Communications in Mathematical Sciences
Volume 17 (2019)
Generalized Kelvin-Voigt equations for nonhomogeneous and incompressible fluids
Pages: 1915 – 1948
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