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| Main Authors: | , , , |
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| Format: | Preprint |
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2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2602.22097 |
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| _version_ | 1866910033087823872 |
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| author | Mokhtari, Yacine Frederick, Christina Yang, Yunan Engquist, Bjorn |
| author_facet | Mokhtari, Yacine Frederick, Christina Yang, Yunan Engquist, Bjorn |
| contents | We study the inverse problem of reconstructing an incompressible velocity field $\boldsymbol{v}$ from observations of the induced magnetic field $\boldsymbol{b}$.
In the presence of a strong, constant background field $\mathbf{F}$, the evolution of the magnetic perturbation $\boldsymbol{b}$ is governed by the linearized induction equation. We analyze the system on both the entire space $Ω= \mathbb{R}^d$ and a periodic domain $Ω= \prod_{i=1}^d [0, L_i)$, which models a homogeneous medium with side lengths $L_i > 0$. We analyze this problem by decomposing it into the injectivity of a parabolic forward map and the solvability of a divergence-free transport sub-problem. On the whole space $\mathbb{R}^d$, we show that the transport sub-problem is well-posed when data is prescribed on a non-characteristic hypersurface transverse to $\mathbf{F}$. On the torus, we establish a sharp uniqueness criterion based on the rational dependence of the ratios $\{F_i/L_i\}_{i=1}^d$ between the background-field components and the corresponding domain periods. Furthermore, we show that for the reconstructed velocity to belong to $L^2$, a sufficient condition is that the background field must satisfy a Diophantine condition. The proof combines injectivity of the parabolic forward map with uniqueness for a steady transport equation along $\mathbf{F}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_22097 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Velocity Reconstruction from Flow-Induced Magnetic Fields Mokhtari, Yacine Frederick, Christina Yang, Yunan Engquist, Bjorn Analysis of PDEs We study the inverse problem of reconstructing an incompressible velocity field $\boldsymbol{v}$ from observations of the induced magnetic field $\boldsymbol{b}$. In the presence of a strong, constant background field $\mathbf{F}$, the evolution of the magnetic perturbation $\boldsymbol{b}$ is governed by the linearized induction equation. We analyze the system on both the entire space $Ω= \mathbb{R}^d$ and a periodic domain $Ω= \prod_{i=1}^d [0, L_i)$, which models a homogeneous medium with side lengths $L_i > 0$. We analyze this problem by decomposing it into the injectivity of a parabolic forward map and the solvability of a divergence-free transport sub-problem. On the whole space $\mathbb{R}^d$, we show that the transport sub-problem is well-posed when data is prescribed on a non-characteristic hypersurface transverse to $\mathbf{F}$. On the torus, we establish a sharp uniqueness criterion based on the rational dependence of the ratios $\{F_i/L_i\}_{i=1}^d$ between the background-field components and the corresponding domain periods. Furthermore, we show that for the reconstructed velocity to belong to $L^2$, a sufficient condition is that the background field must satisfy a Diophantine condition. The proof combines injectivity of the parabolic forward map with uniqueness for a steady transport equation along $\mathbf{F}$. |
| title | Velocity Reconstruction from Flow-Induced Magnetic Fields |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2602.22097 |