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Velocity Reconstruction from Flow-Induced Magnetic Fields

Yacine Mokhtari, Christina Frederick, Yunan Yang, Bjorn Engquist

Abstract

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 $\mathbf{F}$.

Velocity Reconstruction from Flow-Induced Magnetic Fields

Abstract

We study the inverse problem of reconstructing an incompressible velocity field from observations of the induced magnetic field . In the presence of a strong, constant background field , the evolution of the magnetic perturbation is governed by the linearized induction equation. We analyze the system on both the entire space and a periodic domain , which models a homogeneous medium with side lengths . 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 , we show that the transport sub-problem is well-posed when data is prescribed on a non-characteristic hypersurface transverse to . On the torus, we establish a sharp uniqueness criterion based on the rational dependence of the ratios between the background-field components and the corresponding domain periods. Furthermore, we show that for the reconstructed velocity to belong to , 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 .
Paper Structure (9 sections, 9 theorems, 77 equations)

This paper contains 9 sections, 9 theorems, 77 equations.

Table of Contents

  1. Introduction
  2. Problem formulation
  3. Well-posedness of the linearized induction problem
  4. Existence and Uniqueness of Divergence-Free Transport Solutions
  5. Existence and uniqueness in the setting $\Omega=\mathbb{R}^d$
  6. Existence and uniqueness in the setting $\Omega = \mathbb{T}^d_{\boldsymbol{L}}$
  7. Inverse problem
  8. Concluding Remarks and Open Questions
  9. Alternate proof of well-posedness of the forward problem on the torus

Key Result

Lemma 1

Let $\eta>0$ and suppose $\boldsymbol{b}_0 \in H^1$ and $\boldsymbol{h} \in L^2(0,T;H^0)$. Let $\boldsymbol{b}$ be the unique solution of eq:forwardproblem satisfying eq:b_reg and eq:bt_reg. Suppose that the initial data and forcing are divergence-free: in the sense of distributions. Then

Theorems & Definitions (19)

  • Lemma 1
  • proof
  • Corollary 1
  • Corollary 2
  • Remark 1
  • Theorem 1: Existence and uniqueness on $\mathbb{R}^d$
  • proof
  • Theorem 2: Existence and uniqueness on $\mathbb{T}^d_{\boldsymbol{L}}$
  • proof
  • Remark 2
  • ...and 9 more