  In this work we investigate the existence and uniqueness of Struwe-like
solutions for a system of partial differential equations modeling the dynamics
of magnetoviscoelastic fluids. The considered system couples a Navier-Stokes
type equation with a dissipative equation for the deformation tensor and a
Landau-Lifshitz-Gilbert type equation for the magnetization field. The main
purpose is to establish a well-posedness theory in a two-dimensional periodic
domain under standard assumption of critical regularity for the (possibly
large) initial data. We prove that the considered weak solutions are everywhere
smooth, except for a discrete set of time values. The proof of the uniqueness
is based on suitable energy estimates for the solutions within a functional
framework which is less regular than the one of the Struwe energy level. These
estimates rely on several techniques of harmonic analysis and paradifferential
calculus.
