We consider a variational approach to the evolution of a compressible magnetoelastic solid. The energy is non-convex and coupled to a gradient flow equation for the magnetization in the quasi-static setting. The viscous dissipation yields an extended material derivative in the magnetic force balance. I will introduce the model and outline the proof of existence of weak solutions, which is based on De Giorgi's minimizing movements scheme and a representation of the magnetic force balance in terms of the same energy and dissipation potentials as the equation of motion.
This is joint work with Barbora Bene{\v s}ov{\' a}, {\v S}{\' ar}ka Ne{\v c}asov{\' a} and Jan Scherz.