  We prove existence of weak solutions to an evolutionary model derived for
magnetoelastic materials. The model is phrased in Eulerian coordinates and
consists in particular of (i) a Navier-Stokes equation that involves magnetic
and elastic terms in the stress tensor obtained by a variational approach, of
(ii) a regularized transport equation for the deformation gradient and of (iii)
the Landau-Lifshitz-Gilbert equation for the dynamics of the magnetization. The
proof is built on a Galerkin method and a fixed-point argument. It is based on
ideas from F.-H. Lin and the third author for systems modeling the flow of
liquid crystals as well as on methods by G. Carbou and P. Fabrie for solutions
of the Landau-Lifshitz equation.
