Glasnik Matematicki, Vol. 58, No. 1 (2023), 85-99. \( \)

RELATIVE ENERGY INEQUALITY AND WEAK-STRONG UNIQUENESS FOR AN ISOTHERMAL NON-NEWTONIAN COMPRESSIBLE FLUID

Richard Andrášik, Václav Mácha and Rostislav Vodák

Department of Mathematical Analysis and Applications of Mathematics, Faculty of Science, Palacký University Olomouc, 17. listopadu 12, 771 46 Olomouc, Czech Republic
e-mail:andrasik.richard@gmail.com

Institute of Mathematics of the Czech Academy of Sciences, Žitná 25, 115 67 Praha 1, Czech Republic
e-mail:macha@math.cas.cz

Department of Mathematical Analysis and Applications of Mathematics, Faculty of Science, Palacký University Olomouc, 17. listopadu 12, 771 46 Olomouc, Czech Republic
e-mail:rostislav.vodak@gmail.com

Abstract.   Our paper deals with three-dimensional nonsteady Navier-Stokes equations for non-Newtonian compressible fluids. It contains a de­ri­va­tion of the relative energy inequality for the weak solutions to these equations. We show that the standard energy inequality implies the relative energy inequality. Consequently, the relative energy inequality allows us to achieve a weak-strong uniqueness result. In other words, we present that the weak solution of the Navier-Stokes system coincides with the strong solution emanated from the same initial conditions as long as the strong solution exists. For this purpose, a new assumption on the coercivity of the viscous stress tensor was introduced along with two natural examples satisfying it.

2020 Mathematics Subject Classification.   35Q30, 35Q35, 76N06

Key words and phrases.   Compressible Navier-Stokes equations, non-constant viscosity, relative energy inequality, weak-strong uniqueness

Full text (PDF) (access from subscribing institutions only)

https://doi.org/10.3336/gm.58.1.07

References:

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

  2. R. Andrášik and R. Vodák, Rigorous derivation of a 1D model from the 3D non-steady Navier-Stokes equations for compressible nonlinearly viscous fluids, Electron. J. Differential Equations (2018), Paper No. 114, 21.
    MathSciNet

  3. R. Andrášik and R. Vodák, Compressible nonlinearly viscous fluids: asymptotic analysis in a 3D curved domain, J. Math. Fluid Mech. 21 (2019), Paper No. 13, 27.
    MathSciNet    CrossRef

  4. D. Basarić, Existence of dissipative (and weak) solutions for models of general compressible viscous fluids with linear pressure, J. Math. Fluid Mech. 24 (2022), Paper No. 56, 22.
    MathSciNet    CrossRef

  5. P. Bella, E. Feireisl and A. Novotný, Dimension reduction for compressible viscous fluids, Acta Appl. Math. 134 (2014), 111–121.
    MathSciNet    CrossRef

  6. A. Blouza and H. Le Dret, Existence and uniqueness for the linear Koiter model for shells with little regularity, Quart. Appl. Math. 57 (1999), 317–337.
    CrossRef    MathSciNet

  7. J. Březina, O. Kreml and V. Mácha, Dimension reduction for the full Navier-Stokes-Fourier system, J. Math. Fluid Mech. 19 (2017), 659–683.
    CrossRef    MathSciNet

  8. P. G. Ciarlet, Mathematical elasticity. Vol. II, North-Holland Publishing Co., Amsterdam, 1997.
    MathSciNet

  9. P. G. Ciarlet, Mathematical elasticity. Vol. III, North-Holland Publishing Co., Amsterdam, 2000.
    MathSciNet

  10. E. DiBenedetto, Degenerate parabolic equations, Springer-Verlag, New York, 1993.
    MathSciNet    CrossRef

  11. L. Diening and F. Ettwein, Fractional estimates for non-differentiable elliptic systems with general growth, Forum Math. 20 (2008), 523–556.
    MathSciNet    CrossRef

  12. B. Ducomet, Š. Nečasová, M. Pokorný and M. A. Rodríguez-Bellido, Derivation of the Navier-Stokes-Poisson system with radiation for an accretion disk, J. Math. Fluid Mech. 20 (2018), 697–719.
    MathSciNet    CrossRef

  13. R. G. Durán and M. A. Muschietti, The Korn inequality for Jones domains, Electron. J. Differential Equations 2004, No. 127, 10.
    MathSciNet

  14. E. Feireisl, Dynamics of viscous compressible fluids, Oxford University Press, Oxford, 2004.
    MathSciNet

  15. E. Feireisl, B. J. Jin and A. Novotný, Relative entropies, suitable weak solutions, and weak-strong uniqueness for the compressible Navier-Stokes system, J. Math. Fluid Mech. 14 (2012), 717–730.
    MathSciNet    CrossRef

  16. E. Feireisl, X. Liao and J. Málek, Global weak solutions to a class of non-Newtonian compressible fluids, Math. Methods Appl. Sci. 38 (2015), 3482–3494.
    MathSciNet    CrossRef

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

  18. D. Iftimie, G. Raugel and G. R. Sell, Navier-Stokes equations in thin 3D domains with Navier boundary conditions, Indiana Univ. Math. J. 56 (2007), 1083–1156.
    MathSciNet    CrossRef

  19. M. Jurak and J. Tambača, Derivation and justification of a curved rod model, Math. Models Methods Appl. Sci. 9 (1999), 991–1014.
    MathSciNet    CrossRef

  20. M. Jurak and J. Tambača, Linear curved rod model. General curve, Math. Models Methods Appl. Sci. 11 (2001), 1237–1252.
    MathSciNet    CrossRef

  21. M. Kalousek, V. Mácha and Š. Nečasová, Local-in-time existence of strong solutions to a class of the compressible non-Newtonian Navier-Stokes equations, Math. Ann. 384 (2022), 1057–1089.
    MathSciNet    CrossRef

  22. M. A. Krasnosel'skiĭ and J. B. Rutickiĭ, Convex functions and Orlicz spaces, P. Noordhoff Ltd., Groningen, 1961.
    MathSciNet

  23. A. Kufner, O. John and S. Fučík, Function spaces, Noordhoff International Publishing, Leyden; Academia, Prague, 1977.
    MathSciNet

  24. A. E. Mamontov, On the global solvability of the multidimensional Navier-Stokes equations of a nonlinearly viscous fluid. I, Sibirsk. Mat. Zh. 40 (1999), 408–420.
    MathSciNet

  25. A. E. Mamontov, On the global solvability of the multidimensional Navier-Stokes equations of a nonlinearly viscous fluid. II, Sibirsk. Mat. Zh. 40 (1999), 635–649.
    MathSciNet    CrossRef

  26. R. Vodák, Asymptotic analysis of three dimensional Navier-Stokes equations for compressible nonlinearly viscous fluids, Dyn. Partial Differ. Equ. 5 (2008), 299–311.
    MathSciNet    CrossRef

  27. V. V. Zhikov and S. E. Pastukhova, On the solvability of the Navier-Stokes system for a compressible non-Newtonian fluid, Dokl. Akad. Nauk 427 (2009), 303–307.
    MathSciNet    CrossRef
Glasnik Matematicki Home Page