Skip to main content
SpringerLink
Log in

Some nonlocal problems for modified Navier-Stokes equations

  • Published:
Journal of Mathematical Sciences Aims and scope Submit manuscript

Abstract

For the Navier-Stokes equations (2) and (3), modified in the O. A. Ladyzhenskaya sense, and for the Navier-Stokes equations (4), modified in the Kelvin-Voight sense, one investigates the solvability on the semiaxis t≥ 0 of the initial-boundary value problems with right-hand sidesf,f t ε S2 (ℝ+; L2Ω) and one proves the existence of solutions, periodic with respect to t with any period Ω>0, in the case of a right-hand sidef(x, t) εL 2 (Q ω) that is periodic with respect to t with period Ω.

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. O. A. Ladyzhenskaya, “On certain nonlinear problems of the theory of continuous media,” in: Internat. Congress of Mathematicians (Moscow, 1966), Abstracts of Reports, Moscow (1966), p. 149.

  2. O. A. Ladyzhenskaya, “On new equations for the description of the motion of viscous incompressible fluids, and the global solvability of their boundary value problems,” Trudy Mat. Inst. Akad. Nauk SSSR,102, 85–104 (1967).

    Google Scholar 

  3. O. A. Ladyzhenskaya, “On modifications of the Navier-Stokes equations for large gradients of velocities,” Zap. Nauchn. Sem. Leningr. Otd. Mat. Inst.,7, 126–154 (1968).

    Google Scholar 

  4. O. A. Ladyzhenskaya, The Mathematical Problems of the Dynamics of a Viscous Incompressible Fluid [in Russian], Nauka, Moscow (1970).

    Google Scholar 

  5. O. A. Ladyzhenskaya, “Limit states for modified Navier-Stokes equations in three-dimensional space,” Zap. Nauchn. Sem. Leningr. Otd. Mat. Inst.,84, 131–146 (1979).

    Google Scholar 

  6. O. A. Ladyzhenskaya, “On the determination of minimal global attractors for the Navier-Stokes and other partial differential equations,” Usp. Mat. Nauk,42, No. 6, 25–60 (1987).

    Google Scholar 

  7. O. A. Ladyzhenskaya, “On some directions in the research carried out at the Laboratory of Mathematical Physics of the Leningrad Branch of the Institute of Mathematics,” Trudy Mat. Inst. Akad. Nauk SSSR,175, 217–245 (1986).

    Google Scholar 

  8. A. P. Oskolkov, “On a certain nonstationary quasilinear system with a small parameter, that regularizes the system of Navier-Stokes equations,” Probl. Mat. Anal., No. 4, 75–86 (1973).

    Google Scholar 

  9. A. P. Oskolkov, “On certain nonstationary linear and quasilinear systems occurring in the study of the motion of viscous fluids,” Zap. Nauchn. Sem. Leningr. Otd. Mat. Inst.,59, 133–177 (1976).

    Google Scholar 

  10. V. K. Kalantarov, “On the attractors for some nonlinear problems of mathematical physics,” Zap. Nauchn. Sem. Leningr. Otd. Mat. Inst.,152, 50–54 (1986).

    Google Scholar 

  11. N. A. Karazeeva, A. A. Kotsiolis, and A. P. Oskolkov, “On attractors and dynamical systems that can be generated by initial-boundary value problems for the equations of motion of linear viscoelastic fluids,” Preprint LOMI R-10-88, Leningrad (1988).

  12. N. A. Karazeeva, A. A. Kotsiolis, and A. P. Oskolkov, “On dynamical systems generated by initial-boundary value problems for the equations of motion of linear viscoelastic fluids,” Trudy Mat. Inst. Akad. Nauk SSSR,188, 59–87 (1990).

    Google Scholar 

  13. A. A. Kotsiolis, A. P. Oskolkov, and P. D. Shadiev, “Global a priori estimates on the semiaxis t≥0, the asymptotic stability and periodicity with respect to time of the ‘small’ solutions of the equations of motion of Oldroyd and Kelvin-Voight fluids,” Preprint LOMI R-10-89 (1989).

  14. A. A. Kotsiolis, A. P. Oskolkov, and P. D. Shadiev, “A priori estimates on the semiaxis t≥0 for the solutions of the equations of motion of linear viscoelastic fluids with an infinite Dirichlet integral, and their applications,” Zap. Nauchn. Sem. Leningr. Otd. Mat. Inst.,182, 86–101 (1990).

    Google Scholar 

  15. A. P. Oskolkov and R. D. Shadiev, “Nonlocal problems in the theory of equations of motion for Kelvin-Voight fluids,” Zap. Nauchn. Sem. Leningr. Otd. Mat. Inst.,181, 146–185 (1990).

    Google Scholar 

  16. A. P. Oskolkov and R. D. Shadiev, “Nonlocal problems in the theory of equations of motion for Kelvin-Voight fluids. II,” Zap. Nauchn. Sem. Leningr. Otd. Mat. Inst.,185, 111–124 (1990).

    Google Scholar 

  17. R. D. Shadiev, “Some nonlocal problems for the equations of the motion of Kelvin-Voight fluids,” Izv. Akad. Nauk UzSSR, Ser. Fiz.-Mat. Nauk, No. 6, 17–24 (1990).

    Google Scholar 

  18. R. D. Shadiev, “Some nonlocal problems for the Navier-Stokes equations, modified in the O. A. Ladyzhenskaya sense,” Preprint LOMI R-6-90, Leningrad (1990).

  19. J.-L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires, Dunod, Gauthier-Villars, Paris (1969).

    Google Scholar 

  20. M. A. Krasnosel'skii, Topological Methods in the Theory of Nonlinear Integral Equations, Macmillan, New York (1964).

    Google Scholar 

Download references

Authors
  1. A. P. Oskolkov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. R. D. Shadiev
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 188, pp. 105–127, 1991.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Oskolkov, A.P., Shadiev, R.D. Some nonlocal problems for modified Navier-Stokes equations. J Math Sci 70, 1789–1805 (1994). https://doi.org/10.1007/BF02149149

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02149149

Keywords

Search

Navigation