# Well-balanced lattice Boltzmann equation for two-phase flows

Zhaoli Guo<EMAIL_ADDRESS>State Key Laboratory of Coal Combustion,
Huazhong University of Science and Technology, Wuhan 430074, China

###### Abstract

The standard lattice Boltzmann equation (LBE) method usually fails to capture
the physical equilibrium state of a two-phase fluid system, i.e., zero
velocity and constant chemical potential. Consequently, spurious velocities
and inconsistent thermodynamic density properties are frequently encountered
in LBE simulations. In this work, based on a rigorous analysis of the discrete
balance equation of LBE, we identify the structure of the force imbalance due
to discretization errors from different parts. Then a well-balanced LBE is
proposed which can achieve the discrete equilibrium state. The well-balanced
properties of the LBE are confirmed by some numerical tests of a flat
interface problem and a droplet system.

###### pacs:

47.11.-j,47.11.St,47.45.-n,47.61.-k

##  I Introduction

It is a challenging problem for modelling and simulating multi-phase fluid
flows due to the complex interfacial dynamics involving large space and time
scales. Over the past three decades, the lattice Boltzmann equation (LBE)
based on kinetic models has been developed as an efficient method for multi-
phase flows ref:SucciBook ; ref:GuoBook . The kinetic nature of LBE makes it
straightforward to incorporate the intermolecular interactions into the model,
which is recognized as one of the main advantages of LBE that distinguishes it
from other numerical methods. However, the LBE also faces some undesirable
features in simulating multiphase systems. One fundamental problem is that it
can be quite inaccurate at equilibrium state, such that the numerical results
are not compatible with exact physical conditions.

Physically, the chemical potential $\mu$ should be constant and the fluid
velocity ${\bm{u}}$ should be zero everywhere for a two-phase system at
equilibrium, namely it should maintain the equilibrium state

${\cal S}\equiv\\{\mu=\mbox{constant},{\bm{u}}=0\\},$ (1)

from which other state properties, such as interface structure, bulk phase
densities, and pressure distribution, can be deduced. However, there are
numerous examples in the literature that the LBE yields nonzero spurious
velocity (SV) in the vicinity of phase interface ref:Cristea ; ref:Wagner2003
; ref:Lee ; ref:Shan ; ref:Gong2019 ; ref:LeeRev  for a stationary droplet
immersed in another phase, and produces inaccurate density properties such as
coexistence curve and density profiles across interface for an equilibrium
flat interface ref:Zheng2005 ; ref:Wagner2006 ; ref:Kupershtokn2009 ;
ref:HuangHBPRE2011 ; ref:Sun2012 ; ref:Siebert2014 ; ref:LuoKH2015 ;
ref:ChenBX2015 ; ref:ZhengLin2017 ; ref:HuangPRE2019 ; ref:Khar2019 ;
ref:WagnerPRE2020 ; ref:WenPRE2020 . The appearance of SV and inaccurate
density properties suggests that the equilibrium state is not reached by the
corresponding LBE models. In other words, at the discrete level the forces
from different parts of LBE are NOT well balanced. Particularly, the SV is
produced by the non-constant chemical potential which serves as a net driven
force. The SV and inaccurate equilibrium density properties may lead to
unphysical phenomena and numerical instability in some circumstances, and it
is necessary to understand the origin to reduce or remove their effects.

In the past decades, a number of efforts have been attempted to identify the
force imbalance in LBE at equilibrium state. Particularly, Guo et al. made a
rigorous analysis the discrete balance equations of a LBE for a flat interface
problem ref:GuoPRE2011 , and a necessary condition for discrete force balance
was obtained. It was revealed that the condition cannot be achieved by the
standard LBE due to discretization errors in the discretizations of pressure
gradient and interaction force. Following this guidance, Lou and Guo developed
a Lax-Wendroff type LBE (LW-LBE) which can control the force imbalance with an
adjustable Courant-Friedrichs-Levy (CFL) number. However, the force imbalance
cannot be removed completely since the CFL number cannot be zero. Furthermore,
the time step of the LW-LBE is proportional to the CFL number, such that the
computation time can be much increased.

Although the total force imbalance equation was obtained in Ref.
ref:GuoPRE2011 , the detailed discrete errors from different parts are not
identified, and the structure of the net force is unclear. The aim of the
present study is twofold: (i) to give a further theoretical analysis of the
force imbalance property of the standard LBE to identify the detailed errors
from different parts, and (ii) to develop a well-balanced LBE that can achieve
the equilibrium state.

The rest of the paper is organized as follows. In Sec. II, a theoretical
analysis of the balance properties of the standard LBE for a special one-
dimensional (1D) flat interface system is presented, focusing on the structure
of net force due to discretization errors from different parts. Based on the
analysis, a well-balanced LBE is presented and analyzed in Sec. III. Then the
capability of capturing equilibrium state of the new LBE is tested in Sec. IV
by simulating a 1D flat interface and a two-dimensional (2D) droplet system.
Finally a brief summary is presented in Sec. V.

##  II Force imbalance of standard LBE

For simplicity without loss of generality, we consider the following standard
two-dimensional nine-velocity (D2Q9) LBE,

$f_{i}({\bm{x}}+{\bm{c}}_{i}\Delta t,t+\Delta
t)-f_{i}({\bm{x}},t)=\dfrac{1}{\tau}\left(f_{i}-f_{i}^{(eq)}\right)+\Delta
t\left(1-\dfrac{1}{2\tau}\right)F_{i},$ (2)

where $f_{i}({\bm{x}},t)$ is the distribution function for particles moving
with velocity ${\bm{c}}_{i}$ at position ${\bm{x}}$ and time $t$, $\Delta t$
is the time step, and ${\bm{c}}_{i}$ are discrete velocities defined by
${\bm{c}}_{0}=(0,0)$, ${\bm{c}}_{1}=-{\bm{c}}_{3}=c(1,0)$,
${\bm{c}}_{2}=-{\bm{c}}_{4}=c(0,1)$, ${\bm{c}}_{5}=-{\bm{c}}_{7}=c(1,1)$, and
${\bm{c}}_{7}=-{\bm{c}}_{8}=c(-1,1)$, with $c=\Delta x/\Delta t$ being the
lattice speed, and $\Delta x$ and $\Delta t$ the lattice spacing and time
step, respectively; $\tau$ is the dimensionless relaxation time related to the
shear viscosity $\nu$ by $\nu=c_{s}^{2}(\tau-0.5)\Delta t$ with
$c_{s}=c/\sqrt{3}$ a model parameter; The equilibrium distribution function
$f^{(eq)}$ is given by

$f_{i}^{(eq)}=w_{i}\rho\left[1+\frac{{\bm{c}}_{i}\cdot{\bm{u}}}{c_{s}^{2}}-\frac{\left({\bm{c}}_{i}\cdot{\bm{u}}\right)^{2}}{2c_{s}^{4}}+\frac{{\bm{u}}\cdot{\bm{u}}}{2c_{s}^{2}}\right],$
(3)

where the weights are given by $w_{0}=4/9$, $w_{1}=w_{2}=w_{3}=w_{4}=1/9$, and
$w_{5}=w_{6}=w_{7}=w_{8}=1/36$. The density $\rho$ and velocity ${\bm{u}}$ are
defined by

$\rho=\sum_{i=0}^{8}{f_{i}},\quad\rho{\bm{u}}=\sum_{i=0}^{8}{{\bm{c}}_{i}f_{i}}+\dfrac{\Delta
t}{2}{\bm{F}},$ (4)

where ${\bm{F}}$ is the interaction force defined later; The forcing term
$F_{i}$ in Eq. (2) is defined by

$F_{i}=w_{i}\left[\frac{c_{i}\cdot{\bm{F}}}{c_{s}^{2}}+\frac{{\bm{u}}{\bm{F}}:\left({\bm{c}}_{i}{\bm{c}}_{i}-c_{s}^{2}{\bm{I}}\right)}{c_{s}^{4}}\right].$
(5)

Note that the forcing term in ref:GuoBook  is defined as
$F_{i}=({\bm{c}}_{i}-{\bm{u}})\cdot{\bm{F}}f_{i}^{(eq)}/c_{s}^{2}\rho$. The
first and second order moments of the forcing terms of the two terms are the
same, but the third-order moment are different with difference of order
$O(|{\bm{u}}|^{3})$.

The interaction force ${\bm{F}}$ usually depends on the free energy of the
system, and can be written in either pressure form or potential form
ref:Wagner2003 ; ref:Lee ,

${\bm{F}}=\underbrace{\nabla(c_{s}^{2}\rho-
p_{0})-\kappa\nabla\nabla^{2}\rho}_{pressureform}=\underbrace{\nabla(c_{s}^{2}\rho)-\rho\nabla\mu}_{potentialform},$
(6)

where $p_{0}$ is the thermodynamic pressure dependent on the equation of
state, $\kappa$ is a parameter relating to the surface tension,
$\mu=\mu_{0}-\kappa\nabla^{2}\rho$ is the chemical potential with $\mu_{0}$
being the chemical potential in bulk phase. Although the two formulations are
identical mathematically, the performance of their discrete versions may be
different due to discretizations errors. For instance, it was shown that the
use of potential form can reduce the magnitude of SV greatly in both finite-
volume and LBE methods ref:Wagner2003 ; ref:Lee . In the present study we will
also adopt the potential form.

It should be noted that the term $-\rho\nabla\mu$ is the physical
thermodynamic driven force in the two phase system, while the term
$\nabla(c_{s}^{2}\rho)$ in ${\bm{F}}$ is artificial which is used to cancel
the ideal gas pressure induced by the collision-streaming process of the LBE.
To see this more clearly, we write out the hydrodynamic equations derived from
the LBE (2) via the Chapman-Enskog expansion up to second order (i.e., Navier-
Stokes level) ref:GuoBook ,

$\partial_{t}\rho+\nabla\cdot(\rho{\bm{u}})=0,$ (7a)
$\partial_{t}(\rho{\bm{u}})+\nabla\cdot(\rho{\bm{u}}{\bm{u}})=-\nabla(p_{id})+\nabla\cdot\bm{\sigma}+{\bm{F}},$
(7b)

where $p_{id}=c_{s}^{2}\rho$ is the equation of state of ideal gas,
$\sigma_{\alpha\beta}=\rho\nu(\partial_{\alpha}u_{\beta}+\partial_{\beta}u_{\alpha})$
is the viscous stress tensor, with $\nu=c_{s}^{2}(\tau-0.5)\Delta t$. The
first term on the right hand side of Eq. (7b) comes from the $f^{(eq)}$
defined by Eq. (3). Even without the forcing term, the collision-streaming
procedure in the standard LBE with such $f^{(eq)}$ will also yield this term.
In order to recover the desired nonideal equation of state $p_{0}$ or the
chemical potential, it is necessary to include the additional term
$\nabla(c_{s}^{2}\rho)$ in ${\bm{F}}$ to cancel this undesired term. However,
although mathematically the term $-\nabla p_{id}$ and the
$\nabla(c_{s}^{2}\rho)$ in ${\bm{F}}$ can be well balanced, their discrete
versions may be not due to discretization errors.

The above arguments also suggest that it is not adequate to reveal the force
imbalance structure of LBE based on the derived hydrodynamic equations. This
is reasonable, since the Navier-Stokes equations (7) describe just the
asymptotic behavior of the LBE. Therefore, in order to obtain the structure of
force imbalance of LBE, we follow the method in ref:GuoPRE2011  to analyze the
discrete behavior of LBE.

Specifically, we consider a simple liquid-vapor system with a flat phase
interface parallel to the $x$-direction (Fig. 1), where far from the interface
the liquid and vapor phases maintain their bulk densities $\rho_{l}$ and
$\rho_{v}$, respectively. For this unidirectional problem, the density,
velocity, and interaction force depend only on $y$, i.e., $\rho=\rho(y)$,
${\bm{u}}=(u_{x},u_{y})=(0,v)$, and ${\bm{F}}=(F_{x},F_{y})=(0,F)$. For this
system, the equilibrium state gives that $v=0$ and $\mu=\mbox{constant}$, from
which the density distribution can be derived. For the LBE, the distribution
functions can be categorized into three groups, namely,
$A=\\{f_{0},f_{1},f_{3}\\}$, $B=\\{f_{2},f_{5},f_{6}\\}$, and
$C=\\{f_{4},f_{7},f_{8}\\}$. In group $A$, the distribution functions are for
particles moving parallel to the $x$-axis, while those in $B$ and $C$ are for
particles moving upward and downward, respectively.

At steady state, the LBE (2) can be written as

$f_{i}({\bm{x}}+{\bm{c}}_{i}\Delta
t)-f_{i}({\bm{x}})=\dfrac{1}{\tau}\left(f_{i}-f_{i}^{(eq)}\right)+\Delta
t\left(1-\dfrac{1}{2\tau}\right)F_{i}.$ (8)

Note that $f_{i}$ is independent on $x$, and so for the distribution functions
in group $A$, we have

$f_{i}=f_{i}^{(eq)}+\Delta t(\tau-0.5)F_{i},\quad i=0,1,3,$ (9)

from which we can obtain that

$f_{A}=f_{0}+f_{1}+f_{3}=\frac{2}{3}\rho\left(1-\frac{v^{2}}{2c_{s}^{2}}\right)+\frac{1}{3}(1-2\tau)\Delta
t\frac{vF}{c_{s}^{2}}.$ (10)

Then from Eq. (4) we have

$f_{B}+f_{C}=\rho-f_{A},\quad c(f_{B}-f_{c})=\rho v-\dfrac{\Delta t}{2}F,$
(11)

where $f_{B}=f_{2}+f_{5}+f_{6}$ and $f_{C}=f_{4}+f_{7}+f_{8}$. Without lose of
generality, we take the lattice speed $c$ as the velocity unit (i.e., $c=1$).
It is easy to obtain from the above equations that

$\displaystyle f_{B}=\dfrac{\rho}{6}+\dfrac{\rho v}{2}+\frac{\rho
v^{2}}{2}-\dfrac{\Delta t}{4}F+(\tau-0.5)\Delta tFv,$ (12) $\displaystyle
f_{C}=\dfrac{\rho}{6}-\dfrac{\rho v}{2}+\frac{\rho v^{2}}{2}+\dfrac{\Delta
t}{4}F+(\tau-0.5)\Delta tFv.$ (13)

With these results, we multiply Eq. (8) by $c_{iy}$ and take summation over
$i$ to obtain the following discrete momentum balance equation,

$f_{B,j+1}-f_{C,j-1}=\rho_{j}v_{j}+\frac{\Delta t}{2}F_{j},$ (14)

or

$-\dfrac{c_{s}^{2}\left(\rho_{j+1}-\rho_{j-1}\right)}{2\Delta
x}+\dfrac{F_{j+1}+2F_{j}+F_{j-1}}{4}=-\Delta xR(v_{j}),$ (15)

where $j$ is the index of the lattice nodes in $y$-direction, and

$R(v_{j})=\dfrac{\rho_{j+1}v_{j+1}-2\rho_{j}v_{j}+\rho_{j-1}v_{j-1}}{2\Delta
x^{2}}+(2\tau-1)\dfrac{\left(F_{j+1}v_{j+1}-F_{j-1}v_{j-1}\right)}{2\Delta
x}+\dfrac{\rho_{j+1}v_{j+1}^{2}-\rho_{j-1}v_{j-1}^{2}}{2\Delta x}.$ (16)

Note that Eq. (15) represents the total force balance at discrete level, which
can be viewed as a finite-difference scheme of the continuous momentum
equation (7b). This discrete balance equation has been obtained in ref:GuoBook
except for a slight difference in $R(v)$ at the order of $O(v^{3})$. However,
in that work the detailed discretization errors were not identified.

