Communications in Mathematical Sciences

Volume 17 (2019)

Number 7

Generalized Kelvin-Voigt equations for nonhomogeneous and incompressible fluids

Pages: 1915 – 1948



Stanislav N. Antontsev (Centre of Mathematics, Fundamental Applications and Operations Research (CMAFCIO), Universidade de Lisboa, Portugal; and Lavrentyev Institute of Hydrodynamics, SB RAS, Novosibirsk, Russia)

Hermenegildo B. de Oliveira (CMAFCIO, Universidade de Lisboa, Portugal; and Departmento de Matemática, Faculdade de Ciências e Tecnologia, Universidade do Algarve, Faro, Portugal)

Khonatbek Khompysh (Department of Mechanics and Mathematics, Al-Farabi Kazakh National University, Almaty, Kazakhstan)


In this work, we consider the Kelvin–Voigt equations for non-homogeneous and incompressible fluid flows with the diffusion and relaxation terms described by two distinct power-laws. Moreover, we assume that the momentum equation is perturbed by an extra 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. For the associated initial-boundary value problem, we study the existence of weak solutions as well as the large-time behavior of the solutions. In the case the extra term is a sink, we prove the global existence of weak solutions and we establish the conditions for the polynomial time decay and for the exponential time decay of these solutions. If the extra term is a source, we show how the exponents of nonlinearity must interact to ensure the local existence of weak solutions.


Kelvin–Voigt equations, nonhomogeneous and incompressible fluids, power-laws, existence, large-time behavior

2010 Mathematics Subject Classification

35D30, 35Q30, 35Q35, 76D03, 76D05

Copyright © 2019 by S.N. Antontsev, H.B. de Oliveira, and K. Khompysh.

The first author was partially supported by the Russian Federation government, Grant no. 14.W03.31.0002. 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 second author was also supported by the Grant no. SFRH/BSAB/135242/2017 of the Portuguese Foundation for Science and Technology (FCT). The third author was partially supported by the Grant no. AP08052425 of the Ministry of Science and Education of the Republic of Kazakhstan (MES RK), Kazakhstan.

Received 27 December 2018

Accepted 20 June 2019

Published 6 January 2020