Quantitative finite-time approximation of scattering fields for Alfv\'en waves in ideal MHD

Abstract

In this paper, we establish a quantitative finite-time approximation of scattering fields for nonlinear Alfv\'en waves in three-dimensional ideal incompressible magnetohydrodynamics near a strong constant magnetic background. Using the global weighted estimates, we prove that truncating the scattering profiles at time $T$ introduces an error in both the $L^\infty$ norm and the $H^{N_*+1}$ norm of order $$ O\bigl(\varepsilon^2(R+T)^{-\delta}\bigr), $$ where $\varepsilon\in (0,1)$ measures the size of the initial perturbation, $R\ge100$ is the scale parameter in the weighted energy, $\delta\in(0,2/3)$ is the exponent related to weights, and $N_*+1$ is the Sobolev regularity order assumed for the initial data. The pointwise estimate follows from the decay of the nonlocal pressure along nonlinear characteristics, while the Sobolev estimate combines higher-order weighted pressure bounds with estimates for derivatives of the characteristic flows. Numerical experiments measure pointwise and Sobolev pressure-integral tails to verify the quadratic dependence of the initial perturbation amplitude.