The existence and uniqueness of weak solutions to three-dimensional Kelvin-Voigt equations with damping and unbounded delays

There are many results involving PDEs in fluid mechanics with delays and many results about asymptotic behavior to PDEs. Navier-Stokes equations with delays have been studied extensively over the last decades, for their important contributions to understanding fluid motion and turbulence. In this paper we consider the modifications of the three dimensional Navier-Stokes equations: the three dimensional Kelvin-Voigt equations involving damping and unbounded delays in a bounded domain Ω ⊂ R3. The damping term is often introduced to model energy dissipation, which can stabilize the system. We show the existence and uniqueness of weak solutions by the Galerkin approximations method and the energy method.