# A note on the antisymmetry in the speed of a random walk in reversible
dynamic random environment

Oriane Blondel

###### Abstract

In this short note, we prove that $v(-\varepsilon)=-v(\varepsilon)$. Here,
$v(\varepsilon)$ is the speed of a one-dimensional random walk in a dynamic
_reversible_ random environment, that jumps to the right (resp. to the left)
with probability $1/2+\varepsilon$ (resp. $1/2-\varepsilon$) if it stands on
an occupied site, and vice-versa on an empty site. We work in any setting
where $v(\varepsilon),v(-\varepsilon)$ are well-defined, _i.e._ a weak LLN
holds. The proof relies on a simple coupling argument that holds only in the
discrete setting.

##  1 Introduction

We consider the so-called “$\varepsilon$–random walk”: a random walk in one-
dimensional dynamic random environment with two values that jumps to the right
(resp. to the left) with probability $1/2+\varepsilon$ (resp.
$1/2-\varepsilon$) if it stands on an occupied site, and vice-versa on an
empty site (Figure 1).

Figure 1: Jump rates for the $\varepsilon$–random walk.

A lot of energy has been devoted to describing the behavior of this
$\varepsilon$–RW in various random environments, mainly to find whether it
satisfies the usual limit theorems (LLN, CLT…). This is generally a hard
problem, since the environment seen from the $\varepsilon$–RW is highly non-
stationary (even when the environment is). Additionally, it belongs to the
class of nestling random walks, which do not have an a-priori preferred
direction. It is possible to find settings in which the LLN does not hold.
[BHT20, Section 9] proposes an example in which space-time traps lead the
random walk into longer and longer stretches of motion with drifts to the
right or the left. Cases in which the LLN has been shown include perturbative
regimes (small $\varepsilon$) [ABF18], well-known environments like the
exclusion process [HS15, HKT20] or the contact process [MV15], uniform mixing
hypotheses [CZ04, AdHR11, RV13], or fast enough decay of correlations [BHT20].

When the random walk does satisfy a law of large numbers, let us call
$v(\varepsilon)$ its asymptotic speed (we will simply say that
$v(\varepsilon)$ is well-defined). We observed in [ABF16] that, if the
environment is given by a reversible Markov process with positive spectral
gap, and $|\varepsilon|$ is smaller than the spectral gap, $v$ –in addition to
being well-defined– satisfies the antisymmetry property

$v(-\varepsilon)=-v(\varepsilon).$ (1)

This property also holds in higher dimensions for random walks with a certain
symmetry property (Assumption 3 in [ABF16]). Simulations moreover suggest that
this property extends out of the perturbative regime of [ABF16] (where the law
of large numbers was not yet even established; see Figure 3 in [ABF16]). The
proof relies on an expansion of the speed in $\varepsilon$, in which the terms
of even degrees are shown to cancel due to reversibility.

It is worth pointing out that this antisymmetry does not seem to follow from
obvious symmetry properties of the system. Indeed, it is tempting at first
sight to think that the $(-\varepsilon)$–random walk reversed in time should
have the same distribution as the $\varepsilon$–random walk, from which (1)
would immediately follow. But closer inspection reveals that this is not true
and those two processes have very different trajectories in general (see for
instance Figure 4 in [ABF16]). Rather, (1) holds iff the speed of the
$\varepsilon$–random walk is the same in the reversible environment and in its
image under the mirror symmetry $x\mapsto-x$. While this property is obvious
for reversible systems invariant under mirror symmetry (e.g. simple symmetric
exclusion process), there exist reversible, translation invariant processes
which have no mirror-invariance. One that has been extensively studied is the
East model [JE91, FMRT13] , in which particles appear (resp. disappear) at
site $x$ at rate $p$ (resp. $1-p$), but only if $x+1$ is empty. This dynamics
is reversible w.r.t. the product Bernoulli measure of density $p$ (it is easy
to check detailed balance since the constraint required to update $\eta(x)$
does not involve $\eta(x)$), and clearly non mirror-symmetric. In fact, any
kinetically constrained model with non-symmetric constraint would yield an
example where the result below is non-empty.

To state our main result, let $(X^{\varepsilon}_{t})_{t\geq 0}$ be the
trajectory of a $\varepsilon$–RW started from $0$. A proper construction of
this object is given in Section 2.

######  Theorem 1.

Assume the environment is a translation invariant and reversible Markov
process. Moreover, assume that $X^{\varepsilon}$ and $X^{-\varepsilon}$
satisfy a weak law of large numbers, _i.e._ there exist
$v(\varepsilon),v(-\varepsilon)$ such that

$\displaystyle\frac{1}{t}X^{\pm\varepsilon}_{t}\underset{t\rightarrow\infty}{\overset{\mathbb{P}}{\longrightarrow}}v(\pm\varepsilon).$
(2)

Then

$v(-\varepsilon)=-v(\varepsilon).$ (3)

Checking that the weak law of large numbers holds is generally not easy. In
[BHT20], it was established under the condition that the environment has
polynomially decaying covariances with a high enough exponent. This includes
Markov processes with positive spectral gap. The East model [JE91, FMRT13] is
an example of such a model with no mirror symmetry.

One point that the theorem above does not address is whether the limiting
speed is non-zero. This is in general hard to settle. [HS15] gives a nice
example in perturbative settings where the update rates of the environment are
either very large or very small. [BHT20] also offers a sufficient criterion
for non-zero speed, albeit rather limited in its application range. Theorem 1
does say that $v(\varepsilon)$ is non-zero whenever $v(-\varepsilon)$ is,
which is in itself not obvious.

Let us note that, to the statistical physics community, the hidden symmetry of
Theorem 1 in a reversible context may be reminiscent of Gallavotti-Cohen type
results. However, it is unclear what the connection is, if it exists at all.
Indeed, the action functional of the environment seen from the particle does
not seem to give information about the displacement of the particle.

The proof presented here relies on a coupling valid only for the discrete time
$\varepsilon$–RW, which in turn allows to deduce the result for the continuous
time. The coupling relies heavily on the fact that we consider nearest-
neighbor trajectories in dimension $1$. However, as mentioned above, Theorem 1
is expected to hold also in higher dimensions and for random walks allowing
longer range jumps under appropriate symmetry assumptions [ABF16]. Outside the
perturbative region, this is still an open problem.

##  2 Setting and construction of the random walk

We now give explicit constructions of the $\varepsilon$–RW, in discrete and
continuous time. In the following, $\mathbb{L}=\mathbb{R}_{+}$ when we
consider the continuous time setting, $\mathbb{L}=\mathbb{N}$ in the discrete
time setting.

###  2.1 Environment

The environment is given by a collection
$\eta=(\eta_{t}(x))_{(x,t)\in\mathbb{Z}\times\mathbb{L}}$ such that
$\eta_{t}(x)\in\\{0,1\\}$ for all $(x,t)\in\mathbb{Z}\times\mathbb{L}$. We
assume

  1. 1.

(Stationarity) The distribution of the environment is invariant under space-
time shifts, _i.e._

$\forall
x\in\mathbb{Z},t\in\mathbb{L},\quad(\eta_{t+s}(x+\cdot))_{s\in\mathbb{L}}=(\eta_{s}(\cdot))_{s\in\mathbb{L}}\quad\text{in
distribution.}$ (4)

  2. 2.

(Reversibility) The environment is given by a reversible process
$\eta=(\eta_{t})_{t\in\mathbb{L}}$ on $\\{0,1\\}^{\mathbb{Z}}$, _i.e._

$\text{for all
$T\in\mathbb{L}\setminus\\{0\\}$,}\quad(\eta_{T-t})_{t\in[0,T]\cap\mathbb{L}}=(\eta_{t})_{t\in[0,T]\cap\mathbb{L}}\quad\text{in
distribution.}$ (5)

###  2.2 $\varepsilon$–RW in discrete time

Let us consider a collection $U=(U_{x,n})_{x\in\mathbb{Z},n\in\mathbb{N}}$ of
iid $\mathcal{U}([0,1])$ random variables. With the environment $\eta$ and the
collection $U$ we associate a set
$(A^{\varepsilon}_{x,n})_{x\in\mathbb{Z},n\in\mathbb{N}}$ (that will prescribe
the directions taken by the RW) in the following way:

$A^{\varepsilon}_{x,n}=\begin{cases}+1&\text{if }\big{(}\eta_{n}(x)=1\text{
and }U_{x,n}\leq\frac{1}{2}+\varepsilon\big{)}\text{ or
}\big{(}\eta_{n}(x)=0\text{ and }U_{x,n}\leq\frac{1}{2}-\varepsilon\big{)},\\\
-1&\text{else}.\end{cases}$ (6)

For $x\in\mathbb{Z}$, we denote by $(X^{x,\varepsilon}_{n})_{n\in\mathbb{N}}$
the $\varepsilon$–RW started at $x$. It is built iteratively as follows:

  1. 1.

$X^{x,\varepsilon}_{0}=x$;

  2. 2.

if $X^{x,\varepsilon}_{n}=y$,
$X^{x,\varepsilon}_{n+1}=y+A^{\varepsilon}_{x,n}$.

Note that by translation invariance of the environment,
$X^{x,\varepsilon}-x\overset{(d)}{=}X^{0,\varepsilon}$ for all
$x\in\mathbb{Z}$. Also, $X_{n}^{x,\varepsilon}$ has the same parity as $n+x$.

###  2.3 $\varepsilon$–RW in continuous time

We could use a similar construction, but it will be more convenient to alter
it a little. Let $U=(U_{n})_{n\in\mathbb{N}}$ be a collection of iid
$\mathcal{U}([0,1])$ random variables and $T=(T_{n})_{n\in\mathbb{N}}$ a
PPP($1$) on $\mathbb{R}_{+}$. Assume $(U,T,\eta)$ to be independent. The
continuous time $\varepsilon$–RW started at $x$,
$(X^{x,\varepsilon}_{t})_{t\in\mathbb{R}_{+}}$ is built as follows:

  1. 1.

$X^{x,\varepsilon}_{0}=x$;

  2. 2.

if $X^{x,\varepsilon}_{t^{-}}=y$ and $y\notin\\{T_{n},n\in\mathbb{N}\\}$,
$X^{x,\varepsilon}_{t}=y$;

  3. 3.

if $X^{x,\varepsilon}_{t^{-}}=y$ and $t=T_{n}$,

$X^{x,\varepsilon}_{t}=\begin{cases}y+1&\text{if }\big{(}\eta_{t}(y)=1\text{
and }U_{n}\leq\frac{1}{2}+\varepsilon\big{)}\text{ or
}\big{(}\eta_{t}(y)=0\text{ and }U_{n}\leq\frac{1}{2}-\varepsilon\big{)},\\\
y-1&\text{else}.\end{cases}$ (7)

##  3 Proof of Theorem 1 in the discrete setting

A key remark is that the reversibility assumption allows to construct jointly
a $\varepsilon$–RW and a backwards $(-\varepsilon)$–RW, that is a
$(-\varepsilon)$–RW evolving on the reversed environment process
$(\eta_{N-n})_{n\leq N}$ for $N\in\mathbb{N}$.

In order to simplify the notations, let us fix $\varepsilon\in[-1,1]$, and
$N\in\mathbb{N}$. We let $X=X^{0,\varepsilon}$, and for $x\in\mathbb{Z}$ we
let $(\widehat{X}_{n})_{n\leq N}$ be constructed as follows.

  1. 1.

$\widehat{X}_{N}=x$;

  2. 2.

for $n<N$, if $\widehat{X}_{N-n}=y$,
$\widehat{X}_{N-n-1}=y-A^{\varepsilon}_{y,N-n}$.

Note that the construction uses the same collection $A^{\varepsilon}$ as the
construction of $X$.

Reversibility of the environment clearly gives us the following property.

######  Lemma 1.

$(\widehat{X}_{n})_{n\leq N}\overset{(d)}{=}(X^{x,-\varepsilon}_{n})_{n\leq
N}$.

###### Proof.

This follows from the reversibility of $\eta$. ∎

It is also not difficult to check that $\widehat{X}$ and $X$ cannot cross, in
the following sense.

######  Lemma 2.

Let $X$ and $\widehat{X}$ be built using the same collection
$A^{\varepsilon}$. Then, if $\widehat{X}_{0}$ and $X_{0}$ have the same
parity,

$(\widehat{X}_{0}-X_{0})(\widehat{X}_{N}-X_{N})\geq 0.$ (8)

###### Proof.

Without loss of generality, let us assume $\widehat{X}_{N}>X_{N}$. Assume by
contradiction that $X_{0}>\widehat{X}_{0}$. With our parity assumption, for
any $n\leq N$, $\widehat{X}_{n}-X_{n}\in 2\mathbb{Z}$, and that difference
only takes steps in $\\{-2,0,2\\}$. Let us look at the largest $n<N$ such that
$\widehat{X}_{n}=X_{n}$. Let $y=X_{n}$. Under our assumptions, necessarily
$\widehat{X}_{n+1}=y+1$ and $X_{n+1}=y-1$. In particular,
$A^{\varepsilon}_{y,n}=-1$, and therefore $\widehat{X}_{n-1}=y+1\geq X_{n-1}$.
Now, if $\widehat{X}_{n-1}>X_{n-1}$, we reproduce the previous argument to
show $\widehat{X}_{n-2}\geq X_{n-2}$ (unless $n-1$ is zero, in which case we
stop). If $\widehat{X}_{n-1}=X_{n-1}=y+1$, $A^{\varepsilon}_{y+1,n-1}=-1$, and
we also end up with $\widehat{X}_{n-2}=y+2\geq X_{n-2}$. We can reproduce this
argument until we reach time $0$ to get a contradiction.∎

With this non-crossing property, it becomes easy to show the antisymmetry
property.

