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Well-posedness of the transport of normal currents by time-dependent vector fields

Paolo Bonicatto, Giacomo Del Nin

TL;DR

The paper addresses the well-posedness of the Geometric Transport Equation $\frac{d}{dt} T_t + \mathcal{L}_{b_t}T_t = 0$ for currents driven by time-dependent vector fields that are Lipschitz in space and integrable in time. It develops an extension of the decomposability bundle framework to the AC-in-time, Lipschitz-in-space setting via the space-time flow $\Psi(t,x)=(t,\Phi_t^0(x))$, and proves existence and uniqueness of solutions in the class of normal currents, with $T_t=(\Phi^0_t)_* \bar{T}$. A corresponding pushforward formula is established for maps in the class $AC_t\mathrm{Lip}_x$, ensuring robust handling of time-dependent dynamics. The work also exhibits non-uniqueness in the finite-mass regime, highlighting the necessity of restricting to normal currents for well-posedness and clarifying the limits of natural solution concepts in weaker settings. Overall, the results extend previous autonomous-Lipschitz well-posedness to time-dependent fields and deepen the geometric-measure-theoretic understanding of current transport under evolving vector fields.

Abstract

We prove existence and uniqueness for the transport equation for currents (Geometric Transport Equation) when the driving vector field is time-dependent, Lipschitz in space and merely integrable in time. This extends previous work where well-posedness was shown in the case of a time-independent, Lipschitz vector field. The proof relies on the decomposability bundle and requires to extend some of its properties to the class of functions that in one direction are only absolutely continuous, rather than Lipschitz.

Well-posedness of the transport of normal currents by time-dependent vector fields

TL;DR

The paper addresses the well-posedness of the Geometric Transport Equation for currents driven by time-dependent vector fields that are Lipschitz in space and integrable in time. It develops an extension of the decomposability bundle framework to the AC-in-time, Lipschitz-in-space setting via the space-time flow , and proves existence and uniqueness of solutions in the class of normal currents, with . A corresponding pushforward formula is established for maps in the class , ensuring robust handling of time-dependent dynamics. The work also exhibits non-uniqueness in the finite-mass regime, highlighting the necessity of restricting to normal currents for well-posedness and clarifying the limits of natural solution concepts in weaker settings. Overall, the results extend previous autonomous-Lipschitz well-posedness to time-dependent fields and deepen the geometric-measure-theoretic understanding of current transport under evolving vector fields.

Abstract

We prove existence and uniqueness for the transport equation for currents (Geometric Transport Equation) when the driving vector field is time-dependent, Lipschitz in space and merely integrable in time. This extends previous work where well-posedness was shown in the case of a time-independent, Lipschitz vector field. The proof relies on the decomposability bundle and requires to extend some of its properties to the class of functions that in one direction are only absolutely continuous, rather than Lipschitz.

Paper Structure

This paper contains 12 sections, 18 theorems, 103 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries and notation
  3. A lemma on differentiation
  4. Decomposability bundle
  5. Maximal function
  6. Flows
  7. The decomposability bundle for AC Lip functions
  8. Differentiability along the bundle
  9. Approximation with Lipschitz functions and pushforward formula
  10. Existence
  11. Uniqueness
  12. On non-uniqueness for finite mass currents

Key Result

Theorem 1

Let $b:[0,1]\times\mathbb{R}^d\to\mathbb{R}^d$ be a vector field satisfying eq:Lip_assumption. Moreover, let $\bar{T}\in \mathrm{N}_k(\mathbb{R}^d)$. Then there exists a unique family of currents $T_t\in L^\infty([0,1]; \mathrm{N}_k(\mathbb{R}^d))$ that solves More precisely, denoting by $(\Phi^0_t)_t$ the flow of $b$ starting from time $0$, the solution is given by the pushforward $T_t=(\Phi^0_t

Figures (2)

  • Figure 1: Visual representation of the function $L(f,E)$. On the open intervals $I_1$ and $I_2$ we replace the function $f$ (dashed graph) with the linear interpolation between endpoints.
  • Figure 2: We depicted the family $T_t=(e_2+te_1)\delta_{(t\varepsilon,\varepsilon)}$, which solves \ref{['eq:GTE']} (in the classical sense) with initial datum $e_2\delta_{(0,\varepsilon)}$. Assuming the Stability property, the limit as $\varepsilon\to 0$ of this family gives rise to the solution $T^2_t=(e_2+te_1)\delta_0$ with initial datum $e_2\delta_0$. On the other hand, building an approximating family from below, the constant family $T^1_t=e_2\delta_0$ also solves the equation starting from the same initial datum. This shows non-uniqueness in the class of natural solutions $\mathcal{S}_b$ with finite mass as defined in Definition \ref{['def:natural_family']}.

Theorems & Definitions (35)

  • Theorem
  • Theorem
  • Lemma 2.1
  • proof
  • Theorem 2.2: AM
  • Proposition 2.3: Pushforward AM
  • Lemma 2.4
  • Lemma 2.5: The flow is $AC_t\mathop{\mathrm{Lip}}\nolimits_x$
  • proof
  • Lemma 2.6
  • ...and 25 more