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On one problem of dynamics of thermoviscoelastic medium of Oldroyd type

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Abstract

We establish nonlocal existence theorem for the weak solution for an initial-boundary value problem for the dynamic model of thermoviscoelasticity of Oldroyd type in the planar case.

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References

  1. J. G. Oldroyd, Non-Newtonian Flow of Liquids and Solids. Rheology: Theory and Applications, Ed. by F. R. Eirich (AP, New York, 1956), Vol. I.

  2. A. P. Oskolkov, “Initial-Boundary Value Problems for Equations of Motion of Kelvin-Voight Fluids and Oldroyd Fluids,” Trudy MIAN SSSR 2, 137–182 (1989).

    Google Scholar 

  3. S. N. Antontsev, A. V. Kazhihov, and V. N. Monahov, Boundary-Value Problems in Mechanics (Nauka, Noviosibirsk, 1983) [in Russian].

    MATH  Google Scholar 

  4. Yu. Ya. Agranovich and P. E. Sobolevskii, “Investigation of Mathematical Model of a Viscoelastic Fluid,” Dokl. Akad. Nauk UkrSSR. Ser. A, 86(10), 3–6 (1989).

    MathSciNet  Google Scholar 

  5. V. P. Orlov, “On the Strong Solutions of a Regularized Model of a Nonlinear Visco-Elastic Medium,” Matem. Zametki 84(2), 238–253 (2008).

    Article  Google Scholar 

  6. V. P. Orlov, “Strong Solution of Certain Thermoviscoelastic System,” Izv. Vyssh. Uchebn. Zaved. Mat., No. 8, 51–56 (2010) [Russian Mathematics (Iz. VUZ) 54 (8), 41–47 (2010)].

    Google Scholar 

  7. D. Blanchard, N. Bruyère, and O. Guibé, “Existence and Uniqueness of the Solution of a Boussinesq System with Nonlinear Dissipation,” Commun. Pure Appl. Anal. 12(5), 2213–2227 (2013).

    Article  MATH  MathSciNet  Google Scholar 

  8. I. Pawlow and W. Zajaczkowski, “Global Regular Solutions to a Kelvin-Voigt Type Thermoviscoelastic System,” arXiv:1112.3176v1 [math.AP] 267–293 (2011).

    Google Scholar 

  9. L. Consiglieri, “Weak Solutions for a Class of Non-Newtonian Fluids with Energy Transfer,” J. Math. Fluid Mech. 2(3), 267–293 (2000).

    Article  MATH  MathSciNet  Google Scholar 

  10. E. Bonetti and G. Bonfanti, “Existence and Uniqueness of the Solution to a 3D Thermoviscoelastic System,” Electronic J. Diff. Equat., No. 50, 1–15 (2003).

    Google Scholar 

  11. S. G. Krein, Functional Analysis (Nauka, Moscow, 1972) [in Russian].

    Google Scholar 

  12. R. Temam, Navier-Stokes Equations: Theory and Numerical Analysis (North-Holland Publ. Co., Amsterdam-New York-Oxford, 1977;Mir, Moscow, 1981).

    MATH  Google Scholar 

  13. J.-L. Lions, Quelques Methodes de Resolution des Problemes aux Limites non Lineaires (Dunod Gauthiers-Villar, Paris, 1969).

    MATH  Google Scholar 

  14. J. Simon, “Compact Sets in the Space L p(0, T;B),” Ann. Math. Pure Appl. (4) 146, 65–96 (1988).

    Article  Google Scholar 

  15. K. Yosida, Functional Analysis (Springer-Verlag, Berlin-Göttingen-Heidelberg, 1965; Mir, Moscow, 1967).

    Book  MATH  Google Scholar 

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Correspondence to V. P. Orlov.

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Original Russian Text © V.P. Orlov, M.I. Parshin, 2014, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2014, No. 5, pp. 68–74.

Submitted by V.G. Zvyagin

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Orlov, V.P., Parshin, M.I. On one problem of dynamics of thermoviscoelastic medium of Oldroyd type. Russ Math. 58, 57–62 (2014). https://doi.org/10.3103/S1066369X14050089

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