Skip to main content
SpringerLink
Log in

The Weak Solvability of an Inhomogeneous Dynamic Problem for a Viscoelastic Continuum with Memory

  • Research Articles
  • Published:
Functional Analysis and Its Applications Aims and scope

Abstract

The existence of a weak solution to the initial boundary value problem for the equations of motion of a viscoelastic fluid with memory along the trajectories of a nonsmooth velocity field with inhomogeneous boundary condition is proved. The analysis involves Galerkin-type approximations of the original problem followed by the passage to the limit based on a priori estimates. To study the behavior of trajectories of a nonsmooth velocity field, the theory of regular Lagrangian flows is used.

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

References

  1. O. A. Ladyzhenskaya, Mathematical Questions of the Dynamics of Viscous Incompressible Fluid, Nauka, Moscow, 1970 (Russian).

    MATH  Google Scholar 

  2. I. I. Vorovich and V. I. Yudovich, Mat. Sb., 53(95):4 (1961), 393–428.

    Google Scholar 

  3. J. G. Oldroyd, Rheology: Theory and Applications, 1956, 653–682.

    Google Scholar 

  4. A. P. Oskolkov, Tr. Mat. Inst. im. V.A. Steklova, Akad. Nauk SSSR, 179 (1988), 126–164; English transl.:, Proc. Steklov Inst. Math., 179 (1989), 137–182.

    Google Scholar 

  5. Yu. N. Bibikov, A General Course in Ordinary Differential Equations, Izd. Leningrad. Univ., Leningrad, 1981 (Russian).

    MATH  Google Scholar 

  6. V. G. Zvyagin and D. A. Vorotnikov, Topological Approximation Methods for Evolutionary Problems of Nonlinear Hydrodynamics, de Gruyter Series in Nonlinear Analysis and Applications, 12 Walter de Gruyter & Co, Berlin, 2008.

    Book  MATH  Google Scholar 

  7. V. G. Zvyagin and V. P. Orlov, Nonlinear Analysis: TMA, 172 (2018), 73–98.

    Article  Google Scholar 

  8. V. G. Zvyagin and V. P. Orlov, Discrete Contin. Dyn. Syst., Ser. A, 38:12 (2018), 6327–6350.

    Article  Google Scholar 

  9. V. Zvyagin and V. Orlov, Discrete Contin. Dyn. Syst., Ser. B, 23:9 (2018), 3855–3877.

    MathSciNet  Google Scholar 

  10. R. Temam, Navier–Stokes Equations: Theory and Numerical Analysis, North-Holland, Amsterdam, 1979.

    MATH  Google Scholar 

  11. R. J. DiPerna and P.-L. Lions, Invent. Math., 98:3 (1989), 511–547.

    Article  MathSciNet  Google Scholar 

  12. G. Crippa and C. de Lellis, J. Reine Angew. Math., 616 (2008), 15–46.

    MathSciNet  Google Scholar 

Download references

Funding

This work was supported by the Russian Science Foundation under grant no. 22-11-00103, https://rscf.ru/project/22-11-00103/.

Author information

Authors and Affiliations

  1. Voronezh State University, Voronezh, Russia

    V. G. Zvyagin & V. P. Orlov

Authors
  1. V. G. Zvyagin
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. V. P. Orlov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to V. G. Zvyagin.

Additional information

Translated from Funktsional'nyi Analiz i ego Prilozheniya, 2023, Vol. 57, pp. 93–99 https://doi.org/10.4213/faa4034.

Translated by O. V. Sipacheva

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zvyagin, V.G., Orlov, V.P. The Weak Solvability of an Inhomogeneous Dynamic Problem for a Viscoelastic Continuum with Memory. Funct Anal Its Appl 57, 74–79 (2023). https://doi.org/10.1134/S0016266323010082

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0016266323010082

Keywords

Search

Navigation