Figure 1: (Color online) Sketch of the flat interface problem.

.

If the LBE is well-balanced at equilibrium, the equilibrium state ${\cal S}$
should be the solution of Eq. (15). Unfortunately, this does not hold in
general. We will demonstrate this point using the method of proof by
contradiction. If ${\cal S}$ is the solution of (15), then we have
$R(v_{j})=0$ and
$F_{j}=\partial_{y}(c_{s}^{2}\rho)_{j}=c_{s}^{2}\delta_{h}\rho_{j}$, where
$\delta_{h}$ is certain discrete gradient operator, which can be expressed
generally as

$\delta_{h}\rho_{j}=\dfrac{1}{\Delta x}\sum_{k=-r}^{s}{a_{j+k}\rho_{j+k}},$

where $r$ and $s$ are two positive integers, and $a_{i}$ are nonzero constant
coefficients. Equation (15) then leads to

$\dfrac{\rho_{j+1}-\rho_{j-1}}{2\Delta x}+\delta_{h}^{F}\rho_{j}=0,$ (17)

where

$\delta_{h}^{F}\equiv\dfrac{\delta_{h}\rho_{j+1}+2\delta_{h}\rho_{j}+\delta_{h}\rho_{j-1}}{4}.$
(18)

The first and second terms of Eq. (17) are both discretizations of
$\partial_{y}(c_{s}^{2}\rho)$. However, their stencils and truncation errors
are different generally. For instance, if the widely used isotropic central
scheme (ICS) ref:GuoPRE2011  is employed to discretize the gradients in
${\bm{F}}$, we have

$\delta_{h}\rho_{j}=\dfrac{\rho_{j+1}-\rho_{j-1}}{2\Delta x},$ (19)

and the second term of Eq. (17) can be written as

$\delta_{h}^{F}\rho_{j}=\dfrac{\rho_{j+2}+2(\rho_{j+1}-\rho_{j-1})-\rho_{j-2}}{8\Delta
x}\approx\partial_{y}\rho+\dfrac{5\Delta x^{2}}{12}\partial_{y}^{3}\rho.$ (20)

On the other hand, the first term coming from the evolution of LBE can be
written as

$-\delta_{h}^{LB}\rho_{j}=-\dfrac{\rho_{j+1}-\rho_{j-1}}{2\Delta
x}\approx-\partial_{y}\rho-\dfrac{\Delta x^{2}}{6}\partial_{y}^{3}\rho.$ (21)

Obviously, the truncation errors of $\delta_{h}^{LB}\rho_{j}$ (from LBE) and
$\delta_{h}^{F}\rho_{j}$ (from interaction force) cannot be canceled with each
other, except as $\rho$ is a second polynomial of $y$. However, if
$\rho(y)=a_{2}y^{2}+a_{1}y+a_{0}$ with $a_{i}$ being some constants, the
boundary conditions $\rho(-\infty)=\rho_{v}$ and $\rho(\infty)=\rho_{l}$
suggest that $a_{1}=a_{2}=0$, meaning that the density is a constant. This is
clearly impossible for a two phase system with different bulk densities.

Other discretization schemes of $\delta_{h}\rho_{j}$ other than ICS will lead
to different formulation of $\delta_{h}^{F}\rho_{j}$, but the stencil of
$\delta_{h}^{F}\rho_{j}$ is usually larger than that of
$\delta_{h}^{LB}\rho_{j}$, and their truncation errors cannot be canceled with
each other. Therefore, in general the force balance equation (17) does not
hold, and we complete the proof that the equilibrium state ${\cal S}$ cannot
be the solution of the discrete momentum balance equation (15) of the standard
LBE.

A byproduct of the above proof is the structure of the net force due to the
discretization errors in the standard LBE, namely

$\Delta F=-\delta_{h}^{LB}\rho_{j}+\delta_{h}^{F}\rho_{j},$ (22)

which depends on the stencil of $\delta_{h}$. Particularly, the net force with
the ICS discretization is

$\Delta F\approx\dfrac{\Delta x^{2}}{12}\partial_{y}^{3}\rho.$ (23)

suggesting that deviation of the LBE solution from the equilibrium state
${\cal S}$ depends on mesh size and density difference. It is noted that in
some studies high-order discretizations for the interaction ${\bm{F}}$ has
been employed (e.g., ref:Shan ; ref:FromPRE2020 ). Although the high-order
schemes can increase the isotropy of discretization and reduce the magnitude
of SV, but the improvement is limit. This can also be explained from the
structure of net force (22): the force balance cannot be reached by improving
the accuracy of $\delta_{h}^{F}$ solely.

##  III A well-balanced LBE

Based on the above analysis, we now present a well-balanced LBE that can
capture the equilibrium state. Since the different truncation errors in
$\delta_{h}^{LB}\rho$ and $\delta_{h}^{F}\rho$ are the main barrier for the
standard LBE to reach the equilibrium state, it is necessary to remove their
effects in order to capture the correct thermodynamic properties of the two-
phase system. As indicated in Sec. II, $\delta_{h}^{LB}\rho$ comes from the
evolution of LBE with the equilibrium distribution function defined by Eq.
(3), and $\delta_{h}^{F}\rho$ in $F$ is used to cancel it. Therefore, if
$\delta_{h}^{LB}\rho$ does not appear, $\delta_{h}^{F}\rho$ is not required
yet, and the force imbalance in the standard LBE may be avoided.

A more careful examination shows that $\delta_{h}^{LB}\rho$ is induced by the
first term ($\rho$) in $f^{(eq)}$ given by Eq. (3). Therefore, to avoid the
appearing of $\delta_{h}^{LB}\rho$, we redefine the equilibrium distribution
function as

$f_{i}^{(eq)}=\left\\{\begin{array}[]{ll}\rho-(1-w_{0})\rho_{0}-w_{0}\rho\dfrac{{\bm{u}}\cdot{\bm{u}}}{2c_{s}^{2}},&i=0\\\
w_{i}\left\\{\rho_{0}+\rho\left[\dfrac{{\bm{c}}_{i}\cdot{\bm{u}}}{c_{s}^{2}}+\dfrac{\left({\bm{c}}_{i}\cdot{\bm{u}}\right)^{2}}{2c_{s}^{4}}-\dfrac{{\bm{u}}\cdot{\bm{u}}}{2c_{s}^{2}}\right]\right\\},&i\neq
0\end{array}\right.$ (24)

where $\rho_{0}$ is a numerical constant. The choice of $\rho_{0}$ may
influence the stability of the scheme, but does not influence the theoretical
results. In what follows, we simply set $\rho_{0}=0$. Accordingly, the
interaction force is defined now by

${\bm{F}}=-\rho\nabla\mu,$ (25)

and the forcing term $F_{i}$ is modified as

$F_{i}=w_{i}\left[\frac{c_{i}\cdot{\bm{F}}}{c_{s}^{2}}+\frac{{\bm{u}}({\bm{F}}+c_{s}^{2}\nabla\rho):\left({\bm{c}}_{i}{\bm{c}}_{i}-c_{s}^{2}{\bm{I}}\right)}{c_{s}^{4}}+\dfrac{1}{2}\left(\dfrac{c_{i}^{2}}{c_{s}^{2}}-D\right)({\bm{u}}\cdot\nabla\rho)\right],$
(26)

where $D$ is the dimension and $D=2$ for the D2Q9 model. It is easy to verify
the following moments of $f^{(eq)}$ and $F_{i}$ defined above,

$\sum_{i}{f_{i}^{(eq)}}=\rho,\quad\sum_{i}{{\bm{c}}_{i}f_{i}^{(eq)}}=\rho{\bm{u}},\quad\sum_{i}{{\bm{c}}_{i}{\bm{c}}_{i}f_{i}^{(eq)}}=\rho{\bm{u}}{\bm{u}},\quad\sum_{i}{{\bm{c}}_{i}{\bm{c}}_{i}{\bm{c}}_{i}f_{i}^{(eq)}}=c_{s}^{2}\rho[{\bm{u}}\bm{I}],$
(27)
$\sum_{i}F_{i}=0,\quad\sum_{i}{\bm{c}}_{i}F_{i}={\bm{F}},\quad\sum_{i}{\bm{c}}_{i}{\bm{c}}_{i}F_{i}=[{\bm{u}}{\bm{F}}]+c_{s}^{2}\left([{\bm{u}}\nabla\rho]+({\bm{u}}\cdot\nabla\rho)\bm{I}\right),$
(28)

where
$[{\bm{u}}\bm{I}]_{\alpha\beta\gamma}=u_{\alpha}\delta_{\beta\gamma}+u_{\beta}\delta_{\alpha\gamma}+u_{\gamma}\delta_{\alpha\beta}$,
$[{\bm{u}}{\bm{F}}]_{\alpha\beta}=u_{\alpha}F_{\beta}+u_{\beta}F_{\alpha}$,
and
$[{\bm{u}}\nabla\rho]_{\alpha\beta}=u_{\alpha}\partial_{\beta}\rho+u_{\beta}\partial_{\alpha}\rho$.
With these moments, it can be derived the hydrodynamic equations at the
Navier-Stokes level via the Chapman-Enskog expansion,

$\partial_{t}\rho+\nabla\cdot(\rho{\bm{u}})=0,$ (29a)
$\partial_{t}(\rho{\bm{u}})+\nabla\cdot(\rho{\bm{u}}{\bm{u}})=-\rho\nabla\mu+\nabla\cdot\bm{\sigma}^{\prime},$
(29b)

where
$\sigma^{\prime}_{\alpha\beta}=\rho\nu[\partial_{\alpha}u_{\beta}+\partial_{\beta}u_{\alpha}+(\nabla\cdot{\bm{u}})\delta_{\alpha\beta}]$,
which is slightly different from $\sigma_{\alpha\beta}$ of the standard LBE in
the bulk viscosity.

It is clear that the new LBE with the modified $f^{(eq)}$, ${\bm{F}}$, and
$F_{i}$ can yield the desired asymptotic hydrodynamic equations, which can
describe equilibrium state of a two-phase system. To see whether the LBE can
capture the state at the discrete level, we again consider the flat interface
problem as considered in Sec. II. Following the same procedure, we can obtain
that

$\displaystyle f_{A}=\rho(1-v^{2})+(1-2\tau)\Delta tFv-(1-2\tau)\Delta
t\left(F+0.5\delta_{h}\rho\right)$ (30) $\displaystyle f_{B}=\dfrac{\rho
v}{2}+\frac{\rho v^{2}}{2}-\dfrac{\Delta t}{4}F+(\tau-0.5)\Delta
t(F+0.5\delta_{h}\rho)v,$ (31) $\displaystyle f_{C}=-\dfrac{\rho
v}{2}+\frac{\rho v^{2}}{2}+\dfrac{\Delta t}{4}F+(\tau-0.5)\Delta
t(F+0.5\delta_{h}\rho)v,$ (32)

and the momentum balance equation is

$\dfrac{\rho_{j+1}\delta_{h}\mu_{j+1}+2\rho_{j}\delta_{h}\mu_{j}+\rho_{j-1}\delta_{h}\mu_{j-1}}{4}=\Delta
xR^{\prime}(v_{j}),$ (33)

where

$\begin{split}R^{\prime}(v_{j})&=\dfrac{\rho_{j+1}v_{j+1}-2\rho_{j}v_{j}+\rho_{j-1}v_{j-1}}{2\Delta
x^{2}}+(2\tau-1)\dfrac{\left(F_{j+1}v_{j+1}-F_{j-1}v_{j-1}\right)}{2\Delta
x}+\dfrac{\rho_{j+1}v_{j+1}^{2}-\rho_{j-1}v_{j-1}^{2}}{2\Delta x}\\\
&\qquad+(\tau-0.5)\dfrac{(\delta_{h}\rho_{j+1})v_{j+1}-(\delta_{h}\rho_{j-1})v_{j-1}}{2\Delta
x}.\end{split}$ (34)

It is easy to verify that $v_{j}=0$ and $\mu_{j}=\mbox{constant}$ is a
solution of Eq. (33), suggesting that the new LBE can reach the equilibrium
state ${\cal S}$ at discrete level. Therefore, the new LBE (2) together with
Eqs. (24), (25), and (26) can be well balanced at equilibrium, and we will
term it as well-balanced LBE (WB-LBE).

##  IV Numerical results

We now test the well-balanced properties of the WB-LBE by simulating two
cases. The free-energy density functional of the system is chosen to be

${\cal
F}=\int{\left[\psi_{0}(\rho)+\dfrac{1}{2}\kappa|\nabla\rho|^{2}\right]dV},$
(35)

where the bulk energy $\psi_{0}$ takes the double-well form,

$\psi_{0}=\beta(\rho-\rho_{l}^{0})^{2}(\rho-\rho_{v}^{0})^{2},$ (36)

where $\beta$ is a constant, $\rho_{l}^{0}$ and $\rho_{v}^{0}$ are the
densities of liquid and vapor phases at saturation, respectively. The equation
of state and bulk chemical potential are given respectively by

$p_{0}=\dfrac{d\psi_{0}}{d\rho}-\psi_{0},\quad\mu_{0}=\dfrac{d\psi_{0}}{d\rho}.$
(37)

Figure 2: (Color online) Results of the flat interface problem. (a) Time
history of the kinetic energy; (b) Velocity of the standard LBE at steady
state; (c) Velocity of the WB-LBE at steady state.

###  IV.1 1D flat interface

The first test case is a flat interface problem, where the computation domain
is covered by a $N_{x}\times N_{y}=21\times 101$ lattice and the liquid slab
is put in the central region ($25\leq y\leq 75$) and the rest is filled with
the vapor phase. Periodic boundary conditions are employed at the four
boundaries. In the simulations, we choose $\rho_{l}^{0}=1.0$,
$\rho_{v}^{0}=0.2$, $\beta=0.01$, and $\kappa=0.0128$. For this plane
interface problem, the density at equilibrium follows a hyperbolic tangent
profile,

$\rho_{0}(y)=\rho_{v}^{0}+\dfrac{\rho_{l}^{0}-\rho_{v}^{0}}{2}\left[\tanh(2(y-y_{1})/D)-\tanh(2(y-y_{2})/D)\right],$
(38)

where $y_{1}$ and $y_{2}$ are the positions of the lower and upper interfaces,
$D=(\rho_{l}-\rho_{v})\sqrt{8\kappa/\beta}$ characterizes the interface
thickness. The relaxation time $\tau$ is set to be 0.85 in all simulations.
Initially, the density is specified with the theoretical solution (38) with a
small random perturbation, $\rho(y)=(1+\epsilon r)\rho_{0}(y)$, with
$\epsilon=0.01$ and $r$ a random number sampled between $-1$ and 1 from a
uniform distribution.