###### Proof of Theorem 1 (discrete setting).

Without loss of generality, assume by contradiction that
$\delta:=v(\varepsilon)+v(-\varepsilon)>0$. Fix $N$ large enough that

$\displaystyle\mathbb{P}\left(\left|\frac{X_{N}^{\pm\varepsilon}}{N}-v(\pm\varepsilon)\right|\geq\delta/4\right)\leq
1/3.$ (9)

Define $\tilde{x}=\lfloor(v(\varepsilon)-\delta/2)N\rfloor$. If $\tilde{x}$
has the same parity as $N$, let $x=\tilde{x}$. Let $x=\tilde{x}+1$ else.
Define $X,\widehat{X}$ as above. Consider the event

$E=\left\\{\frac{X_{N}}{N}\geq v(\varepsilon)-\delta/4\text{ and
}\frac{\widehat{X}_{0}}{N}-x\geq v(-\varepsilon)-\delta/4\right\\}.$ (10)

By Lemma 1, $\mathbb{P}(E)\geq 1/3$, but by Lemma 2, $E$ is empty, since
$X_{N}>\widehat{X}_{N}=x$ and $X_{0}<\widehat{X}_{0}$ on $E$. We have our
contradiction.

∎

######  Remark 1.

While the non-crossing property is elementary to check in the discrete time
setting, it is not easy to find a coupling with the same property in the
continuous time setting. One could think of associating independent Poisson
clocks to every vertex of $\mathbb{Z}$, a direction $A^{\varepsilon}_{x,n}$
with every clock ring and build $X,\widehat{X}$ in a similar fashion. However,
the jumps starting from neighboring sites would not be simultaneous, and it is
easy to find examples where crossings occur. Fortunately, once we have the
result in the discrete time setting, we do not need to work hard to push it to
the continuous time setting.

##  4 Proof of Theorem 1 in the continuous setting

Recall the construction of Section 2.3. Define for $n\in\mathbb{N}$,
$\sigma_{n}=\eta_{T_{n}}$. Theorem 1 will be proved when we have checked the
three points of the following lemma.

######  Lemma 3.

  1. 1.

The process $\sigma$ is translation invariant and reversible.

  2. 2.

If $X^{\varepsilon}$ is the $\varepsilon$–RW in continuous time defined on
$\eta$ as in Section 2.3, $(X^{\varepsilon}_{T_{n}})_{n\in\mathbb{N}}$ is
equal in distribution to $\widetilde{X}^{\varepsilon}$ the $\varepsilon$–RW in
discrete time defined on $\sigma$.

  3. 3.

$X^{\varepsilon}$ satisfies the weak LLN (2) iff $\widetilde{X}^{\varepsilon}$
does, with the same limit speed $v(\varepsilon)$.

###### Proof of Lemma 3.

  1. 1.

This follows from the translation invariance and reversibility of $\eta$ and
the PPP $T$, along with the independence of $T$ and $\eta$.

  2. 2.

This follows from the independence of $U,T,\eta$, and a construction in the
discrete case analog to the one we use in Section 2.3.

  3. 3.

This is an immediate consequence of the fact that
$T_{n}/n\underset{n\rightarrow\infty}{\longrightarrow}1$ almost surely.

∎

###### Proof of Theorem 1 (continuous setting).

We assume that both $X^{\varepsilon}$ and $X^{-\varepsilon}$ satisfy the weak
LLN (2). By the third point of Lemma 3, $\widetilde{X}^{\varepsilon}$ and
$\widetilde{X}^{-\varepsilon}$ satisfy (2) with respective speeds
$v(\varepsilon),v(-\varepsilon)$. Therefore, the discrete time version of
Theorem 1 applies to $\widetilde{X}^{\varepsilon}$,
$\widetilde{X}^{-\varepsilon}$ and $v(-\varepsilon)=-v(\varepsilon)$.

∎

## References

  * [ABF16] Luca Avena, Oriane Blondel, and Alessandra Faggionato.  A class of random walks in reversible dynamic environments: Antisymmetry and applications to the East model.  Journal of Statistical Physics, 165(1):1–23, Oct 2016.
  * [ABF18] L. Avena, O. Blondel, and A. Faggionato.  Analysis of random walks in dynamic random environments via l2-perturbations.  Stochastic Processes and their Applications, 128(10):3490 – 3530, 2018.
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  * [CZ04] Francis Comets and Ofer Zeitouni.  A law of large numbers for random walks in random mixing environments.  Ann. Probab., 32(1B):880–914, 2004.
  * [FMRT13] A. Faggionato, F. Martinelli, C. Roberto, and C. Toninelli.  The East model: recent results and new progresses.  Markov Process. Relat. Fields, 19(3):407–452, 2013.
  * [HKT20] Marcelo R. Hilário, Daniel Kious, and Augusto Teixeira.  Random walk on the simple symmetric exclusion process.  Commun. Math. Phys., 379(1):61–101, 2020.
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  * [JE91] J. Jäckle and S. Eisinger.  A hierarchically constrained kinetic Ising model.  J.Phys.B, 84:115–124, 1991.
  * [MV15] Thomas Mountford and Maria E. Vares.  Random walks generated by equilibrium contact processes.  Electron. J. Probab., 20:no. 3, 17, 2015.
  * [RV13] Frank Redig and Florian Völlering.  Random walks in dynamic random environments: a transference principle.  Ann. Probab., 41(5):3157–3180, 2013.


# Geometric interpretation of the vanishing Lie Bracket for two-dimensional
rough vector fields

Rebucci<EMAIL_ADDRESS>Max-Planck Institute for Mathematics
in the Sciences  Zizza M<EMAIL_ADDRESS>
###### Abstract

In this paper, we prove that if $X,Y$ are continuous, Sobolev vector fields
with bounded divergence on the real plane and $[X,Y]=0$, then their flows
commute. In particular, we improve the previous result of [11], where the
authors require the additional assumption of the weak Lie differentiability on
one of the two flows. We also discuss possible extensions to the
$\operatorname{BV}$ setting.

Key words: Frobenius Theorem, Lie Bracket, rough vector fields, Regular
Lagrangian Flows.

MSC2020: 35A24, 35D30, 35F35, 35Q49.

##  1 Introduction

The aim of this paper is to extend the classical result on commutativity of
flows of smooth vector fields (in the Euclidean space) to the weak setting of
Regular Lagrangian Flows. To this end, we first recall that, if $X,Y\in
C^{\infty}(\mathbb{R}^{n},\mathbb{R}^{n})\cap
L^{\infty}(\mathbb{R}^{n},\mathbb{R}^{n})$ are two smooth vector fields, then
their flows $\phi^{X}_{t},\phi^{Y}_{s}$ are the solutions of the following
Cauchy Problem

$\begin{cases}\partial_{t}\phi^{X}_{t}(x)=X(\phi^{X}_{t}(x)),\\\
\phi^{X}_{0}(x)=x,\end{cases}$

$\forall x\in\mathbb{R}^{n}$, (and the same for the flow $\phi^{Y}_{s}$). Then
one of the classical results of differential geometry, ultimately leading to
Frobenius theorem, states that

$\phi_{t}^{X}\circ\phi_{s}^{Y}=\phi_{s}^{Y}\circ\phi_{t}^{X},\quad\forall
t,s\in\mathbb{R}\qquad\iff\qquad[X,Y]=0,$ (1.1)

where $[X,Y]$ can be defined pointwise as $[X,Y]=DYX-DXY$. The equivalence in
(1.1) was proved for locally Lipschitz vector fields in [16], and the
extension to Sobolev vector fields was analyzed in [11], where the authors
prove that, if $X,Y\in W^{1,p}_{\text{loc}}(\mathbb{R}^{n},\mathbb{R}^{n})\cap
L^{\infty}(\mathbb{R}^{n},\mathbb{R}^{n})$ with bounded divergence, then

$\phi^{X}_{t}\circ\phi^{Y}_{s}=\phi^{Y}_{s}\circ\phi^{X}_{t},\quad\forall
s,t\in\mathbb{R},\text{ for a.e. }z\in\mathbb{R}^{n}$ (1.2) $\iff$
$[X,Y]=0\text{ and }\phi^{X}_{t}\text{ is weakly (Lie) differentiable w.r.t.
$Y$ with locally bounded derivative.}$

Here, the flows of the two vector fields are Regular Lagrangian Flows as in
[13],[3] (see also Section 2). Moreover, we remark that the condition $\forall
s,t\in\mathbb{R}$, for a.e. $z\in\mathbb{R}^{n}$ in (1.2) means that for every
$s,t$ fixed, there exists $\Omega_{s,t}$ such that
$\mathcal{L}^{2}(\Omega_{s,t}^{c})=0$ and the flows commute in $\Omega_{s,t}$.
Hence, it is not restrictive to consider a bounded time interval, i.e.
$t\in(-T,T)$ for some $T>0$. We finally recall that the notion of weak Lie
differentiability introduced in [11] has to be intended in the following
sense:

######  Definition 1.1.

Let $F\in L^{1}(\mathbb{R}^{n},\mathbb{R}^{n})$ and let $Y\in
L^{\infty}(\mathbb{R}^{n},\mathbb{R}^{n})$ be a vector field with
$\operatorname{div}Y\in L^{\infty}(\mathbb{R}^{n},\mathbb{R})$. We say that
$F$ is weakly (Lie) differentiable w.r.t. $Y$ if there exists $f\in
L^{1}(\mathbb{R}^{n},\mathbb{R}^{n})$ such that for every $\varphi\in
C^{1}_{c}(\mathbb{R}^{n},\mathbb{R}^{n})$ the following holds

$\int_{R^{n}}\left(F(z),D\varphi(z)Y(z)\right)+\operatorname{div}(Y)(z)\left(F(z),\varphi(z)\right)dz=-\int_{\mathbb{R}^{n}}\left(f(z),\varphi(z)\right)dz.$
(1.3)

We will say that $f$ is the _weak (Lie) derivative_ of $F$ in direction of
$Y$. If moreover $f\in L^{\infty}_{\text{loc}}(\mathbb{R}^{n},\mathbb{R}^{n})$
we will say that $F$ is weakly (Lie) differentiable w.r.t. $Y$ with locally
bounded derivative.

In particular, if $F\in W^{1,p}_{\text{loc}}(\mathbb{R}^{n},\mathbb{R}^{n})$,
then $F$ is weakly (Lie) differentiable. Indeed, the weak Lie derivative is a
directional derivative where the direction is given by the vector field $Y$.

We point out that the weak Lie differentiability assumption is a non-trivial
requirement in the non-smooth setting. Indeed, in the case of rough regularity
of the vector field (being either Sobolev or $\operatorname{BV}$), the flow
may not be differentiable, namely may not belong to any
Sobolev/$\operatorname{BV}$ space, as first proved in [14] through a random
construction. We also quote [15], where the author provides an example where
the RLF fails to be Sobolev/$\operatorname{BV}$.

In this paper, we propose a purely geometric approach that allows us to remove
the additional assumption of the weak Lie differentiability in the case of
two-dimensional Hamiltonian vector fields. More precisely, we prove that the
classical equivalence (1.1) holds in dimension $n=2$, with the further
assumption of the continuity of the two vector fields. Moreover, in Section 5
we explain how to generalize one of the implications in (1.1) to the
$\operatorname{BV}$ setting.

We start by describing the heuristics of our approach, in order to show in
particular how to drop the weak Lie differentiability assumption on the flow
of $X$. Our construction draws inspiration from [15]. The core idea is that,
if the vector field $X\in W^{1,p}(\mathbb{R}^{2},\mathbb{R}^{2})\cap
C^{0}(\mathbb{R}^{2},\mathbb{R}^{2})$ is bounded, divergence-free and
uniformly bounded from below, then its flow $\phi^{X}_{t}$ belongs to the
Sobolev space $W^{1,p}$. In particular, the theorem proved in [15] is the
following.

######  Theorem 1.2.

[Theorem 1.4 in [15]] Let $X\in W^{1,p}(\mathbb{R}^{2},\mathbb{R}^{2})$ be a
continuous divergence-free vector field with bounded support and let $\Omega$
be an open ball of radius $R>0$ such that there exists $\delta>0$ and
$\mathbf{e}\in\mathbb{S}^{1}$ for which $X\cdot\mathbf{e}>\delta$ in $\Omega$.
Then for every $t>0$ the Regular Lagrangian Flow of $X$ has a representative
$\phi^{X}_{t}\in C^{0}(\Omega)\cap W^{1,p}(\Omega).$

The proof of the previous result is based on the underlying Hamiltonian
structure of the vector field and therefore the set $\Omega$, which is a ball
in [15], must be simply connected in order to guarantee the existence of a
Lipschitz Hamiltonian $H$ such that $X=\nabla^{\perp}H$. We remark also that,
if we consider $\mathbf{e}$ as in the statement of Theorem 1.2, then we have
$|X|\geq\delta>0$. A version of Theorem 1.2 in the $\operatorname{BV}$ setting
is also available.

In virtue of Theorem 1.2, it is clear that, if $|X|\geq c>0$ in some
$\Omega\in\mathbb{R}^{2}$, and the set $\Omega$ is invariant for the flows
$\phi^{X}_{t},\phi^{Y}_{s}$ (meaning that the flows keep living in the set
$\Omega$), then $\phi^{X}_{t}$ is weakly Lie differentiable in $\Omega$, since
it is Sobolev, and therefore the two flows commute in $\Omega$. On the other
hand, if $|X|=0$, then its flow is the identity map (at least for a.e.
$x\in\mathbb{R}^{2}$) and the commutativity simply follows by proving that
$X\circ\phi^{Y}_{s}=0$ for every $s$, i.e. the vector field $X$ is forced to
vanish along the trajectory of $Y$.

The proof of our main result relies on finding regions of the plane invariant
for the flows $\phi^{X}_{t},\phi^{Y}_{s}$ where the two different behaviours
occur. In this context, the condition $[X,Y]=0$ prescribes very strict
geometric properties on the flows of the two vector fields (see in particular
Equation (1.6) below).

We finally remark that the study of the Lie Bracket for rough vector fields is
of interest in the context of Magneto Hydrodynamics, for example [10].
Furthermore, the investigation of the commutativity of the flows in the non-
smooth setting, together with the notion of Lie bracket for rough vector
fields, arises naturally in various research fields, see for instance
[17],[7].

