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Time periodic solutions of smooth convergent dissipative ε-approximations of the modified Navier-Stikes equations

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Abstract

In this paper, we prove the global existence of time periodic classical solutions v' of dissipative ε-approximations (4)–(6) for the three-dimensional modified Navier-Stokes Eqs. (1)–(3) that satisfy the first boundary condition. We also study the convergence for ε → 0 of solutions {v'} to time periodic classical solutions v of Eqs. (1)–(3). Bibliography: 21 titles.

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Literature Cited

  1. O. A. Ladyzhenskaya,Mathematical Problems in the Dynamics of Viscous Incompressible Fluid [in Russian], 2nd ed., Moscow (1970).

  2. O. A. Ladyzhenskaya, “On some nonlinear problems in the mechanics of solid medium,”Proc. Intern. Math. Cong., Moscow (1968), pp. 560–573.

  3. O. A. Ladyzhenskaya, “On new equations for describing viscous incompressible flows and the solvability in the large of boundary-value problems for them,”Trudy Mat. Inst. AN SSSR,102, 85–104 (1967);Zap. Nauchn. Semin. LOMI,76, 126–154 (1968).

    MATH  Google Scholar 

  4. J. L. Lions,Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéares, Springer-Verlag (1961).

  5. B. Brefort, J. M. Ghidaglia, and R. Temam, “Attractors for the penalized Navier-Stokes equations,”SIAM J. Math. Anal.,19, 1–21 (1988).

    Article  MathSciNet  Google Scholar 

  6. O. A. Ladyzhenskaya and G. A. Seregin, “On a method for approximate solution of initial boundary-value problems for the Navier-Stokes equations,”Zap. Nauchn. Semin. LOMI,197, 87–119 (1992).

    Google Scholar 

  7. J. Hale,Theory of Functional Differential Equations, Appl. Math. Sci.,3, Springer-Verlag, New York (1977).

    Google Scholar 

  8. V. A. Pliss,Nonlocal Problems in the Theory of Oscillations [in Russian], Nauka, Moscow (1964).

    Google Scholar 

  9. A. P. Oskolkov, “Nonlocal problems for the equations of viscoelastic liquids,”Byull. Sib. Mat. Ob.,2, 14–16 (1990).

    Google Scholar 

  10. A. P. Oskolkov and R. D. Shadiev, “Nonlocal problems in the theory of Kelvin-Voight fluids,”Zap. Nauchn. Semin. LOMI,181, 122–163 (1990).

    Google Scholar 

  11. A. P. Oskolkov and R. D. Shadiev, “Some nonlocal problems for the modified Navier-Stokes equations,”Zap. Nauchn. Semin. LOMI,188, 105–127 (1991).

    Google Scholar 

  12. A. P. Oskolkov, “Nonlocal problems for a class of nonlinear operator equations which appear in the theory of equations of S. L. Sobolev type,”Zap. Nauchn. Semin. LOMI,198, 31–48 (1991).

    MATH  Google Scholar 

  13. A. P. Oskolkov, “Nonlocal problems for the equations of Kelvin-Voight fluids,”Zap. Nauchn. Semin. LOMI,197, 120–158 (1992).

    MATH  Google Scholar 

  14. A. P. Oskolkov, A. A. Kotsiolis, and R. D. Shadiev, “Nonlocal problems for some classes of dissipative equations of S. L. Sobolev type,”Zap. Nauchn. Semin. POMI,199, 91–113 (1992).

    Google Scholar 

  15. A. P. Oskolkov, “On some semilinear dissipative systems with small parameter which appear in numerical solution of the Navier-Stokes equations, the equations of Jeffrey-Oldroyd fluids, and Kelvin-Voight fluids,”Zap. Nauchn. Semin. POMI,202, 158–184 (1992).

    MATH  Google Scholar 

  16. A. A. Kotsiolis and A. P. Oskolkov, “The initial boundary-value problem with a free surface condition for the ε-approximations of the Navier-Stokes equations and some their regularizations,”Zap. Nauchn. Semin. POMI,205, 38–70 (1993).

    Google Scholar 

  17. A. P. Oskolkov, “Smooth convergent ε-approximations for the first boundary-value problem for the equations of Kelvin-Voight fluids and Jeffrey-Oldroyd fluids,”Zap. Nauchn. Semin. POMI,215, 76–85 (1994).

    Google Scholar 

  18. A. P. Oskolkov, “Smooth convergent ε-approximations for the first boundary-value problem for the equations of Kelvin-Voight fluids,”Zap. Nauchn. Semin. POMI,225, 117–139 (1994).

    Google Scholar 

  19. A. P. Oskolkov, “Smooth convergent ε-approximations for the first boundary-value problem for the modified Navier-Stokes equations and their attractors,”Zap. Nauchn. Semin. POMI,221, 98–127 (1994).

    Google Scholar 

  20. A. P. Oskolkov, “On some pseudoparabolic systems with small parameter which appear in numerical investigation of the equations of Kelvin-voight fluids,”Theory of Sobolev Type Equations and Applic., Chelyab. Gos. Univ., 113–132 (1994).

  21. A. A. Kotsiolis, “Time periodic solutions of dissipative ε-approximations for the modified Navier-Stokes equations,”Zap. Nauchn. Semin. POMI,209, 108–123 (1994).

    Google Scholar 

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Authors
  1. A. P. Oskolkov
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Dedicated to V. A. Solonnikov on his sixtieth anniversary

Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 213, 1994, pp. 116–130.

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Oskolkov, A.P. Time periodic solutions of smooth convergent dissipative ε-approximations of the modified Navier-Stikes equations. J Math Sci 84, 888–897 (1997). https://doi.org/10.1007/BF02399940

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  • DOI: https://doi.org/10.1007/BF02399940

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