Skip to main content
SpringerLink
Log in

Cauchy problem for the Navier–Stokes–Voigt model governing nonhomogeneous flows

  • Original Paper
  • Published:
Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas Aims and scope Submit manuscript

Abstract

The Navier–Stokes–Voigt model that governs flows with non-constant density of incompressible fluids with elastic properties is considered in the whole space domain \(\mathbb {R}^d\) and in the entire time interval. If \(d\in \{2,\,3,\,4\}\), we prove the existence of weak solutions (velocity, density and pressure) to the associated Cauchy problem. We also analyse some issues of regularity of the weak solutions to the considered problem and the large time behavior in special unbounded domains.

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. Amrouche, C., Berselli, L.C., Lewandowski, R., Duong, N.D.: Turbulent flows as generalized Kelvin–Voigt materials: modeling and analysis. Nonlinear Anal. 196, 111790 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  2. Antontsev, S.N., Kazhikhov, A.V., Monakhov, V.N.: Boundary value problems in mechanics of nonhomogeneous fluids. (Translation from the original Russian edition, Nauka, Novosibirsk, 1983), North-Holland, Amsterdam, (1990)

  3. Antontsev, S.N., de Oliveira, H.B., Khompysh, Kh.: The classical Kelvin–Voigt problem for nonhomogeneous and incompressible fluids: existence, uniqueness and regularity. Nonlinearity 34(5), 3083–3111 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  4. Antontsev, S.N., de Oliveira, H.B., Khompysh, Kh.: Kelvin–Voigt equations perturbed by anisotropic relaxation, diffusion and damping. J. Math. Anal. Appl. 473, 1122–1154 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  5. Antontsev, S.N., de Oliveira, H.B., Khompysh, Kh.: Kelvin–Voigt equations with anisotropic diffusion, relaxation, and damping: blow-up and large time behavior. J. Asymptot. Anal. 121(2), 125–157 (2021)

    MathSciNet  MATH  Google Scholar 

  6. Antontsev, S.N., de Oliveira, H.B., Khompysh, Kh.: Existence and large time behavior for generalized Kelvin–Voigt equations governing nonhomogeneous and incompressible fluids. J. Phys. Conf. Ser. 1268, 012008 (2019)

    Article  Google Scholar 

  7. Antontsev, S.N., de Oliveira, H.B., Khompysh, Kh.: Generalized Kelvin–Voigt equations for nonhomogeneous and incompressible fluids. Commun. Math. Sci. 17(7), 1915–1948 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  8. Bogovskiǐ, M.E., Solutions of some problems of vector analysis, associated with the operators div and grad. (Russian). Theory of cubature formulas and the application of functional analysis to problems of mathematical physics, 5–40, 149. Trudy Sem. S. L. Soboleva, no. 1,: Akad, p. 1980. Otdel., Inst. Mat., Novosibirsk, Nauk SSSR Sibirsk (1980)

  9. Desjardins, B.: Global existence results for the incompressible density-dependent Navier–Stokes equations in the whole space. Differ. Integral Equ. 10(3), 587–598 (1997)

    MathSciNet  MATH  Google Scholar 

  10. Cao, Y.P., Lunasin, E.M., Titi, E.S.: Global well-posedness of the three-dimensional viscous and inviscid simplified Bardina turbulence models. Commun. Math. Sci. 4, 823–848 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  11. Constantin, P., Foias, C.: Navier–Stokes Equations. The University of Chicago Press, Chicago (1988)

    Book  MATH  Google Scholar 

  12. DiPerna, R.J., Lions, P.-L.: Ordinary differential equations, transport theory and Sobolev spaces. Invent. Math. 98(3), 511–547 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  13. Galdi, G.P.: An Introduction to the Mathematical Theory of the Navier–Stokes Equations. Steady-State Problems. Springer, New York (2011)

    Book  MATH  Google Scholar 

  14. Joseph, D.D.: Fluid Dynamics of Viscoelastic Liquids. Springer, New York (1990)

    Book  MATH  Google Scholar 

  15. Kotsiolis, A.A., Oskolkov,A.P., Shadiev, R.: Global a priori estimates on the half-line \(t\ge 0\), asymptotic stability and time periodicity of "small" solutions of equations for the motion of Oldroyd fluids and Kelvin–Voigt fluids (Russian). LOMI Preprints, R-10-89. Akad. Nauk SSSR, Mat. Inst. Leningrad. Otdel., Leningrad, (1989)

  16. Ladyzenskaya, O.A.: On certain nonlinear problems of the theory of continuous media. In International Congress of Mathematicians at Moscow, Abstracts of Reports, 149 (1966)

  17. Ladyzhenskaya, O.A.: On errors in two of my publications on Navier–Stokes equations and their corrections. J. Math. Sci. N. Y. 115(6), 2789–2791 (2003)

