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The Avalos–Triggiani Problem for the Linear Oskolkov System and a System of Wave Equations


The Avalos–Triggiani problem for a system of wave equations and the linear Oskolkov system is investigated. The method proposed by G. Avalos and R. Triggiani is used to prove a theorem on the existence of a unique solution to the Avalos–Triggiani problem. The underlying mathematical model involves the linear Oskolkov system describing the flow of an incompressible viscoelastic Kelvin–Voigt fluid of zero order and a vector wave equation describing a structure immersed in the fluid.

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  1. A. P. Oskolkov, “Unsteady flows of viscoelastic fluids,” Proc. Steklov Inst. Math. 159, 105–134 (1984).

    MATH  Google Scholar 

  2. A. P. Oskolkov, “Some nonstationary linear and quasilinear systems occurring in the investigation of the motion of viscous fluids,” J. Sov. Math. 10, 299–335 (1978).

    Article  Google Scholar 

  3. O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow, 2nd ed. (Gordon and Breach, New York, 1969; Nauka, Moscow, 1970).

  4. A. P. Oskolkov, “Theory of Voight fluids,” J. Sov. Math. 21, 818–821 (1983).

    Article  Google Scholar 

  5. A. P. Oskolkov, “Initial-boundary value problems for equations of motion of Kelvin–Voight fluids and Oldroyd fluids,” Proc. Steklov Inst. Math. 179, 137–182 (1989).

    MATH  Google Scholar 

  6. A. P. Oskolkov, M. M. Achmatov, and A. A. Cotsiolis, “On the equations of motion of linear viscoelastic fluids and the equations of filtration of fluids with delay,” Zap. Nauchn. Semin. LOMI Akad. Nauk SSSR 163, 132–136 (1987).

    MATH  Google Scholar 

  7. A. P. Oskolkov, “Nonlocal problems for one class of nonlinear operator equations that arise in the theory of Sobolev type equations,” J. Sov. Math. 64, 724–735 (1993).

    MathSciNet  Article  Google Scholar 

  8. G. A. Sviridyuk and T. G. Sukacheva, “Phase spaces of a class of operator semilinear equations of Sobolev type,” Differ. Equations 26 (2), 188–195 (1990).

    MathSciNet  MATH  Google Scholar 

  9. G. A. Sviridyuk and T. G. Sukacheva, “On the solvability of a nonstationary problem describing the dynamics of an incompressible viscoelastic fluid,” Math. Notes 63 (3), 388–395 (1998).

    MathSciNet  Article  Google Scholar 

  10. A. O. Kondyukov and T. G. Sukacheva, “Phase space of the initial–boundary value problem for the Oskolkov system of nonzero order,” Comput. Math. Math. Phys. 55 (5), 823–828 (2015).

    MathSciNet  Article  Google Scholar 

  11. A. O. Kondyukov and T. G. Sukacheva, “A non-stationary model of the incompressible viscoelastic Kelvin–Voigt fluid of nonzero order in the magnetic field of the Earth,” Bull. South Ural State Univ. Ser. Math. Model. Program. Comput. Software 12 (3), 42–51 (2019).

    MATH  Google Scholar 

  12. G. Avalos, I. Lasiecka, and R. Triggiani, “Higher regularity of a coupled parabolic-hyperbolic fluid-structure interactive system,” Georgian Math. J. 15 (3), 403–437 (2008).

    MathSciNet  Article  Google Scholar 

  13. G. Avalos and R. J. Triggiani, “Backward uniqueness of the s.c. semigroup arising in parabolic-hyperbolic fluid–structure interaction,” Differ. Equations 245, 737–761 (2008).

    MathSciNet  Article  Google Scholar 

  14. K. V. Vasyuchkova, N. A. Manakova, and G. A. Sviridyuk, “Some mathematical models with a relatively bounded operator and additive 'white noise',” Bull. South Ural State Univ. Ser. Math. Model. Program. Comput. Software 10 (4), 5–14 (2017).

    MATH  Google Scholar 

  15. G. A. Sviridyuk, A. A. Zamyshlyaeva, and S. A. Zagrebina, “Multipoint initial-final value for one class of Sobolev type models of higher order with additive ‘white noise’,” Bull. South Ural State Univ. Ser. Math. Model. Program. Comput. Software 11 (3), 103–117 (2018).

    MATH  Google Scholar 

  16. A. Favini, S. A. Zagrebina, and G. A. Sviridyuk, “Multipoint initial-final value problems for dynamical Sobolev-type equations in the space of noises,” Electron. J. Differ. Equations 2018 (128), 1–10 (2018).

    MathSciNet  MATH  Google Scholar 

  17. O. A. Oleinik, “On a system of equations in boundary layer theory,” USSR Comput. Math. Math. Phys. 3 (3), 650–673 (1963).

    MathSciNet  Article  Google Scholar 

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This work was supported in part by the Ministry of Science and Higher Education of the Russian Federation, grant no. FENU-2020-0022 (2020072GZ).

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Correspondence to G. A. Sviridyuk or T. G. Sukacheva.

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Translated by I. Ruzanova

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Sviridyuk, G.A., Sukacheva, T.G. The Avalos–Triggiani Problem for the Linear Oskolkov System and a System of Wave Equations. Comput. Math. and Math. Phys. 62, 427–431 (2022).

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  • Avalos–Triggiani problem
  • incompressible viscoelastic fluid
  • linear Oskolkov system