# Lagrangian solutions to the Vlasov-Poisson
system with $L^{1}$ density

Anna Bohun  Anna Bohun, Departement Mathematik und Informatik, Universität
Basel, Rheinsprung 21, CH-4051, Basel, Switzerland<EMAIL_ADDRESS>,
François Bouchut  Université Paris-Est, Laboratoire d’Analyse et de
Mathématiques Appliquées (UMR 8050), CNRS, UPEM, UPEC, F-77454, Marne-la-
Vallée, France<EMAIL_ADDRESS>and  Gianluca Crippa  Gianluca
Crippa, Departement Mathematik und Informatik, Universität Basel, Rheinsprung
21, CH-4051, Basel, Switzerland<EMAIL_ADDRESS>
###### Abstract.

The recently developed theory of Lagrangian flows for transport equations with
low regularity coefficients enables to consider non BV vector fields. We apply
this theory to prove existence and stability of global Lagrangian solutions to
the repulsive Vlasov-Poisson system with only integrable initial distribution
function with finite energy. These solutions have a well-defined Lagrangian
flow. An a priori estimate on the smallness of the superlevels of the flow in
three dimensions is established in order to control the characteristics.

###### Key words and phrases:

Vlasov-Poisson system, Lagrangian flows, non BV vector fields, superlevels,
weakly convergent initial data

##  1\. Introduction

We consider the Cauchy problem for the classical Vlasov-Poisson system

$\partial_{t}f+v\cdot\nabla_{x}f+E\cdot\nabla_{v}f=0,$ (1.1)
$f(0,x,v)=f^{0}(x,v)\,,$ (1.2)

where $f(t,x,v)\geq 0$ is the distribution function, $t\geq 0$,
$x,v\in\mathbb{R}^{N}$, and

$E(t,x)=-\nabla_{x}U(t,x)$ (1.3)

is the force field. The potential $U$ satisfies the Poisson equation

$-\Delta_{x}U=\omega(\rho(t,x)-\rho_{b}(x)),$ (1.4)

with $\omega=+1$ for the electrostatic (repulsive) case, $\omega=-1$ for the
gravitational (attractive) case, and where the density $\rho$ of particles is
defined through

$\rho(t,x)=\int_{\mathbb{R}^{N}}f(t,x,v)dv,$ (1.5)

and $\rho_{b}\geq 0$, $\rho_{b}\in L^{1}(\mathbb{R}^{N})$ is an autonomous
background density. Since we are in the whole space, the relation (1.3)
together with the Poisson equation (1.4) yield the equivalent relation

$E(t,x)=\frac{\omega}{|S^{N-1}|}\frac{x}{|x|^{N}}*(\rho(t,x)-\rho_{b}(x)),$
(1.6)

where the convolution is in the space variable.

The Vlasov-Poisson system has been studied for long. Existence of local in
time smooth solutions in dimension $N=3$ has been obtained in [15] after the
results of [7]. Global smooth solutions have been proved to exist in [21] (and
simultaneously in [17] with a different method), with improvements on the
growth in time in [22, 23]. These solutions need a sufficiently smooth initial
datum $f^{0}$. In particular, the following theorem is a classical result due
to Pfaffelmoser [21].

######  Theorem 1.1.

If $N=3$, let $f^{0}$ be a non-negative $C^{1}$ function of compact support
defined on $\mathbb{R}^{6}$. Then there are a non-negative $f\in
C^{1}(\mathbb{R}^{7})$ and $U\in C^{2}(\mathbb{R}^{4})$, which tends to zero
at infinity for every fixed $t$, satisfying equations (1.1)-(1.5) with
$\rho_{b}=0$. For each fixed $t$, the function $f(t,x,v)$ has compact support.
The solution is determined uniquely by the initial datum $f^{0}$.

In a different spirit, global weak solutions were proved to exist in [6, 12,
14], with only $f^{0}\in L^{1}(\mathbb{R}^{6})$, $f^{0}\log^{+}f^{0}\in
L^{1}$, $|v|^{2}f^{0}\in L^{1}$, $E^{0}\in L^{2}$ (and $\rho_{b}=0$,
$\omega=+1$). Related results with weak initial data have been obtained in
[20, 16, 24].

In this paper we would like mainly to extend the existence result of [12] to
initial data in $L^{1}$ with finite energy (in the repulsive case
$\omega=+1$), avoiding the $L\log^{+}\\!L$ assumption. Our existence result is
Theorem 8.4. It involves a well-defined flow. Even weaker solutions were
considered in [25, 18, 19], where the distribution function is a measure.
However, these solutions do not have well-defined characteristics.

Our approach uses the theory of Lagrangian flows for transport equations with
vector fields having weak regularity, developed in [13, 1, 4, 11, 5], and
recently in [10, 2, 8]. It enables to consider force fields that are not in
$W^{1,1}_{loc}$, nor in $BV_{loc}$. In this context we prove stability results
with strongly or weakly convergent initial distribution function. The flow is
proved to converge strongly anyway. Our main results were announced in [9].
Related results can be found in [3].

##  2\. Conservation of mass and energy

We would like here to recall some basic identities related to the VP system.
Integrating (1.1) with respect to $v$ and noting that the last term is in
$v$-divergence form we obtain the local conservation of mass

$\partial_{t}\rho(t,x)+{\rm{div}}_{x}({J}(t,x))=0,$ (2.1)

where the current ${J}$ is defined by

${J}(t,x)=\int_{\mathbb{R}^{N}}vf(t,x,v)\,dv.$ (2.2)

Integrating again with respect to $x$, we obtain the global conservation of
mass

$\frac{d}{dt}\iint\limits_{\mathbb{R}^{N}\times\mathbb{R}^{N}}f(t,x,v)dxdv=\frac{d}{dt}\int\limits_{\mathbb{R}^{N}}\rho(t,x)dx=0.$
(2.3)

Multiplying (1.1) by $\frac{|v|^{2}}{2}$, integrating in $x$ and $v$, we get
after integration by parts in $v$

$\frac{d}{dt}\iint\limits_{\mathbb{R}^{N}\times\mathbb{R}^{N}}\frac{|v|^{2}}{2}f(t,x,v)dxdv-\iint\limits_{\mathbb{R}^{N}\times\mathbb{R}^{N}}E\cdot
vf\,dxdv=0.$ (2.4)

Using (1.6) and (2.1), one has

$\partial_{t}E+\sum_{k=1}^{N}\frac{\partial}{\partial
x_{k}}\left(\frac{\omega}{|S^{N-1}|}\frac{x}{|x|^{N}}*{J}_{k}\right)=0,$ (2.5)

or in other words

$\partial_{t}E=\omega\nabla_{x}(-\Delta_{x})^{-1}{\rm{div}}_{x}{J},$ (2.6)

which means that $\partial_{t}E$ is the the gradient component of $-\omega{J}$
(Helmholtz projection). We deduce that

$\int_{\mathbb{R}^{N}}E\cdot\partial_{t}E\,dx=-\omega\int_{\mathbb{R}^{N}}E\cdot{J}\,dx.$
(2.7)

Using (2.2) in (2.4), we obtain the conservation of energy

$\frac{d}{dt}\left[\iint\limits_{\mathbb{R}^{N}\times\mathbb{R}^{N}}\frac{|v|^{2}}{2}f(t,x,v)dxdv+\frac{\omega}{2}\int\limits_{\mathbb{R}^{N}}|E(t,x)|^{2}dx\right]=0\,.$
(2.8)

The total conserved energy is the sum of the kinetic energy and of the
potential energy multiplied by the factor $\omega=\pm 1$. In particular, in
the electrostatic case $\omega=+1$ we deduce from (2.8) a uniform bound in
time on both the kinetic and the potential energy, assuming that they are
finite initially. In the gravitational case $\omega=-1$ it is not possible to
exclude that the individual terms of the kinetic and potential energy become
unbounded in finite time, while the sum remains constant. Indeed it is known
that it does not happen in three dimensions as soon as $f^{0}$ is sufficiently
integrable, but we cannot exclude this a priori for only $L^{1}$ solutions.

