The Inverse Problem of Velocity Reconstruction from Flow-Induced Magnetic Fields: Stability and Uniqueness

The reconstruction of velocity fields from flow-induced magnetic measurements is a fundamental inverse problem in magnetohydrodynamics, with broad applications in geophysics, astrophysics, and metallurgy. To address the nonlinearity of the governing equations, the magnetic induction equation is linearized around a known, space-dependent background magnetic field. This reveals a coupled parabolic–hyperbolic system in which magnetic diffusion interacts with velocity advection along the background field lines. By decoupling these mechanisms into a two-stage framework, sufficient conditions are established for both the uniqueness and stability of the velocity recovery. The main result proves that the non-uniqueness arising from unobservable velocity perturbations parallel to the background field is eliminated when the velocity trace on the inflow boundary is prescribed alongside magnetic observations. The analysis yields global uniqueness and a quantitative stability estimate of hybrid type: a Lipschitz contribution from the inflow trace combined with a weaker modulus from the parabolic observation, reflecting the mixed parabolic–hyperbolic structure of the problem. These theoretical findings are validated through three-dimensional numerical simulations that demonstrate the robustness of the decoupled reconstruction approach under noisy conditions.