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Existence of weak solution to initial-boundary value problem for finite order Kelvin–Voigt fluid motion model

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Abstract

The paper is devoted to the proof of a weak solution existence for the Kelvin–Voigt fluid motion model of an arbitrary finite order. First, for the model under consideration, using the Laplace transform, the stress tensor deviator is expressed from the rheological relation and substituted into the system of fluid motion equations. Then, for the resulting system of equations, an initial-boundary value problem is posed, a definition of a weak solution is given, and its existence is proved. For the proof, it is considered some problem approximating the original one and proved its solvability based on the Leray–Schauder theorem. Finally, based on a priori estimates of solutions, it is shown that the sequence of solutions for the approximation problem converges weakly to the solution of the original problem as the approximation parameter tends to zero.

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Acknowledgements

The study was supported by the Russian Science Foundation, grant No 23-21-00091, https://rscf.ru/project/23-21-00091/

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Author notes

  1. Anastasiia Ustiuzhaninova contributed equally to this work.

Authors and Affiliations

  1. Research Institute of Mathematics, Voronezh State University, Universitetskaya sq. 1, Voronezh, 394018, Russia

    Mikhail Turbin & Anastasiia Ustiuzhaninova

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  1. Mikhail Turbin
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  2. Anastasiia Ustiuzhaninova
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Correspondence to Mikhail Turbin.

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Turbin, M., Ustiuzhaninova, A. Existence of weak solution to initial-boundary value problem for finite order Kelvin–Voigt fluid motion model. Bol. Soc. Mat. Mex. 29, 54 (2023). https://doi.org/10.1007/s40590-023-00526-y

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