Note that the assumption $E^{0}\in L^{2}$ is satisfied in 3 dimensions as soon
as $\rho^{0}-\rho_{b}\in L^{6/5}$. However, in one or two dimensions, for
$E^{0}$ to be in $L^{2}$ it is necessary that $\int(\rho^{0}-\rho_{b})dx=0$,
as is easily seen in Fourier variable. It is also necessary that
$\rho^{0}-\rho_{b}$ has enough decay at infinity. Thus in one or two
dimensions, in order to have finite energy, $\rho_{b}$ cannot be zero
identically.

##  3\. Regularity of the force field for $L^{1}$ densities

###  3.1. Singular integrals

######  Definition 3.1.

A function $K$ is a _singular kernel of fundamental type_ in $\mathbb{R}^{N}$
if the following properties hold:

  1. (1)

$K|_{\mathbb{R}^{N}\setminus\\{0\\}}\in
C^{1}(\mathbb{R}^{N}\setminus\\{0\\})$.

  2. (2)

There exists a constant $C_{0}\geq 0$ such that

$\displaystyle\begin{array}[]{ccc}|K(x)|\leq\frac{C_{0}}{|x|^{N}},&&x\in\mathbb{R}^{N}\setminus\\{0\\}.\end{array}$
(3.1)

  3. (3)

There exists a constant $C_{1}\geq 0$ such that

$\displaystyle\begin{array}[]{ccc}|\nabla
K(x)|\leq\frac{C_{1}}{|x|^{N+1}},&&x\in\mathbb{R}^{N}\setminus\\{0\\}.\end{array}$
(3.2)

  4. (4)

There exists a constant $A_{2}\geq 0$ such that

$\left|\int_{R_{1}<|x|<R_{2}}K(x)dx\right|\leq A_{2},$ (3.3)

for every $0<R_{1}<R_{2}<\infty$.

######  Theorem 3.2 (Calderón-Zygmund).

A singular kernel of fundamental type $K$ has an extension as a distribution
on $\mathbb{R}^{N}$ (still denoted by $K$), unique up to a constant times
Dirac delta at the origin, such that $\hat{K}\in L^{\infty}(\mathbb{R}^{N})$.
Define

$Su=K*u,\quad\textrm{for }u\in L^{2}(\mathbb{R}^{N}),$ (3.4)

in the sense of multiplication in the Fourier variable. Then we have the
estimates for $1<p<\infty$

$\begin{array}[]{cc}||Su||_{L^{p}(\mathbb{R}^{N})}\leq
C_{N,p}(C_{0}+C_{1}+||\hat{K}||_{L^{\infty}})||u||_{L^{p}(\mathbb{R}^{N})},&u\in
L^{p}\cap L^{2}(\mathbb{R}^{N}).\end{array}$ (3.5)

If $K$ is a singular kernel of fundamental type, we call the associated
operator $S$ a singular integral operator on $\mathbb{R}^{N}$. We define then
the Fréchet space
$\mathscr{R}(\mathbb{R}^{N})=\cap_{m\in{\mathbb{N}},\,1<p<\infty}W^{m,p}(\mathbb{R}^{N})$
and its dual
$\mathscr{R}^{\prime}(\mathbb{R}^{N})\subset\mathscr{S}^{\prime}(\mathbb{R}^{N})$,
where $\mathscr{S}^{\prime}(\mathbb{R}^{N})$ is the space of tempered
distributions on $\mathbb{R}^{N}$. Since all singular integral operators are
bounded on $\mathscr{R}(\mathbb{R}^{N})$, by duality we can define the
operator $S$ also
$\mathscr{R}^{\prime}(\mathbb{R}^{N})\to\mathscr{R}^{\prime}(\mathbb{R}^{N})$.
In particular it enables to define $Su$ for $u\in L^{1}(\mathbb{R}^{N})$ or
for $u$ a measure. The result $Su$ is in
$\mathscr{R}^{\prime}(\mathbb{R}^{N})\subset\mathscr{S}^{\prime}(\mathbb{R}^{N})$.

###  3.2. The split vector field

Let $\rho(t,x)\in L^{\infty}((0,T);L^{1}(\mathbb{R}^{N}))$. We denote by

$b(t,x,v)=(b_{1},b_{2})(t,x,v)=(v,E(t,x))=\Bigl{(}v,-\omega\nabla_{x}(-\Delta_{x})^{-1}(\rho(t,x)-\rho_{b}(x))\Bigr{)}$
(3.6)

the associated vector field on
$(0,T)\times\mathbb{R}^{N}\times\mathbb{R}^{N}$. Then the Vlasov equation can
be written in the form of the transport equation
$\partial_{t}f+b\cdot\nabla_{x,v}f=0$. In the following subsections we
establish bounds on the vector field $b$.

###  3.3. Local integrability

For $L^{1}$ densities, we have the weak estimates from the Hardy-Littlewood-
Sobolev inequality:

$\begin{array}[]{l}\displaystyle\hphantom{=\
}\left|\left|\left|\nabla(-\Delta)^{-1}(\rho(t,x)-\rho_{b}(x))\right|\right|\right|_{M^{\frac{N}{N-1}}(\mathbb{R}^{N})}\\\
\displaystyle\leq\left|\left|\left|{\frac{1}{|S^{N-1}|}|x|^{1-N}}*|\rho(t,x)-\rho_{b}(x)|\right|\right|\right|_{M^{\frac{N}{N-1}}(\mathbb{R}^{N})}\\\
\displaystyle\leq
c_{N}||\rho(t,x)-\rho_{b}(x)||_{L^{1}(\mathbb{R}^{N})},\end{array}$ (3.7)

where
$|||u|||_{M^{p}(\mathbb{R}^{N})}\equiv\sup_{\gamma>0}\gamma\mathop{\mathscr{L}^{N}}(\\{x\in\mathbb{R}^{N}\,s.t.\,|u(x)|>\gamma\\})^{1/p}$.
It follows that

$|||E|||_{L^{\infty}((0,T);M^{\frac{N}{N-1}}(\mathbb{R}^{N}))}\leq
c_{N}||\rho-\rho_{b}||_{L^{\infty}((0,T);L^{1}(\mathbb{R}^{N}))},$ (3.8)

and using the inclusion $M^{\frac{N}{N-1}}(\mathbb{R}^{N})\subset
L^{p}_{\rm{loc}}(\mathbb{R}^{N})$ for $1\leq p<\frac{N}{N-1}$ we conclude that
$b\in
L^{\infty}((0,T);L^{p}_{\rm{loc}}(\mathbb{R}^{N}_{x}\times\mathbb{R}^{N}_{v}))$
for any $1\leq p<\frac{N}{N-1}$, since $v\in
L^{p}_{\rm{loc}}(\mathbb{R}^{N}_{x}\times\mathbb{R}^{N}_{v})$ for any $p$.

