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Monotone Mappings and Flows of Viscous Media

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Abstract

We establish solvability of the boundary-value problems describing stationary and periodic flows of viscoplastic media. In the case of stationary flows we study the question of convergence of the Galerkin method. For the problem of periodic flows we prove a version of the second Bogolyubov theorem.

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References

  1. Mosolov P. P. and Myasnikov V. P., Mechanics of Rigid-Plastic Media [in Russian], Nauka, Moscow (1981).

    Google Scholar 

  2. Gajewski H., Gröger K., and Zacharias K., Nonlinear Operator Equations and Operator Differential Equations [Russian translation], Mir, Moscow (1978).

    Google Scholar 

  3. Krasnosel'ski\(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\imath } \) M. A. and Zabre\(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\imath } \)ko P. P., Geometric Methods of Nonlinear Analysis [in Russian], Nauka, Moscow (1975).

    Google Scholar 

  4. Borisovich Yu. G., Gel'man B. D., Myshkis A. D., and Obukhovski\(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\imath } \) V. V., “Topological methods in the fixed-point theory of multivalued maps, ” Uspekhi Mat. Nauk, 35, No. 1, 59-126 (1980).

    Google Scholar 

  5. Pokhozhaev S. I., “On solvability of nonlinear equations with odd operators, ” Funktsional. Analiz i Plilozhen., 1, No. 3, 66-73 (1967).

    Google Scholar 

  6. Browder F. E., “Nonlinear elliptic boundary value problems and the generalized topological degree, ” Bull. Amer. Math. Soc., 76, No. 5, 999-1005 (1970).

    Google Scholar 

  7. Skrypnik I. V., Methods for Studying Nonlinear Elliptic Boundary Value Problems [in Russian], Nauka, Moscow (1990).

    Google Scholar 

  8. Bobylev N. A., Emel'yanov S. V., and Korovin S. K., Geometric Methods in Variational Problems [in Russian], Magistr, Moscow (1998).

    Google Scholar 

  9. Klimov V. S., “On the problem of periodic solutions of operator differential inclusions, ” Izv. Akad. Nauk SSSR Ser. Mat., 53, No. 2, 309-327 (1989).

    Google Scholar 

  10. Klimov V. S., “Topological characteristics of nonsmooth functionals, ” Izv. Akad. Nauk SSSR Ser. Mat., 62, No. 5, 117-134 (1998).

    Google Scholar 

  11. Klimov V. S., “Bounded solutions to differential inclusions with a homogeneous principal part, ” Izv. Ross. Akad. Nauk, 64, No. 4, 109-130 (2000).

    Google Scholar 

  12. Klimov V. S., “An infinite-dimensional version of Morse theory for Lipschitz functionals, ” Mat. Sb., 193, No. 6, 105-122 (2002).

    Google Scholar 

  13. Klimov V. S. and Semko E. R., “Multivalued superposition operators in the spaces of measurable functions, ” submitted to VINITI, No. 2934-B91.

  14. Zabre\(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\imath } \)ko P. P. and Kovalenok A. P., “Calculation of the index of a singular point of a pseudomonotonic field, ” Tr. Inst. Mat. NAN Belarusi, 1, 107-124 (1998).

    Google Scholar 

  15. Bogolyubov N. N. and Mitropol'ski\(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\imath } \) Yu. A., Asymptotic Methods in the Theory of Nonlinear Oscillations [in Russian], Nauka, Moscow (1974).

    Google Scholar 

  16. Godunov S. K., Elements of Continuum Mechanics [in Russian], Nauka, Moscow (1978).

    Google Scholar 

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

    Google Scholar 

  18. Lions J.-L., Some Methods for Solving Nonlinear Boundary Value Problems [Russian translation], Mir, Moscow (1972).

    Google Scholar 

  19. Litvinov V. G., Motion of a Nonlinear-Viscous Fluid [in Russian], Nauka, Moscow (1982).

    Google Scholar 

  20. Temam R., Navier-Stokes Equations. Theory and Numerical Analysis [Russian translation], Mir, Moscow (1981).

    Google Scholar 

  21. Yudovich V. I., “An example of the onset of a secondary stationary or periodic ow under the loss of stability of a laminar ow of a viscous incompressible uid, ” Prikl. Mat. i Mekh., 29, No. 3, 453-467 (1965).

    Google Scholar 

  22. Klimov V. S., “A regularization method for the evolution problems of the mechanics of viscoplastic media, ” Mat. Zametki, 62, No. 4, 483-493 (1997).

    Google Scholar 

  23. Klimov V. S., “Bounded and almost periodic solutions of the differential inclusions of parabolic type, ” Differentsial'nye Uravneniya, 28, No. 6, 1049-1056 (1992).

    Google Scholar 

  24. Devaut G. and Lions J.-L., Inequalities in Mechanics and Physics [Russian translation], Mir, Moscow (1980).

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Authors and Affiliations

  1. Yaroslavl'' State University, Russia

    V. S. Klimov

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  1. V. S. Klimov
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Klimov, V.S. Monotone Mappings and Flows of Viscous Media. Siberian Mathematical Journal 45, 1063–1074 (2004). https://doi.org/10.1023/B:SIMJ.0000048921.07543.17

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  • DOI: https://doi.org/10.1023/B:SIMJ.0000048921.07543.17

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