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

TL;DR

This paper analyzes the long-term behavior of a 3-D nonlinear magnetoelasticity model with small initial data, proving global regularity and scattering using vector-field methods and energy functionals.

Contribution

It introduces a novel approach to establish asymptotic behavior and decay estimates for a complex dispersive magnetoelasticity system with small data.

Findings

Proved global regularity for small initial data.

Established scattering and decay estimates for the model.

Analyzed asymptotic behavior in mass and energy spaces.

Abstract

In this article, we consider the evolutionary model for magnetoelasticity with 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 to 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.