The time history of the system kinetic energy,
$E(t)=\tfrac{1}{2}\int{\rho|{\bm{u}}|^{2}d{\bm{x}}}$, are shown in Fig. 2(a).
For the standard LBE, the kinetic energy reaches to a nearly constant value at
the order of $10^{-12}$. On the other, the kinetic energy from the WB-LBE
deceases to about $10^{-30}$. The velocity profiles of both LBE models are
shown in Fig. 2(b) and 2(c). It can be seen that the standard LBE predicts a
velocity field of order $10^{-8}$, which clearly demonstrate the existence of
SV; while for the WB-LBE, the SV is removed up to the machine accuracy. The
density and chemical potential distributions are shown in Fig. 3. The density
profile predicted by the WB-LBE is nearly indistinguishable from the
theoretical one, while that predicted by the standard LBE shows clearly
deviations, particularly in the vapor phase. The chemical potential predicted
by the WB-LBE is nearly constant, while that predicted by the standard LBE
varies at the order of $10^{-3}$. The above results clearly show that the
present WB-LBE can reach the equilibrium state for this flat interface
problem, while the standard one fails, which is consistent with the
theoretical analysis.

Figure 3: (Color online) Density (a) and chemical potential (b) at the steady
state of the flat interface problem.

###  IV.2 2D droplet

The second test case is a 2D droplet immersed in its vapor in a square domain
of size $100\times 100$. The radius of the droplet is 25, and is initialized
at the center of domain. Periodic boundary conditions are applied to all
boundaries, and the simulation parameters are the same as used in the 1D flat
interface problem. Figure 4 presents the evolutions of the system kinetic
energy, and the velocity fields predicted by the standard LBE and WB-LBE at
steady state. It is clear that the WB-LBE can produce negligible SV, while the
SV in standard LBE is quite large.

The density and chemical potential structure are shown in Fig. 5. It can be
seen that from Fig. 5(a) that the density profile from the WB-LBE agree well
with the initial profile, but the bulk liquid and vapor densities are slightly
larger than the corresponding saturation values. The increase is due to the
curvature of the droplet, which can be obtained from the bulk pressure values
determined by the Laplace’s lawref:Lee ; ref:LouPRE . However, the bulk
density of the vapor phase predicted by the standard LBE deviates from the
theoretical ones greatly, and the interface thickness becomes wider than the
theoretical values. Figure 5(b) suggests that the WB-LBE predicted a nearly
constant chemical potential, while the standard LBE gives a nonuniform one
with variation of order $10^{-3}$, particularly near the interface as shown in
5(c).

Figure 4: (Color online) (a) Time history of the kinetic energy; (b) Velocity
field of the standard LBE. The maximum magnitude is $4.78\times 10^{-4}$; (c)
Velocity field of the standard LBE. The maximum magnitude is $8.63\times
10^{-15}$. The red circle is the density contour plot at
$(\rho_{l}+\rho_{v})/2$.

Figure 5: (Color online) Results of the 2D droplet problem at steady state.
Density (a) and chemical potential (b) profiles across the droplet center; (c)
The distribution of chemical potential of the standard LBE.

##  V Summary

The capability of capturing physical equilibrium state of a two-phase fluid
system is a fundamental requirement of LBE. In this work a rigorous analysis
is made to track the origin of force imbalance in the standard LBE at discrete
level. The detailed structure of the net force is derived. It is shown that
the net force comes from the discretization errors of two parts: the gradient
of $-p_{id}$ coming from the evolution of LBE, and the term
$\nabla({\bm{c}}_{s}^{2}\rho)$ in the interaction force. Generally, the
stencils of these two discretizations are different, and the truncation errors
cannot be canceled with each other. This net force drives the system to
deviate the physical equilibrium state, producing the spurious velocity and
non-constant chemical potential or unphysical density profiles.

Motivated by the theoretical analysis, a well-balanced LBE is proposed which
has the same algorithm structure as the standard LBE but avoids the
incompatible discretizations. Mathematical analysis shows that the LBE can
achieve the force balance condition (well-balanced), and can capture the
equilibrium state at discrete level. The numerical tests of a flat interface
and a droplet show that the spurious velocity can be removed, and consistent
interface profile and bulk densities can be captured, confirming the well-
balanced properties of the model.

It is noted that the strategy adopted in the present well-balanced LBE can
also be employed in other LBE models, such as pseudo-potential and phase-field
models.

###### Acknowledgements.

This work was supported by the National Science Foundation of China (Grant No.
51836003).

## References

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$\begin{split}d_{t_{0}}(p,p_{\ell+1})&\leq\sum_{k=1}^{\ell}d_{t_{0}}(p_{k},p_{k+1})\leq\sum_{k=1}^{\ell}d_{t_{n}}(p_{k},p_{k+1})+\varepsilon/2\\\
&\leq L_{d_{t_{n}}}(\gamma_{t_{n}}|_{[p,p_{\ell+1}]})+\varepsilon/2\leq
L_{d_{t_{n}}}(\gamma_{t_{n}})+\varepsilon/2\\\ &\leq
d_{t_{n}}(p,q)+\varepsilon\leq d_{t_{0}}(p,q)+2\varepsilon\leq
D+2,\end{split}$

where the first inequality is a consequence of the triangle inequality; the
second is a consequence of (44) and the fact that the chain of points has
cardinality not greater than $m$; the third inequality is a consequence of the
definition of length; the fifth is a consequence of (50); and the sixth is a
consequence of (45). Now, arguing exactly as before, we can show that
$p_{\ell+1}\in\overline{U}_{k_{\ell+1}}$ with $k_{\ell+1}\in\\{1,\dots,m\\}$
different from all $k_{1},\dots,k_{\ell}$.

Now to obtain (49) one argues exactly as before. Namely, for $p,q\in K$, and
$n\geq N$ we fix a chain of points $p=p_{1},\dots,p_{i}=q$ inductively
constructed as above, and we repeat the estimate (48) exchanging the roles of
$d_{t_{n}}$ and $d_{t_{0}}$. Hence, (45) and (49) give the sought
contradiction with (43), thus proving (41).

Taking into account the definition of pointed Gromov–Hausdorff convergence,
see [6, page 272], to prove (42) it is sufficient to use (41) and that, if we
fix $x\in X$, we have that, for every $t_{n}\to t_{0}$ and for every $R$,

$\limsup_{n\to+\infty}\operatorname{\mathrm{diam}}_{d_{t_{0}}}B_{d_{t_{n}}}(x,R)<+\infty.$

The previous inequality is a direct consequence of a slight variation of the
second argument above. Indeed, arguing as before, one can show that for every
sequence $t_{n}\to 0$ and every $R>0$ there exists $N$ sufficiently big such
that for every $n\geq N$ we have

$\bar{B}_{d_{t_{n}}}(x,R)\subseteq\bar{B}_{d_{t_{0}}}(x,R+1).$

∎

##  4\. Examples

In this section we discuss several examples in which we can apply our main
convergence result Theorem 1.4.

In Section 4.1 we use Theorem 1.4 to directly prove that the asymptotic cone
of the Riemannian Heisenberg group is the sub-Riemannian Heisenberg group, see
Proposition 4.1. The same reasoning can be easily generalized to arbitrary
Carnot groups.

In Section 4.2 we state and prove Mitchell’s Theorem in the sub-Finsler
cathegory for a continuously varying norm on the manifold, see Theorem 1.5. We
give for granted the construction of privileged coordinates and of the
nilpotent approximation, for which we refer the reader to standard and well-
established references, see [4], [7, Section 2.1], [1, Sections
10.4-10.5-10.6], or the recent [13]. Hence we exploit Theorem 1.4 to directly
prove the final convergence part of Mitchell’s Theorem in such a general
setting.

In Section 4.3 we use Theorem 1.4 to directly prove Theorem 1.6. Namely, we
prove that on a connected Lie group the CC distances associated to bracket-
generating sub-spaces and norms that converge are uniformly convergent on
compact subsets. The latter result has been used in the very recent [8].

In Section 4.4 we record a general approximation theorem for sub-Finsler
distances associated to converging vector fields on a manifold. Notice that
Theorem 4.5 can be used to produce Finsler approximation of sub-Finsler
manifolds.

###  4.1. Asymptotic cone of the Riemannian Heisenberg group

The first application we discuss is the well-known fact that the asymptotic
cone of the Riemannian Heisenberg group is the sub-Riemannian Heisenberg
group.

Using exponential coordinates of the first kind, we identify the first
Heisenberg group $\mathbb{H}^{1}$ with the manifold $\mathbb{R}^{3}$ endowed
with left-invariant frame

$X=\partial_{x}-\frac{y}{2}\partial_{z},\qquad
Y=\partial_{y}+\frac{x}{2}\partial_{z},\qquad Z=\partial_{z}.$

Let $\langle\cdot,\cdot\rangle$ be the left-invariant Riemannian tensor on
$\mathbb{H}^{1}$ that makes the above frame orthonormal, and $d_{R}$ the
corresponding distance. Let $d_{sR}$ be the sub-Riemannian distance defined by
$X,Y$, namely

$d_{sR}(p,q):=\inf_{\gamma(0)=p,\gamma(1)=q}\left\\{\int_{0}^{1}\sqrt{a_{1}(t)^{2}+a_{2}(t)^{2}}\,\mathrm{d}t:\gamma^{\prime}(t)=a_{1}(t)X_{1}|_{\gamma(t)}+a_{2}(t)X_{2}|_{\gamma(t)}\right\\},$

for every $p,q\in M$, where the infimum is taken over absolutely continuous
curves $\gamma$.

######  Proposition 4.1.

The asymptotic cone of $(\mathbb{H}^{1},d_{R})$ is $(\mathbb{H}^{1},d_{sR})$.

###### Proof.

Define $\mathbb{E}=\mathbb{R}^{3}$, $\|\cdot\|$ the Euclidean norm on
$\mathbb{E}$ and, for $\varepsilon\in\mathbb{R}$,
$\mathtt{f}_{\varepsilon}\in\mathrm{Lip}\Gamma(\mathbb{E}^{*}\otimes
T\mathbb{H}^{1})$ to be

$\mathtt{f}_{\varepsilon}(p)(v)=v_{1}X|_{p}+v_{2}Y|_{p}+\varepsilon
v_{3}Z|_{p}.$

Let $d_{\varepsilon}:=d_{(\mathtt{f}_{\epsilon},\|\cdot\|)}$. Then, , we have
that $d_{1}=d_{R}$ and $d_{0}=d_{sR}$. Since $\mathtt{f}_{0}$ is totally non-
holonomic, see Proposition 2.12, we obtain from Theorem 1.4(iv) that
$d_{\varepsilon}\to d_{0}$ as $\varepsilon\to 0$, uniformly on compact subsets
of $\mathbb{H}^{1}\times\mathbb{H}^{1}$.

What is only left to show is that $(\mathbb{H}^{1},d_{\varepsilon})$ is
isometric to $(\mathbb{H}^{1},\varepsilon d_{R})$. Notice that $\varepsilon
d_{R}$ is the Riemannian distance defined by the orthonormal frame
$(X/\varepsilon,Y/\varepsilon,Z/\varepsilon)$. Notice also that the map
$\delta_{\varepsilon}(x,y,z):=(\varepsilon x,\varepsilon y,\varepsilon^{2}z)$
satisfies

$\displaystyle(\delta_{\varepsilon})_{*}X/\varepsilon$
$\displaystyle=X=\mathtt{f}_{\varepsilon}(\cdot)(1,0,0),$
$\displaystyle(\delta_{\varepsilon})_{*}Y/\varepsilon$
$\displaystyle=Y=\mathtt{f}_{\varepsilon}(\cdot)(0,1,0),$
$\displaystyle(\delta_{\varepsilon})_{*}Z/\varepsilon$
$\displaystyle=\varepsilon Z=\mathtt{f}_{\varepsilon}(\cdot)(0,0,1).$

It follows that $\delta_{\varepsilon}:(\mathbb{H}^{1},\varepsilon
d_{R})\to(\mathbb{H}^{1},d_{\varepsilon})$ is an isometry. ∎

######  Remark 4.2.

Mutatis mutandis the statement of Proposition 4.1 works, with the same proof,
for arbitrary Carnot groups. Actually, with some additional work, one could
also recover the well-known fact that the asymptotic cone of a sub-Finsler
nilpotent Lie group is a sub-Finsler Carnot group.

###  4.2. Tangents to sub-Finsler manifolds

In this section we discuss the celebrated Mitchell’s Theorem. We take for
granted the existence of a system of priviliged coordinates, for which we
refer the reader, e.g., to the complete discussion in [7, Section 2.1]. We
stress that, up to the authors’ knowledge, this is the first time that such a
theorem is stated in the generality of sub-Finsler cathegory. Standard
references for the sub-Riemannian Mitchell’s Theorem are [7, Theorem 2.5],
[12, 4], and [2, Sections 10-4-10.5-10.6]. Our aim, here, is to give a
complete and detailed proof of the final convergence part of the proof.

Hereafter we follow the terminology of Jean’s book [7, Section 2.1]. We fix a
smooth manifold $M^{m}$ of dimension $m\in\mathbb{N}$, $k\in\mathbb{N}$, and
$k$ smooth vector fields $X_{1},\dots,X_{k}$. We say that the family
$\mathcal{X}:=\\{X_{1},\dots,X_{k}\\}$ is a non-holonomic system if it is
bracket-generating. Notice that we do not assume that the rank of
$\mathcal{X}$ is constant.

Let us fix $\mathbb{E}:=\mathbb{R}^{k}$, with canonical basis
$\\{e_{1},\dots,e_{k}\\}$, and let us fix $N:M\times\mathbb{E}\to[0,+\infty)$
a continuously varying family of norms on $\mathbb{E}$. Attached to the
previously defined non-holonomic system $\mathcal{X}$ and to $N$ there is a
notion of a sub-Finsler metric $|\cdot|_{\mathcal{X},N}$, see (4). In the
specific case in which $N(p,\cdot)$ is the sandard Euclidean norm for every
$p\in M$, the previous sub-Finsler metric is exactly the one considered in
Jean’s book [7, Equation (1.4)].

The sub-Finsler metric $|\cdot|_{(\mathcal{X},N)}$ gives raise to a length
distance $d_{(\mathcal{X},N)}$, see (5) and compare with [7, Definition 1.3].
Notice that, by definition, and by exploiting Lemma 3.7(d) and Theorem
1.4(ii), we get that any two distances $d_{(\mathcal{X},N_{1})}$ and
$d_{(\mathcal{X},N_{2})}$ are locally bi-Lipschitz equivalent. As a
consequence we stress that the notion of non-holonomic order of a smooth
function/vector field at a point $p\in M$, see [7, Section 2.1], can be
equivalently given by using any of such distances.