#### The Hamiltonian case

In the first part of our work, namely Section 3, we restrict ourselves to the
case of Hamiltonian dynamics, i.e. we assume $\operatorname{div}X=0$,
$\operatorname{div}Y=0$, $n=2$. Our main contribution here is the following
result.

######  Proposition 1.3.

Let $X,Y\in W^{1,p}_{loc}(\mathbb{R}^{2},\mathbb{R}^{2})\cap
C^{0}(\mathbb{R}^{2},\mathbb{R}^{2})$ be two bounded vector fields and assume
that $\operatorname{div}X=\operatorname{div}Y=0$ almost everywhere. Then

$\qquad[X,Y]=0\iff\phi_{t}^{X}\circ\phi_{s}^{Y}=\phi_{s}^{Y}\circ\phi_{t}^{X},\quad\forall
t,s\in\mathbb{R},\quad\text{for a.e. in }\mathbb{R}^{2}.$ (1.4)

The main idea behind the proof of Proposition 1.3 is to exploit the well-known
fact that, if $X\in L^{\infty}(\mathbb{R}^{2},\mathbb{R}^{2})$ with
$\operatorname{div}X=0$, then there exists a Lipschitz Hamiltonian
$H:\mathbb{R}^{2}\rightarrow\mathbb{R}$ such that

$X=\nabla^{\perp}H=(-\partial_{2}H,\partial_{1}H).$

Then, given two smooth Hamiltonian vector fields
$X=\nabla^{\perp}H,Y=\nabla^{\perp}K$, we have

$[X,Y]=0\iff\\{H,K\\}=\text{const},$

where $\\{\cdot,\cdot\\}$ denotes the Poisson Bracket. In the two dimensional
case, the vanishing of the Lie bracket corresponds in particular to the
following property:

$X\cdot Y^{\perp}=\text{const}.$ (1.5)

Indeed, the following equation holds [5]:

$[X,Y]=\operatorname{div}YX-\operatorname{div}XY-\nabla^{\perp}(X\cdot
Y^{\perp}).$ (1.6)

In Section 3, we show that the same equivalence between the vanishing of the
Lie bracket and property (1.5) holds if we understand the Lie bracket in (1.6)
in the distributional sense. Hence, we are left with proving that $X\cdot
Y^{\perp}=\text{const}\Rightarrow\phi_{t}^{X}\circ\phi_{s}^{Y}=\phi_{s}^{Y}\circ\phi_{t}^{X}$,
for every $t,s\in\mathbb{R}$, for a.e. in $x\in\mathbb{R}^{2}$ and therefore
we can study separately the two different cases $X\cdot Y^{\perp}\not=0$, and
$X\cdot Y^{\perp}=0$. In the first case, the commutativity of the flows is an
almost direct consequence of the results proved in [15] (see also [11,
Corollary 4.2]) for which we need the continuity assumption. On the contrary,
the proof in the second case $X\cdot Y^{\perp}=0$ is more delicate and relies
on the geometry of the two vector fields. To be more precise, we rely on the
observation that the condition $X\cdot Y^{\perp}=0$ implies that the level
sets of the two Hamiltonians $H,K$ coincide, so that we can reduce the study
of the commutativity of flows on each level set, in the spirit of what was
done in [1]. In particular, the strict geometry of the two vector fields
allows us to extend the result in [11] by means of purely geometric
considerations, in order to recover the classical equivalence.

#### The Nearly incompressible case

In the second part of our work (Section 4), we assume the more general
hypothesis $\operatorname{div}X,\operatorname{div}Y\in
L^{\infty}(\mathbb{R}^{2},\mathbb{R})$. A key observation in our analysis is
that classical results state that, if we assume that $X$, $Y$ have bounded
divergence, then they are nearly incompressible in the sense of the following
definition.

######  Definition 1.4 (Nearly incompressible).

A bounded vector field $X\in
L^{1}_{\text{loc}}(\mathbb{R}^{n},\mathbb{R}^{n})\cap
W^{1,p}_{\text{loc}}(\mathbb{R}^{n},\mathbb{R}^{n})$ is called _nearly
incompressible_ if there exists a positive constant $C>0$ and a density
$\rho:I\times\Omega\rightarrow\mathbb{R}$ such that

$\partial_{t}\rho+\operatorname{div}(\rho X)=0\quad\text{in
}\mathcal{D}^{\prime}(I\times\Omega),$

with $\rho\in[C^{-1},C]$ $\mathcal{L}^{n}$-a.e. $(t,x)\in I\times\Omega$.

It is clear that divergence-free vector fields are nearly incompressible. We
recall that nearly incompressible vector fields were introduced in the study
of the hyperbolic system of conservation laws named after Keyfitz and Kranzer
(for a systematic treatement of this topic, we refer the reader to [12]).

Also in this more general setting, Equation (1.6) provides strong geometric
insights on the flows of the two vector fields. Indeed, under the hypothesis
$[X,Y]=0$, the vector field

$W\doteq\operatorname{div}YX-\operatorname{div}XY\in L^{\infty}$

is a Hamiltonian vector field, with Lipschitz Hamiltonian given by

$H_{W}=X\cdot Y^{\perp},$

where $Y^{\perp}$ denotes $(-Y_{2},Y_{1})$, namely the counterclockwise
rotation of the vector field $Y$. As in the previous case of Hamiltonian
vector fields, there is a clear distinction between the two behaviors
occurring in the regions $\\{X\cdot Y^{\perp}=0\\}$ and $\\{X\cdot
Y^{\perp}\not=0\\}$. To study the behaviour of the flows of $X,Y$, we prove
two key results. The first one (Proposition 1.5 below) states that the flows
of the two vector fields $X,Y$ remain confined in the regions where $H_{W}$ is
bounded, namely the closed sets
$\overline{\Omega}_{\kappa}=H_{W}^{-1}\left[{C}^{-1}\kappa,C\kappa\right]$.
Indeed,

######  Proposition 1.5.

Let $X,Y\in W^{1,p}(\mathbb{R}^{2},\mathbb{R}^{2})\cap
L^{\infty}(\mathbb{R}^{2},\mathbb{R}^{2})$ with bounded divergence. Moreover,
assume that $[X,Y]=0$. Then the following equation holds

$X\cdot Y^{\perp}\circ\phi^{X}_{t}=\text{det}D\phi^{X}_{t}X\cdot
Y^{\perp}\quad\text{a.e.},$ (1.7)

(and the same for the flow $\phi^{Y}_{s}$). In particular, if $X\cdot
Y^{\perp}>0$, then there holds

$\frac{1}{C}X\cdot Y^{\perp}\leq X\cdot Y^{\perp}\circ\phi^{X}_{t}\leq CX\cdot
Y^{\perp}\qquad\text{a.e.,}$

for some positive constant $C>0$.

The proof of the previous proposition is a consequence of Equation (3.4) below
and of the boundedness of $\text{det}D\phi^{X}_{t}$, namely

$\frac{1}{C}\leq\text{det}D\phi^{X}_{t}\leq C,\quad\text{a.e.}.$ (1.8)

The second result (Proposition 1.7) states that the two vector fields $X,Y$
are steady nearly incompressible in the sets $\Omega_{\kappa}$. We recall that

######  Definition 1.6 (Steady nearly incompressible).

A bounded vector field $X\in
L^{1}_{\text{loc}}(\mathbb{R}^{n};\mathbb{R}^{n})$ is called _steady nearly
incompressible_ in $\Omega$ if there exists a positive constant $C>0$ and a
time-independent density $\rho:\Omega\rightarrow\mathbb{R}$ such that

$\operatorname{div}(\rho X)=0\quad\text{in }\mathcal{D}^{\prime}(\Omega),$

with $\rho\in[C^{-1},C]$ $\mathcal{L}^{n}$-a.e. $x\in\Omega$.

Then the following Proposition holds

######  Proposition 1.7.

Let $X,Y\in L^{\infty}(\mathbb{R}^{2},\mathbb{R}^{2})\cap
W^{1,p}(\mathbb{R}^{2},\mathbb{R}^{2})$ with bounded divergence such that
$[X,Y]=0$. Then for every $\kappa\in\mathbb{R}\setminus\\{{0}\\}$, the vector
fields $X\llcorner_{\Omega_{\kappa}},Y\llcorner_{\Omega_{\kappa}},$ are steady
nearly incompressible. Moreover, they have the same Lipschitz and bounded
density $\rho_{\kappa}:\Omega_{\kappa}\rightarrow\mathbb{R}$ given by the
function

$\rho_{\kappa}=\frac{1}{H_{W}},$

that is

$\operatorname{div}(\rho_{\kappa}(X-Y))=0,\quad\text{in
}\mathcal{D}^{\prime}(\Omega_{\kappa}).$

This proposition allows us to study the problem of commutativity of the flows
separately in the two regions $\Omega=\cup_{\kappa\not=0}\Omega_{\kappa}$ and
in $\\{X\cdot Y^{\perp}=0\\},$ similarly to what previously done in the case
of Hamiltonian vector fields. Also in this case, we expect that in the set
$\Omega$ the flows of the two vector fields are differentiable in the spirit
of [15], while in the complementary set $\\{X\cdot Y^{\perp}=0\\}$ we exploit
the geometry of the trajectories. We are now in a position to state the main
result of this paper, namely

######  Theorem 1.8.

Let $X,Y\in W^{1,p}(\mathbb{R}^{2},\mathbb{R}^{2})\cap
C^{0}(\mathbb{R}^{2},\mathbb{R}^{2})$ be two bounded vector fields with
$\operatorname{div}X,\operatorname{div}Y\in
L^{\infty}(\mathbb{R}^{2},\mathbb{R}^{2}).$ Then we have

$\qquad[X,Y]=0\iff\phi_{t}^{X}\circ\phi_{s}^{Y}=\phi_{s}^{Y}\circ\phi_{t}^{X},\quad\forall
t,s\in\mathbb{R},\quad\text{for a.e. in }\mathbb{R}^{2}.$

It is clear that Proposition 1.3 follows immediately once we have established
Theorem 1.8. However, we prefer to analyze first the Hamiltonian case in order
to present the main ideas of the proof more clearly. Moreover, the Hamiltonian
structure allows us to propose in Section 3 an alternative proof of the
commutativity of the flows which relies on the analysis of level sets of
Hamiltonians.

To summarize the above, the general idea is that, in the regions where $X\cdot
Y^{\perp}\not=0$, one of the two vector field plays the role of the direction
$\mathbf{e}$ in Theorem 3.6 and therefore we have the differentiability
properties of the two flows. In the points where the direction $\mathbf{e}$
cannot be found, we exploit the fact that the trajectories of the two vector
fields are the same. In particular, we remark that the non-symmetric
assumption ’$\phi^{X}_{t}$ is weakly Lie differentiable w.r.t. $Y$’ (without
asking anything on the flow of the vector field $Y$) is in fact symmetric,
since in the same region $X\cdot Y^{\perp}\not=0$ the flow of the vector field
$Y$ is weakly Lie differentiable w.r.t. $X$.

We conclude this paragraph by emphasizing that, since $[X,Y]=0$ imposes strict
geometric conditions on the flows of the two vector fields, we expect that
also in higher dimension the geometry of trajectories could be analyzed to
improve the result in [11], since analogous of Equation (1.6) are available.
On the other hand, it is not clear whether in higher dimension we can still
expect Sobolev regularity results for the flows under study. Equation (1.6)
turned out to be a surprisingly rich source of information about the geometric
properties of the two vector fields, and we expect that it could be further
investigated for our understanding of the underlying geometric structure given
by the vanishing Lie bracket.

#### Extension to $\operatorname{BV}$ setting

In the final part of this work, namely Section 5, we briefly discuss how to
generalize the previous results to $\operatorname{BV}$ vector fields, since
the results of the previous discussion holds true also in this case. In our
opinion, the extension to the $\operatorname{BV}$ setting is not trivial, and
relies mainly on the following definition of distributional Lie Bracket.

######  Definition 1.9 (Distributional Lie bracket).

Let $X$,
$Y\in\operatorname{BV}_{\text{loc}}(\mathbb{R}^{n},\mathbb{R}^{n})\cap
L^{\infty}(\mathbb{R}^{n},\mathbb{R}^{n})$ be vector fields with bounded
divergence. Then we define the (distributional) Lie bracket $[X,Y]$ by
setting, for every $\varphi\in C^{1}_{c}(\mathbb{R}^{n},\mathbb{R}^{n})$,

$\int_{\mathbb{R}^{n}}([X,Y],\varphi)dz\doteq\int_{\mathbb{R}^{n}}\left((X,D\varphi
Y)+\operatorname{div}Y(X,\varphi)-(Y,D\varphi
X)-\operatorname{div}X(Y,\varphi)\right)dz.$ (1.9)

We remark that the Lie bracket in (1.9) is well-defined as we assume that the
vector fields $X$ and $Y$ have bounded divergence. Moreover, we emphasize that
our definition of Lie bracket is consistent with the classical one if $X,Y$
are smooth vector fields and reduces to the one in [11] if we identify $X$ and
$Y$ with their precise representative, by Vol’pert chain rule formula (see
Sections 2 and 5). Owing to Definition 1.9, we prove the following
proposition, which extends to the $\operatorname{BV}$ setting one of the
implications in (1.1).

######  Proposition 1.10.

Let $X$,
$Y\in\operatorname{BV}_{\text{loc}}(\mathbb{R}^{n},\mathbb{R}^{n})\cap
L^{\infty}(\mathbb{R}^{n},\mathbb{R}^{n})$ be vector fields with bounded
divergence and let $\phi_{t}^{X}$, $\phi_{s}^{Y}$ be their Regular Lagrangian
Flows. Then the following implication holds true

$\phi_{t}^{X}\circ\phi_{s}^{Y}=\phi_{s}^{Y}\circ\phi_{t}^{X},\quad\forall
t,s\in\mathbb{R},\quad\text{a.e. in
}\mathbb{R}^{n}\quad\Rightarrow\quad[X,Y]=0.$ (1.10)

The proof of Proposition 1.10 is obtained by simply adapting the method used
in [11]. In contrast to the Sobolev case studied in [11], it is our belief
that it is not straightforward to prove the missing implication in Proposition
1.10. Indeed, a key point in order to show that the vanishing Lie bracket
implies that the two flows commute, one would need to evaluate the function

$t\rightarrow[X,Y]\circ\phi^{X}_{t},$

and to use the hypothesis on vanishing Lie Bracket on the trajectories. In the
$\operatorname{BV}$ setting, where we need a distributional definition of Lie
Bracket, it is not clear to us how to give meaning to the previous function.
For this reason, the geometric approach we follow in Sections 3 and 4 can not
be straightforwardly generalized to the $\operatorname{BV}$ setting (it is in
particular not possible to employ Theorem 1.2 to deal with the case $X\cdot
Y^{\perp}\neq 0$). This constitutes a key difference between the Sobolev and
the $\operatorname{BV}$ case and therefore we left the other implication as an
open problem.

