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Investigation of the Weak Solvability of One Viscoelastic Fractional Voigt Model

Author

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  • Andrey Zvyagin

    (Mathematical Department, Voronezh State University, Voronezh 394018, Russia
    These authors contributed equally to this work.)

  • Ekaterina Kostenko

    (Mathematical Department, Voronezh State University, Voronezh 394018, Russia
    These authors contributed equally to this work.)

Abstract

This article is devoted to the investigation of the weak solvability to the initial boundary value problem, which describes the viscoelastic fluid motion with memory. The memory of the fluid is considered not at a constant position of the fluid particle (as in most papers on this topic), but along the trajectory of the fluid particle (which is more physical). This leads to the appearance of an unknown function z , which is the trajectory of fluid particles and is determined by the velocity v of a fluid particle. However, in this case, the velocity v belongs to L 2 ( 0 , T ; V 1 ) , which does not allow the use of the classical Cauchy Problem solution. Therefore, we use the theory of regular Lagrangian flows to correctly determine the trajectory of the particle. This paper establishes the existence of weak solutions to the considered problem. For this purpose, the topological approximation approach to the study of mathematical hydrodynamics problems, constructed by Prof. V. G. Zvyagin, is used.

Suggested Citation

  • Andrey Zvyagin & Ekaterina Kostenko, 2023. "Investigation of the Weak Solvability of One Viscoelastic Fractional Voigt Model," Mathematics, MDPI, vol. 11(21), pages 1-23, October.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:21:p:4472-:d:1269440
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    References listed on IDEAS

    as
    1. D. A. Vorotnikov & V. G. Zvyagin, 2004. "On the existence of weak solutions for the initial-boundary value problem in the Jeffreys model of motion of a viscoelastic medium," Abstract and Applied Analysis, Hindawi, vol. 2004, pages 1-15, January.
    2. Yang, Xuehua & Wu, Lijiao & Zhang, Haixiang, 2023. "A space-time spectral order sinc-collocation method for the fourth-order nonlocal heat model arising in viscoelasticity," Applied Mathematics and Computation, Elsevier, vol. 457(C).
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