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A general Frobenius' Theorem via the Transport of Currents

Paolo Bonicatto

TL;DR

The paper extends Frobenius' theorem to settings where one flow is Lipschitz and the transported object is a (possibly singular) 1-current, by linking the Lie bracket to the Lie derivative of currents via two evolution equations: the Vector Advection Equation $\frac{d}{dt}v_t + [v_t,b]=0$ and the Geometric Transport Equation $\frac{d}{dt}T_t + \mathcal{L}_b T_t =0$. It develops a representation framework using Regular Lagrangian Flows, the decomposability bundle, and current calculus to obtain a generalized Frobenius-type commutativity result, valid even for normal currents, and provides a Lagrangian interpretation through Smirnov representations. A key contribution is the identification of a precise passage from VAE to GTE by introducing the density $\varrho$ and showing $T_{\varrho v}$ is transported by $b$, which yields explicit formulas for the evolved vector fields, e.g. $v_t(x) = (\nabla X_t \cdot \overline{v})(X_{-t}(x))$. The work also recasts Alfvén’s theorem in magnetohydrodynamics as a time-dependent Frobenius-type result, clarifying when magnetic field lines are frozen into a fluid flow and how compressibility affects the field strength via the density factor $\varrho$.

Abstract

A classical result in Differential Geometry states that the flows of two smooth vector fields commute if and only if their Lie Bracket vanishes. In this work, we extend this result to a more general setting where one of the vector fields is bounded and Lipschitz, while the other may be a singular vector-valued measure, i.e. a normal 1-current. This result is achieved via the study of two distinct evolutionary PDEs describing the transport of vector quantities (the Vector Advection Equation and the Geometric Transport Equation). Furthermore, we show that a celebrated theorem by Alfvén in Magnetohydrodynamics can be interpreted as a suitable time-dependent version of Frobenius' Theorem. Our approach builds on recent advances concerning the Geometric Transport Equation for currents [5, 6].

A general Frobenius' Theorem via the Transport of Currents

TL;DR

The paper extends Frobenius' theorem to settings where one flow is Lipschitz and the transported object is a (possibly singular) 1-current, by linking the Lie bracket to the Lie derivative of currents via two evolution equations: the Vector Advection Equation and the Geometric Transport Equation . It develops a representation framework using Regular Lagrangian Flows, the decomposability bundle, and current calculus to obtain a generalized Frobenius-type commutativity result, valid even for normal currents, and provides a Lagrangian interpretation through Smirnov representations. A key contribution is the identification of a precise passage from VAE to GTE by introducing the density and showing is transported by , which yields explicit formulas for the evolved vector fields, e.g. . The work also recasts Alfvén’s theorem in magnetohydrodynamics as a time-dependent Frobenius-type result, clarifying when magnetic field lines are frozen into a fluid flow and how compressibility affects the field strength via the density factor .

Abstract

A classical result in Differential Geometry states that the flows of two smooth vector fields commute if and only if their Lie Bracket vanishes. In this work, we extend this result to a more general setting where one of the vector fields is bounded and Lipschitz, while the other may be a singular vector-valued measure, i.e. a normal 1-current. This result is achieved via the study of two distinct evolutionary PDEs describing the transport of vector quantities (the Vector Advection Equation and the Geometric Transport Equation). Furthermore, we show that a celebrated theorem by Alfvén in Magnetohydrodynamics can be interpreted as a suitable time-dependent version of Frobenius' Theorem. Our approach builds on recent advances concerning the Geometric Transport Equation for currents [5, 6].
Paper Structure (15 sections, 16 theorems, 142 equations)

This paper contains 15 sections, 16 theorems, 142 equations.

Table of Contents

  1. Introduction
  2. Notation and preliminaries
  3. Currents
  4. Decomposability bundle
  5. Flows of Lipschitz fields
  6. Regular Lagrangian Flows
  7. Geometric Transport Equation
  8. A lemma on absolutely continuous functions
  9. The Vector Advection Equation and the Geometric Transport Equation
  10. On the Lie derivative and the Lie Bracket of smooth vector fields
  11. Vector Advection Equation
  12. From the Vector Advection Equation to the Geometric Transport Equation
  13. Frobenius' Theorem
  14. Extension to normal 1-currents
  15. Alfvén Theorem as a time-dependent Frobenius' Theorem

Key Result

Proposition 1.1

If $b \in \mathop{\mathrm{Lip}}\nolimits(\mathbb{R}^d;\mathbb{R}^d)$ is a bounded velocity field, and $v \in \mathrm{C}_t^0(\mathrm{W}_x^{1,1})$ is a solution to eq:VAE_intro starting from $v_0 := \overline{v}$, then the representation formula eq:stellina_intro holds true for all $t \in (0,1)$ and $

Theorems & Definitions (30)

  • Proposition 1.1: Representation formula for \ref{['eq:VAE_intro']}
  • Theorem : Generalised Frobenius' Theorem
  • Lemma 2.1
  • Theorem 2.2
  • Theorem 2.3
  • proof : Proof of Theorem \ref{['thm:duhamel']}
  • Lemma 2.4
  • Lemma 3.1
  • proof
  • Remark 3.2
  • ...and 20 more