We define the Lipschitz-vector-field structure on $M$ modelled by
$\mathbb{R}^{k}$

(53) $\widetilde{\mathtt{f}}_{1}(p)(e_{i}):=X_{i}|_{p},$

for every $p\in M$, and $i=1,\dots,k$. The distance
$d_{(\widetilde{\mathtt{f}}_{1},N)}$ as in Definition 3.6, coincides with the
length distance induced by $|\cdot|_{\mathcal{X},N}$ as above, by virtue of
Corollary 3.19.

###### Proof of Theorem 1.5.

Let $o\in M$ be as in the statement. There exists a neighborhood $U\subseteq
M^{m}$ of $o$ and a neighborhood $V\subseteq\mathbb{R}^{m}$ of $0$ such that
$(x_{1},\dots,x_{m}):U\to V$ is a system of priviliged coordinates, see [7,
Definition 2.5] and [7, pages 22-23]. Hereafter we will identify $U\subseteq
M$ with $V\subseteq\mathbb{R}^{m}$ by means of the coordinates
$(x_{1},\dots,x_{m})$.

Being $w_{i}$ the weights defined as in [7, page 20], for $0<\varepsilon\leq
1$ we define the diffeomorphism
$\delta_{\varepsilon}:\delta_{\varepsilon}^{-1}V\to V$ as follows

$\delta_{\varepsilon}(x_{1},\dots,x_{m}):=(\varepsilon^{w_{1}}x_{1},\dots,\varepsilon^{w_{j}}x_{j},\dots,\varepsilon^{w_{m}}x_{m}).$

Let us define, for every $\varepsilon>0$, every $i=1,\dots,k$, and every
$p\in\delta_{\varepsilon}^{-1}V\subseteq\mathbb{R}^{m}$, the Lipschitz-vector-
fields structure

(54)
$\mathtt{f}_{\varepsilon}(p)(e_{i}):=\varepsilon(D\delta_{\varepsilon}|_{p})^{-1}[X_{i}|_{\delta_{\varepsilon}p}].$

Notice that $\delta_{\varepsilon}^{-1}V$ invades $\mathbb{R}^{m}$ as
$\varepsilon\to 0$.

Notice, moreover, that when $\varepsilon=1$ we are defining, by means of (54),
the Lipschitz-vector-field structure $\widetilde{\mathtt{f}}_{1}$ restricted
to $U$, namely $\mathtt{f}_{1}=(\widetilde{\mathtt{f}}_{1})|_{U}$, cf. (53).
In addition, as a consequence of Lemma 3.7(d), we can find $U^{\prime}\Subset
U$ sufficiently small such that

(55) $(U^{\prime},d_{(\mathtt{f}_{1},N)})\quad\text{is isometric
to}\quad(U^{\prime},d_{(\widetilde{\mathtt{f}}_{1},N)}).$

Let $\\{\hat{X}_{1},\dots,\hat{X}_{k}\\}$ be the homogeneous nilpotent
approximation of $\\{X_{1},\dots,X_{k}\\}$ at $o$ associated to the
coordinates $(x_{1},\dots,x_{m})$, see [7, Definition 2.7]. Notice that, for
every $i=1,\dots,k$, $\hat{X}_{i}$ are polynomial vector fields on
$\mathbb{R}^{m}$. Let us define, for every $p\in\mathbb{R}^{m}$, and every
$i=1,\dots,k$, the Lipschitz-vector-fields structure

$\mathtt{f}_{0}(p)(e_{i}):=\hat{X}_{i}|_{p}.$

Notice that the family $\\{\hat{X}_{1},\dots,\hat{X}_{k}\\}$ is a bracket-
generating family of vector fields on $\mathbb{R}^{m}$, cf. [7, Lemma 2.1,
(i)], and hence $\mathtt{f}_{0}$ is essentially non-holonomic, see Proposition
2.12.

Moreover, by the definitions of homogeneous nilpotent approximation, of
$\mathtt{f}_{\varepsilon}$, and of the maps $\delta_{\varepsilon}$ the
following holds: for every $\varepsilon>0$, $\mathtt{f}_{\varepsilon}$ is a
Lipschitz-vector-field structure defined on $\delta_{\varepsilon}^{-1}V$
modelled by $\mathbb{R}^{k}$, and

$\mathtt{f}_{\varepsilon}\to\mathtt{f}_{0},\qquad\text{as $\varepsilon\to
0$},$

in the sense of Definition 2.2.

Let us now define, for every $0<\varepsilon\leq 1$,
$N_{\varepsilon}:\delta_{\varepsilon}^{-1}V\times\mathbb{E}\to[0,+\infty)$ as

(56) $N_{\varepsilon}(p,v):=N(\delta_{\varepsilon}p,v),$

for every $p\in\delta_{\varepsilon}^{-1}V$, and $v\in\mathbb{E}$. We stress
that $N_{1}\equiv N$ on $V\times\mathbb{E}$. It is readily seen that
$N_{\varepsilon}$ converge, as $\varepsilon\to 0$, uniformly on compact
subsets of $\mathbb{R}^{m}\times\mathbb{E}$ to
$N_{0}:\mathbb{R}^{m}\times\mathbb{E}\to[0,+\infty]$ defined as

$N_{0}(p,v):=N(o,v),$

for every $p\in\mathbb{R}^{m}$ and $v\in\mathbb{E}$.

In case $o$ is a regular point then the metric space
$(\mathbb{R}^{m},d_{(\mathtt{f}_{0},N_{0})})$ is isometric to a sub-Finsler
Carnot group, cf. [7, Lemma 2.3]. In the general case,
$(\mathbb{R}^{m},d_{(\mathtt{f}_{0},N_{0})})$ is isometric to a quotient of a
sub-Finsler Carnot group by one of its closed subgroups, cf. [7, Theorem 2.6].
In any of such alternatives, we have that
$(\mathbb{R}^{m},d_{(\mathtt{f}_{0},N_{0})})$ is boundedly compact since it is
a locally compact, homogeneous, length space.

Hence, an application of Theorem 1.4(iv) gives that

(57) $d_{(\mathtt{f}_{\varepsilon},N_{\varepsilon})}\to_{\varepsilon\to
0}d_{(\mathtt{f}_{0},N_{0})},$

uniformly on compact subsets of $\mathbb{R}^{m}\times\mathbb{R}^{m}$. Notice
that the structures $\mathtt{f}_{\varepsilon}$ are not defined on the entire
$\mathbb{R}^{m}$, but just on $\delta_{\varepsilon}^{-1}V$. Anyway, we can
still apply Theorem 1.4 since for every compact set
$K\subseteq\mathbb{R}^{m}\times\mathbb{R}^{m}$ there exists $\varepsilon_{0}$
such that
$K\subseteq\delta_{\varepsilon}^{-1}V\times\delta_{\varepsilon}^{-1}V$ for
every $\varepsilon<\varepsilon_{0}$. Moreover, as a consequence of the last
part of the proof of Lemma 3.25, one also has that, for any $R>0$, there
exists $\varepsilon_{0}$ small enough such that
$\overline{B}_{R}^{d_{(\mathtt{f}_{\varepsilon},N_{\varepsilon})}}(0)$ are
contained in a common compact set of $\mathbb{R}^{m}$ for every
$\varepsilon<\varepsilon_{0}$; and moreover the Gromov–Hausdorff distance
between $\overline{B}_{R}^{d_{(\mathtt{f}_{\varepsilon},N_{\varepsilon})}}(0)$
and $\overline{B}_{R}^{d_{(\mathtt{f}_{0},N_{0})}}(0)$ converges to $0$ as
$\varepsilon\to 0$, cf. (57).

We now claim that for every $\varepsilon$ and every
$p,q\in\delta_{\varepsilon}^{-1}V$, the following equality holds

(58)
$d_{(\mathtt{f}_{\varepsilon},N_{\varepsilon})}(p,q)=\varepsilon^{-1}d_{(\mathtt{f}_{1},N_{1})}(\delta_{\varepsilon}p,\delta_{\varepsilon}q).$

Indeed, let us fix $\varepsilon>0$, and let us take
$p,q\in\delta_{\varepsilon}^{-1}V$. For every curve
$\gamma:[0,1]\to\mathbb{R}^{m}$, whose image is contained in
$\delta_{\varepsilon}^{-1}V$, and such that

(59)
$\begin{cases}\gamma^{\prime}(t)&=\mathtt{f}_{\varepsilon}(\gamma(t),u(t))\\\
\gamma(0)&=p,\end{cases}$

for almost every $t\in[0,1]$, we let
$\gamma_{\varepsilon}:=\delta_{\varepsilon}\circ\gamma$. We notice that
$\gamma_{\varepsilon}$ has support contained in $V$ and, from the definition
of $\mathtt{f}_{\varepsilon}$, we have

(60)
$\begin{cases}\gamma_{\varepsilon}^{\prime}(t)&=\mathtt{f}_{1}(\gamma_{\varepsilon}(t),\varepsilon
u(t)),\qquad\text{for almost every $t\in[0,1]$},\\\
\gamma_{\varepsilon}(0)&=\delta_{\varepsilon}p.\end{cases}$

Since we have that $\gamma_{\varepsilon}\subseteq V$, by taking into account
the previous computation, the definition of the norms $N_{\varepsilon}$, see
(56), and the definition of the distance, see (24), we finally get (58) for
every $p,q\in\delta_{\varepsilon}^{-1}V$.

The latter reasoning implies that for every $R>0$ there exists
$\varepsilon_{0}$ small enough such that for every
$\varepsilon<\varepsilon_{0}$ one has

(61)
$\left(\overline{B}_{R}^{d_{(\mathtt{f}_{\varepsilon},N_{\varepsilon})}}(0),d_{(\mathtt{f}_{\varepsilon},N_{\varepsilon})}\right)\quad\text{is
isometric
to}\quad\left(\overline{B}_{R}^{\varepsilon^{-1}d_{(\mathtt{f}_{1},N)}}(0),\varepsilon^{-1}d_{(\mathtt{f}_{1},N_{1})}\right),$

and the isometry is given by $\delta_{\varepsilon}$.

Thus, the latter, together with the convergence in (57), and (55), directly
implies that the Gromov–Hausdorff tangent of
$(M,d_{(\widetilde{\mathtt{f}}_{1},N)})$ at $o$ is $\mathbb{R}^{n}$ equipped
with the sub-Finsler distance induced by the vector fields
$\hat{X}_{1},\dots,\hat{X}_{k}$ and the norm $N(o,\cdot)$, which is what we
wanted. ∎

######  Remark 4.3.

We stress that our convergence result Theorem 1.4 holds in the Lipschitz
cathegory, provided the essentialy non-holonomicity of the limit. Hence,
whenever one has some analogues of priviliged coordinates while dealing with
less regular vector fields, the proof of Theorem 1.5 is very likely to be
adapted.

###  4.3. Left-invariant CC distances on Lie groups

Let $\mathbb{G}$ be a connected Lie group, and let $\mathfrak{g}$ be its Lie
algebra. Given a vector subspace $\mathcal{H}\subseteq\mathfrak{g}$ of
$\mathfrak{g}$, and a norm $b$ on $\mathcal{H}$, we associate to
$(\mathcal{H},b)$ a left-invariant sub-Finsler structure $(\mathcal{D},b)$ as
in (6). Moreover, we define $d^{(\mathcal{H},b)}$ as in (7).

Let us denote by $k$ the dimension of $\mathcal{H}$. Choose a basis
$\\{v_{1},\dots,v_{k}\\}$ of $\mathcal{H}$ and define $X_{i}$ to be the left-
invariant extension of $v_{i}$, for every $i=1,\dots,k$. Let
$\mathbb{E}:=\mathbb{R}^{k}$ with the canonical basis
$\\{e_{1},\dots,e_{k}\\}$. Define a Lipschitz-vector-field structure
$\mathtt{f}$ modelled by $\mathbb{R}^{k}$ as

$\mathtt{f}_{p}(e_{i}):=X_{i}(p),\quad\text{for all $p\in\mathbb{G}$, and
$i=1,\dots,k$}.$

Define the continuously varying norm
$N:\mathbb{G}\times\mathbb{E}\to\mathbb{R}$ as

$N(p,w):=b\left(\sum_{i=1}^{k}w_{i}v_{i}\right),\quad\text{for all
$p\in\mathbb{G}$, and $w=\sum_{i=1}^{k}w_{i}e_{i}\in\mathbb{E}$}.$

By virtue of Corollary 3.19 we deduce that

(62) $d_{(\mathtt{f},N)}(p,q)=d^{(\mathcal{H},b)}(p,q),\quad\text{for all
$p,q\in\mathbb{G}$}.$

We now give the proof of Theorem 1.6.

###### Proof of Theorem 1.6.

Let us choose a bracket-generating basis $\\{v_{1},\dots,v_{k}\\}$ of
$\mathcal{H}$. By the fact that $\mathcal{H}_{n}\to\mathcal{H}$, we have that,
for every $n\in\mathbb{N}$, there exists a basis
$\\{v_{1}^{n},\dots,v_{k}^{n}\\}$ of $\mathcal{H}_{n}$ such that $v_{i}^{n}\to
v_{i}$ as $n\to+\infty$ for every $i=1,\dots,k$. Let $X_{i}$ be the left-
invariant extension of $v_{i}$ for every $i=1,\dots,k$, and let $X_{i}^{n}$ be
the left-invariant extension of $v_{i}^{n}$ for every $i=1,\dots,k$, and every
$n\in\mathbb{N}$. Let us fix $\mathbb{E}:=\mathbb{R}^{k}$ with basis
$\\{e_{1},\dots,e_{k}\\}$.

For every $n\in\mathbb{N}$, we define the Lipschitz-vector-field structure
$\mathtt{f}_{n}$ modelled by $\mathbb{R}^{k}$ as

$(\mathtt{f}_{n})_{p}(e_{i}):=X_{i}^{n}(p),\qquad\text{for all
$p\in\mathbb{G}$, for all $i=1,\dots,k$,}$

and the Lipschitz-vector-field structure $\mathtt{f}$ modelled by
$\mathbb{R}^{k}$ as

$\mathtt{f}_{p}(e_{i}):=X_{i}(p),\qquad\text{for all $p\in\mathbb{G}$, for all
$i=1,\dots,k$.}$

By the convergence $v_{i}^{n}\to v_{i}$ in $\mathfrak{g}$, we have that
$\mathtt{f}_{n}\to\mathtt{f}$ in the sense of Definition 2.2.

For every $n\in\mathbb{N}$ we define the continuously varying norm
$N_{n}:\mathbb{G}\times\mathbb{E}$ as

$N_{n}(p,w):=b_{n}\left(\sum_{i=1}^{k}w_{i}v_{i}\right),\quad\text{for all
$p\in\mathbb{G}$, and $w=\sum_{i=1}^{k}w_{i}e_{i}\in\mathbb{E}$},$

and the continuously varying norm $N$ as

$N(p,w):=b\left(\sum_{i=1}^{k}w_{i}v_{i}\right),\quad\text{for all
$p\in\mathbb{G}$, and $w=\sum_{i=1}^{k}w_{i}e_{i}\in\mathbb{E}$}.$

By the fact that $b_{n}\to b$ uniformly on compact sets it follows that
$N_{n}\to N$ uniformly on compact sets.