###  3.4. Spatial regularity

Since $b_{1}=v$ is smooth, the only non-trivial gradient is the one of
$b_{2}=E$, indeed the differential matrix of the vector field is given by

$Db=\left(\begin{matrix}D_{x}b_{1}&D_{v}b_{1}\\\
D_{x}b_{2}&D_{v}b_{2}\end{matrix}\right)=\left(\begin{matrix}0&{\rm Id}\\\
D_{x}E&0\end{matrix}\right).$ (3.9)

We have by (3.6)

$(D_{x}E)_{ij}\equiv\partial_{x_{j}}E_{i}=-\omega\partial^{2}_{x_{i}x_{j}}((-\Delta_{x})^{-1}(\rho-\rho_{b}))\quad\mbox{for
}1\leq i,j\leq N.$ (3.10)

It is well-known that the operator
$\partial^{2}_{x_{i}x_{j}}(-\Delta_{x})^{-1}$ is a singular integral operator.
Its kernel is

$K_{ij}(x)=-\frac{1}{|S^{N-1}|}\frac{\partial}{\partial_{x_{j}}}\left(\frac{x_{i}}{|x|^{N}}\right),$
(3.11)

it is given outside of the origin by

$\displaystyle
K_{ij}(x)=\frac{1}{|S^{N-1}|}\left(N\frac{x_{i}x_{j}}{|x|^{N+2}}-\frac{\delta_{ij}}{|x|^{N}}\right),\quad\mbox{
for }x\in\mathbb{R}^{N}\backslash\\{0\\}.$ (3.12)

The kernel satisfies the conditions of Subsection 3.1, and
$\hat{K}_{ij}(\xi)=-\xi_{i}\xi_{j}/|\xi|^{2}$. Thus (each component of)
$D_{x}E$ is a singular integral of an
$L^{\infty}((0,T);L^{1}(\mathbb{R}^{N}))$ function.

###  3.5. Time regularity

According to (2.6), $\partial_{t}E$ is a singular integral of the current
${J}$ defined by (2.2). Using the bounds available for solutions with finite
mass and energy

$||f(t,\cdot)||_{L^{1}(\mathbb{R}^{N}_{x}\times\mathbb{R}^{N}_{v})},\iint|v|^{2}f(t,x,v)dxdv\leq
C,$ (3.13)

and since $|v|\leq 1+|v|^{2}$, we get that ${J}\in
L^{\infty}((0,T);L^{1}(\mathbb{R}^{N}_{x}))$. Hence $\partial_{t}E$ is a
singular integral of an $L^{\infty}((0,T);L^{1}(\mathbb{R}^{N}_{x}))$
function. In particular,

$\partial_{t}E\in L^{\infty}((0,T);\mathscr{S}^{\prime}(\mathbb{R}^{N})).$
(3.14)

##  4\. Lagrangian flows

Suppose that $f$ is a smooth solution to (1.1)-(1.5) with $f(0,x,v)=f^{0}$.
Then $f$ is constant along the characteristics $(X(s,t,x,v),V(s,t,x,v))$,
which solve the system of equations

$\begin{cases}\displaystyle\frac{dX}{ds}(s,t,x,v)=V(s,t,x,v),\\\ \\\
\displaystyle\frac{dV}{ds}(s,t,x,v)=E(s,X(s,t,x,v),V(s,t,x,v)),\end{cases}$
(4.1)

with initial data $X(t,t,x,v)=x$ and $V(t,t,x,v)=v$. Thus the solution can be
expressed as $f(t,x,v)=f^{0}(X(0,t,x,v),V(0,t,x,v))$.

In order to extend this notion of characteristics to non-smooth solutions, we
define regular Lagrangian flows, which are defined in an almost everywhere
sense.

######  Definition 4.1.

Let $b\in L^{1}_{\rm{loc}}([0,T]\times\mathbb{R}^{2N};\mathbb{R}^{2N})$, and
$t\in[0,T)$. A map $Z:[t,T]\times\mathbb{R}^{2N}\rightarrow\mathbb{R}^{2N}$ is
a regular Lagrangian flow starting at time $t$ for the vector field $b$ if

  * (1)

for a.e. $z\in\mathbb{R}^{2N}$ the map $s\mapsto Z(s,z)$ is an absolutely
continuous integral solution of $\dot{\beta}(s)=b(s,\beta(s))$ for $s\in[t,T]$
with $\beta(t)=z$.

  * (2)

There exists a constant $L$ independent of $s$ such that

$\mathscr{L}^{2N}(Z(s,\cdot)^{-1}(A))\leq L\mathscr{L}^{2N}(A)$

for every Borel set $A\subset\mathbb{R}^{2N}$.

The constant $L$ is called the compressibility constant of $Z$.

######  Definition 4.2.

Define the sublevel of the flow as the set

$G_{\lambda}=\\{z\in\mathbb{R}^{2N}:|Z(s,z)|\leq\lambda\text{ for almost all
}s\in[t,T]\\}.$ (4.2)

##  5\. Stability estimate for Lagrangian flows

We summarize the main result from [10] in the following regularity setting of
the vector field in arbitrary dimension. We say that a vector field $b$
satisfies (R1) if $b$ can be decomposed as

$\frac{b(t,z)}{1+|z|}=\tilde{b}_{1}(t,z)+\tilde{b}_{2}(t,z)$ (5.1)

where $\tilde{b}_{1}\in L^{1}((0,T);L^{1}(\mathbb{R}^{2N}))$,
$\tilde{b}_{2}\in L^{1}((0,T);L^{\infty}(\mathbb{R}^{2N}))$.

We assume also that $b$ satisfies (R2): for every $j=1,\ldots,2N$,

$\partial_{z_{j}}b=\displaystyle\sum_{k=1}^{m}S_{jk}g_{jk}$ (5.2)

where $S_{jk}$ are singular integrals of fundamental type on $\mathbb{R}^{2N}$
and $g_{jk}\in L^{1}((0,T);L^{1}(\mathbb{R}^{2N}))$. Moreover, we assume
condition (R3), that is

$b\in L^{p}_{\rm{loc}}([0,T]\times\mathbb{R}^{2N}),\qquad\mbox{ for some
}p>1.$ (5.3)

We recall the following stability theorem from [10], where we denote by
$B_{r}$ the ball with center $0$ and radius $r$ in $\mathbb{R}^{2N}$.