####  1.0.1 Outline of the paper

The structure of the paper is the following. Section 2 is devoted to
preliminaries: we collect some known results regarding $\operatorname{BV}$
functions, we introduce the notion of regular Lagrangian flow and we recall
the structure of level sets of Lipschitz functions with compact support and
gradient with bounded variation. In Section 3, we prove Proposition 1.3
through a dimension-reduction argument in the spirit of [1]. In Section 4 we
generalize the previous arguments to the case of rough vector fields with
bounded divergence and we prove our main result, Theorem 1.8. Finally, some
comments on the extension to the case of $\operatorname{BV}$ vector fields are
presented in Section 5.

Acknowledgements. The authors acknowledge Riccardo Tione for suggesting the
problem and Riccardo Tione, Giacomo del Nin and Elio Marconi for many useful
discussions and insights on the topic.

##  2 Preliminaries

In this section, we collect some preliminary results that will be used
throughout the paper. We first give a quick overview of $\operatorname{BV}$
functions, Regular Lagrangian Flows and Commutator estimates. Finally, in the
last subsection, we give a characterization of Level sets of Lipschitz
functions with compact support.

###  2.1 BV functions

In this subsection, we recall some known results concerning functions of
bounded variation. For a more comprehensive treatment of the topic, we refer
the reader to [4]. Let $u\in\operatorname{BV}(\Omega,\mathbb{R}^{n})$ and
$Du\in\mathcal{M}_{b}(\Omega)^{n\times n}$ the $n\times n$-valued measure
representing its distributional derivative. We recall the canonical
decomposition of the measure $Du$

$Du=D^{\textrm{a}}u+D^{\textrm{s}}u=D^{\textrm{a}}u+D^{\mathrm{c}}u+D^{\mathrm{j}}u,$

where $D^{\textrm{a}}u$ is the absolutely continuous part of $Du$ with respect
to the $n$-dimensional Lebesgue measure $\mathcal{L}^{n}$ and
$D^{\textrm{s}}u$ is the singular part of $Du$ with respect to
$\mathcal{L}^{n}$. Moreover, we split $D^{\textrm{s}}u$ into the jump part
$D^{s}u$ and the Cantor part $D^{\mathrm{c}}u$. Since $D^{a}u$ is an
absolutely continuous function, we can write it as $D^{a}u=\nabla
u\mathcal{L}^{n}$, i.e. we denote by $\nabla
u=(\partial_{i}u^{j})\in\mathbb{R}^{n\times n}$ the density of $D^{a}u$ with
respect to $\mathcal{L}^{n}$. We finally recall a structure theorem for
$\operatorname{BV}$ vector fields from [4].

######  Theorem 2.1.

Let $u\in\operatorname{BV}(\mathbb{R}^{n},\mathbb{R}^{n})$. Then there exists
a partition of $\mathbb{R}^{n}=S_{u}\cup J_{u}\cup D_{u}$ such that the
following properties hold true:

  * 1.

$\mathcal{H}^{n-1}(S_{u})=0$;

  * 2.

$J_{u}$ is the union of the images of at most countably many Lipschitz curves
and there exist Borel functions $\nu:J_{u}\rightarrow\mathbb{S}^{n-1}$,
$u^{-}$, $u^{+}:J_{u}\rightarrow\mathbb{R}^{n}$ such that, for every
$\overline{x}\in J_{u}$, we have

$\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{B_{r}^{-}(\overline{x})}|u(x)-u^{-}(\overline{x})|dx=o(1),\quad\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{B_{r}^{+}(\overline{x})}|u(x)-u^{+}(\overline{x})|dx=o(1),\quad\text{as
$r\to 0$},$

where $B_{r}^{-}(\overline{x})\doteq\\{x\in B_{r}:(x-\overline{x},\nu)<0\\}$
and $B_{r}^{+}(\overline{x})\doteq\\{x\in B_{r}:(x-\overline{x},\nu)>0\\}$;

  * 3.

for every $\overline{x}\in D_{u}$, there exists $\tilde{u}(\overline{x})$ such
that

$\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{B_{r}(\overline{x})}|u(x)-\tilde{u}(\overline{x})|dx=o(1),\quad\text{as
$r\to 0$}.$

Taking advantage of the previous theorem, it is possible to assign a value to
$u$ at every point $x\notin S_{u}\setminus J_{u}$. More precisely, the precise
representative of $u\in\operatorname{BV}(\Omega,\mathbb{R}^{n})$ is the Borel
function $u^{*}$ such that $u^{*}(x)=\tilde{u}(x)$ if $x\in D_{u}$ and
$u^{*}(x)=(u^{+}(x)+u^{-}(x))/2$ if $x\in J_{u}$. The precise representative
allows us to state the general Leibniz rule for the product of two
$\operatorname{BV}$ functions. More precisely, if $u$,
$v\in\operatorname{BV}(\mathbb{R}^{n},\mathbb{R}^{n})\cap
L^{\infty}(\mathbb{R}^{n},\mathbb{R}^{n})$, then
$uv\in\operatorname{BV}(\mathbb{R}^{n},\mathbb{R}^{n})$ with

$D(uv)=v^{*}Du+u^{*}Dv,$ (2.1)

see [18] or [4, Theorem 3.96].

###  2.2 Regular Lagrangian Flows

Throughout this paper, we will consider autonomous vector fields
$X,Y:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ in the space
$\operatorname{BV}_{\text{loc}}(\mathbb{R}^{n},\mathbb{R}^{n})\cap
L^{\infty}(\mathbb{R}^{n},\mathbb{R}^{n})$ (in short
$X,Y\in\operatorname{BV}_{\text{loc}}\cap L^{\infty}$) with
$\operatorname{div}X,\operatorname{div}Y\in
L^{\infty}(\mathbb{R}^{n},\mathbb{R})$. We first recall that, when the
velocity fields $X,Y$ are Lipschitz, then their _flows_ are well-defined in
the classical sense, i.e. they are the maps
$\phi^{X}_{t},\phi^{Y}_{s}:\mathbb{R}\times\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$
satisfying

$\begin{cases}\partial_{t}\phi^{X}_{t}(x)=X(\phi^{X}_{t}(x)),&\\\
\phi^{X}_{0}(x)=x,\end{cases}$ (2.2)

(and same for $\phi^{Y}_{s}$). When dealing with rough vector fields, the
notion of pointwise uniqueness of (2.2) is not any more the appropriate one
and we only have uniqueness “in the selection sense”, as encoded in the
following definition.

######  Definition 2.2.

Let $X\in\operatorname{BV}_{\text{loc}}\cap L^{\infty}$. A map
$\phi^{X}_{t}:\mathbb{R}\times\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ is a
_Regular Lagrangian Flow_ (RLF) for the vector field $X$ if

  1. 1.

for a.e. $x\in\mathbb{R}^{2}$ the map $t\rightarrow X_{t}(x)$ is an absolutely
continuous integral solution of

$\begin{cases}\partial_{t}x(t)=X(x(t));&\\\ x(0)=x.\end{cases}$ (2.3)

  2. 2.

there exists a positive constant $C$ independent of $t$ such that

$\mathcal{L}^{2}(X_{t}^{-1}(A))\leq C\mathcal{L}^{2}(A),\quad\forall
A\in\mathcal{B}(\mathbb{R}^{2}).$ (2.4)

We recall that Regular Lagrangian flows were introduced in [13], where the
authors prove their existence and uniqueness for vector fields $X\in
W^{1,p}(\mathbb{R}^{n},\mathbb{R}^{n})$, $p\geq 1$, with bounded divergence.
The theory was subsequently extended to $\operatorname{BV}$ vector fields with
bounded divergence in [3]. Finally, uniqueness results in the more general
class of nearly incompressible vector fields were established in [6]. Here we
briefly recall the stability result for RLFs of $\operatorname{BV}$ vector
fields, that is

######  Theorem 2.3 (Stability, Theorem 6.3 [2]).

Let $X_{n},X\in\operatorname{BV}_{\text{loc}}\cap L^{\infty}$ with
$\operatorname{div}(X_{n})$ with bounded divergence and let
$\phi^{X^{n}},\phi^{X}$ be their corresponding Regular Lagrangian Flows.
Assume that

$||X_{n}-X||_{L^{1}}\to 0\quad\text{as
}n\to\infty,\quad\|\operatorname{div}X_{n}\|_{\infty}\leq C,\quad\forall n,$

then

$\lim_{n\to\infty}\int_{\mathbb{R}^{n}}\sup_{t}|\phi^{X^{n}}_{t}(x)-\phi^{X}_{t}(x)|dx=0.$

From now on we denote by $\phi_{t}^{X}(\cdot)_{\sharp}\mathcal{L}^{d}$ the
push-forward of the Lebesgue measure on $\mathbb{R}^{d}$ via the function
$\phi_{t}^{X}(\cdot)$. We observe, as already pointed out in [12], that
Condition $1.$ in Definition 2.2 is equivalent to

$\phi_{t}^{X}(\cdot)_{\sharp}\mathcal{L}^{d}\ll\mathcal{L}^{d}$

for every $t\geq 0$. Then, for every RLF, there exists a $\xi\in
L^{\infty}([-T,T]\times\mathbb{R}^{n})$, for every $T>0$, such that
$\phi_{t}^{X}(\cdot)_{\sharp}\mathcal{L}^{d}=\xi\mathcal{L}^{d}$. The function
$\xi$ will be called density of the flow $\phi_{t}^{X}$ and, by definition, it
satisfies the following “change of variables” identity

$\int_{\mathbb{R}^{n+1}}\varphi(t,\phi^{X}_{-t}(z))dzdt=\int_{\mathbb{R}^{n+1}}\varphi(t,z)\xi(t,z)dzdt,\quad\forall\varphi\in\mathcal{C}_{c}(\mathbb{R}^{n+1}).$
(2.5)

In the smooth setting, clearly $\xi=\text{det}D\phi^{X}_{t}.$ Moreover,
$\xi\in L^{\infty}([-T,T]\times\mathbb{R}^{n})$ for every $T>0$ and it
satisfies the analogous of Liouville Theorem, that is

$\partial_{t}\xi=\operatorname{div}(X)\circ\phi^{X}_{t}\xi\quad\text{distributionally}.$
(2.6)

See [11] for the details, where the Stability Theorem is used to obtain this
result.

###  2.3 Structure of level sets of $\text{Lip}_{c}$ functions

In this subsection, we consider $X\in
L^{\infty}(\mathbb{R}^{2};\mathbb{R}^{2})\cap\operatorname{BV}_{\text{loc}}(\mathbb{R}^{2})$
with $\operatorname{div}X=0$ a.e. and we additionally assume that $X$ has
compact support (this is not restrictive since we consider bounded vector
fields). Then there exists $H\in\text{Lip}_{c}(\mathbb{R}^{2},\mathbb{R})$
such that $X=\nabla^{\perp}H$, where
$\nabla^{\perp}=(-\partial_{2},\partial_{1})$ is the orthogonal gradient.
Before we give the characterization of the level sets of $H$ (the results we
state here are from [9] as in [8]), we recall that a curve $\Gamma$ in
$\mathbb{R}^{2}$ is the image of a continuous, non-constant
$\gamma:[a,b]\rightarrow\mathbb{R}^{2}$, thus it is a compact connected set.
We say that $\Gamma$ is a simple closed curve if there exists a
parametrization $\gamma:[a,b]\rightarrow\mathbb{R}^{2}$, injective on $[a,b)$
such that $\gamma(a)=\gamma(b)$.

######  Theorem 2.4.

We consider $X\in
L^{\infty}(\mathbb{R}^{2};\mathbb{R}^{2})\cap\operatorname{BV}_{\text{loc}}(\mathbb{R}^{2})$
with $\operatorname{div}X=0$ a.e. and with compact support. Then there exists
a set $C\subset H(\mathbb{R}^{2})$ with
$\mathcal{L}^{1}(H(\mathbb{R}^{2})\setminus C)=0$ and such that for every
$h\in C$ the following properties hold:

  * •

$\mathcal{H}^{1}(H^{-1}(h))<\infty$;

  * •

there exists $N=N(h)\in\mathbb{N}$ and $C_{i}(h)$ with $i\in\\{1,2,\dots,N\\}$
disjoint simple closed curves such that

$H^{-1}(h)=\cup_{i=1}^{N}C_{i}(h).$

Furthermore, there exist Lipschitz parametrizations
$\gamma_{i}(h):[0,\ell_{i}(h)]\rightarrow\mathbb{R}^{2}$ with
$\left|\frac{d}{dt}({\gamma_{i}(h)})_{t}\right|=1$ $\mathcal{L}^{1}$-a.e. $t$
and $\gamma_{i}(h)([0,\ell_{i}(h)])=C_{i}(h).$

Moreover, for every $\epsilon>0$ there exists $C_{\epsilon}\subset
H(\mathbb{R}^{2})$, and a constant $c>0$ such that
$\mathcal{L}^{1}(H(\mathbb{R}^{2})\setminus C_{\epsilon})<\epsilon$ and for
every $h\in C_{\epsilon}$ it holds

$\forall x\in H^{-1}(h)\cap D_{X},\qquad|X(x)|\geq c.$

######  Corollary 2.5.

In the assumption of the previous theorem, there exists a set
$\tilde{E}_{X}\subset H(\mathbb{R}^{2})$ with
$\mathcal{L}^{1}(H(\mathbb{R}^{2})\setminus\tilde{E}_{X})=0$ such that for
every $h\in\tilde{E}_{X}$, there exists $c_{h}>0$ such that $|X(x)|>c_{h}$ for
every $x\in\\{H=h\\}$.

