This paper aims to establish existence and uniqueness results of weak
and strong solution to the three-dimensional periodic
magnetization-variables formulation to Navier-Stokes equations with
damping term. Authors in precedent works addressed the question as to
whether this model and similar ones without damping term possess a weak
solution. In this vein, considering a damping term in the
magnetization-variable formulation turned to be consequential as it
enforces existence and uniqueness results. Energy methods, compactness
methods are the main tools.