######  Theorem 5.1.

Let $b$, $\bar{b}$ be two vector fields satisfying (R1), $b$ satisfying also
(R2), (R3). Fix $t\in[0,T)$ and let $Z$ and $\bar{Z}$ be regular Lagrangian
flows starting at time $t$ associated to $b$ and $\bar{b}$ respectively, with
compression constants $L$ and $\bar{L}$. Then the following holds.
For every $\gamma>0$, $r>0$ and $\eta>0$ there exist $\lambda>0$ and
$C_{\gamma,r,\eta}>0$ such that

$\mathscr{L}^{2N}\left(B_{r}\cap\\{|Z(s,\cdot)-\bar{Z}(s,\cdot)|>\gamma\\}\right)\leq
C_{\gamma,r,\eta}||b-\bar{b}||_{L^{1}((0,T)\times B_{\lambda})}+\eta$

for all $s\in[t,T]$. The constants $\lambda$ and $C_{\gamma,r,\eta}$ also
depend on

  * •

The equi-integrability in $L^{1}((0,T);L^{1}(\mathbb{R}^{2N}))$ of $g_{jk}$
coming from (R2),

  * •

The norms of the singular integral operators $S_{jk}$ from (R2),

  * •

The norm $||b||_{L^{p}((0,T)\times B_{\lambda})}$ corresponding to (R3),

  * •

The $L^{1}(L^{1})$ and $L^{1}(L^{\infty})$ norms of the decompositions of $b$
and $\bar{b}$ in (R1),

  * •

The compression constants $L$ and $\bar{L}$.

We would like now to state a variant of this theorem, where (R1) and (R2) are
replaced by (R1a) and (R2a).

We consider the following weakened assumption (R1a): for all regular
Lagrangian flow $Z:[t,T]\times\mathbb{R}^{2N}\rightarrow\mathbb{R}^{2N}$
relative to $b$ starting at time $t$ with compression constant $L$, and for
all $r,\lambda>0$,

$\mathscr{L}^{2N}(B_{r}\setminus G_{\lambda})\leq g(r,\lambda),\quad\mbox{with
}g(r,\lambda)\to 0\text{ as }\lambda\to\infty\mbox{ at fixed }r,$ (5.4)

where $G_{\lambda}$ denotes the sublevel of the flow $Z$, defined in (4.2).

We next consider a splitting of the variables,
$\mathbb{R}^{2N}_{z}=\mathbb{R}^{N}_{x}\times\mathbb{R}^{N}_{v}$, and we
denote $Z(t,x,v)=(X,V)(t,x,v)$. Noticing the special form (3.6), we assume the
condition that $b$ satisfies (R2a):

$b(t,x,v)=(b_{1},b_{2})(t,x,v)=(b_{1}(v),b_{2}(t,x)),$ (5.5)

with

$b_{1}\in\textrm{Lip}(\mathbb{R}^{N}_{v}),$ (5.6)

and where $b_{2}$ involves singular kernels only in the first set of variables
$x$, that is for every $j=1,...,N$,

$\partial_{x_{j}}b_{2}=\displaystyle\sum_{k=1}^{m}S_{jk}g_{jk},$ (5.7)

where $S_{jk}$ are singular integrals of fundamental type on $\mathbb{R}^{N}$
and $g_{jk}\in L^{1}((0,T);L^{1}(\mathbb{R}^{N}))$.

######  Theorem 5.2.

Let $b$, $\bar{b}$ be two vector fields satisfying (R1a), $b$ satisfying also
(R2a), (R3). Fix $t\in[0,T)$ and let $Z$ and $\bar{Z}$ be regular Lagrangian
flows starting at time $t$ associated to $b$ and $\bar{b}$ respectively, with
compression constants $L$ and $\bar{L}$, and sublevels $G_{\lambda}$ and
$\bar{G}_{\lambda}$. Then the following holds.
For every $\gamma>0$, $r>0$ and $\eta>0$, there exist $\lambda>0$ and
$C_{\gamma,r,\eta}>0$ such that

$\mathscr{L}^{2N}\left(B_{r}\cap\\{|Z(s,\cdot)-\bar{Z}(s,\cdot)|>\gamma\\}\right)\leq
C_{\gamma,r,\eta}||b-\bar{b}||_{L^{1}((0,T)\times B_{\lambda})}+\eta$

for all $s\in[t,T]$. The constants $\lambda$ and $C_{\gamma,r,\eta}$ also
depend on

  * •

The equi-integrability in $L^{1}((0,T);L^{1}(\mathbb{R}^{N}))$ of $g_{jk}$
coming from (R2a),

  * •

The norms of the singular integral operators $S_{jk}$ from (R2a),

  * •

The Lipschitz constant of $b_{1}$ from (R2a),

  * •

The norm $||b||_{L^{p}((0,T)\times B_{\lambda})}$ corresponding to (R3),

  * •

The rate of decay of $\mathscr{L}^{2N}(B_{r}\setminus{G}_{\lambda})$ and
$\mathscr{L}^{2N}(B_{r}\setminus\bar{G}_{\lambda})$ from (R1a),

  * •

The compression constants $L$ and $\bar{L}$.

###### Proof.

We summarize the modifications of the proof of Theorem 5.1 from [10]. The main
added difficulty is the singular integral operators on $\mathbb{R}^{N}_{x}$
instead of $\mathbb{R}^{2N}$. More general situations with singular integrals
of measures instead of $L^{1}$ functions are considered in [8]. Let
$z=(x,v)\in\mathbb{R}^{N}\times\mathbb{R}^{N}$. We estimate the quantity

$\Phi_{\delta}(s)=\int\limits_{B_{r}\cap
G_{\lambda}\cap\widebar{G}_{\lambda}}\log\left(1+\frac{|Z(s,z)-\widebar{Z}(s,z)|}{\delta}\right)dz\,.$
(5.8)

Differentiating with respect to time, we get

$\displaystyle\Phi_{\delta}^{\prime}(s)\leq$
$\displaystyle\int\limits_{B_{r}\cap
G_{\lambda}\cap\widebar{G}_{\lambda}}\frac{|b(s,Z(s,z))-\widebar{b}(s,\widebar{Z}(s,z))|}{\delta+|Z(s,z)-\widebar{Z}(s,z)|}dz$
$\displaystyle\leq$
$\displaystyle\frac{\widebar{L}}{\delta}||b(s,\cdot)-\widebar{b}(s,\cdot)||_{L^{1}(B_{\lambda})}$
$\displaystyle+\int\limits_{B_{r}\cap
G_{\lambda}\cap\widebar{G}_{\lambda}}\min\biggl{\\{}\frac{|b(s,Z(s,z))|+|b(s,\widebar{Z}(s,z))|}{\delta},$
$\displaystyle\frac{|b_{1}(s,Z(s,z))-b_{1}(s,\widebar{Z}(s,z))|}{|Z(s,z)-\widebar{Z}(s,z)|}+\frac{|b_{2}(s,Z(s,z))-b_{2}(s,\widebar{Z}(s,z))|}{|Z(s,z)-\widebar{Z}(s,z)|}\biggr{\\}}dz.$

