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

2025/02/03 by Mengni Li, Li, Mengni
Mathematics · #Gas Dynamics and Kinetic Theory #Navier-Stokes equation solutions

paper · pdf · doi:10.48550/arxiv.2502.01139

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 x₁-axis in 3D thin domains Ω_δ=ℝ²×(-δ,δ) with δ∈(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 ℝ² 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.

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