Asymptotic behavior of 3-D evolutionary model of Magnetoelasticity for small data

In this article, we consider the evolutionary model for magnetoelasticity with vanishing external magnetic field and vanishing viscosity/damping, which is a nonlinear dispersive system. The global regularity and scattering of the evolutionary model for magnetoelasticity under small size of initial data is proved. Our proof relies on the idea of vector-field method due to the quasilinearity and the presence of convective term. A key observation is that we construct a suitable energy functional including the mass quantity, which enable us provide a good decay estimates for Schr\"odinger flow. In particular, we establish the asymptotic behavior in both mass and energy spaces for Schr\"odinger map, not only for gauged equation.