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Kelvin-Voigt equations for incompressible and nonhomogeneous fluids with anisotropic viscosity, relaxation and damping

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Abstract

In this work, we consider the nonlinear initial-boundary value problem posed by the Kelvin-Voigt equations for non-homogeneous and incompressible fluid flows with fully anisotropic diffusion, relaxation and damping. Moreover, we assume that the momentum equation is perturbed by a damping term which, depending on whether its signal is positive or negative, may account for the presence of a source or a sink within the system. In the particular case of considering this problem with a linear and isotropic relaxation term, we prove the existence of global and local weak solutions for the associated initial-boundary value problem supplemented with no-slip boundary conditions. When the damping term describes a sink, we establish the conditions for the polynomial time decay or for the exponential time decay of these solutions.

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Acknowledgements

The first author was supported by the Lavrenty’ev Institute of Hydrodynamics of the Siberian Branch RAS (project no. III.22.4.2, Analysis of mathematical models of continua with singularities, discontinuities and intrinsic inhomogeneities), Novosibirsk, Russia. Both first and second authors were partially supported by the Project UID/MAT/04561/2019 of the Portuguese Foundation for Science and Technology (FCT), Portugal. The third author was supported by the Grant no. AP08052425 of the Ministry of Science and Education of the Republic of Kazakhstan (MES RK), Kazakhstan.

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Authors and Affiliations

  1. CMAFCIO - Universidade de Lisboa, Lisbon, Portugal

    S. N. Antontsev & H. B. de Oliveira

  2. Lavrentyev Institute of Hydrodynamics, SB RAS, Novosibirsk, Russia

    S. N. Antontsev

  3. FCT - Universidade do Algarve, Faro, Portugal

    H. B. de Oliveira

  4. Al-Farabi Kazakh National University, Almaty, Kazakhstan

    Kh. Khompysh

  5. CFM - Universidade Federal de Santa Catarina, Florianopolis, Brazil

    H. B. de Oliveira

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  1. S. N. Antontsev
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  2. H. B. de Oliveira
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  3. Kh. Khompysh
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Correspondence to Kh. Khompysh.

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Antontsev, S.N., de Oliveira, H.B. & Khompysh, K. Kelvin-Voigt equations for incompressible and nonhomogeneous fluids with anisotropic viscosity, relaxation and damping. Nonlinear Differ. Equ. Appl. 29, 60 (2022). https://doi.org/10.1007/s00030-022-00794-z

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