Global weak attractors in the dynamics of bodies with vector-type microstructure

We investigate the dynamics of bodies with vector-type microstructure. We consider linear constitutive relations and a nonlinear coupling between macroscopic and microscopic motions, determined by gyroscopic-type inertia. Based on an existence result obtained in the presence of viscous-type stress components, we determine the existence of a global attractor; its weak nature derives from the lack of uniqueness determined by the nonlinear coupling.