Compressible Navier-Stokes-Landau-Lifshitz-Gilbert system: derivations and well-posedness

In this paper, we first derive the compressible Navier-Stokes/Landau-Lifshitz-Gilbert (NS-LLG) model for magnetoelastic materials via the energetic variational approach (EnVarA). It is important to emphasize that the manner in which the evolution of magnetoelastic materials is influenced by the fluid motion--specifically through the deformation gradient--determines the kinematics of the magnetization and consequently leads to distinct governing equations. Subsequently, we establish the local-in-time existence of solutions to the compressible NS-LLG system under finite initial energy. Finally, near the constant equilibrium for magnetoelasticity in the absence of an external magnetic field, we reformulate the evolutionary model, which allows an additional dissipative term to be identified from the elastic stress. Based on this reformulation, we justify the global well-posedness of the evolutionary magnetoelasticity system with zero external magnetic field, provided the initial data are sufficiently small. In particular, when the magnetic field $M$ vanishes, this model reduces to the viscoelastic model. Our results significantly relax the previous initial data requirements, only assume the most basic structural condition $\rho_{0} \operatorname{det} F_{0} = 1$.