Using the special form of $b$ from (R2a), we obtain

$\displaystyle\Phi_{\delta}^{\prime}(s)\leq$
$\displaystyle\frac{\widebar{L}}{\delta}||b(s,\cdot)-\widebar{b}(s,\cdot)||_{L^{1}(B_{\lambda})}$
(5.9) $\displaystyle+\int\limits_{B_{r}\cap
G_{\lambda}\cap\widebar{G}_{\lambda}}\min\biggl{\\{}\frac{|b(s,Z(s,z))|+|b(s,\widebar{Z}(s,z))|}{\delta},$
$\displaystyle\quad\frac{|b_{1}(V(s,z))-b_{1}(\widebar{V}(s,z))|}{|V(s,z)-\widebar{V}(s,z)|}+\frac{|b_{2}(s,X(s,z))-b_{2}(s,\widebar{X}(s,z))|}{|X(s,z)-\widebar{X}(s,z)|}\biggr{\\}}dz$
$\displaystyle\leq$
$\displaystyle\frac{\widebar{L}}{\delta}||b(s,\cdot)-\widebar{b}(s,\cdot)||_{L^{1}(B_{\lambda})}+\textrm{Lip}(b_{1})\mathscr{L}^{2N}(B_{r})$
$\displaystyle+\int\limits_{B_{r}\cap
G_{\lambda}\cap\widebar{G}_{\lambda}}\min\biggl{\\{}\frac{|b(s,Z(s,z))|+|b(s,\widebar{Z}(s,z))|}{\delta},$
$\displaystyle\mkern
150.0mu\frac{|b_{2}(s,X(s,z))-b_{2}(s,\widebar{X}(s,z))|}{|X(s,z)-\widebar{X}(s,z)|}\biggr{\\}}dz.$

Using assumption (R2a), we can use the estimate of [10] on the difference
quotients of $b_{2}$,

$\frac{|b_{2}(s,X(s,z))-b_{2}(s,\widebar{X}(s,z))|}{|X(s,z)-\widebar{X}(s,z)|}\leq\mathscr{U}(s,X(s,z))+\mathscr{U}(s,\widebar{X}(s,z)),$
(5.10)

where $\mathscr{U}(s,.)\in M^{1}(\mathbb{R}^{N})$ for fixed $s$ is indeed
given by

$\mathscr{U}(s,\cdot)=\sum_{j=1}^{N}\sum_{k=1}^{m}M_{j}(S_{jk}g_{jk}(s,.)),$
(5.11)

with $M_{j}$ a smooth maximal operator on $\mathbb{R}^{N}$. Next, we can
define the function

$\mathscr{Z}(s,x,v)=\mathscr{U}(s,x)\mathbbm{1}_{(x,v)\in\widebar{B_{\lambda}}},$
(5.12)

and we notice that since the above integrals are over
$G_{\lambda}\cap\widebar{G}_{\lambda}$, we can replace the right-hand side of
(5.10) by $\mathscr{Z}(s,Z(s,z))+\mathscr{Z}(s,\widebar{Z}(s,z))$. Then, for
given $\varepsilon>0$, we can decompose $g_{jk}=g_{jk}^{1}+g_{jk}^{2}$, with
$\|g_{jk}^{1}\|_{L^{1}((0,T)\times\mathbb{R}^{N})}\leq\varepsilon$ and
$\|g_{jk}^{2}\|_{L^{2}((0,T)\times\mathbb{R}^{N})}\leq C_{\varepsilon}$,
$g_{jk}^{2}$ having support in a set $A_{\varepsilon}$ of finite measure. This
gives rise to two type of terms $\mathscr{Z}^{1}$ and $\mathscr{Z}^{2}$. Since
all the $\mathscr{Z}$ terms have compact support in $v$, this allows to
perform the same estimates as in [10]. We finally get

$\displaystyle\mathscr{L}^{2N}(B_{r}\cap\\{|Z(\tau,\cdot)-\widebar{Z}(\tau,\cdot)|>\gamma\\})$
(5.13) $\displaystyle\leq$
$\displaystyle\frac{1}{\log(1+\gamma/\delta)}\int_{t}^{\tau}\Phi_{\delta}^{\prime}(s)\,ds+\mathscr{L}^{2N}(B_{r}\setminus
G_{\lambda})+\mathscr{L}^{2N}(B_{r}\setminus\widebar{G}_{\lambda}).$

By choosing $\lambda$ large first, then $\varepsilon$ small, and finally
$\delta$ small, we conclude the proof as in [10]. ∎

##  6\. Control of superlevels

In order to apply Theorem 5.2, we need to satisfy (R1a). Therefore, we seek an
upper bound on the size of $B_{r}\setminus G_{\lambda}.$

###  6.1. The case of low space dimension

We first recall the following lemma from [10].

######  Lemma 6.1.

Let $b:(0,T)\times\mathbb{R}^{2N}\rightarrow\mathbb{R}^{2N}$ be a vector field
satisfying (R1). Then $b$ satisfies (R1a), where the function $g(r,\lambda)$
depends only on
$L,||\tilde{b}_{1}||_{L^{1}((0,T);L^{1}(\mathbb{R}^{2N}))},||\tilde{b}_{2}||_{L^{1}((0,T);L^{\infty}(\mathbb{R}^{2N}))}$.

This lemma allows us to control the superlevels of $b$ in $1$ or $2$
dimensions.

######  Proposition 6.2.

Let $b$ be the vector field in (3.6), with $E\in
L^{\infty}((0,T);L^{2}(\mathbb{R}^{N}))$. For $N=2$ or $N=1$, $b$ satisfies
(R1), hence also (R1a).

###### Proof.

It is clear that

$\frac{v}{1+|x|+|v|}\in L^{\infty}_{t}(L^{\infty}_{x,v}),$ (6.1)

and

$\frac{E(t,x)}{1+|x|+|v|}=\frac{E(t,x)}{1+|x|+|v|}\mathbbm{1}_{|v|\leq|E(t,x)|}+\frac{E(t,x)}{1+|x|+|v|}\mathbbm{1}_{|v|>|E(t,x)|}\equiv\tilde{E_{1}}+\tilde{E_{2}}.$
(6.2)

Clearly $\tilde{E_{2}}\in L^{\infty}_{t}(L^{\infty}_{x,v})$, and if $N=2$,
$\tilde{E_{1}}\in L^{\infty}((0,T);L^{1}_{x,v})$ since

$\begin{array}[]{l}\displaystyle\iint\limits_{\mathbb{R}^{2}\times\mathbb{R}^{2}}\frac{|E(t,x)|}{1+|x|+|v|}\mathbbm{1}_{|v|\leq|E(t,x)|}dxdv\\\
\displaystyle\leq\int\limits_{\mathbb{R}^{2}}|E(t,x)|\biggl{(}\int\limits_{|v|\leq|E(t,x)|}\frac{1}{|v|}dv\biggr{)}dx=2\pi\int\limits_{\mathbb{R}^{2}}|E(t,x)|^{2}dx.\end{array}$
(6.3)

In the case $N=1$, we have directly that $E(t,x)/(1+|v|)\in
L^{\infty}((0,T);L^{2}_{x,v})$. ∎

###  6.2. The case of three space dimensions

The condition (R1) being not satisfied in $3$ dimensions, we need an estimate
on $|Z|$ in order to control the superlevels. For getting this we integrate in
space a function growing slower at infinity than $\log(1+|Z|)$ (this
corresponding to the case (R1)).

