Global classical solutions to an evolutionary model for magnetoelasticity

In this paper, we first prove the local-in-time existence of the evolutionary model for magnetoelasticity with finite initial energy by employing the nonlinear iterative approach given cite{Jiang-Luo-2019-SIAM} to deal with the geometric constraint $M in mathbb{S}^{d-1}$ the Landau-Lifshitz-Gilbert (LLG) equation. Inspired by cite{Lin-Liu-Zhang-CPAM2005, Lin-Zhang-2008-CPAM}, we reformulate the evolutionary model for magnetoelasticity with vanishing external magnetic field $H_{ext}$, so that a further dissipative term will be sought from the elastic stress. We thereby justify the global well-posedness to the evolutionary model for magnetoelasticity with zero external magnetic field under small size of initial data.