Hence we showed that $(\mathtt{f}_{n},b_{n})\to(\mathtt{f},b)$ as
$n\to+\infty$. Let us finally check that the remaining hypotheses of Theorem
1.4 are met. Indeed, $\mathtt{f}$ is essentially non-holonomic due to
Proposition 2.12. Moreover, $(\mathbb{G},d^{(\mathcal{H},b)})$ is boundedly
compact because, by homogeneity and Theorem 1.4(ii), there exists
$\varepsilon>0$ such that for every $p\in\mathbb{G}$ the closed ball
$\overline{B}_{d^{(\mathcal{H},b)}}(p,\varepsilon)$ is compact. Hence an
application of Theorem 1.4(iv), together with the equality (62), gives the
sought conclusions. ∎

######  Remark 4.4.

As a special case of Theorem 1.6, when $b_{n}\equiv b$ is a norm coming from a
scalar product, it follows that the Condition 3.9 conjectured in [8] is always
true.

###  4.4. Limit of sub-Finsler distances on a manifold

In this section we prove a general convergence result that is a consequence of
Theorm 1.4.

######  Theorem 4.5.

Let $M^{m}$ be a smooth connected manifold of dimension $m$. Let
$k\in\mathbb{N}$. For every $\lambda\in[0,1)$ consider
$\mathcal{X}^{\lambda}:=\\{X_{1}^{\lambda},\dots,X_{k}^{\lambda}\\}$ a family
of smooth vector-fields such that

  1. (1)

$X_{i}^{\lambda}$ are locally equi-Lipschitz for every $i=1,\dots,k$, and
every $\lambda\in[0,1)$;

  2. (2)

$X_{i}^{\lambda}\to X_{i}^{0}$ uniformly on compact sets as $\lambda\to 0$,
for every $i=1,\dots,k$.

Let us assume that $\\{X_{1}^{0},\dots,X_{k}^{0}\\}$ is a bracket-generating
set of vector fields.

Let $\mathbb{E}:=\mathbb{R}^{k}$ with basis $\\{e_{1},\dots,e_{k}\\}$ and, for
every $\lambda\in[0,1)$, let $N_{\lambda}:M\times\mathbb{E}\to[0,+\infty)$ be
a continuously varying norm. Assume that $N_{\lambda}\to N_{0}$ uniformly on
compact sets. For each $\lambda$, let $|\cdot|_{\lambda}$ the sub-Finsler
metric defined by

$|v|_{\lambda}=\inf\left\\{N_{\lambda}\left(p,\sum_{j=1}^{k}x_{j}e_{j}\right):\
v=\sum_{j=1}^{k}x_{j}X_{j}^{\lambda}(p)\right\\}$

for every $v\in T_{p}M$, and $p\in M$. Let $d_{\lambda}$ be the sub-Finsler
distance associated to the sub-Finsler metric $|\cdot|_{\lambda}$ (cf. (5)).
Hence, $d_{\lambda}\to d_{0}$ locally uniformly on compact sets of $M\times
M$, as $\lambda\to 0$.

Moreover, if $(M,d_{0})$ is a complete metric space, we have that
$d_{\lambda}\to d_{0}$ uniformly on compact sets of $M\times M$, as
$\lambda\to 0$, and for every $p\in M$, we have that
$(M,d_{\lambda},p)\to(M,d_{0},p)$ in the pointed Gromov–Hausdorff topology as
$\lambda\to 0$.

###### Proof.

For every $\lambda\in[0,1)$, let
$\mathtt{f}_{\lambda}\in\mathrm{Lip}\Gamma(\mathbb{E}^{*}\otimes TM)$ be
defined as

$\mathtt{f}_{\lambda}(p)(e_{i}):=X_{i}^{\lambda}(p),\quad\text{for all $p\in
M$, and for every $i=1,\dots,k$.}$

From the hypotheses we get that
$(\mathtt{f}_{\lambda},N_{\lambda})\to(\mathtt{f}_{0},N_{0})$ as $\lambda\to
0$, where the convergence of the first components has to be intended in the
sense of Definition 2.2, and the convergence of the second components has to
be intended in the uniform sense on compact sets. Moreover, as a consequence
of Corollary 3.19, we have that
$d_{\lambda}=d_{(\mathtt{f}_{\lambda},N_{\lambda})}$ for every
$\lambda\in[0,1)$. Hence the results follow from Theorem 1.4(iii) and Theorem
1.4(iv). ∎

##  Appendix A Gronwall Lemma

######  Lemma A.1.

Let $\Omega\subset\mathbb{R}^{n}$, and
$X,Y:\Omega\times[0,T]\to\mathbb{R}^{n}$. Let $\|\cdot\|$ be the standard
Euclidean norm on $\mathbb{R}^{n}$. Fix $o\in\Omega$.

Suppose that there are $E,K>0$ be such that for all $p,q\in\Omega$ and all
$t\in[0,T]$

$\|X(p,t)-Y(q,t)\|\leq E+K\|p-q\|.$

Let $\gamma,\eta:[0,T]\to\Omega$ be two absolutely continuous curves such that
$\gamma(0)=\eta(0)=o$, $\gamma^{\prime}(t)=X(\gamma(t),t)$, and
$\eta^{\prime}(t)=Y(\eta(t),t)$ for almost every $t\in[0,T]$.

Then

$\|\gamma(t)-\eta(t)\|\leq E\frac{e^{Kt}-1}{K},\qquad\forall t\in[0,T].$

###### Proof.

Define $f(t):=\|\gamma(t)-\eta(t)\|$. Notice that $f:[0,T]\to\mathbb{R}$ is
absolutely continuous and $f(0)=0$. Moreover, for almost every $t\in[0,T]$ we
have

$2f(t)f^{\prime}(t)=\frac{\,\mathrm{d}}{\,\mathrm{d}t}(f(t)^{2})=\\\
=2\langle\gamma(t)-\eta(t),\gamma^{\prime}(t)-\eta^{\prime}(t)\rangle\leq
2\cdot\|\gamma(t)-\eta(t)\|\cdot\|\gamma^{\prime}(t)-\eta^{\prime}(t)\|=\\\
=2f(t)\cdot\|\gamma^{\prime}(t)-\eta^{\prime}(t)\|\leq
2f(t)\cdot\|X(\gamma(t),t)-Y(\eta(t),t)\|\leq\\\ \leq
2f(t)\cdot(E+K\|\gamma(t)-\eta(t)\|)\leq 2f(t)\cdot(E+Kf(t)).$

So, whenever $f(t)\neq 0$ we have

$f^{\prime}(t)\leq E+Kf(t).$

Let $g(t):=e^{-Kt}f(t)$. Then, whenever $g(t)\neq 0$, i.e., whenever $f(t)\neq
0$, we have

$g^{\prime}(t)=-Ke^{-Kt}f(t)+e^{-Kt}f^{\prime}(t)\leq-
Ke^{-Kt}f(t)+e^{-Kt}(E+Kf(t))=e^{-Kt}E.$

We claim

(63) $g(t)\leq\int_{0}^{t}e^{-Ks}E\,\mathrm{d}s,$

for almost all $t\in[0,T]$. Indeed, if $g(t)=0$, then there is nothing to show
because the right-hand side is positive. If $g(t)>0$ instead, since $g$ is
absolutely continuous, there is a maximal $\hat{t}<t$ such that
$g(\hat{t})=0$, and we have

$g(t)=g(\hat{t})+\int_{\hat{t}}^{t}g^{\prime}(s)\,\mathrm{d}s\leq\int_{\hat{t}}^{t}e^{-Ks}E\,\mathrm{d}s\leq\int_{0}^{t}e^{-Ks}E\,\mathrm{d}s,$

and (63) is proved.

Finally we obtain

$e^{-Kt}f(t)=g(t)\leq\int_{0}^{t}e^{-Ks}E\,\mathrm{d}s=\frac{E}{K}(1-e^{-Kt})$

hence the thesis. ∎

##  Appendix B Lemma about open mappings

We recall a few facts about degree theory. The $n$-th homology group
$H_{n}(\mathbb{S}^{n})$ of the $n$-dimensional sphere $\mathbb{S}^{n}$ is
isomorphic to $\mathbb{Z}$. If $\phi:\mathbb{S}^{n}\to\mathbb{S}^{n}$ is
continuous, then it induces a group morphism
$\phi_{*}:H_{n}(\mathbb{S}^{n})\to H_{n}(\mathbb{S}^{n})$ of the form
$\phi_{*}(x)=ax$ for some $a\in\mathbb{Z}$. This coefficient $a$ is called the
_degree_ of $\phi$ and it is denoted by $\deg(\phi)$.

######  Lemma B.1.

Let $\Omega\subset\mathbb{R}^{m}$ be an open set containing $0$. Let
$f_{\infty}:\Omega\to\mathbb{R}^{m}$ be a continuous map with
$f_{\infty}(0)=0$. Suppose that there is $r>0$ such that
$B(0,r)\Subset\Omega$, $0\notin f_{\infty}(\partial B(0,r))$, and the map

$\phi:\mathbb{S}^{m-1}\to\mathbb{S}^{m-1},\qquad\phi(x):=\frac{f_{\infty}(rx)}{|f_{\infty}(rx)|}$

has nonzero degree. Let $f_{n}:\Omega\to\mathbb{R}^{m}$ be continuous
functions that converge uniformly on $\Omega$ to $f_{\infty}$.

Then there exist $\delta>0$ and $N\in\mathbb{N}$ such that for all $n\geq N$,
we have

$B(0,\delta)\subset f_{n}(\Omega).$

###### Proof.

Let $B:=B(0,r)\Subset\Omega$ and, for every $n\in\mathbb{N}\cup\\{\infty\\}$,
define $S_{n}:=f_{n}(\partial B)$. Since $S_{\infty}$ is compact and it does
not contain $0$ by assumption, we have that $\delta:=d(S_{\infty},0)>0$.
Hence, there exists $N^{\prime}$ such that for $n\geq N^{\prime}$ we have
$\|f_{n}-f_{\infty}\|_{L^{\infty}(\Omega)}\leq\delta/2$. This means that for
every $x\in\partial B$ and every $n\geq N^{\prime}$, we have

$|f_{n}(x)|\geq|f_{\infty}(x)|-|f_{n}(x)-f_{\infty}(x)|\geq\delta-\delta/2=\delta/2,$

from which we deduce that $d(S_{n},0)\geq\delta/2$ for every $n\geq
N^{\prime}$. Hence, for every $n\in\mathbb{N}\cup\\{\infty\\}$ such that also
$n\geq N^{\prime}$, we can define $g_{n}:\partial B\to\mathbb{S}^{m-1}$ as

$g_{n}(x):=\frac{f_{n}(x)}{|f_{n}(x)|}.$

From the hypothesis we have that $g_{n}\to g_{\infty}$ uniformly on $\partial
B$. Hence, for some $N\geq N^{\prime}$, we have that, for every $n\geq N$, the
map

$\alpha_{n,t}(x):=\frac{tg_{n}(x)+(1-t)g_{\infty}(x)}{|tg_{n}(x)+(1-t)g_{\infty}(x)|}$

is a homotopy from $\alpha_{n,1}=g_{n}$ to $\alpha_{n,0}=g_{\infty}$ and thus
$\deg(g_{n})=\deg(g_{\infty})=i(f_{\infty},0)\neq 0$.

Now we claim that

(64) $B(0,\delta/2)\subset f_{n}(B)\qquad\forall n\geq N.$

Fix $n\geq N$ and suppose by contradiction that (64) is not true. Then there
exists $p\in B(0,\delta/2)\setminus f_{n}(B)$. For every $0\leq\eta\leq 1$ let
us define $\phi_{\eta}:\partial B\to\mathbb{S}^{m-1}$ as

$\phi_{\eta}(x):=\frac{f_{n}(\eta x)-p}{|f_{n}(\eta x)-p|},$

which is well defined since $p\notin f_{n}(B)$. Clearly, as $\eta$ varies from
$0$ to $1$, the maps $\phi_{\eta}$ are an homotopy between the constant map
$\phi_{0}\equiv p/|p|$ and $\phi_{1}$: therefore, $\deg\phi_{1}=0$. Moreover,
for every $0\leq\eta\leq 1$, we can define $\psi_{\eta}:\partial
B\to\mathbb{S}^{m-1}$ as

$\psi_{\eta}(x):=\frac{f_{n}(x)-\eta p}{|f_{n}(x)-\eta p|}.$

Notice that $\psi_{\eta}$ is well-defined because
$|f_{n}(x)|\geq\delta/2>\eta|p|$ for every $0\leq\eta\leq 1$ and for every
$x\in\partial B$. Since the maps $\psi_{\eta}$ are an homotopy from
$\psi_{0}=g_{n}$ to $\psi_{1}=\phi_{1}$, we obtain $\deg
g_{n}=\deg(\phi_{1})=0$, which is in contradiction with the fact that
$\deg(g_{n})\neq 0$ for every $n\geq N$. Therefore, (64) must be true, and it
directly implies the assertion. ∎

In order to apply the previous result, the following lemma gives a criterium
to check the hypothesis of Lemma B.1.

######  Lemma B.2.

Let $\Omega\subset\mathbb{R}^{m}$ be an open set containing $0$, and let
$f:\Omega\to\mathbb{R}^{m}$ a topological embedding with $f(0)=0$. Let $r>0$
such that $B(0,r)\Subset\Omega$ and define
$\phi:\mathbb{S}^{m-1}\to\mathbb{S}^{m-1}$ as

$\phi(x)=\frac{f(rx)}{|f(rx)|}.$

Then $\deg(\phi)\in\\{-1,1\\}$.

###### Proof.

The function $\phi$ is the composition

$\mathbb{S}^{m-1}\overset{rx}{\longrightarrow}\Omega\setminus\\{0\\}\overset{f}{\longrightarrow}f(\Omega)\setminus\\{0\\}\hookrightarrow\mathbb{R}^{m}\setminus\\{0\\}\overset{x/|x|}{\longrightarrow}\mathbb{S}^{m+1}.$

Notice that the maps $x\mapsto rx$ and $x\mapsto\frac{x}{|x|}$ are retracts of
$\mathbb{R}^{m}$ to $\mathbb{S}^{m-1}$, and thus they induce isomorphisms
between the homology groups. Since $f$ is an embedding and $f(0)=0$, also $f$
induces an isomorphism between the homology groups of $\Omega\setminus\\{0\\}$
and those of $f(\Omega)\setminus\\{0\\}$. Finally, since $f(\Omega)$ is open,
every homology class of $\mathbb{R}^{m}$ has a representative inside
$f(\Omega)$ and thus the immersion $f(\Omega)\hookrightarrow\mathbb{R}^{m}$
defines a surjective morphism of the corresponding homology groups.