######  Proposition 6.3.

Let $b$ be as in (3.6) with $N=3$, $E\in
L^{\infty}((0,T);L^{2}(\mathbb{R}^{N}))$, satisfying (2.6) with ${J}\in
L^{\infty}((0,T);L^{1}(\mathbb{R}^{N}))$. Furthermore, assume that
$\omega=+1$, $\rho\geq 0$, and $\rho_{b}\in L^{1}\cap L^{p}(\mathbb{R}^{3})$
for some $p>3/2$. Then (R1a) holds, where the function $g$ depends only on
$L$, $T$, $||E||_{L^{\infty}_{t}(L^{2}_{x})}$,
$||{J}||_{L^{\infty}_{t}(L^{1}_{x})}$,
$\|(-\Delta)^{-1}\rho_{b}\|_{L^{\infty}}$, and one has $g(r,\lambda)\to 0$ as
$\lambda\to\infty$ at fixed $r$.

###### Proof.

Step 1.) Let
$Z:[t,T]\times\mathbb{R}^{3}\times\mathbb{R}^{3}\rightarrow\mathbb{R}^{3}\times\mathbb{R}^{3}$
be a regular Lagrangian flow relative to $b$ starting at time $t$, with
compression constant $L$ and sublevel $G_{\lambda}$. Denoting $Z=(X,V)$, we
have the ODEs

$\displaystyle\begin{cases}&\dot{X}(s,x,v)=V(s,x,v),\\\
&\dot{V}(s,x,v)=E(s,X(s,x,v)).\end{cases}$ (6.4)

Recalling that $E=-\nabla_{x}U$, one has

$\displaystyle\partial_{s}\frac{|V(s,x,v)|^{2}}{2}=V(s,x,v)\cdot\partial_{s}V(s,x,v)=E(s,X(s,x,v))\cdot\partial_{s}X(s,x,v)$
$\displaystyle=-\partial_{s}[U(s,X(s,x,v))]+\partial_{t}U(s,X(s,x,v)).$ (6.5)

This computation is indeed related to the form of the Hamiltonian for (1.1),
${\mathscr{H}}=|v|^{2}/2+U(t,x)$.

We are going to bound the superlevels of $V(s,x,v)$. We claim that

$\iint_{B_{r}}\sup_{s\in[t,T]}\left(1+\log\Bigl{(}1+|V(s,x,v)|^{2}/2\Bigr{)}\right)^{\alpha}dxdv\leq
A,$ (6.6)

where $0<\alpha<1/3$, and for some constant $A$ depending on $L$, $T$, $r$,
$\alpha$, and on the norms $||E||_{L^{\infty}_{t}(L^{2}_{x})}$,
$||{J}||_{L^{\infty}_{t}(L^{1}_{x})}$,
$\|(-\Delta)^{-1}\rho_{b}\|_{L^{\infty}}$. Assume for the moment that this
holds. From the lower bound

$\begin{array}[]{l}\displaystyle\hphantom{\geq\
}\iint_{B_{r}}\sup_{s\in[t,T]}\left(1+\log\Bigl{(}1+|V(s,x,v)|^{2}/2\Bigr{)}\right)^{\alpha}dxdv\\\
\displaystyle\geq\mathscr{L}^{6}(B_{r}\setminus\widetilde{G}_{\lambda})(1+\log(1+\lambda^{2}/2))^{\alpha},\end{array}$
(6.7)

with $\widetilde{G}_{\lambda}$ the sublevel of $V$, we get that

$\mathscr{L}^{6}(B_{r}\setminus\widetilde{G}_{\lambda})\leq\frac{A}{(1+\log(1+\lambda^{2}/2))^{\alpha}}.$
(6.8)

Next, we remark that by the first equation in (6.4), whenever
$(x,v)\in\widetilde{G}_{\lambda}$ one has $|X(s,x,v)|\leq|x|+|s-t|\lambda$,
and $|Z(s,x,v)|\leq|x|+(1+T)\lambda$. Thus for $\lambda>r$, one has
$B_{r}\setminus G_{\lambda}\subset
B_{r}\setminus\widetilde{G}_{(\lambda-r)/(1+T)}$, which enables to conclude
the proposition (for $\lambda\leq r$ we can just bound
$\mathscr{L}^{6}(B_{r}\setminus G_{\lambda})$ by $\mathscr{L}^{6}(B_{r})$).
Step 2.) By Step 1, it is enough to prove that we have a decomposition

$\left(1+\log\left(1+\frac{|V(s,\cdot,\cdot)|^{2}}{2}\right)\right)^{\alpha}\leq
f_{1}+f_{2}\in
L^{1}(\mathbb{R}^{3}_{x}\times\mathbb{R}^{3}_{v})+L^{\infty}(\mathbb{R}^{3}_{x}\times\mathbb{R}^{3}_{v}),$
(6.9)

for $(x,v)\in B_{r}$, where $f_{1}$, $f_{2}$ are independent of $s\in[t,T]$.
Let

$\beta(y)=\left(1+\log\left(1+y\right)\right)^{\alpha},\quad\mbox{for }y\geq
0.$ (6.10)

Then

$\begin{array}[]{c}\displaystyle\beta^{\prime}(y)=\frac{\alpha(1+\log(1+y))^{\alpha-1}}{1+y},\\\
\displaystyle
0<-\beta^{\prime\prime}(y)\leq\frac{(1+\log(1+y))^{\alpha-1}}{(1+y)^{2}}.\end{array}$
(6.11)

Using (6.2), we compute

$\displaystyle\partial_{s}\left[\beta\left(\frac{|V(s,x,v)|^{2}}{2}\right)\right]$
(6.12) $\displaystyle=$
$\displaystyle\Bigl{(}-\partial_{s}[U(s,X(s,x,v))]+\partial_{t}U(s,X(s,x,v))\Bigr{)}\beta^{\prime}\left(\frac{|V(s,x,v)|^{2}}{2}\right)$
$\displaystyle=$
$\displaystyle-\partial_{s}\left[U(s,X(s,x,v))\beta^{\prime}\left(\frac{|V(s,x,v)|^{2}}{2}\right)\right]$
$\displaystyle+U(s,X(s,x,v))\beta^{\prime\prime}\left(\frac{|V(s,x,v)|^{2}}{2}\right)V(s,x,v)\cdot
E(s,X(s,x,v))$
$\displaystyle+\partial_{t}U(s,X(s,x,v))\beta^{\prime}\left(\frac{|V(s,x,v)|^{2}}{2}\right).$

Thus, integrating between $t$ and $s$,

$\displaystyle\left(1+\log\left(1+\frac{|V(s,x,v)|^{2}}{2}\right)\right)^{\alpha}$
(6.13) $\displaystyle=$ $\displaystyle-\frac{\alpha
U(s,X(s,x,v))}{\left(1+\frac{|V(s,x,v)|^{2}}{2}\right)\left(1+\log\left(1+\frac{|V(s,x,v)|^{2}}{2}\right)\right)^{1-\alpha}}$
$\displaystyle+\frac{\alpha
U(t,x)}{\left(1+\frac{|v|^{2}}{2}\right)\left(1+\log\left(1+\frac{|v|^{2}}{2}\right)\right)^{1-\alpha}}+\left(1+\log\left(1+\frac{{|v|^{2}}}{2}\right)\right)^{\alpha}$
$\displaystyle+\int^{s}_{t}\left\\{U(\tau,X(\tau,x,v))V(\tau,x,v)\cdot
E(\tau,X(\tau,x,v))\beta^{\prime\prime}\left(\frac{|V(\tau,x,v)|^{2}}{2}\right)\right.$
$\displaystyle\left.+\partial_{t}U(\tau,X(\tau,x,v))\beta^{\prime}\left(\frac{|V(\tau,x,v)|^{2}}{2}\right)\right\\}d\tau.$