######  Remark 2.6.

We remark that a slightly different characterization is needed for vector
fields that do not have compact support, where we lose the property for the
level sets to be finite union of simple closed curves. In this paper we omit
this description in order to avoid further technicalities.

We recall that, under this assumption, the Hamiltonian structure is compatible
with the flow, namely

######  Theorem 2.7 (Theorem 2.6 in [8]).

Let $X=\nabla^{\perp}H$ be a $\operatorname{BV}$ vector field. If
$\phi^{X}_{t}:\mathbb{R}^{2}\rightarrow\mathbb{R}^{2}$ is its unique RLF, then
for $\mathcal{L}^{2}$-a.e. $x\in\mathbb{R}^{2}$ and for every
$t\in\mathbb{R}$,

$H(\phi^{X}_{t}(x))=H(x).$

In particular, if we fix a representative for the RLF of $X$, it holds that
for every $h\in E_{X}$, for every $x\in C_{i}(h)$, where $C_{i}(h)\in
H^{-1}(h)$ is some cycle, and $\gamma_{i}(h)$ is the parametrization of
Theorem 2.4, then $\phi^{X}_{t}(x)=\gamma_{i}(h)(\bar{\tau})$, where
$\bar{\tau}\in[0,\ell_{i}(h)]$ is uniquely determined by

$t=N\int_{0}^{\ell_{i}(h)}\frac{1}{|X(\gamma_{i}(h)(\tau))|}d\tau+\int_{\gamma_{i}(h)^{-1}(x)}^{\bar{\tau}}\frac{1}{|X(\gamma_{i}(h)(\tau))|}d\tau,$
(2.7)

where $N\in\mathbb{N}$.

##  3 Hamiltonian vector fields

In this section, we consider two autonomous divergence-free vector fields in
the real plane. The analysis in this setting is facilitated by the Hamiltonian
structure of the two vector fields. Indeed, if $X,Y\in
L^{\infty}(\mathbb{R}^{2};\mathbb{R}^{2})$ with
$\operatorname{div}X=\operatorname{div}Y=0$ a.e., then there exist two
Lipschitz Hamiltonians $H,K:\mathbb{R}^{2}\rightarrow\mathbb{R}$ such that

$X=\nabla^{\perp}H,\qquad Y=\nabla^{\perp}K,$ (3.1)

where $\nabla^{\perp}=(-\partial_{2},\partial_{1})$ is the orthogonal
gradient. Here the main property that we want to exploit, requiring the
Sobolev regularity of the vector fields, is the fact that the trajectories of
particles advected by $X$ (resp. $Y$) are contained in the level sets of the
Hamiltonian $H$ (resp. $K$). Indeed, if $X,Y$ have $W^{1,p}_{\text{loc}}$
regularity in space, the Hamiltonians are preserved by their RLFs (see
Definition 2.2), namely

$H\circ\phi^{X}_{t}=\text{const},\qquad K\circ\phi^{Y}_{s}=\text{const},$

and the RLFs are uniquely determined by

$\phi^{X}_{t}(x)=x,\quad\forall x\in H^{-1}(S_{H}),$

where $S_{H}$ denotes the singular values of $H$, namely

$S_{H}=\\{h\in\mathbb{R}:\exists x\in\mathbb{R}^{2}:H(x)=h,X(x)=0\\},$

which is a closed set, under the assumption that $X$ is continuous with
compact support. Clearly, the same holds for the Hamiltonian $K$ along the
flow of the vector field $Y$ (under the same continuity assumption on the
vector field $Y$). Moreover, the flow $\phi^{X}_{t}$ (resp. $\phi_{s}^{Y}$) is
the unique solution of $\partial_{t}\phi^{X}_{t}=X(\phi^{X}_{t})$ (resp.
$\partial_{s}\phi^{Y}_{t}=Y(\phi^{Y}_{s})$), understood as a 1-dimensional
problem on the level sets of the Hamiltonian $H$, see in particular Formula
(2.7).

The aim of this section is to prove the following proposition.

######  Proposition 3.1.

Let $X,Y\in W^{1,p}_{loc}(\mathbb{R}^{2},\mathbb{R}^{2})\cap
C^{0}(\mathbb{R}^{2},\mathbb{R}^{2})$ be two bounded vector fields and assume
that $\operatorname{div}X=\operatorname{div}Y=0$ almost everywhere. Then

$[X,Y]=0\quad\Rightarrow\quad\phi^{X}_{t}\circ\phi^{Y}_{s}=\phi^{Y}_{s}\circ\phi^{X}_{t},\quad\forall
t,s\in\mathbb{R},\quad\mathcal{L}^{2}\text{-a.e. }x\in\mathbb{R}^{2}.$ (3.2)

This result extends in particular [11, Corollary 4.2] to vector fields that
can vanish on the real plane, so that the classical equivalence is
established. As explained above, the main property used here is the following
well-known fact in Hamiltonian dynamics

$[X,Y]=0\quad\iff\\{H,K\\}=\text{const},$ (3.3)

where $\\{\cdot,\cdot\\}$ denotes the Poisson Bracket. Such formula holds also
in this weak context, as proved in Lemma 3.3. In the two-dimensional case,
condition (3.3) is equivalent to the following property of the two vector
fields:

$X\cdot Y^{\perp}=\text{const}.$

######  Remark 3.2.

We point out that the condition $X\cdot Y^{\perp}=\text{const}$ can be written
as $\text{det}(\nabla K,\nabla H)=\text{const},$ connecting this problem to
differential inclusions.

Before proving Proposition 3.1, we give the following general result

######  Lemma 3.3.

Let $X,Y\in W^{1,p}_{\text{loc}}(\mathbb{R}^{2})\cap
L^{\infty}(\mathbb{R}^{2})$ with $\operatorname{div}X,\operatorname{div}Y\in
L^{\infty}(\mathbb{R}^{2})$. Then

$[X,Y]=\operatorname{div}YX-\operatorname{div}XY-\nabla^{\perp}(X\cdot
Y^{\perp}),\quad\text{distributionally}$ (3.4)

In particular, if $\operatorname{div}X=\operatorname{div}Y=0$ then

$[X,Y]=0\qquad\text{iff}\qquad X\cdot Y^{\perp}=\text{const}.$

###### Proof.

Let $\varphi\in C^{1}_{c}(\mathbb{R}^{2})$. Then, by definition (see Equation
(1.9)), the distributional Lie Bracket reads as

$\displaystyle\int_{\mathbb{R}^{2}}([X,Y],\varphi)dz$
$\displaystyle=\int_{\mathbb{R}^{2}}(X,D\varphi Y)-(Y,D\varphi
X)+\operatorname{div}Y(X,\varphi)-\operatorname{div}X(Y,\varphi)dz.$

In the general case, we have

$\displaystyle\int_{\mathbb{R}^{2}}(X,D\varphi Y)-(Y,D\varphi X)dz$
$\displaystyle=\int_{\mathbb{R}^{2}}(Y_{1}X_{1}\partial_{1}\varphi_{1}+X_{1}Y_{2}\partial_{2}\varphi_{1}+Y_{1}X_{2}\partial_{1}\varphi_{2}+Y_{2}X_{2}\partial_{2}\varphi_{2})dz$
$\displaystyle-\int_{\mathbb{R}^{2}}(X_{1}Y_{1}\partial_{1}\varphi_{1}+Y_{1}X_{2}\partial_{2}\varphi_{1}+X_{1}Y_{2}\partial_{1}\varphi_{2}+X_{2}Y_{2}\partial_{2}\varphi_{2})dz$
$\displaystyle=\int_{\mathbb{R}^{2}}(X_{1}Y_{2}\partial_{2}\varphi_{1}+Y_{1}X_{2}\partial_{1}\varphi_{2}-Y_{1}X_{2}\partial_{2}\varphi_{1}-X_{1}Y_{2}\partial_{1}\varphi_{2})dz$
$\displaystyle=\int_{\mathbb{R}^{2}}(X\cdot
Y^{\perp})(-\partial_{2}\varphi_{1}+\partial_{1}\varphi_{1})dz,$

which immediately gives the statement. ∎

######  Remark 3.4.

We observe that Equation (3.4) and Lemma 3.3 hold also in the
$\operatorname{BV}$ setting.

######  Remark 3.5.

As anticipated before, the condition $[X,Y]=0$ for Hamiltonian vector fields
corresponds to $\\{H,K\\}=\text{const}$. In particular, this formula holds in
any even dimension, giving strong geometric properties to the flows of
Hamiltonian vector fields (that is, the Lie bracket of two Hamiltonian vector
fields is itself a Hamiltonian vector field).

###  3.1 Analysis of $X\cdot Y^{\perp}=\text{const}$

This subsection is devoted to the proof of Proposition 3.1, which in turn
implies Proposition 1.3 in virtue of [11, Theorem 1.1]. Thus, we are just left
with proving that the vanishing of the Lie bracket implies the commutativity
of flows in the case where the two vector fields $X,Y$ are Hamiltonian and
continuous with Sobolev regularity. As explained above, this boils down to
discussing the two separate cases $X\cdot Y^{\perp}\not=0$ and $X\cdot
Y^{\perp}=0$. In particular, if $X\cdot Y^{\perp}\not=0$, under the assumption
of continuity of the two vector fields, the commutativity is a corollary of
the result in [15], that we recall here for the reader’s convenience as stated
in [11], namely

######  Theorem 3.6 ([15]).

Let $X\in W^{1,p}_{\text{loc}}(\mathbb{R}^{2},\mathbb{R}^{2})\cap
L^{\infty}(\mathbb{R}^{2},\mathbb{R}^{2})$ be a continuous vector field with
zero divergence. If $X\not=0$, then its RLF $\phi^{X}_{t}\in
L^{\infty}_{\text{loc}}(\mathbb{R},W^{1,p}_{\text{loc}}(\mathbb{R}^{2},\mathbb{R}^{2})).$

We remark that the continuity of the vector field is necessary to ensure that
$X$ is uniformly bounded from below. In the case $X\cdot Y^{\perp}=1$, the
assumptions of the previous theorem are trivially satisfied, as condition
$X\cdot Y^{\perp}=1$ implies that there exists a positive constant $c>0$ such
that $|X|\geq c$. Indeed,

$1\leq|X||Y|\quad\longrightarrow\quad|X|\geq\frac{1}{\|Y\|_{\infty}},$

and therefore Theorem 3.6 guarantees the differentiability of the flow
$\phi^{X}_{t}$, which in turn implies the weak Lie differentiability
(Definition 1.1).

According to the previous arguments, from now on we will analyze the more
delicate case $X\cdot Y^{\perp}=0$ everywhere. In this case, thanks to the
boundedness of the vector fields, we can assume without loss of generality
that $X,Y$ have compact support, in order to make the analysis more
transparent.

We first recall that $S_{H},S_{K}$ are the sets of singular values of $H,K$
respectively, while $R_{H},R_{K}$, defined as
$R_{H}=H(\mathbb{R}^{2})\setminus S_{H}$, $R_{K}=K(\mathbb{R}^{2})\setminus
S_{K}$, are the sets of regular values. The identity $X\cdot Y^{\perp}=0$, at
the level of Hamiltonians of the two vector fields, translates into the
property that the level sets of the two Hamiltonians $H,K$ coincide. This has
to be intended in the following sense: consider a simple closed curve
$C_{i}(h)\subset\\{H=h\\}$ with $h\in\tilde{E}_{X}\subset R_{H}$ (recall the
definition of the set $\tilde{E}_{X}$ from Corollary 2.5), and assume that
$Y(x)\not=0$ for $x\in\\{H=h\\}$. Then there exists $k\in\tilde{E}_{Y}$ such
that $C_{i}(h)\subset\\{K=k\\}.$ This crucial property of the level sets of
the Hamiltonians will be clarified even further in the proofs of the upcoming
propositions.

With these considerations at hand, we aim at giving a partition of the space
$\mathbb{R}^{2}$ into sets where the commutativity of flows can be proved
independently and with different ideas, up to $\mathcal{L}^{2}$-negligible
sets. Before starting our analysis, we recall that:

$\phi^{X}_{t}(x)=x,\quad\forall x\in
H^{-1}(S_{H}),\quad\phi^{Y}_{s}(x)=x,\quad\forall x\in K^{-1}(S_{K}).$ (3.5)

Moreover, the RLFs $\phi^{X}_{t}$, $\phi^{Y}_{s}$ are integral solution of the
Cauchy Problem (2.3) for a.e. initial data $x\in A\subset\mathbb{R}^{2}$,
$\mathcal{L}^{2}(A^{c})=0$. For our analysis, we will limit the study to the
set $A$. We recall that in this work the commutativity of flows has to be
intended in the following weak sense: we say that the flows $\phi_{t}^{X}$ and
$\phi_{s}^{Y}$ commute if, for every $s,t$ fixed, there exists $\Omega_{s,t}$
such that $\mathcal{L}^{2}(\Omega_{s,t}^{c})=0$ and the flows commute in
$\Omega_{s,t}$. However, in this section, we rely on a stronger notion of
commutativity, namely we we prove that there exists a set $\Omega$ with
$\mathcal{L}^{2}(\Omega^{c})=0$ such that

$\phi^{X}_{t}\circ\phi^{Y}_{s}=\phi^{Y}_{s}\circ\phi^{X}_{t},\quad\forall
t,s\in\mathbb{R},\forall x\in\Omega.$ (3.6)

We recall the following property of the Structure Theorem 2.4:

$\exists\tilde{E}_{X}\subset H(\mathbb{R}^{2}),\text{ with
}\mathcal{L}^{1}(H(\mathbb{R}^{2})\setminus\tilde{E}_{X})=0:\quad\forall
h\in\tilde{E}_{X},X(x)\not=0,\quad\forall x\in\\{H=h\\}.$

Moreover, for such $h$, $\\{H=h\\}$ is finite union of simple closed curves
$C_{i}(h)$, which are the connected components of the level set $\\{H=h\\}$.