We conclude that the induced group morphism
$\phi_{*}:H_{n}(\mathbb{S}^{m-1})\to H_{n}(\mathbb{S}^{m-1})$ is a surjective
group morphism from $\mathbb{Z}$ to $\mathbb{Z}$ and thus
$\deg(\phi)\in\\{-1,1\\}$. ∎

The following lemma should be compared with [11, Proposition 3.5]. It gives a
quantitative open mapping theorem for $C^{2}$ functions, where the bounds
depend explictly on the first and second derivatives of the function. A non-
quantitative statement for $C^{1}$ functions is in [11, Proposition 3.5]. Even
if its proof is rather standard, we record it here for the reader’s
convenience. We give the statement for an arbitrary Banach space, and later we
will apply it with $\mathbb{B}=\mathbb{R}^{k}$.

######  Lemma B.3.

Let $(\mathbb{B},\|\cdot\|)$ be a Banach space. For every $\ell,L>0$ there
exist $C_{1}:=1/(2\ell)$ and $C_{2}:=1/(2\ell L)$ such that the following
holds. Let $f:B(x_{0},r)\subset\mathbb{B}\to\mathbb{B}$ be a $C^{2}$ map such
that

  * •

The linear map $Df_{x_{0}}$ is an isomorphism with
$\|\left(Df_{x_{0}}\right)^{-1}\|\leq\ell$,

  * •

We have $\|(D^{2}f)_{x}\|\leq L$ for every $x\in B(x_{0},r)$.

Hence

(65) $B(f(x_{0}),C_{1}\rho)\subseteq f(B(x_{0},\rho)),\qquad\text{for all
$\rho\in(0,\min\\{r,C_{2}\\})$.}$

###### Proof.

Let us denote for simplicity $A:=Df_{x_{0}}$. Define, for $x\in B(0,r)$,

$\widetilde{f}(x):=A^{-1}(f(x+x_{0})-f(x_{0}))-x.$

Notice that $\widetilde{f}(0)=0$, $(D\widetilde{f})_{0}=0$, and

(66) $(D\widetilde{f})_{x}=A^{-1}((Df)_{x+x_{0}}-A).$

Notice that, given $\rho\leq r$, for every $x\in B(0,\rho)$ we have

$\|(Df)_{x+x_{0}}-A\|\leq\sup_{x\in B(0,\rho)}\|(D^{2}f)_{x+x_{0}}\|\|x\|\leq
L\rho.$

Since $\|A^{-1}\|\leq\ell$, the previous inequality together with (66) gives
that for every $x\in B(0,\rho)$, with $\rho\leq r$, the following holds

$\|(D\widetilde{f})_{x}\|\leq\ell L\rho.$

The latter implies that for every $x,z\in B(0,\rho)$, with $\rho\leq r$, we
have

$\|\widetilde{f}(x)-\widetilde{f}(z)\|\leq\ell L\rho|x-z|.$

Let us now prove (65). Fix $\rho<\min\\{1/(2\ell L),r\\}$. Take an arbitrary
$y\in\overline{B}(0,(1-\ell L\rho)\rho)$. Due to the previous inequality, the
fact $\widetilde{f}(0)=0$, and the triangle inequality, the function

$x\mapsto-\widetilde{f}(x)+y$

maps $\overline{B}(0,\rho)$ into $\overline{B}(0,\rho)$. Moreover, the latter
function is a contraction, then it has a fixed point. Namely, for every $y\in
B(0,(1-\ell L\rho)\rho)$, there exists $x\in B(0,\rho)$ such that

$f(x+x_{0})=f(x_{0})+Ay$

Now we claim that the set made by $Ay$’s, when $y$ runs in $B(0,(1-\ell
L\rho)\rho)$, contains the ball $B(0,(2\ell)^{-1}\rho)$ . Indeed, if $\eta\in
B(0,(2\ell)^{-1}\rho)$ we claim that we can take $y=A^{-1}\eta$. Indeed,
$A^{-1}\eta\in B(0,(2\ell)^{-1}\rho\ell)$, since $\|A^{-1}\|\leq\ell$, and
$2^{-1}\rho<(1-\ell L\rho)\rho$ by how we chose $\rho$. ∎

##  Appendix C A shorter proof of the convergence result for smooth vector
fields

In this section we offer a shorter proof of Theorem 1.4 in the case in which
the vector fields are smooth. For the ease of notation, let us introduce the
following terminology, useful for the discussion of this section. Let us fix
$\mathbb{E}:=\mathbb{R}^{k}$, for $k\in\mathbb{N}$, with canonical basis
$\\{e_{1},\dots,e_{k}\\}$.

Let $M$ be a smooth manifold. Let

$\mathtt{f}:M\times\mathbb{\mathbb{E}}\to TM,$

be a smooth morphism of bundles. Notice that in particular we have that
$\mathtt{f}(p,\cdot):\mathbb{E}\to T_{p}M$ is a linear map for every $p\in M$,
and moreover, for every $i=1,\dots,k$, we have that $\mathtt{f}(\cdot,e_{i})$
is a smooth vector field. Let

$N:M\times\mathbb{E}\to[0,+\infty),$

be a continuous function such that $N(p,\cdot)$ is a norm for every $p\in M$.

Any couple $(\mathtt{f},N)$ satisfying the two above conditions will be called
CC-bundle structure, and it induces an energy function $\mathtt{J}$, a length
functional $\ell$ and a distance $d_{(\mathtt{f},N)}$ as discussed in
Definition 3.3, and Definition 3.6.

######  Definition C.1 (Continuosuly varying CC-bundle structure).

Let $\Lambda$ be a compact space, which will be called set of parameters. Let
$M$ be a smooth manifold endowed with a Riemannian metric $\rho$. Endow $TM$
with the bundle metric induced by $\rho$.

Let $\mathtt{f}:\Lambda\times M\times\mathbb{E}\to TM$ and $N:\Lambda\times
M\times\mathbb{E}\to[0,+\infty)$ be such that for every $\lambda\in\Lambda$ we
have that $(\mathtt{f}_{\lambda},N_{\lambda})$ is a CC-bundle structure, where
$\mathtt{f}_{\lambda}:=\mathtt{f}(\lambda,\cdot,\cdot)$ and
$N_{\lambda}:=N(\lambda,\cdot,\cdot)$. We say that the family
$\\{(\mathtt{f}_{\lambda},N_{\lambda})\\}_{\lambda\in\Lambda}$ is a
continuously varying CC-bundle structure if

  1. (1)

In coordinates around every point $p\in M$, all the partial derivatives in $q$
of $X_{\lambda}^{i}(q):=\mathtt{f}_{\lambda}(q,e_{i})$ of order at most two
are continuous in $(\lambda,q)$, for every $i=1,\dots,k$;

  2. (2)

$N\in C^{0}(\Lambda\times M\times\mathbb{E})$ and $\mathtt{f}\in
C^{0}(\Lambda\times M\times\mathbb{E})$;

From item (1) above we consequently have the following. For every compact
$K_{1}\subseteq M$, and every compact $K_{2}\subseteq\Lambda\times\mathbb{E}$
there exists $L$ such that for every $(\lambda,v)\in K_{2}$ the vector field

(67) $K_{1}\ni p\mapsto\mathtt{f}(\lambda,p,v)\in TM$

is $L$-Lipschitz with respect to the Riemannian distances.

We now prove the following theorem, that is essentially a restatement of
Theorem 1.4(iv) for smooth vector fields.

######  Theorem C.2.

Let $\Lambda$ be a compact space, and let
$\\{(\mathtt{f}_{\lambda},N_{\lambda})\\}_{\lambda\in\Lambda}$ be a
continuously varying CC-bundle structure on a smooth manifold $M$. Let
$d_{\lambda}:=d_{(\mathtt{f}_{\lambda},N_{\lambda})}$ for $\lambda\in\Lambda$.
Let $\lambda_{0}\in\Lambda$ be such that
$\\{\mathtt{f}(\lambda_{0},M,e_{i})\\}_{i=1,\dots,k}$ is bracket-generating,
and assume that the metric space $(M,d_{\lambda_{0}})$ is boundedly compact.

Then $d_{\lambda}\to d_{\lambda_{0}}$ uniformly on compact sets of $M$ as
$\lambda\to\lambda_{0}$.

We give a direct proof of the previous theorem using the following crucial
lemma, that contains some ideas already used in Lemma 3.21, and Proposition
3.22. We stress that the proof given below is more direct because we are
essentially able to bypass the use of Theorem 2.5.

######  Lemma C.3.

Let $\Lambda$ be a compact space, and let
$\\{(\mathtt{f}_{\lambda},N_{\lambda})\\}_{\lambda\in\Lambda}$ be a
continuously varying CC-bundle structure on a smooth manifold $M$. Let
$d_{\lambda}:=d_{(\mathtt{f}_{\lambda},N_{\lambda})}$ for $\lambda\in\Lambda$.
Assume $\\{\mathtt{f}(\lambda_{0},M,e_{i})\\}_{i=1,\dots,k}$ is bracket-
generating for some $\lambda_{0}\in\Lambda$.

For every compact set $K\subseteq M$ and every Riemannian metric $\rho$ on $M$
there exist a compact neighborhood $I_{\lambda_{0}}\subseteq\Lambda$ of
$\lambda_{0}$ and a continuous nondecreasing function
$\beta:(0,+\infty)\to(0,+\infty)$, with $\lim_{s\to 0^{+}}\beta(s)=0$, such
that

(68) $d_{\lambda}(p,q)\leq\beta(\rho(p,q)),\qquad\text{for all $p,q\in K$, and
$\lambda\in I_{\lambda_{0}}$}.$

###### Proof.

Fix such $\rho$ and $K$. Let us denote, for $\lambda\in\Lambda$ and $p\in M$,

$X_{i}^{\lambda}(p):=\mathtt{f}(\lambda,p,e_{i}),$

where $\\{e_{1},\dots,e_{k}\\}$ is the standard basis of
$\mathbb{E}=\mathbb{R}^{k}$.

We know that $\\{X_{i}^{\lambda_{0}}\\}_{i=1}^{k}$ is a bracket-generating set
of vector fields on $M$. Hence, there exists a compact neighborhood $I_{1}$ of
$\lambda_{0}$ such that $\\{X_{i}^{\lambda}\\}_{i=1}^{k}$ is a bracket-
generating set of vector fields on $B_{\rho}(K,1)$.

Claim 1. For every $x\in K$, for every $\lambda^{\prime}\in I_{1}$, and for
every $\varepsilon>0$ there exist $\delta>0$ and a compact neighborhood
$I_{\lambda^{\prime}}\subseteq I_{1}$ of $\lambda^{\prime}$ such that

(69) $B_{\rho}(x,\delta)\subseteq
B_{d_{\lambda}}(x,\varepsilon),\quad\text{for all $\lambda\in
I_{\lambda^{\prime}}$.}$

Let us prove the claim. Let us fix $x\in K$ and $\lambda^{\prime}\in I_{1}$
from now on. Hence, since $\\{X_{i}^{\lambda^{\prime}}\\}_{i=1}^{k}$ are
bracket-generating on $B_{\rho}(K,1)$, the following holds, due to [1, Lemma
3.33]. For every $0<\eta<1$, there exist
$\\{i_{1},\dots,i_{n}\\}\subseteq\\{1,\dots,k\\}$ and
$\hat{t}:=(\hat{t}_{1},\dots,\hat{t}_{n})$ such that $n:=\dim M$,
$|\hat{t}|<\eta$, and

$\Phi:(t_{1},\dots,t_{n})\mapsto\Phi_{X_{i_{n}}^{\lambda^{\prime}}}^{t_{n}}\circ\dots\circ\Phi_{X_{i_{1}}^{\lambda^{\prime}}}^{t_{1}}(x),$

has a regular point at $\hat{t}$. Hence the map

$(t_{1},\dots,t_{n})\mapsto\Phi^{-\hat{t}_{1}}_{X_{i_{1}}^{\lambda^{\prime}}}\circ\dots\circ\Phi^{-\hat{t}_{n}}_{X_{i_{n}}^{\lambda^{\prime}}}\circ\Phi_{X_{i_{n}}^{\lambda^{\prime}}}^{t_{n}}\circ\dots\circ\Phi_{X_{i_{1}}^{\lambda^{\prime}}}^{t_{1}}(x),$

has a regular point at $\hat{t}$ and sends $\hat{t}$ to $x$. Now, let
$\omega:=\omega_{1}\dots\omega_{D}$ be a word of $D$ letters such that it
contains as a subword every string of $2n$ elements chosen among
$\\{1,\dots,k\\}$. Notice now that the map

(70)
$\Psi:(\lambda,t_{1},\dots,t_{D})\mapsto\Phi_{X_{i_{\omega_{D}}}^{\lambda}}^{t_{D}}\circ\dots\circ\Phi_{X_{i_{\omega_{1}}}^{\lambda}}^{t_{1}}(x),$

together with the maps $D_{T}\Psi,D^{2}_{T}\Psi$ \- where $D_{T}$ denotes the
differential with respect to the components in $\mathbb{R}^{D}$ \- are
continuous and well defined on $I_{2}\times\overline{B}_{|\cdot|_{1}}(0,\xi)$,
where $\overline{B}_{|\cdot|_{1}}(0,\xi)$ is the ball in $\mathbb{R}^{D}$ with
respect to the $\ell_{1}$-norm $|\cdot|_{1}$ centered at $0$ and with a
sufficiently small radius $\xi$, and $I_{2}\subseteq I_{1}$ is a sufficiently
small compact neighborhood of $\lambda^{\prime}$. The last assertion is a
consequence of an iterated application of Gronwall’s Lemma (see Lemma A.1),
and the fact that we have the continuity property in Definition C.1(1).

Define the compact set $\widetilde{K}:=\Psi(I_{2}\times\overline{B}(0,\xi))$
in $M$. Hence, by continuity of the norm $N$, there exists $L>0$ such that

(71) $N(\lambda,p,v)\leq L|v|_{1},\quad\text{for all $\lambda\in I_{2}$,
$p\in\widetilde{K}$, and $v\in\mathbb{E}$}.$

Let us now conclude the proof of the claim. Fix $\varepsilon>0$. Let

$\nu:=\min\\{\xi/2,\varepsilon/(2DL)\\},$

where $\xi$ is defined above. Notice that, by what we noticed above, we have
that there exists $\widetilde{t}\in\mathbb{R}^{D}$ with
$|\widetilde{t}|_{1}<\nu$ such that $\Psi(\lambda^{\prime},\cdot)$ has a
regular point at $\widetilde{t}$ and $\Psi(\lambda^{\prime},\widetilde{t})=x$.
In addition to this, $\Psi(\lambda,t)\to\Psi(\lambda^{\prime},t)$ as
$\lambda\to\lambda^{\prime}$, uniformly when
$t\in\overline{B}_{|\cdot|_{1}}(0,\xi)\subseteq\mathbb{R}^{N}$, and the same
convergence holds with $D_{T}\Psi,D^{2}_{T}\Psi$. The last assertion is a
consequence of the fact that the maps $\Psi,D_{T}\Psi,D^{2}_{T}\Psi$ are
continuous, and thus uniformly continuous on compact sets.