Step 3.) Since $E(t,\cdot)\in L^{2}(\mathbb{R}^{3})$, we have by the Sobolev
embedding that $U(t,\cdot)\in L^{6}(\mathbb{R}^{3})$. Thus clearly

$\frac{U(t,x)}{1+\frac{|v|^{2}}{2}}\in
L^{6}(\mathbb{R}^{3}_{x}\times\mathbb{R}^{3}_{v})\subset
L^{1}(\mathbb{R}^{3}_{x}\times\mathbb{R}^{3}_{v})+L^{\infty}(\mathbb{R}^{3}_{x}\times\mathbb{R}^{3}_{v}).$
(6.14)

Next, since $\omega=+1$ and $\rho\geq 0$, one has $U=U_{\rho}-U_{\rho_{b}}$,
with $U_{\rho}\geq 0$. Thus $U\geq-\|U_{\rho_{b}}\|_{L^{\infty}}$. Thus the
first three terms in the expansion (6.13) are upper bounded in
$L^{1}(\mathbb{R}^{3}_{x}\times\mathbb{R}^{3}_{v})+L^{\infty}(\mathbb{R}^{3}_{x}\times\mathbb{R}^{3}_{v})$.
It remains to estimate the integral. We can bound it by $\Phi_{1}+\Phi_{2}$,
with

$\Phi_{1}:=\int^{T}_{t}\left|U(\tau,X(\tau,x,v))V(\tau,x,v)\cdot
E(\tau,X(\tau,x,v))\beta^{\prime\prime}\left(\frac{|V(\tau,x,v)|^{2}}{2}\right)\right|d\tau,$
(6.15)
$\Phi_{2}:=\int^{T}_{t}\left|\partial_{t}U(\tau,X(\tau,x,v))\beta^{\prime}\left(\frac{|V(\tau,x,v)|^{2}}{2}\right)\right|d\tau.$
(6.16)

Note that $\Phi_{1}$, $\Phi_{2}$ are independent of $s$. We estimate
$\Phi_{1}$ in $L^{3/2}(\mathbb{R}^{3}_{x}\times\mathbb{R}^{3}_{v})\subset
L^{1}(\mathbb{R}^{3}_{x}\times\mathbb{R}^{3}_{v})+L^{\infty}(\mathbb{R}^{3}_{x}\times\mathbb{R}^{3}_{v})$.
Passing the $L^{3/2}$ norm under the integral and changing
$(X(\tau,x,v),V(\tau,x,v))$ to $(x,v)$, this gives (up to a factor $L$)

$\displaystyle\int\limits_{\mathbb{R}^{3}}\int\limits_{\mathbb{R}^{3}}\left|\frac{U(\tau,x)v\cdot
E(\tau,x)}{\left(1+\frac{|v|^{2}}{2}\right)^{2}\left(1+\log\left(1+\frac{|v|^{2}}{2}\right)\right)^{1-\alpha}}\right|^{3/2}dxdv$
$\displaystyle\leq\int\limits_{\mathbb{R}^{3}}|U(\tau,x)E(\tau,x)|^{3/2}dx\int\limits_{\mathbb{R}^{3}}\frac{2^{3/4}\,dv}{\left(1+\frac{|v|^{2}}{2}\right)^{9/4}\left(1+\log\left(1+\frac{|v|^{2}}{2}\right)\right)^{3(1-\alpha)/2}}$
$\displaystyle\leq
c||U(\tau,\cdot)||_{L^{6}(\mathbb{R}^{3})}^{3/2}||E(\tau,\cdot)||_{L^{2}(\mathbb{R}^{3})}^{3/2}.$

Thus $\Phi_{1}\in L^{3/2}(\mathbb{R}^{3}_{x}\times\mathbb{R}^{3}_{v})$.
Step 4.) For $\Phi_{2}$, we notice that $E$ satisfies (2.6) and
$E=-\nabla_{x}U$, thus

$\partial_{t}U=-\omega(-\Delta_{x})^{-1}{\rm{div}}_{x}{J}.$ (6.17)

Since ${J}(\tau,\cdot)\in L^{1}(\mathbb{R}^{3}_{x})$, we deduce by the Hardy
Littlewood Sobolev inequality that

$|||\partial_{t}U(\tau,\cdot)|||_{M^{3/2}(\mathbb{R}^{3})}\leq
c||{J}(\tau,\cdot)||_{L^{1}(\mathbb{R}^{3})}.$ (6.18)

Therefore, we estimate

$\displaystyle\left|\left|\left|\frac{\partial_{t}U(\tau,X(\tau,x,v))}{\left(1+\frac{|V(\tau,x,v)|^{2}}{2}\right)\left(1+\log\left(1+\frac{|V(\tau,x,v)|^{2}}{2}\right)\right)^{1-\alpha}}\right|\right|\right|_{M^{3/2}(\mathbb{R}^{3}_{x}\times\mathbb{R}^{3}_{v})}$
(6.19) $\displaystyle\leq
L^{2/3}\left|\left|\left|\frac{\partial_{t}U(\tau,x)}{\left(1+\frac{|v|^{2}}{2}\right)\left(1+\log\left(1+\frac{|v|^{2}}{2}\right)\right)^{1-\alpha}}\right|\right|\right|_{M^{3/2}(\mathbb{R}^{3}_{x}\times\mathbb{R}^{3}_{v})}$
$\displaystyle\leq
L^{2/3}\left|\left|\left|\frac{\partial_{t}U(\tau,x)}{\left(1+\frac{|v|^{2}}{2}\right)\left(1+\log\left(1+\frac{|v|^{2}}{2}\right)\right)^{1-\alpha}}\right|\right|\right|_{L^{3/2}(\mathbb{R}_{v}^{3};M^{3/2}(\mathbb{R}_{x}^{3}))}$
$\displaystyle\leq
L^{2/3}|||\partial_{t}U(\tau,\cdot)|||_{M^{3/2}(\mathbb{R}_{x}^{3})}\left(\int\limits_{\mathbb{R}^{3}}\frac{dv}{\left(1+\frac{|v|^{2}}{2}\right)^{3/2}\left(1+\log\left(1+\frac{|v|^{2}}{2}\right)\right)^{3(1-\alpha)/2}}\right)^{2/3}$
$\displaystyle\leq c||{J}(\tau,\cdot)||_{L^{1}(\mathbb{R}^{3})},$

where the last integral is convergent since $3(1-\alpha)/2>1$. From the
inclusion $M^{3/2}(\mathbb{R}^{3}_{x}\times\mathbb{R}^{3}_{v})\subset
L^{1}(\mathbb{R}^{3}_{x}\times\mathbb{R}^{3}_{v})+L^{\infty}(\mathbb{R}^{3}_{x}\times\mathbb{R}^{3}_{v})$,
and integrating (6.16) over $B_{r}$, we get (6.9) as desired. ∎

##  7\. Renormalized solutions and Lagrangian solutions

We recall the different notions of weak solutions for the Vlasov-Poisson
system. We shall always assume that $1\leq N\leq 3$, and we consider an
initial datum $f^{0}\in L^{1}(\mathbb{R}^{N}_{x}\times\mathbb{R}^{N}_{v})$,
$f^{0}\geq 0$. We introduce first renormalized solutions, following [12, 14].

