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

Xiaonan Hao, Jiaxi Huang, Ning Jiang, Lifeng Zhao

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 Schrödinger flow. In particular, we establish the asymptotic behavior in both mass and energy spaces for Schrödinger map, not only for gauged equation.

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

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 Schrödinger flow. In particular, we establish the asymptotic behavior in both mass and energy spaces for Schrödinger map, not only for gauged equation.
Paper Structure (17 sections, 18 theorems, 209 equations)

This paper contains 17 sections, 18 theorems, 209 equations.

Table of Contents

  1. Introduction
  2. Modeling of magnetoelasticity
  3. The main results
  4. An overview of the paper
  5. Reformulation and function spaces
  6. Reformulation of \ref{['ori_sys']}
  7. Function spaces and bootstrap proposition
  8. Decay estimates
  9. Energy estimates
  10. Higher-order energy estimates
  11. Lower-order energy estimates
  12. Estimates of the Weighted Norms
  13. Weighted estimates of elastic equations
  14. Weighted estimates of profile for Schrödinger flow
  15. Proofs of the main theorems
  16. ...and 2 more sections

Key Result

Theorem 1.1

Let $d=2,3$ be the dimensions, $Q\in {\mathbb S}^2$ be a fixed unit vector, and let initial data $(u_0,F_0,\phi_0)\in H^3\times H^3\times H^{4}_Q$. Then the Cauchy problem ori_sys with constraints constraints1-re admits a unique local solution $(u,F,\phi)$ on $[0,T]$ satisfying where the time interval $T$ depends on initial data $\lVert u_0\rVert_{H^3},\lVert F_0\rVert_{H^3}$ and $\lVert \phi_0\r

Theorems & Definitions (39)

  • Theorem 1.1: Local well-posedness
  • Theorem 1.2: Global regularity and scattering for small data in 3-D
  • Remark 1.2.1
  • Remark 1.2.2
  • Lemma 2.1: Bounds for $\psi$
  • proof
  • Theorem 2.2
  • Proposition 2.3: Bootstrap Proposition
  • Lemma 3.1
  • proof
  • ...and 29 more