We are now in a position to start our analysis. We partition the space
$\mathbb{R}^{2}$ as follows:

$\mathbb{R}^{2}=\left(\\{X\not=0\\}\cap
E_{X}\right)\cup\left(\\{X\not=0\\}\cap E_{X}^{c}\right)\cup\\{X=0\\},$

where the set $E_{X}=H^{-1}(\tilde{E}_{X})$. We define in the same way
$\tilde{E}_{Y}$ and $E_{Y}$. By Coarea formula, it holds that
$\mathcal{L}^{2}(\\{X\not=0\\}\cap E_{X}^{c})=0$, since

$\int_{\\{X\not=0\\}\cap E_{X}^{c}}|\nabla
H|dx=\int_{\mathbb{R}}\mathcal{H}^{1}(\\{X\not=0\\}\cap E_{X}^{c}\cap
H^{-1}(t))dt.$ (3.7)

The RHS is clearly zero by definition of $E_{X}$ and $|\nabla H|\not=0$ in the
LHS since $\nabla^{\perp}H=X$. Owing to this property, we give the following

######  Corollary 3.7.

The space $\mathbb{R}^{2}$ can be partitioned as follows

$\left[\left(\\{X\not=0\\}\cap E_{X}\right)\cap\left(\\{Y\not=0\\}\cap
E_{Y}\right)\right]\cup\\{X=0\\}\cup\\{Y=0\\}\cup N,$ (3.8)

where $\mathcal{L}^{2}(N)=0$ and $E_{X},E_{Y}$ are the spaces defined above.

###### Proof.

The proof follows easily from set-theoretical considerations and (3.7). ∎

In virtue of the previous corollary, we are left with proving separately the
commutativity of the flows on the two non-negligible sets in (3.8), as done in
the two upcoming propositions.

######  Proposition 3.8.

In the set $\\{X=0\\}\cup\\{Y=0\\}$ the flows commute, in the sense that there
exists $\Omega^{\prime}\subset\\{X=0\\}\cup\\{Y=0\\}$ with
$\mathcal{L}^{2}(\Omega^{\prime c})=0$ such that

$\phi^{X}_{t}\circ\phi^{Y}_{s}=\phi^{Y}_{s}\circ\phi^{X}_{t},\quad\forall
t,s\in\mathbb{R},\forall x\in\Omega^{\prime}.$

###### Proof.

We fix $x\in\\{X=0\\}\cup\\{Y=0\\}$ and w.l.o.g. we can assume $X(x)=0$. If
$Y(x)=0$, then the flows $\phi^{X}_{t},\phi^{Y}_{s}$ clearly commute for every
$t,s$ being the flow maps the identity. Hence, we assume that $X(x)=0$ and
$Y(x)\not=0$. By the same reasoning as in (3.7), we first observe that it
holds that $\mathcal{L}^{2}(\\{x:X(x)=0,Y(x)\not=0,x\in E_{Y}^{c}\\})=0$, by
the monotonicity of the Lebesgue measure.

Thus, we are left with analyzing the case $X(x)=0,Y(x)\not=0$, and $x\in
E_{Y}$. Since $x\in E_{Y}$, by definition of $E_{Y}$, we have that
$Y(y)\not=0$ for every $y\in\\{K=K(x)\\}$. We now denote by $\mathcal{C}_{k}$
the connected component of $\\{K=K(x)\\}$ containing both $x$ and $y$ (we
recall that $\mathcal{C}_{k}$ is a simple closed curve by Theorem 2.4). If
$X(y)=0$ for every $y\in\mathcal{C}_{k}$, then in particular
$X\circ\phi^{Y}_{s}(x)=0$ for every $s\in\mathbb{R}$, thus the commutativity
of flows would follow immediately. Observe that there exists $h\in
H(\mathbb{R}^{2})$ such that $\mathcal{C}_{k}\subset\\{H=h\\},$ since $X\cdot
Y^{\perp}=0$. As a consequence, if there exists $y\in\mathcal{C}_{k}$ such
that $X(y)\not=0$, then it must be that $y\in\\{X\not=0\\}\cap E_{X}^{c}$ (if
not, we would have $X(x)\not=0$ by definition of $E_{X}$). Observe that

$\mathcal{L}^{2}(E)=\mathcal{L}^{2}(\\{y\in H^{-1}(h):\exists
x:X(x)=0,X(y)\not=0,\exists\mathcal{C}_{k}\quad\text{connected component:
}x,y\in\mathcal{C}_{k}\\})=0,$

since $E$ is a subset of $\\{X\not=0\\}\cap E_{X}^{c}$, which we proved to be
$\mathcal{L}^{2}$-negligible. Notice that, since $X$ is continuous and
$X(y)\not=0$, for the connected component $\mathcal{C}_{k}$ satisfying the
previous property, there exists
$\gamma_{k}:[-T_{k},T_{k}]\rightarrow\mathbb{R}^{2}$ with
$\gamma_{k}([-T_{k},T_{k}])\subset\mathcal{C}_{k}$ and $\gamma_{k}(0)=y$ such
that $X(\gamma_{k}(t))\not=0$ for all $t\in[-T_{k},T_{k}]$. In terms of flows,
this means that there exists $q\in\mathbb{Q}$ rational number such that
$X(\phi^{Y}_{q}(x))\not=0$. Therefore

$\\{x:X(x)=0,Y(x)\not=0,x\in
E_{Y}\\}\subset\cup_{q\in\mathbb{Q}}(\phi^{Y}_{q})^{-1}(E).$

Since $Y$ is incompressible and $\mathcal{L}^{2}(E)=0$, this immediately gives
that $\mathcal{L}^{2}(\\{x:X(x)=0,Y(x)\not=0,x\in E_{Y}\\})=0$, which
concludes the proof.

∎

######  Remark 3.9.

Since the previous proof only relies on geometric considerations on level sets
and does not use the regularity of the vector fields $X,Y$, it is in
particular valid also in the $\operatorname{BV}$ setting. However, we remark
that in the case of Sobolev regularity, the previous result is an easy
corollary of Gronwall’s lemma. Indeed, we have

$\displaystyle\frac{d}{ds}|X\circ\phi^{Y}_{s}|^{2}={2}\langle
X\circ\phi^{Y}_{s},DX(\phi^{Y}_{s})Y\circ\phi^{Y}_{s}\rangle={2}\langle
X\circ\phi^{Y}_{s},DY(\phi^{Y}_{s})X\circ\phi^{Y}_{s}\rangle\leq{2}|X\circ\phi^{Y}_{s}|^{2}|DY(\phi^{Y}_{s})|,$

where the second inequality follows from $[X,Y]=0$. In particular, the
previous inequality and Gronwall’s lemma imply that $X(\phi^{Y}_{s}(x))=0$ if
$X(x)=0$ and one concludes the same result as before. We also remark that the
previous inequality holds also for vector fields with bounded divergence, so
that the vector field remains zero on the trajectories of the second vector
field.

In order to prove Proposition 3.1, we are now left with proving the following
result.

######  Proposition 3.10.

In the set $\left(\\{X\not=0\\}\cap E_{X}\right)\cap\left(\\{Y\not=0\\}\cap
E_{Y}\right)$ the two flows commute, in the sense of (3.6).

###### Proof.

We fix a point $x\in\left(\\{X\not=0\\}\cap
E_{X}\right)\cap\left(\\{Y\not=0\\}\cap E_{Y}\right)$, then, by definition of
the two sets, there exist $c_{H(x)}>0$, $c_{K(x)}>0$ such that $|X(y)|\geq
c_{H(x)}>0$ for every $y\in\\{H=H(x)\\}$ and $|Y(y)|\geq c_{K(x)}>0$ for every
$y\in\\{K=K(x)\\}$. In particular, there exists a simple closed curve $C$ with
$\phi^{X}_{t}(x),\phi^{Y}_{s}(x)\in C$ for every $t,s\in\mathbb{R}$ and
therefore we can restrict ourselves to study the commutativity of the flows
along each curve $C$. We observe that, since the flows of the two vector
fields live in the same curve, we can define a function $\tau=\tau(t,y)$ time-
periodic satisfying

$\phi^{X}_{t}(y)=\phi^{Y}_{\tau(t,y)}(y),\quad\forall y\in C.$

In particular,

$\phi^{X}_{t}\circ\phi^{Y}_{s}(y)=\phi^{Y}_{\tau(t,\phi^{Y}_{s}(y))}\circ\phi^{Y}_{s}(y)=\phi^{Y}_{\tau(t,\phi^{Y}_{s}(y))+s}(y)\quad\forall
s,t\in\mathbb{R},\forall y\in C,$

while

$\phi^{Y}_{s}\circ\phi^{X}_{t}(y)=\phi^{Y}_{s+\tau(t,y)}(y)\quad\forall
s,t\in\mathbb{R},\forall y\in C,$

which follow by the semigroup properties of the flows, being the two vector
fields autonomous. We claim that $\tau\circ\phi^{Y}_{s}=\text{const}$, which
gives a proof of the commutativity of the two flows along each curve $C$.
Observe that, since $X\cdot Y^{\perp}=0$, in a $\epsilon$-neighbourhood
$C^{\epsilon}$ of $C$ it holds that

$X(x)=\alpha_{\epsilon}(x)Y(x),$

where $\alpha_{\epsilon}\doteq\frac{X\cdot Y}{|Y|^{2}}$, defined for every
$x\in C_{\epsilon}$, is continuous, bounded and $\alpha_{\epsilon}\geq c>0.$
Being $X,Y\in W^{1,p}_{\text{loc}}(C^{\epsilon})$, also $\alpha_{\epsilon}$
inherits the same regularity. In particular, since
$\operatorname{div}(\alpha_{\epsilon}Y)=\operatorname{div}X=0$, we get that
$\nabla\alpha_{\epsilon}\cdot Y=0$ a.e. and

$\frac{d}{ds}\alpha_{\epsilon}\circ\phi^{Y}_{s}(x)=(\nabla\alpha_{\epsilon}\cdot
Y)\circ\phi^{Y}_{s}=0.$

In particular, thanks to the continuity assumption, the map
$\alpha_{\epsilon}$ is constant along the trajectories of the vector field $X$
(up to a set of measure zero). Then the function
$\tau(t,y)=\alpha_{\epsilon}(y){t}$ is constant along $\phi^{Y}_{s}$ and by
Formula 2.7 we find, for every $y\in C$,

$\displaystyle
t=N\int_{0}^{\ell(C)}\frac{1}{|X(\gamma(s))|}ds+\int_{\gamma^{-1}(y)}^{\bar{\tau}}\frac{1}{|X(\gamma(s))|}ds$
$\displaystyle=\frac{1}{\alpha_{\epsilon}}\left({N}\int_{0}^{\ell(C)}\frac{1}{|Y(\gamma(s))|}ds+\int_{\gamma^{-1}(y)}^{\bar{\tau}}\frac{1}{|Y(\gamma(s))|}ds\right)=\tau(t,y),$

where $\gamma(s)$ is the Lipschitz parametrization of the curve $C$ given by
Theorem 2.4 and $\gamma(\bar{\tau})=\phi^{X}_{t}(y)=\phi^{Y}_{\tau(t,y)}(y)$.
∎

##  4 Nearly incompressible vector fields

Here we extend the ideas of the previous section to the more general setting
of continuous vector fields $X,Y\in W^{1,p}(\mathbb{R}^{2},\mathbb{R}^{2})$
with bounded divergence. We recall that, thanks to the boundedness of the
divergence, we can assume that the two vector fields $X,Y$ are nearly
incompressible (see, for instance, [6]) in the sense of Definition 1.4. We
rewrite the equation defining the Hamiltonian vector field $W$, which follows
by Equation (3.4) with $[X,Y]=0$:

$W\doteq\operatorname{div}YX-\operatorname{div}XY\in L^{\infty}.$

Clearly, it follows immediately from (3.4) that $W$ is a Hamiltonian vector
field, with Hamiltonian given by the function

$H_{W}=X\cdot Y^{\perp}.$

In particular, being $W\in L^{\infty}$, the scalar product $X\cdot Y^{\perp}$
has a Lipschitz continuous representative. We start with the following

######  Proposition 4.1.

Let $X,Y\in W^{1,p}(\mathbb{R}^{2},\mathbb{R}^{2})\cap
L^{\infty}(\mathbb{R}^{2},\mathbb{R}^{2})$ with bounded divergence such that
$[X,Y]=0$. Then

$X\cdot Y^{\perp}\circ\phi^{X}_{t}=\xi(t)X\cdot Y^{\perp},$ (4.1)

(and the same for the flow $\phi^{Y}_{s}$), where $\xi$ denotes the density of
$(\phi^{X}_{t})_{\sharp}\mathcal{L}^{2}$ w.r.t. the Lebesgue measure (see
Subsection 2.2). In particular, if $X\cdot Y^{\perp}\geq 0$, then we have

$\frac{1}{C}X\cdot Y^{\perp}\leq X\cdot Y^{\perp}\circ\phi^{X}_{t}\leq CX\cdot
Y^{\perp}\quad\text{a.e.,}$ (4.2)

where $C$ is the constant in (1.8).

###### Proof.

We derive the function $X\cdot Y^{\perp}\circ\phi^{X}_{t}$ and we find that

$\frac{d}{dt}X\cdot Y^{\perp}\circ\phi^{X}_{t}=(\nabla(X\cdot Y^{\perp})\cdot
X)\circ\phi^{X}_{t}=(\nabla^{\perp}(X\cdot Y^{\perp})\cdot
X^{\perp})\circ\phi^{X}_{t}=\operatorname{div}X\circ\phi^{X}_{t}(X\cdot
Y^{\perp})\circ\phi^{X}_{t},$

where we have used Equation (3.4). On the other side, by Liouville Theorem
(see the discussion in the Preliminaries 2)

$\partial_{t}\xi=\operatorname{div}(X)\circ\phi^{X}_{t}\xi.$

Therefore the equality follows, since the two functions satisfy the same ODE
(with the same initial data). Inequality (4.2) is a direct consequence of
Equation (1.8). ∎

######  Remark 4.2.

We point out that these computations hold also in the $\operatorname{BV}$
setting.

