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Equivalence of Weak Solvability of Initial-Boundary Value Problems for the Jeffries-Oldroyd Model and one Integro-Differential System with Memory

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Abstract

The equivalence of weak solvability of initial boundary value problems for the Jeffries-Oldroyd model and one integro-differential system with memory is established. The proofs of the statements are essentially based on the properties of regular Lagrangian flows.

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References

  1. Vorotnikov, D.A., Zvyagin, V.G. “Review of results and open problems on mathematical models of Jeffries-type viscoelastic media”, Vestn. VSU 2, 30–50 (2009).

    MATH  Google Scholar 

  2. Temam, R. The Navier-Stokes Equations. Theory and Numerical Analysis (Mir, Moscow, 1987) [in Russian].

    Google Scholar 

  3. Zvyagin, V.G., Vorotnikov, D.A. Topological Approximation Methods for Evolutionary Problems of Nonlinear Hydrodynamics (De Gruyter Series in Nonlinear Anal. and Appl. (12), Walter de Gruyter, Berlin-New York, 2008).

    Google Scholar 

  4. Zvyagin, V.G., Orlov, V.P. “Solvability of one non-Newtonian fluid dynamics model with memory”, Nonlinear Anal.: TMA 172, 79–98 (2018).

    Article  MathSciNet  Google Scholar 

  5. Zvyagin, A.V., Zvyagin, V.G., Polyakov, D.N. “Dissipative solvability of an alpha model of fluid flow with memory”, Computational Mathematics and Mathematical Physics 59, 1185–1198 (2019).

    Article  MathSciNet  Google Scholar 

  6. Orlov, V.P., Sobolevskii, P.E. “On mathematical models of a viscoelasticity with a memory”, Diff. Integral Equat. 4 (1), 103–115 (1991).

    MathSciNet  MATH  Google Scholar 

  7. DiPerna, R.J., Lions, P.L. “Ordinary differential equations, transport theory and Sobolev spaces”, Invent. Math. 98, 511–547 (1989).

    Article  MathSciNet  Google Scholar 

  8. Ambrosio, L. “Transport equation and Cauchy problem for BV vector fields”, Invent. Math. 158, 227–260 (2004).

    Article  MathSciNet  Google Scholar 

  9. Crippa, G., de Lellis, C. “Estimates and regularity results for the diPerna-Lions flow”, J. Reine Angew. Math. 6 (6), 15–46 (2008).

    MathSciNet  MATH  Google Scholar 

  10. Vorotnikov, D.A., Zvyagin, V.G. “On convergence of solutions of a regularized problem for the equations of motion of a viscoelastic Jeffries medium to solutions of the original problem”, Fund. Appl. Math. 11 (4), 49–63 (2005).

    Google Scholar 

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Funding

The work of the first author was supported by the Ministry of Science and Higher Education of the Russian Federation (Project FZGU-2020-0035), Lemma 1. The work of the second author was supported by a grant from the Russian Science Foundation (Project no. 19-11-00146), Theorem 4, and the Russian Foundation for Basic Researches (Project no. 20-01-00051), Theorem 5.

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Authors and Affiliations

  1. Voronezh State University, 1 Universitetskaya pl., Voronezh, 394018, Russia

    V. G. Zvyagin, V. P. Orlov & A. S. Arsentiev

Authors
  1. V. G. Zvyagin
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  2. V. P. Orlov
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  3. A. S. Arsentiev
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Corresponding authors

Correspondence to V. G. Zvyagin, V. P. Orlov or A. S. Arsentiev.

Additional information

Russian Text © The Author(s), 2020, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2020, No. 6, pp. 79–85.

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Zvyagin, V.G., Orlov, V.P. & Arsentiev, A.S. Equivalence of Weak Solvability of Initial-Boundary Value Problems for the Jeffries-Oldroyd Model and one Integro-Differential System with Memory. Russ Math. 64, 69–74 (2020). https://doi.org/10.3103/S1066369X20060109

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  • DOI: https://doi.org/10.3103/S1066369X20060109

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