On the dynamics of magneto-elastic interactions: existence of solutions and limit behaviors

The dynamics of magneto-elastic materials is described by a nonlinear system which couples the equations of the magnetization and the displacements. We study the three-dimensional case and establish the existence of weak solutions. Our starting point is the Gilbert-Landau-Lifschitz equation introduced for describing the dynamics of micro-magnetic processes. Three terms of the total free energy are taken into account: the exchange energy, the elastic energy and the magneto-elastic energy usually adopted for cubic crystals, neglecting, in this approach, the contributions due to the anisotropy and the demagnetization effects. The existence theorem for the proposed differential system is proved combining the Faedo-Galerkin approximations and the penalty method (FGP method). The asymptotic behavior are deduced from compactness properties.