl. 83, 243–275 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  5. Demoulini, S., Stuart, D.M.A., Tzavaras, A.E.: Weak-strong uniqueness of dissipative measure-valued solutions for polyconvex elastodynamics. Arch. Ration. Mech. Anal. 205(3), 927–961 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  6. DiPerna, R.J.: Measure-valued solutions to conservation laws. Arch. Ration. Mech. Anal. 88(3), 223–270 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  7. DiPerna, R.J., Majda, A.J.: Oscillations and concentrations in weak solutions of the incompressible fluid equations. Comm. Math. Phys. 108(4), 667–689 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  8. Feireisl, E.: Dynamics of viscous compressible fluids. Oxford University Press, Oxford (2004)

    MATH  Google Scholar 

  9. Feireisl, E., Hošek, R., Maltese, D.,  Novotný, A.: Convergence and error estimates for bounded numerical solutions of the barotropic Navier-Stokes system. Numer. Methods Partial Differ. Equ. (2015, to appear)

  10. Feireisl, E.: Bum Ja Jin, and A. Novotný. Relative entropies, suitable weak solutions, and weak-strong uniqueness for the compressible Navier-Stokes system. J. Math. Fluid Mech. 14, 712–730 (2012)

    Google Scholar 

  11. 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  MathSciNet  MATH  Google Scholar 

  12. Feireisl, E., Vasseur, A.: New perspectives in fluid dynamics: mathematical analysis of a model proposed by Howard Brenner. In New directions in mathematical fluid mechanics, Adv. Math. Fluid Mech., pages 153–179. Birkhäuser Verlag, Basel (2010)

  13. Fjordholm, U. S., Käppeli, R., Mishra, S., Tadmor, E.: Construction of approximate entropy measure valued solutions for hyperbolic systems of conservation laws. Preprint (2014)

  14. Gallouët, R., Herbin, T., Maltese, D., Novotný, A.: Error estimate for a numerical approximation to the compressible barotropic Navier-Stokes equations. IMA J. Numer. Anal., To appear (2015)

  15. Germain, P.: Weak-strong uniqueness for the isentropic compressible Navier-Stokes system. J. Math. Fluid Mech. 13(1), 137–146 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  16. Gwiazda, P.: On measure-valued solutions to a two-dimensional gravity-driven avalanche flow model. Math. Methods Appl. Sci. 28(18), 2201–2223 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  17. Gwiazda, P., Świerczewska-Gwiazda, A., Wiedemann, E.: Weak-strong uniqueness for measure-valued solutions of some compressible fluid models. Nonlinearity 28(11), 3873–3890 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  18. Karlsen, K.H., Karper, T.K.: A convergent mixed method for the Stokes approximation of viscous compressible flow. IMA J. Numer. Anal. 32(3), 725–764 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  19. Karper, T.K.: A convergent FEM-DG method for the compressible Navier-Stokes equations. Numer. Math. 125(3), 441–510 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  20. Lions, P.-L.: Mathematical topics in fluid dynamics, vol. 2. Compressible models. Oxford Science Publication, Oxford (1998)

    Google Scholar 

  21. Málek, J., Nečas, J. Rokyta, M., R\(\mathring{{\rm u}}\)žička M.: Weak and measure-valued solutions to evolutionary PDE’s. Chapman and Hall, London (1996)

  22. A. Mellet and A. Vasseur. Existence and uniqueness of global strong solutions for one-dimensional compressible Navier-Stokes equations. SIAM J. Math. Anal. 39(4):1344–1365 (2007/08)

  23. Nečas, J., Novotný, A., Šilhavý, M.: Global solution to the compressible isothermal multipolar fluid. J. Math. Anal. Appl. 162(1), 223–241 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  24. Nečas, J., Šilhavý, M.: Viscous multipolar fluids. Quart. Appl. Math. 49, 247–266 (1991)

    MathSciNet  MATH  Google Scholar 

  25. Neustupa, J.: Measure-valued solutions of the Euler and Navier-Stokes equations for compressible barotropic fluids. Math. Nachr. 163, 217–227 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  26. Öttinger, H.C., Struchtrup, H., Liu, M.: Inconsistency of a dissipative contribution to the mass flux in hydrodynamics. Phys. Rev. E 80, 056303 (2009)

    Article  Google Scholar 

  27. Pedregal, P.: Parametrized measures and variational principles. Birkhäuser, Basel (1997)

    Book  MATH  Google Scholar 

  28. Plotnikov, P.I., Weigant, W.: Isothermal Navier-Stokes equations and Radon transform. SIAM J. Math. Anal. 47(1), 626–653 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  29. Prodi, G.: Un teorema di unicità per le equazioni di Navier-Stokes. Ann. Mat. Pura Appl. 48, 173–182 (1959)

    Article  MathSciNet  MATH  Google Scholar 

  30. J. Serrin. The initial value problem for the Navier-Stokes equations. In Nonlinear Problems (Proc. Sympos., Madison, Wis., 1962), pages 69–98. Univ. of Wisconsin Press, Madison, Wisconsin (1963)

  31. Sun, Y., Wang, C., Zhang, Z.: A Beale-Kato-Majda criterion for the 3-D compressible Navier-Stokes equations. J. Math. Pures Appl. 95(1), 36–47 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  32. Valli, A., Zaja̧czkowski, M.: Navier–Stokes equations for compressible fluids: Global existence and qualitative properties of the solutions in the general case. Commun. Math. Phys. 103:259–296 (1986)

Download references

Author information

Authors and Affiliations

  1. Institute of Mathematics of the Academy of Sciences of the Czech Republic, Žitná 25, 115 67, Praha 1, Czech Republic

    Eduard Feireisl

  2. Institute of Applied Mathematics and Mechanics, University of Warsaw, Banacha 2, 02-097, Warszawa, Poland

    Piotr Gwiazda & Agnieszka Świerczewska-Gwiazda

  3. Institute for Applied Mathematics, Leibniz University Hannover, Welfengarten 1, 30167, Hannover, Germany

    Emil Wiedemann

Authors
  1. Eduard Feireisl
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Piotr Gwiazda
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Agnieszka Świerczewska-Gwiazda
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Emil Wiedemann
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Emil Wiedemann.

Additional information

Communicated by C. De Lellis.

The research of E.F. leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013)/ERC Grant Agreement 320078. The Institute of Mathematics of the Academy of Sciences of the Czech Republic is supported by RVO:67985840. The research of A.Ś-G and P.G. has received funding from the National Science Centre, Poland, 2014/13/B/ST1/03094.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Feireisl, E., Gwiazda, P., Świerczewska-Gwiazda, A. et al. Dissipative measure-valued solutions to the compressible Navier–Stokes system. Calc. Var. 55, 141 (2016). https://doi.org/10.1007/s00526-016-1089-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00526-016-1089-1

Mathematics Subject Classification

Search

Navigation