Since $\widetilde{t}$ is a regular point for $\Psi(\lambda^{\prime},\cdot)$,
we can find an $n$-dimensional subspace $\Pi$ of $\mathbb{R}^{D}$ such that
$\Psi(\lambda^{\prime},\cdot)$ restricted to
$B(\widetilde{t},\nu)\cap(\widetilde{t}+\Pi)$ is a local diffeomorphism around
$\widetilde{t}$. Moreover, there exists a neighborhood of $\lambda^{\prime}$,
which we call $I_{\lambda^{\prime}}$, such that the maps
$\Psi(\lambda,\cdot)$, restricted to
$B(\widetilde{t},\nu)\cap(\widetilde{t}+\Pi)$, satisfy the bounds in the
hypotheses of Lemma B.3 uniformly on $\lambda\in I_{\lambda^{\prime}}$, and
where $\widetilde{t}$ here is the $x_{0}$ there. Notice that the bounds of
Lemma B.3 hold uniformly on $\lambda\in I_{\lambda^{\prime}}$ for some
neighborhood $I_{\lambda^{\prime}}$ of $\lambda^{\prime}$, due to the
continuity of $D_{T}\Psi,D^{2}_{T}\Psi$ discussed above.

Hence, by applying Lemma B.3, we have that there exists $\delta>0$ and
$I_{\lambda^{\prime}}$ a neighborhood of $\lambda^{\prime}$ such that

$B_{\rho}(x,\delta)\subseteq\Psi(\lambda,\widetilde{t}+B_{|\cdot|_{1}}(0,\nu)),\quad\text{for
all $\lambda\in I_{\lambda^{\prime}}$}.$

Since
$\widetilde{t}+B_{|\cdot|_{1}}(0,\nu)\subseteq\overline{B}_{|\cdot|_{1}}(0,\xi)$,
we have that all the concatenation of the curves associated to
$\Psi(\lambda,\widetilde{t}+B(0,\nu))$ is in $K$. Since the estimate (71)
holds, and since we also have
$\widetilde{t}+B_{|\cdot|_{1}}(0,\nu)\subseteq\overline{B}_{|\cdot|_{1}}(0,\varepsilon/(DL))$,
we get that the concatenation of the curves associated to $\Psi(\lambda,t)$
(whose controls can be written explicitely as in (8)), for every
$t\in\widetilde{t}+\overline{B}_{|\cdot|_{1}}(0,\nu)$ has length
$\leq\varepsilon$ for every $\lambda\in I_{\lambda^{\prime}}$. Hence the
sought claim (69) holds.

From Claim 1 and a routine compatness argument, as already done at the end of
Lemma 3.21, we have that for every $\varepsilon>0$ there exists $\delta>0$
such that for every $x\in K$ and every $\lambda\in I_{1}$ we have

(72) $B_{\rho}(x,\delta)\subseteq B_{d_{\lambda}}(x,\varepsilon).$

From (72) the proof of the lemma follows with the following argument. Let
$\tilde{K}$ be a path-connected compact set containing $K$. For instance,
$\tilde{K}$ can be chosen to be a closed $\rho$-balls of sufficiently large
radius.

The inequality in (68) is trivially satisfied if we define $\beta$ as

$\displaystyle\beta(s)$
$\displaystyle=\sup\\{d_{\lambda}(p,q):p,q\in\tilde{K},\lambda\in I_{1},\
\rho(p,q)\leq s\\},$

for every $s\in(0,+\infty)$.

From (72) we get that for every $\varepsilon>0$ there is $\delta>0$ such that
$\beta(\delta)<\varepsilon$. In particular, this implies $\lim_{s\to
0^{+}}\beta(s)=0$.

We also claim that $\sup_{s>0}\beta(s)<\infty$. Indeed, fix $\delta>0$ such
that $\beta(\delta)\leq 1$, and let $B_{1},\dots,B_{N}$ be a collection of
$\rho$-balls of radius less than $\delta/2$ that covers $\tilde{K}$. Since
$\tilde{K}$ is path-connected, given $p,q\in\tilde{K}$, there is a sequence
$p=p_{0},p_{1},\dots,p_{m}=q$ with $m\leq 2N$ such that for every
$i\in\\{0,\dots,m-1\\}$ there is $j\in\\{1,\dots,N\\}$ such that
$p_{i},p_{i+1}\in B_{j}$. Therefore, for every $\lambda\in I_{1}$,

$d_{\lambda}(p,q)\leq\sum_{i=0}^{m-1}d_{\lambda}(p_{i},p_{i+1})\leq
2N\beta(\delta)\leq 2N.$

Therefore, $\sup_{s>0}\beta(s)\leq 2n$. ∎

We now give the proof of Theorem C.2. The strategy is different with respect
to the proof of Theorem 1.4(iv). There, we first proved the local uniform
convergence, relying on the relaxation result in Proposition 3.20, and then we
upgrade it to a uniform convergence on compact sets thanks to Lemma 3.25.
Here, instead, we directly obtain the uniform convergence on compact sets by
making a careful use of Gronwall’s Lemma A.1.

###### Proof of Theorem C.2.

We embed $M$ smoothly isometrically into some $\mathbb{R}^{N}$, on which we
denote with $|\cdot|$ the standard norm. Let us fix a compact set $K$ and a
Riemannian metric $\rho$ on $M$. Notice that on every compact set of $M$,
$\rho$ and $|\cdot|$ are biLipschitz equivalent. Let us fix
$0<\varepsilon<1/2$.

By continuity, there exists a constant $C>0$ such that
$d_{\lambda_{0}}(p,q)\leq C$ for every $p,q\in K$. Let
$K^{\prime}:=\overline{B}_{\lambda_{0}}(K,C+1)$ the closed tubular
neighborhood of $K$ of radius $C+1$. Since $(M,d_{\lambda_{0}})$ is boundedly
compact, we deduce that $K^{\prime}$ is compact.

Let $\beta$ be the function, and $I_{\lambda_{0}}$ be the compact neighborhood
of $\lambda_{0}$, associated to $K^{\prime}$ given by Lemma C.3. We have that,
for some $\vartheta>0$, $\rho(p,q)\leq\vartheta|p-q|$ for every $p,q\in
K^{\prime}$. Thus, up to renaming $\beta$, for every $p,q\in K^{\prime}$, and
for every $\lambda\in I_{\lambda_{0}}$,

$d_{\lambda}(p,q)\leq\beta(|p-q|)\leq\beta(\operatorname{\mathrm{diam}}_{|\cdot|}K^{\prime}).$

Since $N(\lambda,p,\cdot)$ is a norm for every $\lambda\in I_{\lambda_{0}}$
and every $p\in M$, and since $N$ is continuous, we get that there exists a
compact set $K^{\prime\prime}\subseteq\mathbb{E}$ such that

(73) $\text{if
$N(\lambda,x,v)\leq\beta(\operatorname{\mathrm{diam}}_{|\cdot|}K^{\prime})+1$
for some $\lambda\in I_{\lambda_{0}}$ and $x\in K^{\prime}$, then $v\in
K^{\prime\prime}$}.$

Moreover, by definition of continuously varying CC-structures, (67), we have
that there exists $L>0$ such that for every $\lambda\in I_{\lambda_{0}}$ and
$v\in K^{\prime\prime}$ the map

$K^{\prime}\ni p\mapsto\mathtt{f}(\lambda,p,v),$

is $L$-lipschitz.

Because of continuity of the functions $N$ and $\mathtt{f}$ we get that there
exist $0<\delta_{1}<\varepsilon$ and a compact neighborhood
$I^{\prime}_{\lambda_{0}}\subseteq I_{\lambda_{0}}$ of $\lambda_{0}$ with

(74) $|N(\lambda_{0},x,v)-N(\lambda,y,v)|<\varepsilon,\quad\text{for all
$\lambda\in I^{\prime}_{\lambda_{0}}$, $x\in K^{\prime}$, $v\in
K^{\prime\prime}$, $y\in\overline{B}_{|\cdot|}(x,\delta_{1})$},$

and

(75) $|\mathtt{f}(\lambda_{0},x,v)-\mathtt{f}(\lambda,x,v)|<a,\quad\text{for
all $\lambda\in I^{\prime}_{\lambda_{0}}$, $x\in K^{\prime}$, $v\in
K^{\prime\prime}$,}$

where $a$ is chosen such that $a\frac{e^{L}-1}{L}<\delta_{1}$. We now prove
the following claim.

Claim 1. For every $\lambda\in I^{\prime}_{\lambda_{0}}$ and every $p,q\in K$,
we have

$d_{\lambda_{0}}(p,q)\leq d_{\lambda}(p,q)+2\varepsilon+\beta(\varepsilon).$

Fix $p,q,\lambda$ as in the claim. Up to reparametrization, we can take a
curve $\gamma_{\lambda}$ connecting $p$ and $q$ such that
$\gamma_{\lambda}^{\prime}=\mathtt{f}(\lambda,\gamma_{\lambda},u_{\lambda})$
and

(76) $N(\lambda,\gamma_{\lambda}(t),u_{\lambda}(t))\leq
d_{\lambda}(p,q)+\varepsilon,\quad\text{for a.e. $t\in[0,1]$}.$

Let $B:=\overline{B}_{\lambda_{0}}(p,d_{\lambda_{0}}(p,q))$. Notice that
$B\subseteq K^{\prime}$. Define

$\overline{t}:=\max\\{t\in[0,1]:\gamma_{\lambda}(s)\in B\,\,\forall
s\in[0,t]\\}.$

Denote $q^{\prime}_{\lambda}:=\gamma_{\lambda}(\overline{t})$ and notice that
$d_{\lambda_{0}}(p,q^{\prime}_{\lambda})=d_{\lambda_{0}}(p,q)$. Moreover
notice that $(\gamma_{\lambda})|_{[p,q_{\lambda}^{\prime}]}\subseteq
K^{\prime}$. Take now $\gamma_{\lambda,0}$ such that
$\gamma_{\lambda,0}^{\prime}=\mathtt{f}(\lambda_{0},\gamma_{\lambda,0},u_{\lambda})$
and $\gamma_{\lambda,0}(0)=p$. Call
$\overline{q}_{\lambda}:=\gamma_{\lambda,0}(\overline{t})$. Notice that as a
consequence of Gronwall’s Lemma, see Lemma A.1, up to taking a slightly
smaller neighborhood $I^{\prime}_{\lambda_{0}}$, the curve
$\gamma_{\lambda,0}$ is defined up to time $\overline{t}$. We will use the
same argument below.

We now want to estimate $|\overline{q}_{\lambda}-q^{\prime}_{\lambda}|$. From
(76), (73), and the fact that $\gamma_{\lambda}([0,\overline{t}])\subseteq
K^{\prime}$ we get that $u_{\lambda}(t)\in K^{\prime\prime}$ for a.e.
$t\in[0,\overline{t}]$. Hence we can estimate, for every $x,y\in K^{\prime}$
and a.e. $t\in[0,\overline{t}]$,

(77)
$\begin{split}|\mathtt{f}(\lambda,x,u_{\lambda}(t))-\mathtt{f}(\lambda_{0},y,u_{\lambda}(t))|&\leq|\mathtt{f}(\lambda,x,u_{\lambda}(t))-\mathtt{f}(\lambda_{0},x,u_{\lambda}(t))|\\\
&+|\mathtt{f}(\lambda_{0},x,u_{\lambda}(t))-\mathtt{f}(\lambda_{0},y,u_{\lambda}(t))|\\\
&\leq a+L|x-y|.\end{split}$

Hence Gronwall Lemma in A.1 applied on $K^{\prime}$ directly implies that

(78) $|\gamma_{\lambda}(t)-\gamma_{\lambda,0}(t)|\leq
a\frac{e^{Lt}-1}{L}<\delta_{1}<\varepsilon,\quad\text{for a.e.
$t\in[0,\overline{t}]$},$

and moreover that $(\gamma_{\lambda,0})|_{[0,\overline{t}]}\subseteq
K^{\prime}$. Now let us conclude the estimate of Claim 1. We have

(79)
$\begin{split}d_{\lambda_{0}}(p,q)&=d_{\lambda_{0}}(p,q_{\lambda}^{\prime})\leq
d_{\lambda_{0}}(p,\overline{q}_{\lambda})+d_{\lambda_{0}}(\overline{q}_{\lambda},q^{\prime}_{\lambda})\\\
&\leq\int_{0}^{\overline{t}}N(\lambda_{0},\gamma_{\lambda,0}(s),u_{\lambda}(s))\,\mathrm{d}s+\beta(|\overline{q}_{\lambda}-q^{\prime}_{\lambda}|)\\\
&\leq\int_{0}^{\overline{t}}N(\lambda,\gamma_{\lambda}(s),u_{\lambda}(s))\,\mathrm{d}s+\varepsilon+\beta(\varepsilon)\\\
&\leq\int_{0}^{1}N(\lambda,\gamma_{\lambda}(s),u_{\lambda}(s))\,\mathrm{d}s+\varepsilon+\beta(\varepsilon)\\\
&\leq d_{\lambda}(p,q)+2\varepsilon+\beta(\varepsilon),\end{split}$

where we are using (78), (74), and (76). We thus obtained the sought claim.

Claim 2. For every $\lambda\in I^{\prime}_{\lambda_{0}}$ and every $p,q\in K$,
we have

$d_{\lambda}(p,q)\leq d_{\lambda_{0}}(p,q)+\varepsilon+\beta(\varepsilon).$

Fix $p,q,\lambda$ as in the claim. Up to reparametrization, for every
$0<\varepsilon<1$ we can take a curve $\gamma$ connecting $p$ and $q$ such
that $\gamma^{\prime}=\mathtt{f}(\lambda_{0},\gamma,u)$ and

(80) $N(\lambda_{0},\gamma(t),u(t))\leq
d_{\lambda_{0}}(p,q)+\varepsilon,\quad\text{for a.e. $t\in[0,1]$}.$

Notice that $\gamma\subseteq K^{\prime}$. Take now $\gamma_{\lambda}$ such
that $\gamma_{\lambda}^{\prime}=f(\lambda,\gamma_{\lambda},u)$ and
$\gamma_{\lambda}(0)=p$. Again, as a consequence of Gronwall Lemma, the curve
$\gamma_{\lambda}$ is defined up to time $t=1$. Call
$\overline{q}_{\lambda}:=\gamma_{\lambda}(1)$.