For every $\kappa\in\mathbb{R}\setminus\\{0\\}$, we define the sets
$\Omega_{\kappa}=H_{W}^{-1}\left({C}^{-1}\kappa,C\kappa\right),$ if $\kappa>0$
and $\Omega_{\kappa}=H_{W}^{-1}\left({C}\kappa,C^{-1}\kappa\right),$ if
$\kappa<0$. Clearly, $\Omega_{\kappa}$ is the union of countably many
connected components. In the sequel, when we will refer to $\Omega_{\kappa}$,
for simplicity we will refer to each single connected component. By inequality
(4.2), we have that, if $X\cdot Y^{\perp}(x)=\kappa$, then
$\phi^{X}_{t}(x)\in\overline{\Omega}_{\kappa}$.

We are now in a position to prove the main proposition of this section, that
is

######  Proposition 4.3.

Let $X,Y\in L^{\infty}(\mathbb{R}^{2},\mathbb{R}^{2})\cap
W^{1,p}(\mathbb{R}^{2},\mathbb{R}^{2})$ with bounded divergence such that
$[X,Y]=0$. Then for every $\kappa\in\mathbb{R}\setminus\\{{0}\\}$, the vector
fields $X\llcorner_{\Omega_{\kappa}},Y\llcorner_{\Omega_{\kappa}},$ are steady
nearly incompressible. Moreover, they have the same Lipschitz and bounded
density $\rho_{\kappa}:\Omega_{\kappa}\rightarrow\mathbb{R}$ given by the
function

$\rho_{\kappa}=\frac{1}{H_{W}},$

that is

$\operatorname{div}(\rho_{\kappa}(X-Y))=0,\quad\text{in
}\mathcal{D}^{\prime}(\Omega_{\kappa}).$

###### Proof.

Since we are in the set $\Omega_{\kappa}$, the function $\rho_{\kappa}$ of the
statement is well-defined, bounded (from above and below) and Lipschitz, being
$X\cdot Y^{\perp}$ Lipschitz. In particular, by Equation (3.4) with $[X,Y]=0$,
it holds

$\nabla^{\perp}(X\cdot Y^{\perp})\cdot Y^{\perp}=\operatorname{div}Y(X\cdot
Y^{\perp}),$

and therefore

$\displaystyle\operatorname{div}(\rho_{\kappa}Y)$
$\displaystyle=\operatorname{div}\left(\frac{1}{H_{W}}Y\right)$
$\displaystyle=\frac{1}{H_{W}}\operatorname{div}Y-\frac{1}{H_{W}^{2}}\nabla\left(X\cdot
Y^{\perp}\right)\cdot Y$
$\displaystyle=\frac{1}{H_{W}}\operatorname{div}Y-\frac{1}{H_{W}^{2}}\nabla^{\perp}\left(X\cdot
Y^{\perp}\right)\cdot Y^{\perp}$
$\displaystyle=\frac{1}{H_{W}}\operatorname{div}Y-\frac{1}{H_{W}^{2}}\operatorname{div}YH_{W}=0.$

The same computation holds for the vector field $X$, being

$\nabla^{\perp}(X\cdot Y^{\perp})\cdot X^{\perp}=-\operatorname{div}X(Y\cdot
X^{\perp})=\operatorname{div}X(X\cdot Y^{\perp}).$

∎

We define now the set

$\Omega=\cup_{\kappa\in\mathbb{Q}\setminus\\{0\\}}\Omega_{\kappa},$

which is an open set such that $\Omega^{c}=\\{H_{W}=0\\}$. Observe that,
thanks again to inequality (4.2), if $x\in\Omega$, then
$\phi^{X}_{t}(x)\in\Omega$ (and the same for the flow of the vector field
$Y$), so that the set $\Omega$ is an invariant set for the two flows.

Taking advantage of the previous results, we can easily adapt the ideas of
[15] to prove the following

######  Proposition 4.4.

Let $X,Y$ be vector fields satisfying the hypotheses of the two previous
propositions. Moreover, we assume that they are continuous. Then, for every
$t,s$ the RLFs of $X,Y$ have representative $\phi^{X}_{t},\phi^{Y}_{s}\in
W^{1,p}(\Omega)\cap\mathcal{C}^{0}(\Omega).$

###### Proof.

We fix $\kappa\in\mathbb{R}\setminus\\{0\\}$ and we consider $\rho_{\kappa}$
as in the statement of Proposition 4.3. Then, as $\rho_{\kappa}$ is Lipschitz
and bounded, the vector field $\rho_{\kappa}X$ belongs to the space
$W^{1,p}_{\text{loc}}(\mathbb{R}^{2},\mathbb{R}^{2})$ and is continuous on the
set $\Omega_{\kappa}$. By Diperna-Lions Theory there exists a unique RLF

$\phi^{\kappa}_{t}\doteq\phi_{t}^{\rho_{\kappa}X}$

solving the Cauchy Problem

$\begin{cases}\partial_{t}\phi^{\kappa}_{t}(x)=(\rho_{\kappa}X)(\phi^{\kappa}_{t}(x)),\\\
\phi^{\kappa}_{t=0}(x)=x.\end{cases}$

By [15], $\phi^{\kappa}_{t}\in W^{1,p}_{\text{loc}}(\Omega_{\kappa})$. Indeed,
we first consider a countable cover of $\Omega_{\kappa}$ by open balls, namely
$\Omega_{\kappa}\subset\cup_{j}B_{r_{j}}(x_{j})$, where $r_{j}>0$ and
$x_{j}\in\Omega_{\kappa}$. The countable cover can be chosen in such a way
that $X\cdot Y^{\perp}\in(C^{-1}\kappa-\delta,C\kappa+\delta),$ for some
$\delta>0$ sufficiently small, thanks to the continuity of the two vector
fields. Finally, we denote by $\Omega^{\delta}_{\kappa}$ the set
$H_{W}^{-1}((C^{-1}\kappa-\delta,C\kappa+\delta))$. To prove the assertion we
just need to show that $\phi^{\kappa}_{t}\in W^{1,p}(B_{r_{j}}(x_{j}))$. This
clearly holds true, since in each ball $B_{r_{j}}(x_{j})$ there exists a
Hamiltonian $H_{j}$ such that $\rho_{\kappa}X=\nabla^{\perp}H_{j}$. We then
denote
$\tilde{\Omega}_{n}=H_{\kappa}^{-1}(\\{h:\min_{H_{j}^{-1}(h)}|\rho_{\kappa}X|>\frac{1}{n}\\}).$
In particular, since $Y$ is bounded, we have

$\bar{B}_{r_{j}}(x_{j})\subset\tilde{\Omega}_{n}.$

The sets $\tilde{\Omega}_{n}$ are those defined in [15] (Section 3), where the
flow has been proved to be Sobolev and therefore we have proved that
$\phi^{\kappa}_{t}\in W^{1,p}(B_{r_{j}}(x_{j}))$. We now claim that, if we
define $\tau_{\kappa}(t,x)\doteq t\rho_{k}(x)$ (which is a Lipschitz function
on $\Omega^{\delta}_{\kappa}$), we have that

$\phi^{X}_{\tau(t,x)}=\phi^{\kappa}_{t},$ (4.3)

(where $\phi^{X}_{t}$ is the unique RLF of $X$), since

$\dot{\phi}^{X}_{\tau(t,x)}=X(\phi^{X}_{\tau(t,x)})\dot{\tau}(t,x)_{x=\phi^{X}_{\tau(t,x)}}=\rho_{\kappa}(\phi^{X}_{\tau(t,x)})X(\phi^{X}_{\tau(t,x)}),$

so that by uniqueness of RLFs the previous identity (Equation (4.3)) holds.
This in particular gives that $\phi^{X}_{t}\in
W^{1,p}_{\text{loc}}(\Omega_{\kappa}),$ and thus $\phi^{X}_{t}\in
W^{1,p}_{\text{loc}}(\Omega).$ ∎

We are finally ready to prove our main theorem:

######  Theorem 4.5.

Let $X,Y\in W^{1,p}(\mathbb{R}^{2},\mathbb{R}^{2})\cap
C^{0}(\mathbb{R}^{2},\mathbb{R}^{2})$ be two bounded vector fields with
$\operatorname{div}X,\operatorname{div}Y\in
L^{\infty}(\mathbb{R}^{2},\mathbb{R}^{2}).$ Then it holds

$\qquad[X,Y]=0\iff\phi_{t}^{X}\circ\phi_{s}^{Y}=\phi_{s}^{Y}\circ\phi_{t}^{X},\quad\forall
t,s\in\mathbb{R},\quad\text{for a.e. in }\mathbb{R}^{2}.$

###### Proof of Theorem 4.5.

Since $\phi^{X}_{t},\phi^{Y}_{s}\in W^{1,p}(\Omega)$, by the result in [11] we
have the commutativity of the two flows in that set, being invariant for the
flows (Proposition 1.5). In particular, if $\mathcal{L}^{2}(\Omega^{c})=0$,
the proof is concluded.

If not, we observe that $W=0$ a.e. in the set $\Omega^{c}$, (since the
gradient of Sobolev functions is strongly local), which in particular implies
that

$\operatorname{div}XY=\operatorname{div}YX\quad\text{a.e. in }\Omega^{c},$
(4.4)

We prove now the commutativity of flows in the open set $\\{X=0\\}^{c}$ (while
in the set $\\{X=0\\}$ it is a consequence of Gronwall’s lemma as done in
Remark 3.9). The condition $X\cdot Y^{\perp}=0$ implies that in the set
$\\{X=0\\}^{c}$ there exists a continuous function $\alpha$ with $Y=\alpha X$
and $\alpha\in W^{1,p}_{\text{loc}}(\\{X=0\\}^{c})$ defined as:

$\alpha(x)=\frac{X\cdot Y}{|X|^{2}}.$

Moreover, equation (4.4), implies that $(\nabla\alpha\cdot X)X=0$, which in
turn yields $\nabla\alpha\cdot X=0$ a.e. in the set
$\\{Y=0\\}^{c}\cup\\{X=0\\}^{c}$.

Always by a Gronwall-type argument, condition $\nabla\alpha\cdot X=0$,
together with the continuity of the function $\alpha$, implies that the the
function $\alpha$ is constant along the trajectories of the vector field $X$
(up to a set of measure zero). Finally we observe that, if $t$ is fixed, then

$\displaystyle\phi^{X}_{t}(x)$
$\displaystyle=x+\int_{0}^{t}X(\phi^{X}_{\tau}(x))d\tau=x+\frac{1}{\alpha}\int_{0}^{t}\alpha
X(\phi^{X}_{\tau}(x))d\tau=x+\frac{1}{\alpha}\int_{0}^{t}Y(\phi^{X}_{\tau}(x))d\tau$
$\displaystyle=x+\int_{0}^{\frac{t}{\alpha}}Y(\phi^{X}_{\alpha\tau}(x))d\tau,$

which yields $\phi^{X}_{t}=\phi^{Y}_{\frac{t}{\alpha}}$. The commutativity
follows by the same argument of the proof of Proposition 3.10. ∎

##  5 Some remarks on BV vector fields

This section is devoted to the proof of Proposition 1.10, which we carry out
by adapting the one in [11] to our case. We first briefly discuss the
definition of Lie bracket we rely on in this work and we show that this
definition is still suitable if we consider $\operatorname{BV}$ vector fields.
More precisely, we point out that, if we identify $X,Y$ with their precise
representative, the Lie bracket defined in 1.9 reduces to the distributional
one given in [11], namely

$[X,Y]\doteq DYX-DXY,\quad\text{distributionally}.$

Indeed, by the Leibniz rule (2.1), we can rewrite the first term in (1.9) as
follows

$(X,D\varphi
Y)=\sum_{i,j=1}^{n}\partial_{j}\left(X^{i}\varphi^{i}Y^{j}\right)-(X^{*},\varphi)\operatorname{div}Y-(DXY^{*},\varphi),$

where $\varphi$ is a smooth test function. Integrating the previous expression
and keeping in mind that $X=X^{*}$ $\mathcal{L}^{n}$-a.e., we obtain

$\int_{\mathbb{R}^{n}}(X,D\varphi
Y)dz=-\int_{\mathbb{R}^{n}}(X,\varphi)\operatorname{div}Ydz-\sum_{i,j=1}^{n}\int_{\mathbb{R}^{n}}\varphi^{i}(Y^{*})^{j}dD_{j}X^{i}.$

Similarly, we have

$\int_{\mathbb{R}^{n}}(Y,D\varphi
X)dz=-\int_{\mathbb{R}^{n}}(Y,\varphi)\operatorname{div}Xdz-\sum_{i,j=1}^{n}\int_{\mathbb{R}^{n}}\varphi^{i}(X^{*})^{j}dD_{j}Y^{i},$

and therefore

$\int_{\mathbb{R}^{n}}([X,Y],\varphi)dz=\sum_{i,j=1}^{n}\int_{\mathbb{R}^{n}}\varphi^{i}(X^{*})^{j}dD_{j}Y^{i}-\sum_{i,j=1}^{n}\int_{\mathbb{R}^{n}}\varphi^{i}(Y^{*})^{j}dD_{j}X^{i}.$

For this reason, in the following, we will always indentify $X$ and $Y$ with
their precise representatives.

