Decay characterization of solutions to the Navier-Stokes-Voigt equations in terms of the initial datum

The Navier-Stokes-Voigt equations are a regularization of the Navier-Stokes equations that share some of its asymptotic and statistical properties and have been used in direct numerical simulations of the latter. In this article we characterize the decay rate of solutions to the Navier-Stokes-Voigt equations in terms of the decay character of the initial datum and study the long time behaviour of its solutions by comparing them to solutions to the linear part.Comment: 13 page

Discontinuous Galerkin approximations for an optimal control problem of three-dimensional Navier-Stokes-Voigt equations

We analyze a fully discrete scheme based on the discontinuous (in time) Galerkin approach, which is combined with conforming finite element subspaces in space, for the distributed optimal control problem of the three-dimensional Navier-Stokes-Voigt equations with a quadratic objective functional and box control constraints. The space-time error estimates of order $O(\sqrt{\tau}+h)$, where $\tau$ and $h$ are respectively the time and space discretization parameters, are proved for the difference between the locally optimal controls and their discrete approximations.Comment: 28 page