Ideal MHD. Part II: Rigidity from infinity for ideal Alfvén waves in 3D thin domains

TL;DR

This work proves a rigidity-from-infinity phenomenon for ideal Alfvén waves in 3D thin domains $Ω_δ=\mathbb{R}^2\times(-δ,δ)$ with slip boundaries and a strong horizontal magnetic field. By developing δ-uniform weighted energy estimates that track wave centers via a position parameter, the authors construct scattering fields and show that vanishing scattering fields imply vanishing initial data, i.e., rigidity from infinity. They further establish that 3D thin-domain Alfvén waves along the horizontal direction converge to 2D Alfvén waves on $\mathbb{R}^2$ as $δ\to0$, and that the 2D rigidity result aligns with Li–Yu’s 2D theory. The analysis combines nonlinear energy methods, pressure deconvolution via Green’s functions, and a careful treatment of thin-domain geometry to connect 3D thin-domain behavior with its 2D limit, providing a rigorous bridge between scattering theory and dimensional reduction in MHD.

Abstract

This paper concerns the rigidity from infinity for Alfvén waves governed by ideal incompressible magnetohydrodynamic equations subjected to strong background magnetic fields along the $x_1$-axis in 3D thin domains $Ω_δ=\mathbb{R}^2\times(-δ,δ)$ with $δ\in(0,1]$ and slip boundary conditions. We show that in any thin domain $Ω_δ$, Alfvén waves must vanish identically if their scattering fields vanish at infinities. As an application, the rigidity of Alfvén waves in $Ω_δ$, propagating along the horizontal direction, can be approximated by the rigidity of Alfvén waves in $\mathbb{R}^2$ when $δ$ is sufficiently small. Our proof relies on the uniform (with respect to $δ$) weighted energy estimates with a position parameter in weights to track the center of Alfvén waves. The key issues in the analysis include dealing with the nonlinear nature of Alfvén waves and the geometry of thin domains.

Figures (4)

• Figure 1: $\Omega_\delta=\mathbb{R}^2\times(-\delta,\delta)$
• Figure 2: Infinities and scattering fields
• Figure 3: The position parameter in rigidity
• Figure 4: Relations among rigidity theorems

Theorems & Definitions (55)

• Lemma 2.1: Sobolev lemma
• Lemma 2.2: Properties of weights
• Lemma 2.3: Weighted div-curl lemma
• proof
• Lemma 2.4: Weighted div-curl lemma with higher order derivatives
• proof
• Theorem 2.5: Uniform weighted energy estimates in $\Omega_{\delta}$; adapted from Theorem 1.1 in Xu for the study of rigidity
• Remark 2.6
• Corollary 2.7: Uniform weighted energy estimates in $\Omega_1$
• Theorem 2.8: Asymptotics of the global solution from $\Omega_{\delta}$ to $\mathbb{R}^2$ as $\delta$ goes to zero; extracted from Theorem 1.3 in Xu