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Nonlocal problems for one class of nonlinear operator equations that arise in the theory of sobolev type equations

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Abstract

Let

, be Hilbert spaces forming a sequence of compact imbeddingsH 3 ⊂H 2 ⊂H 1 ⊂H 0. Consider the nonlinear equation

in H2 and suppose that conditions (8)–(12) and (15) hold for the operators A and K(u) and the external force F(t). Four nonlocal problems (cf. 1–4) are studied for such equations. Examples are given [cf. (2)–(6)] for nonlinear dissipative equations of Sobolev type that are reducible to an abstract non-linear equation [cf. (7)–(12), (15)].

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Literature cited

  1. O. A. Ladyzhenskaya, Mathematical Problems in the Dynamics of a Viscous Incompressible Fluid [in Russian], 2nd rev. aug. ed., Nauka, Moscow (1970); English transl. of 1st edn., The Mathematical Theory of Viscous Incompressible Flow, Gordon and Breach, New York (1963); rev. 1969.

    Google Scholar 

  2. O. A. Ladyzhenskaya, “On some nonlinear problems in the mechanics of continuous media,” in: Intern. Congr. Math., Abstracts of Invited Lectures [in Russian], Moscow (1966), p. 149.

  3. O. A. Ladyzhenskaya, “On new equations for the description of motion of viscous incompressible fluids and on the solvability in the large of boundary value problems for them,” Trudy Mat. Inst. AN SSSR,102, 85–104 (1967); Zap. Nauchn. Sem. Leningr. Otd. Mat. Inst. AN SSSR,7, 126–164 (1968).

    MATH  Google Scholar 

  4. O. A. Ladyzhenskaya, “On the limit states for modified Navier-Stokes equations in a three-dimensional space,” Zap. Nauchn. Sem. Leningr. Otd. Mat. Inst. AN SSSR,84, 131–146 (1979).

    MATH  MathSciNet  Google Scholar 

  5. O. A. Ladyzhenskaya, “On some directions of investigations carried out in the LOMI Mathematical Physics Laboratory,” Trudy Mat. Inst. AN SSSR,175, 217–245 (1986).

    MathSciNet  Google Scholar 

  6. O. A. Ladyzhenskaya, “On the determination of minimal global attractors for the Navier-Stokes and other partial differential equations,” Uspekhi Mat. Nauk,42, No. 6, 25–60 (1987); English transl. in Russian Math. Surveys,42 (1987).

    MathSciNet  Google Scholar 

  7. S. L. Sobolev, “On one new problem in mathematical physics,” Izv. Akad. Nauk SSSR, Ser. Mat.,18, No. 1, 3–50 (1954).

    MathSciNet  Google Scholar 

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

    Google Scholar 

  9. A. P. Oskolkov, “On one time-dependent quasilinear system with a small parameter regularizing the Navier-Stokes equations,” in: Problems in Mathematical Analysis [in Russian], No. 4, 78–87, Leningrad State University, Leningrad (1973).

    Google Scholar 

  10. A. P. Oskolkov, “The uniqueness and solvability in the large of boundary value problems for the equations of motion of aqueous solutions of polymers,” Zap. Nauchn. Sem. Leningr. Otd. Mat. Inst. AN SSSR,38, 98–136 (1973); English transl. in J. Soviet Math.,8, No. 4 (1977).

    MATH  MathSciNet  Google Scholar 

  11. A. P. Oskolkov, “Some time-dependent linear and quasilinear systems that arise in the study of the motion of viscous fluids,” Zap. Nauchn. Sem. Leningr. Otd. Mat. Inst. AN SSSR,59, 133–177 (1976); English transl. in J. Soviet Math.,10, No. 2 (1978).

    MATH  MathSciNet  Google Scholar 

  12. A. P. Oskolkov, “Toward the theory of Voight fluids,” Zap. Nauchn. Sem. Leningr. Otd. Mat. Inst. AN SSSR,96, 233–236 (1980); English transl. in J. Soviet Math.,21, No. 5 (1983).

    MATH  MathSciNet  Google Scholar 

  13. A. P. Oskolkov, “On time-dependent flows of viscoelastic fluids,” Trudy Mat. Inst. Steklov,159, 101–130 (1983); English transl. in Proc. Steklov Inst. Math.,159, No. 2 (1984).

    MathSciNet  Google Scholar 

  14. A. P. Oskolkov, “Initial-boundary value problems for the equations of motion of Kelvin-Voight fluids and Oldroyd fluids,” Trudy Mat. Inst. Steklov,179, 126–164 (1988); English transl. in Proc. Steklov Inst. Math.,179, No. 2 (1989).

    MathSciNet  Google Scholar 

  15. N. A. Karazeeva, A. A. Kot-siolis, and A. P. Oskolkov, “On attractors and dynamical systems generated by initial-boundary value problems for the equations of motion of linear viscoelastic fluids,” Preprint No. R-10-88, Leningr. Branch, Steklov Math. Inst., Leningrad (1988); Trudy Mat. Inst. Steklov,188, 60–87 (1990); English transl. in Proc. Stekov Inst. Math.,188, No. 3, 73–108 (1991).

    Google Scholar 

  16. A. A. Kot-siolis, A. P. Oskolkov, and R. D. Shadiev, “A priori estimates on the semi-axis t ≥ 0 for 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. AN SSSR,182, 86–101 (1990).

    Google Scholar 

  17. A. P. Oskolkov and R. D. Shadiev, “Nonlocal problems in the theory of the equations of motion of Kelvin-Voight fluids,” Zap. Nauchn. Sem. Leningr. Otd. Mat. Inst. AN SSSR,181, 122–163 (1990);185, 134–148 (1990).

    Google Scholar 

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Authors
  1. A. P. Oskolkov
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Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 198, pp. 31–48, 1991.

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Oskolkov, A.P. Nonlocal problems for one class of nonlinear operator equations that arise in the theory of sobolev type equations. J Math Sci 64, 724–735 (1993). https://doi.org/10.1007/BF02988478

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