Local well-posedness of magnetoelastic equations: without viscosity and damping

In this article, we focus on the magnetoelastic equations without viscosity or damping, which govern the coupled phenomena of material deformation under magnetic fields and magnetization evolution induced by mechanical stresses. However, the disappearance of viscosity and damping will bring great difficulties to the well-posedness analysis of the magnetoelastic system. We first establish the local existence of solutions through the parabolic regularization method, supported by a systematic analysis of the vanishing structure and geometric structure inherent to the system. Furthermore, the uniqueness is rigorously demonstrated through an innovative application of the parallel transport method, which precisely tracks magnetization dynamics in the coupled field-stress environment. These results establish the well-posedness of the magnetoelastic system for large initial data completely.