    Article  MathSciNet  Google Scholar 

  18. Ladyzenskaya, O.A., Solonnikov, V.A.: Unique solvability of an initial-and boundary-value problem for viscous incompressible nonhomogeneous fluids. J. Soviet Math. 9, 697–749 (1978)

    Article  MATH  Google Scholar 

  19. Levant, B., Ramosa, F., Titi, E.S.: On the statistical properties of the 3D incompressible Navier–Stokes-Voigt model. Commun. Math. Sci. 8(1), 277–293 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  20. Lewandowski, R., Berselli, L.C.: On the Bardina’s model in the whole space. J. Math. Fluid Mech. 20(3), 1335–1351 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  21. Lewandowski, R., Layton, W.: On a well-posed turbulence model. Discrete Contin. Dyn. Syst. Ser. B 6(1), 111–128 (2006)

    MathSciNet  MATH  Google Scholar 

  22. Lions, P.-L.: Mathematical Topics in Fluid Mechanics, Volume 1: Incompressible Models. Clarendon Press, Oxford (1996)

  23. Maz’ya, V.: Sobolev Spaces with Applications to Elliptic Partial Differential Equations. Springer, Heidelberg (2011)

    Book  MATH  Google Scholar 

  24. Obukhovskiǐ, V., Zecca, P., Zvyagin, V.G.: Optimal feedback control in the problem of the motion of a viscoelastic fluid. Topol. Methods Nonlinear Anal. 23(2), 323–337 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  25. Oskolkov, A.P.: The uniqueness and global solvability of boundary-value problems for the equations of motion for aqueous solutions of polymers. J. Math. Sci. 8, 427–455 (1977)

    Article  MATH  Google Scholar 

  26. Oskolkov, A.P.: Nonlocal problems for equations of Kelvin–Voigt fluids and their \(\epsilon \)-approximations. J. Math. Sci. 87, 3393–3408 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  27. Pavlovsky, V.A.: On the theoretical description of weak water solutions of polymers. Dokl. Akad. Nauk SSSR 200(4), 809–812 (1971)

    Google Scholar 

  28. Pileckas, K.: On spaces of solenoidal vectors (Russian). Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. LOMI. 96, 237–239 (1980)

    MathSciNet  Google Scholar 

  29. Ramos, F., Titi, E.S.: Invariant measures for the 3D Navier–Stokes–Voigt equations and their Navier–Stokes limit. Discrete Contin. Dyn. Syst. 28(1), 375–403 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  30. Simon, J.: Nonhomogeneous viscous incompressible fluids: existence of velocity, density, and pressure. SIAM J. Math. Anal. 21(5), 1093–1117 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  31. Simon, J.: Compact sets in the space \(L^p(0, T;B)\). Ann. Mater. Pura Appl. 146(4), 65–96 (1987)

    MATH  Google Scholar 

  32. Zvyagin, V.G., Orlov, V.P.: On weak solutions of the equations of motion of a viscoelastic medium with variable boundary. Bound. Value Probl. 3, 215–245 (2005)

    MathSciNet  MATH  Google Scholar 

  33. Zvyagin, V.G., Turbin, M.V.: The study of initial-boundary value problems for mathematical models of the motion of Kelvin–Voigt fluids. J. Math. Sci. 168, 157–308 (2010)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The first author was supported by the Lavrenty’ev Institute of Hydrodynamics of the Siberian Branch RAS (project no. III.22.4.2, Analysis of mathematical models of continua with singularities, discontinuities and intrinsic inhomogeneities), Novosibirsk, Russia. Both first and second authors were also partially supported by the Portuguese Foundation for Science and Technology, Portugal, under the project: UIDB/04561/2020.

Author information

Authors and Affiliations

  1. CMAFcIO-Universidade de Lisboa, Lisbon, Portugal

    S. N. Antontsev & H. B. de Oliveira

  2. Lavrentyev Institute of Hydrodynamics, SB RAS, Novosibirsk, Russia

    S. N. Antontsev

  3. FCT-Universidade do Algarve, Faro, Portugal

    H. B. de Oliveira

  4. CFM-Universidade Federal de Santa Catarina, Florianópolis, Brazil

    H. B. de Oliveira

Authors
  1. S. N. Antontsev
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. H. B. de Oliveira
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to S. N. Antontsev.

Additional information

Dedicated to Professor Ildefonso Díaz on the occasion of his 70th birthday.

Publisher's Note

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

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Antontsev, S.N., de Oliveira, H.B. Cauchy problem for the Navier–Stokes–Voigt model governing nonhomogeneous flows. Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat. 116, 158 (2022). https://doi.org/10.1007/s13398-022-01300-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s13398-022-01300-x

Keywords

Mathematics Subject Classification

Search

Navigation