We finally remark that, in the smooth setting, the commutativity of the flows
$\phi_{t}^{X}$, $\phi_{s}^{Y}$ is equivalent to the following property

$D\phi_{t}^{X}(z)Y(z)=Y(\phi_{t}^{X}(z)),\quad\forall
z\in\mathbb{R}^{n},t\in\mathbb{R},$ (5.1)

where the equality is obtained deriving in time both left and right hand side
and observing that, thanks to the assumption $[X,Y]=0$, they satisfy the same
differential equation with the same initial data. In the $\operatorname{BV}$
setting, it is not at all clear the meaning that should be given to this
equivalence. As in [11], Equation (5.1) motivates the definition of the
following distribution, already introduced in [11]:

$T_{t}[X,Y](\varphi)\doteq\int_{\mathbb{R}^{n}}(Y(\phi_{t}^{X}),\varphi)+(\phi_{t}^{X},D\varphi
Y)+\operatorname{div}Y(\varphi,\phi_{t}^{X})dz,\quad\varphi\in
C_{c}^{\infty}(\mathbb{R}^{n},\mathbb{R}^{n})$ (5.2)

which clearly equals $Y(\phi_{t}^{X}(z))-D\phi_{t}^{X}(z)Y(z)$ if $X$ and $Y$
are smooth vector fields. We consider also the following family of
distributions as in [11]

$T_{t,s}[X,Y](\varphi)\doteq\int_{\mathbb{R}^{n}}(Y(\phi_{t}^{X}\circ\phi_{s}^{Y}),\varphi)+(\phi_{t}^{X}\circ\phi_{s}^{Y},D\varphi
Y)+\operatorname{div}Y(\varphi,\phi_{t}^{X}\circ\phi_{s}^{Y})dz,\quad\varphi\in
C_{c}^{\infty}(\mathbb{R}^{n},\mathbb{R}^{n})$ (5.3)

which represents the function
$Y(\phi_{t}^{X}\circ\phi_{s}^{Y})(z)-D(\phi_{t}^{X}\circ\phi_{s}^{Y})(z)Y(z)$
in the smooth setting. The distributions defined in (5.2) and (5.3) play a
crucial role in our analysis since they serve as an intermediate step in
proving the following proposition.

######  Proposition 5.1.

Then the following hold:

$\phi^{X}_{t}\circ\phi^{Y}_{s}=\phi^{Y}_{s}\circ\phi^{X}_{t},\quad\forall
t,s\in\mathbb{R}\text{ and for a.e. }x\in\mathbb{R}^{n}\implies
T_{t,s}=0\quad\forall t,s\in\mathbb{R}\implies T_{t}=0\quad\forall
t\in\mathbb{R}.$ (5.4)

In addition, $T_{t}=0$ for every $t\in\mathbb{R}$ implies $[X,Y]=0$.

###### Proof.

This part of the proof is identical to the one in [11] and we rewrite it for
completeness. The only changes are in the proof of the last implication, where
we make use of the definition of distributional Lie Bracket. We first point
out that, if $T_{t,s}=0$ for every $t,s\in\mathbb{R}$, then it is sufficient
to evaluate $T_{t,s}$ at $s=0$ to obtain $T_{t}=0$ for every $t\in\mathbb{R}$.

Proof of $\phi_{t}^{X}\circ\phi_{s}^{Y}=\phi_{s}^{Y}\circ\phi_{t}^{X},\forall
t,s\,\implies\,T_{t,s}=0,\forall t,s$. As a first step, we aim at proving that

$\phi_{t}^{X}\circ\phi_{s}^{Y}=\phi_{s}^{Y}\circ\phi_{t}^{X},\forall
t,s\,\implies\,\partial_{s}(\phi_{t}^{X}\circ\phi_{s}^{Y})=Y(\phi_{t}^{X}\circ\phi_{s}^{Y}),\,\text{for
a.e. $s\in\mathbb{R}$}.$ (5.5)

Indeed, if
$\phi_{t}^{X}\circ\phi_{s}^{Y}=\phi_{s}^{Y}\circ\phi_{t}^{X},\forall t,s$,
then $s\mapsto\phi_{t}^{X}\circ\phi_{s}^{Y}$ is an absolutely continuous
function which satisfies

$\partial_{s}(\phi_{t}^{X}\circ\phi_{s}^{Y})=\partial_{s}(\phi_{s}^{Y}\circ\phi_{t}^{X})=Y(\phi_{s}^{Y}\circ\phi_{t}^{X})=Y(\phi_{t}^{X}\circ\phi_{s}^{Y}),\quad\text{for
a.e. $s\in\mathbb{R}$}.$

We use this equivalence (5.5) to prove the desired claim. We first observe
that equation
$\partial_{s}(\phi_{t}^{X}\circ\phi_{s}^{Y})=Y(\phi_{t}^{X}\circ\phi_{s}^{Y})$
needs to be understood in the following sense

$-\int_{\mathbb{R}^{n+1}}\left(\partial_{s}\varphi(s,z),\phi_{t}^{X}\circ\phi_{s}^{Y}\right)\,ds\,dz=\int_{\mathbb{R}^{n+1}}\left(Y(\phi_{t}^{X}\circ\phi_{s}^{Y}),\varphi(s,z)\right)\,ds\,dz,\quad\forall\varphi\in
C_{c}^{\infty}(\mathbb{R}^{n+1}).$

We now perform the change of variable $\tilde{z}=\phi_{s}^{Y}(z)$ and rewrite
the previous equation as follows

$\displaystyle-\int_{\mathbb{R}^{n+1}}\left(\partial_{s}\varphi(s,\phi_{-s}^{Y}(\tilde{z})),\phi_{t}^{X}(\tilde{z})\right)\,\xi(-s,\tilde{z})\,ds\,d\tilde{z}=\int_{\mathbb{R}^{n+1}}\left(Y(\phi_{t}^{X}\circ\phi_{s}^{Y}),\varphi(s,z)\right)\,ds\,dz,$
(5.6)

We now exploit the change of variable formula (2.5) to compute

$\int_{\mathbb{R}^{n+1}}\left(\partial_{s}\left(\varphi(s,\phi_{-s}^{Y}(\tilde{z}))\right),\phi_{t}^{X}(\tilde{z})\right)\,\xi(-s,\tilde{z})\,ds\,d\tilde{z}=\int_{\mathbb{R}^{n+1}}\left(\varphi(s,\phi_{-s}^{Y}(\tilde{z})),\phi_{t}^{X}(\tilde{z})\right)\operatorname{div}Y\circ\phi_{-s}^{Y}\,\xi(-s,\tilde{z})\,ds\,.$

Taking advantage of the previous equation, together with

$\partial_{s}\left(\varphi(s,\phi_{-s}^{Y}(\tilde{z}))\right)=\partial_{s}\varphi(s,\phi_{-s}^{Y}(\tilde{z}))-D\varphi(s,\phi_{-s}^{Y}(\tilde{z}))Y(\phi_{-s}^{Y}(\tilde{z})),$

we can rewrite (5.6) as follows

$\displaystyle-\int_{\mathbb{R}^{n+1}}\left(D\varphi(s,\phi_{-s}^{Y}(\tilde{z}))Y\left(\phi_{-s}^{Y}(\tilde{z})\right),\phi_{t}^{X}(\tilde{z})\right)\,\xi(-s,\tilde{z})\,ds\,d\tilde{z}$
$\displaystyle\qquad-\int_{\mathbb{R}^{n+1}}\left(\varphi(s,\phi_{-s}^{Y}(\tilde{z})),\phi_{t}^{X}(\tilde{z})\right)\operatorname{div}Y\circ\phi_{-s}^{Y}\,\xi(-s,\tilde{z})\,ds\,d\tilde{z}=\int_{\mathbb{R}^{n+1}}\left(Y(\phi_{t}^{X}\circ\phi_{s}^{Y}),\varphi(s,z)\right)\,ds\,dz.$

Performing back the change of variable, we obtain

$\displaystyle-\int_{\mathbb{R}^{n+1}}\left(D\varphi(s,z)Y,\phi_{t}^{X}\circ\phi_{s}^{Y}\right)\,ds\,dz$
$\displaystyle\qquad-\int_{\mathbb{R}^{n+1}}\left(\varphi(s,z),\phi_{t}^{X}\circ\phi_{s}^{Y}\right)\operatorname{div}Y\,ds\,dz=\int_{\mathbb{R}^{n+1}}\left(Y(\phi_{t}^{X}\circ\phi_{s}^{Y}),\varphi(s,z)\right)\,ds\,dz.$

If we now choose test functions of the form $\varphi=\alpha(s)\beta(z)$, the
previous equality is equivalent to state that for every $\beta\in
C_{c}^{\infty}(\mathbb{R})$, $t\in\mathbb{R}$ and for a.e. $s\in\mathbb{R}$,
there holds

$\displaystyle-\int_{\mathbb{R}^{n}}\left(D\beta(z)Y,\phi_{t}^{X}\circ\phi_{s}^{Y}\right)\,dz-\int_{\mathbb{R}^{n}}\left(\beta(z),\phi_{t}^{X}\circ\phi_{s}^{Y}\right)\operatorname{div}Y\,dz=\int_{\mathbb{R}^{n}}\left(Y(\phi_{t}^{X}\circ\phi_{s}^{Y}),\beta(z)\right)\,dz.$
(5.7)

We finally prove that (5.7) holds true for every $s\in\mathbb{R}$. To this
end, we first observe that the terms that do not involve $\operatorname{div}Y$
are absolutely continuous in $s$, as can be seen performing again the change
of variable $\tilde{z}=\phi_{s}^{Y}(z)$ and using the dominated convergence
theorem. At the same time, the term involving $\operatorname{div}Y$ in
equality (5.7) is also continuous in $s$, as, for every $\epsilon>0$, we can
rewrite it as

$\displaystyle-\int_{\mathbb{R}^{n}}\left(\beta(z),\phi_{t}^{X}\circ\phi_{s}^{Y}\right)\operatorname{div}Y\,dz=$
$\displaystyle\qquad-\int_{\mathbb{R}^{n}}\left(\beta(z),\left(\phi_{t}^{X}\ast\rho_{\epsilon}\right)\circ\phi_{s}^{Y}\right)\operatorname{div}Y\,dz-\int_{\mathbb{R}^{n}}\left(\beta(z),\left(\phi_{t}^{X}-\phi_{t}^{X}\ast\rho_{\epsilon}\right)\circ\phi_{s}^{Y}\right)\operatorname{div}Y\,dz,$

where $\rho_{\epsilon}$ denotes, as above, a mollifier. We now point out that
the first term in the previous equality is continuous in $s$, while the second
term converges to zero (as $\epsilon\to 0$) locally uniformly in $s$. Thus,
equality (5.7) holds true for every $s\in\mathbb{R}$. We eventually conclude
observing that (5.7) is equivalent to $T_{t,s}\equiv 0$ for every
$t,s\in\mathbb{R}$.

Proof of $T_{t}=0,\forall t\,\Rightarrow\,[X,Y]=0.$ We aim at showing that

$\frac{d}{dt}T_{t}[X,Y]\Big{|}_{t=0}=[X,Y],$ (5.8)

which clearly implies the desired implication. To this end, we first observe
that, performing the change of variable $\tilde{z}=\phi_{t}^{X}(z)$ and
exploiting again (2.5), we can write

$\displaystyle\frac{d}{dt}\int_{\mathbb{R}^{n+1}}(Y(\phi_{t}^{X}(z)),\varphi(s,z))\,dz\,ds=$
$\displaystyle-\int_{\mathbb{R}^{n+1}}(Y(\tilde{z}),D\varphi(s,\phi_{-t}^{X}(\tilde{z}))X(\phi_{-t}^{X}(\tilde{z})))\,\xi(-t,\tilde{z})\,d\tilde{z}\,ds$
$\displaystyle\quad-\int_{\mathbb{R}^{n+1}}(Y(\tilde{z}),\varphi(s,\phi_{-t}^{X}(\tilde{z})))\operatorname{div}X(\phi_{-t}^{X}(\tilde{z}))\,\xi(-t,\tilde{z})\,d\tilde{z}\,ds.$

for every $\varphi\in C_{c}^{\infty}(\mathbb{R}^{n+1},\mathbb{R}^{n})$.
Performing back the change of variable in the previous integrals, we infer
that, for every $\varphi\in
C_{c}^{\infty}(\mathbb{R}^{n+1},\mathbb{R}^{n+1})$, there holds

$\displaystyle\frac{d}{dt}\int_{\mathbb{R}^{n+1}}(Y(\phi_{t}^{X}),\varphi)dzds=$
$\displaystyle-\int_{\mathbb{R}^{n+1}}(Y(\phi_{t}^{X}(z)),D\varphi(z)X(z))dzds-\int_{\mathbb{R}^{n+1}}(Y(\phi_{t}^{X}(z)),\varphi(z))\operatorname{div}Xdzds.$
(5.9)

In particular, with the choice of functions $\varphi(s,z)=\alpha(s)\beta(z)$
as done in [11], for every
$\beta\in\mathcal{C}^{\infty}_{c}(\mathbb{R}^{n},\mathbb{R}^{n})$, it holds
that

$\displaystyle\frac{d}{dt}\int_{\mathbb{R}^{n}}(Y(\phi_{t}^{X}),\beta)dz=$
$\displaystyle-\int_{\mathbb{R}^{n}}(Y(\phi_{t}^{X}(z)),D\beta(z)X(z))dz-\int_{\mathbb{R}^{n}}(Y(\phi_{t}^{X}(z)),\beta(z))\operatorname{div}Xdz.$
(5.10)

We next observe that, for any test function $\varphi\in
C^{\infty}_{c}(\mathbb{R}^{n},\mathbb{R}^{n})$, we can differentiate (5.2)
under the integral sign with respect to $t$ and exploit (5.10) to obtain

$\begin{split}\frac{d}{dt}T_{t}[X,Y](\varphi)=-\int_{\mathbb{R}^{n}}(Y(\phi_{t}^{X}),D\varphi
X)dz-\int_{\mathbb{R}^{n}}(Y(\phi_{t}^{X}),\varphi)\operatorname{div}Xdz\\\
+\int_{\mathbb{R}^{n}}(X(\phi_{t}^{X}),D\varphi
Y)dz+\int_{\mathbb{R}^{n}}\operatorname{div}Y(\varphi,X(\phi_{t}^{X}))dz.\end{split}$
(5.11)

Since $T_{t}[X,Y]=0$ for every $t$, Claim (5.8) then follows by evaluating
(5.11) at $t=0$ and by keeping in mind our definition of Lie bracket in (1.9).
This concludes the proof of Proposition 5.1. ∎

We conclude by pointing out that the expression (5.11) should encode, in our
opinion, the distributional version of the function $[X,Y]\circ\phi^{X}_{t}$,
that we leave as an open question.

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