We now want to estimate $|\overline{q}_{\lambda}-q|$. Arguing verbatim as
before we obtain

(81) $|\gamma_{\lambda}(t)-\gamma(t)|\leq
a\frac{e^{Lt}-1}{L}<\delta_{1}<\varepsilon,\quad\text{for a.e. $t\in[0,1]$},$

and moreover $\gamma_{\lambda}\subseteq K^{\prime}$. Now let us conclude the
estimate of Claim 2. We have

(82) $\begin{split}d_{\lambda}(p,q)&\leq
d_{\lambda}(p,\overline{q}_{\lambda})+d_{\lambda}(\overline{q}_{\lambda},q)\\\
&\leq\int_{0}^{1}N(\lambda,\gamma_{\lambda}(s),u(s))\,\mathrm{d}s+\beta(|\overline{q}_{\lambda}-q|)\\\
&\leq\int_{0}^{1}N(\lambda_{0},\gamma(s),u(s))\,\mathrm{d}s+\varepsilon+\beta(\varepsilon)\\\
&\leq d_{\lambda_{0}}(p,q)+2\varepsilon+\beta(\varepsilon),\end{split}$

where we are using (81), (74), and (80). Thus we obtained the sought claim.

From Claim 1 and Claim 2 jointly with the fact that $\beta(\varepsilon)\to 0$
as $\varepsilon\to 0$ we get the proof of the Theorem. ∎

## References

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# Speech Wikimedia: A 77 Language Multilingual Speech Dataset

Rafael Mosquera Gómez  Julian Eusse  Juan Ciro  Daniel Galvez  Ryan Hileman
Kurt Bollacker  David Kanter

###### Abstract

The Speech Wikimedia Dataset is a publicly available compilation of audio with
transcriptions extracted from Wikimedia Commons. It includes 1780 hours (195
GB) of CC-BY-SA licensed transcribed speech from a diverse set of scenarios
and speakers, in 77 different languages. Each audio file has one or more
transcriptions in different languages, making this dataset suitable for
training speech recognition, speech translation, and machine translation
models.

Machine Learning, ICML

##  1 Introduction

Speech research has witnessed remarkable advancements in recent years, largely
driven by the availability of vast amounts of data. Tasks such as automatic
speech recognition (ASR), speaker recognition, and speech translation have
reached robust results even in the presence of background noise, jargon, and
different accents. Nevertheless, one of the fundamental challenges in speech
research is the scarcity of multilingual datasets.

In this paper, we introduce the Speech Wikimedia Dataset, a supervised dataset
consisting of 1780 hours of audio files, each with one one or more
transcripts. The last point bears repeating: much (approximately 25%) of the
dataset has multiple associated transcripts, each having its own language,
which is rare in datasets sourced from the internet. The dataset is
specifically curated from Wikimedia Commons to address some of the key
challenges in the speech recognition, machine translation, and speech
translation spaces, particularly the need for diverse multilingual data and
appropriate licensing for academic and commercial usage.

The paper is organized as follows. In section 2, we describe the dataset
itself and training tasks that it can support; in section 3, we describe
related work; in section 4, we describe some limitations of the dataset today;
in section 5, we conclude with some future work that this dataset can enable.

##  2 Dataset Description

To construct the Speech Wikimedia dataset, we downloaded raw video and audio
from Wikimedia Commons (“https://commons.wikimedia.org”), which allows only
content that is ”free” (wik, ). For our purposes, this means data under CC-BY
or CC-BY-SA license, or otherwise public domain. After downloading, the data
was converted to 16kHz monochannel FLAC using ffmpeg. The data is uploaded to
Huggingface at https://huggingface.co/datasets/MLCommons/speech-wikimedia

We give statistics for three possible tasks that this dataset can be used for:
speech recognition, speech translation, and machine translation.

###  2.1 Licensing Information

Given that all data is public domain, CC-BY-licensed, or CC-BY-SA licensed, we
are licensing the dataset as CC-BY-SA. Following the requirements imposed by
the CC-BY and CC-BY-SA licenses of our sources, accreditation is provided in
the linked credits.json file.

###  2.2 Audio with Subtitles in the Same Language for Speech Recognition Task

In order to determine the amount of data available for the ASR task we used
only those audios and transcriptions where the language coincided. Since
Wikimedia Commons’s transcripts’ filenames contain the language of the
contained text, we simply extracted the languages from these filenames. For
example, the file “Elephants_Dream.ogv.en.srt” is in English, as indicated by
the “.en.” substring.

Given that we didn’t initially have the audio language for each file,
Whisper’s (Radford et al., 2022) language detection pipeline was used. A total
of 77 different languages were detected, with English, Dutch, German, Russian,
and Spanish being the most common. 69.07% of the 1780 hour dataset comprises
audio-transcription pairs in the same language. We present the number of
transcribed hours of each language in Table 1 in the appendix.

###  2.3 Audio with Subtitles in Different Languages for Speech Translation
Task

For speech translation, we focused on audio-transcript pairs that had
different languages, which corresponds to 31% of the rest of the 1780 hours.
After filtering audios and transcriptions with unknown languages, we were left
with a total of 628.8 hours of audio with transcripts in different languages.
We present the hours of audio for the 20 most common language pairs in Table 2
in the appendix.

###  2.4 Transcript Language Pairs (Bitexts) for Machine Translation Task

While the speech translation task relies on audio from one language with a
transcription from another language, machine translation focuses on pure text
translation. We find paired texts by enumerating all pairs of transcripts
associated with a single audio. This is interesting in particular because
approximately 10.93% of the audio files have transcripts in at least three
languages.

In Table 3 in the appendix, we present total hours, number of words and
occurrences of different pairs of languages for the 20 most common language
pairs in the dataset.

###  2.5 Topic Distribution and Audio Content

We were also interested in determining which topics were covered across the
dataset. In order to analyze this, we ran a zero shot classifier (Maiya, 2020)
with labels ranging different topics, and recorded the hours of audio for each
of the topics. Results are depicted in Table 4 in the Appendix section.
Popular topics were current events, history, and general non-fiction
references.

Based on listening to several files, we also discovered that several audios
are public speeches, music, and clearly pronounced single words, probably for
pronunciation dictionaries like wiktionary.

##  3 Related Work

In this section we provide an overview of previous, similar datasets.

Mozilla Common Voice (Ardila et al., 2020) is a CC0-licensed 17,690-hour
public domain corpus of single speaker read speech in 108 languages created by
volunteers. In contrast, Speech Wikimedia has much more diverse audio sources.

Multilingual Librispeech (Pratap et al., 2020) is a CC-BY dataset of 50,500
hours of transcribed read speech in eight languages; 6,000 of its hours are
non-English. Meanwhile, our dataset contains 77 languages, and the majority of
the data is also in English.

VoxPopuli (Wang et al., 2021) is CC0-licensed dataset containing an
unsupervised set of 400,000 hours in 23 languages, and 1,500 hours of
transcribed audio in 15 languages. Like our dataset, it also contains a subset
suitable for a speech translation task.

Multilingual Spoken Words Corpus (Mazumder et al., 2021) is a CC-BY licensed
6,000-hour dataset, containing more than 340,000 keywords in 50 different
languages. It is for training keyword spotting models, not speech recognition
models.

opensubtitles (Lison & Tiedemann, 2016) is a machine translation dataset
containing 1,782 language pairings extracted from movie subtitles in 62
languages. Given the data source, it is not licensed for commercial usage. In
contrast, the Speech Wikimedia Dataset has 929 language pairings from 77
languages.

##  4 Limitations

The raw data is available publicly online on Hugging Face as mentioned before;
however, this data is not yet processed via forced alignment of audio to
transcript and bitext word alignment for transcript to transcript, and thus
not able to be used immediately in training models.

We removed all video data when converting to FLAC. In future work, this data
could be helpful for a multimodal task.

While collecting this dataset, we realized that there is also a collection of
audio data in Wikimedia Commons without any transcripts. We have not explored
this subset and have not made it available at this time, however.

Given the small size of the dataset, we are not providing a training-test
split.

##  5 Conclusions

We introduce the Speech Wikimedia Dataset, a collection of audio files with
transcriptions in multiple languages extracted form Wikimedia Commons. The
dataset encompasses over 1,780 hours of transcribed speech in multiple
languages. The CC-BY-SA license enables commercial usage. This is the first
non-read multilingual speech dataset allowing for commercial usage that we are
aware of other than VoxPopuli.

## References

  * (1) Wikimedia commons:licensing.  https://commons.wikimedia.org/wiki/Commons:Licensing.  Accessed: 2023-05-23.
  * Ardila et al. (2020) Ardila, R., Branson, M., Davis, K., Henretty, M., Kohler, M., Meyer, J., Morais, R., Saunders, L., Tyers, F. M., and Weber, G.  Common voice: A massively-multilingual speech corpus, 2020.
  * Lison & Tiedemann (2016) Lison, P. and Tiedemann, J.  OpenSubtitles2016: Extracting large parallel corpora from movie and TV subtitles.  In _Proceedings of the Tenth International Conference on Language Resources and Evaluation (LREC’16)_ , pp. 923–929, Portorož, Slovenia, May 2016. European Language Resources Association (ELRA).  URL https://aclanthology.org/L16-1147.
  * Maiya (2020) Maiya, A. S.  ktrain: A low-code library for augmented machine learning.  _CoRR_ , abs/2004.10703, 2020.  URL https://arxiv.org/abs/2004.10703.
  * Mazumder et al. (2021) Mazumder, M., Chitlangia, S., Banbury, C., Kang, Y., Ciro, J. M., Achorn, K., Galvez, D., Sabini, M., Mattson, P., Kanter, D., Diamos, G., Warden, P., Meyer, J., and Reddi, V. J.  Multilingual spoken words corpus.  In _Thirty-fifth Conference on Neural Information Processing Systems Datasets and Benchmarks Track (Round 2)_ , 2021.  URL https://openreview.net/forum?id=c20jiJ5K2H.
  * Pratap et al. (2020) Pratap, V., Xu, Q., Sriram, A., Synnaeve, G., and Collobert, R.  ISCA, oct 2020.  doi: 10.21437/interspeech.2020-2826.  URL https://doi.org/10.21437%2Finterspeech.2020-2826.
  * Radford et al. (2022) Radford, A., Kim, J. W., Xu, T., Brockman, G., McLeavey, C., and Sutskever, I.  Robust speech recognition via large-scale weak supervision, 2022.
  * Wang et al. (2021) Wang, C., Rivière, M., Lee, A., Wu, A., Talnikar, C., Haziza, D., Williamson, M., Pino, J. M., and Dupoux, E.  Voxpopuli: A large-scale multilingual speech corpus for representation learning, semi-supervised learning and interpretation.  _CoRR_ , abs/2101.00390, 2021.  URL https://arxiv.org/abs/2101.00390.

##  Appendix A Figures

Table 1: Automatic Speech Recognition Task
| Hours of audio
---|---
Language |
English (en) | 1488.765773
Dutch (nl) | 22.167223
German (de) | 12.658670
French (fr) | 7.163889
Russian (ru) | 6.985941
Spanish (es) | 6.184720
Latin (la) | 3.066669
Polish (pl) | 3.045028
Japanese (ja) | 2.216300
Bengali (bn) | 2.126192
Swedish (sv) | 1.468487
Chinese (zh) | 1.456599
Italian (it) | 1.419221
Portuguese (pt) | 1.344584
Welsh (cy) | 1.141955
Basque (eu) | 1.008435
Hindi (hi) | 0.886795
Arabic (ar) | 0.572991
Ukrainian (uk) | 0.441770
Slovenian (sl) | 0.377644
Korean (ko) | 0.367545
Hebrew (he) | 0.238240
Indonesian (id) | 0.207363
Thai (th) | 0.177196
Catalan (ca) | 0.161531
Greek (el) | 0.160628
Danish (da) | 0.150981
Persian (fa) | 0.132622
Vietnamese (vi) | 0.131922
Marathi (mr) | 0.124219
Punjabi (pa) | 0.090774
Malayalam (ml) | 0.078354
Telugu (te) | 0.065369
Kannada (kn) | 0.033602
Hungarian (hu) | 0.030055
Estonian (et) | 0.029325
Turkish (tr) | 0.024743
Finnish (fi) | 0.022719
Czech (cs) | 0.021120
Telugu (tl) | 0.016138
Romanian (ro) | 0.015280
Slovak (sk) | 0.000766
Tamil (ta) | 0.000364
Table 2: Speech Translation Task
Audio Language | Transcript Language | Duration(hours)
---|---|---
English | Spanish | 67.115705
English | Arabic | 43.398845
English | French | 38.163062
English | Portuguese | 30.952778
English | Dutch | 24.165356
English | German | 23.678866
English | Italian | 23.442334
English | Russian | 15.557022
Dutch | English | 14.409074
English | Polish | 12.865772
Latin | English | 11.722308
English | Chinese | 11.182589
Hindi | English | 10.256298
English | Turkish | 9.471801
English | Japanese | 8.782186
Welsh | English | 8.761795
English | Vietnamese | 6.731008
Russian | English | 6.037366
Dutch | Russian | 5.438943
Table 3: Transcript Language Pairs Statistics
Language Pair | Total Hours | Source Language Token Count | Target Language Token Count | Bitexts
---|---|---|---|---
English-Spanish | 135.989042 | 481391.0 | 486965.0 | 629
English-French | 85.782796 | 262040.0 | 255998.0 | 343
English-Portuguese | 57.887887 | 200853.0 | 194911.0 | 197
English-Russian | 55.501208 | 149706.0 | 119638.0 | 348
German-English | 55.449766 | 149499.0 | 168156.0 | 394
Spanish-Portuguese | 54.185978 | 200486.0 | 191356.0 | 166
Spanish-French | 51.878961 | 178583.0 | 182886.0 | 213
English-Dutch | 49.582888 | 182302.0 | 166567.0 | 164
English-Italian | 47.008579 | 131800.0 | 125312.0 | 200
French-Portuguese | 38.802198 | 147000.0 | 138013.0 | 146
Arabic-English | 38.239120 | 106115.0 | 136589.0 | 182
German-Spanish | 36.046692 | 110171.0 | 127857.0 | 211
Arabic-Spanish | 34.548516 | 109102.0 | 136234.0 | 139
Arabic-French | 34.121227 | 110543.0 | 138088.0 | 134
German-French | 33.843628 | 94528.0 | 111353.0 | 204
French-Italian | 33.791085 | 117284.0 | 113286.0 | 150
Spanish-Italian | 33.368969 | 117633.0 | 109450.0 | 162
Arabic-Portuguese | 29.675284 | 96408.0 | 113835.0 | 98
German-Italian | 28.917809 | 85169.0 | 96215.0 | 154
French-Russian | 27.784403 | 80372.0 | 63862.0 | 155
Table 4: Distribution of topics and their durations
Topic | Duration(hours)
---|---
current events | 641.809422
other | 406.496021
history | 154.644719
health | 151.203017
general reference | 114.263664
society | 95.286515
political | 46.760335
technology | 46.079005
number | 45.406569
business | 40.940404
science | 34.944243
culture | 27.520957
languages | 26.272240
city | 24.718620
location | 15.258170
software | 8.970613
geography | 8.372475
animal | 7.417614
religion | 7.382679
philosophy | 6.863642
art | 5.678405
entertainment | 5.076250
mathematics | 2.186313
crypto | 1.731447
gaming | 1.252531
engineering | 0.154807