######  Definition 7.1.

We say that $f\in
L^{\infty}((0,T);L^{1}(\mathbb{R}^{N}_{x}\times\mathbb{R}^{N}_{v}))$, $f\geq
0$, is a solution to the Vlasov equation (1.1) in the renormalized sense if
for all test functions $\beta\in C^{1}([0,\infty))$ with $\beta$ bounded, we
have that

$\partial_{t}\beta(f)+v\cdot\nabla_{x}\beta(f)+{\rm{div}}_{v}\Bigl{(}E(t,x)\beta(f)\Bigr{)}=0,$
(7.1)

in
$\mathscr{D}^{\prime}((0,T)\times\mathbb{R}^{N}_{x}\times\mathbb{R}^{N}_{v})$.

We next introduce the notion of Lagrangian solutions.

######  Definition 7.2.

Let be given a vector field $b(t,x,v)=(v,E(t,x))$ as in (3.6) for some
$\rho\in L^{\infty}((0,T);L^{1}(\mathbb{R}^{N}))$, $\rho\geq 0$, and
$\rho_{b}\in L^{1}(\mathbb{R}^{N})$. We assume that $E\in
L^{\infty}((0,T);L^{2}(\mathbb{R}^{N}))$, and that (2.6) holds with ${J}\in
L^{\infty}((0,T);L^{1}(\mathbb{R}^{N}))$. We assume furthermore that either
$N=1$ or $2$, or $N=3$ and $\omega=+1$, $\rho_{b}\in L^{p}(\mathbb{R}^{3})$
for some $p>3/2$. We consider regular Lagrangian flows $Z$ as in Definition
4.1, except that now $s\in[0,T]$ instead of $s\in[t,T]$ (forward-backward
flow), and with compression constant $L$ independent of $t\in[0,T]$. Then
according to Subsections 3.3, 3.4, Proposition 6.2, Proposition 6.3, the
vector field $b$ satisfies assumptions (R1a), (R2a), (R3). Therefore, Theorem
5.2 yields the uniqueness of the forward-backward regular Lagrangian flow
$Z=(X,V)$. The whole theory of [10] then applies indeed, with very little
modifications in the proofs. In particular there is existence and uniqueness
of the forward-backward regular Lagrangian flow, with compression constant
$1$, and stability. We can thus define in accordance with [10] a Lagrangian
solution $f$ to the Vlasov equation (1.1) by

$f(t,x,v)=f^{0}\Bigl{(}X(s=0,t,x,v),V(s=0,t,x,v)\Bigr{)},\qquad\mbox{for all
}t\in[0,T],$ (7.2)

for arbitrary $f^{0}\in L^{1}(\mathbb{R}^{N}_{x}\times\mathbb{R}^{N}_{v})$. It
verifies in particular $f\in
C([0,T];L^{1}(\mathbb{R}^{N}_{x}\times\mathbb{R}^{N}_{v}))$, and it is indeed
also a renormalized solution.

######  Definition 7.3.

We define a Lagrangian solution to the Vlasov-Poisson system as a couple
$(f,E)$ such that

  1. (1)

$f\in C([0,T];L^{1}(\mathbb{R}^{N}_{x}\times\mathbb{R}^{N}_{v}))$, $f\geq 0$,
$|v|^{2}f\in
L^{\infty}((0,T);L^{1}(\mathbb{R}^{N}_{x}\times\mathbb{R}^{N}_{v}))$,

  2. (2)

$E(t,x)$ is given by the convolution (1.6) with $\rho(t,x)=\int f(t,x,v)dv$,
$\rho_{b}\in L^{1}(\mathbb{R}^{N})$, $\rho_{b}\geq 0$ (and if $N=3$,
$\omega=+1$, $\rho_{b}\in L^{p}(\mathbb{R}^{3})$ for some $p>3/2$),

  3. (3)

$E\in L^{\infty}((0,T);L^{2}(\mathbb{R}^{N}_{x}))$,

  4. (4)

The relation (2.6) holds with ${J}(t,x)=\int vf(t,x,v)dv$,

  5. (5)

$f$ is a Lagrangian solution to the Vlasov equation, in the sense of (7.2).

##  8\. Existence of Lagrangian solutions

###  8.1. Compactness

In this subsection we prove two compactness results, Theorems 8.2 and 8.3, for
families of Lagrangian solutions to the Vlasov-Poisson system, with strongly
or weakly convergent initial data.

######  Lemma 8.1.

Let $g(x)=\frac{x}{|x|^{N}}$ for $x\in\mathbb{R}^{N}$, and denote by
$\tau_{h}g(x)=g(x+h)$. Then for any $1<p<\frac{N}{N-1}$,

$\|\tau_{h}g(x)-g(x)\|_{L^{p}(\mathbb{R}^{N})}\leq c|h|^{\alpha},$ (8.1)

with $\alpha=1-N+N/p>0$, and where $c$ depends on $N,p$.

###### Proof.

Fix $h\in\mathbb{R}^{N}$, $h\not=0$. For $|x|>2|h|$, we have for all
$0\leq\theta\leq 1$, $|x+\theta h|\geq|x|-\theta|h|>|x|/2$, thus we have

$\begin{array}[]{l}\displaystyle|\tau_{h}g(x)-g(x)|\leq|h|\sup_{0\leq\theta\leq
1}|\nabla g(x+\theta h)|\\\
\displaystyle\hphantom{|\tau_{h}g(x)-g(x)|}\leq|h|\sup_{0\leq\theta\leq
1}\frac{c_{N}}{|x+\theta h|^{N}}\leq c_{N}\frac{|h|}{|x|^{N}}.\end{array}$
(8.2)

Then we estimate

$\displaystyle\int\limits_{|x|>2|h|}\frac{|h|^{p}}{|x|^{Np}}dx=c_{N}|h|^{p}\int\limits^{\infty}_{2|h|}r^{N-1-Np}dr=c_{N}|h|^{p}\frac{(2|h|)^{N-Np}}{Np-N}=c_{N,p}|h|^{p-Np+N}.$

Next, for $|x|\leq 2|h|$, we write

$|\tau_{h}g(x)-g(x)|\leq\frac{1}{|x+h|^{N-1}}+\frac{1}{|x|^{N-1}},$ (8.3)

and clearly

$\begin{array}[]{l}\displaystyle\int\limits_{|x|\leq
2|h|}\left(\frac{1}{|x+h|^{(N-1)p}}+\frac{1}{|x|^{(N-1)p}}\right)dx\\\
\displaystyle\leq 2\int\limits_{|y|\leq
3|h|}\frac{dy}{|y|^{(N-1)p}}=c_{N}\int\limits^{3|h|}_{0}r^{N-Np+p-1}dr=c_{N,p}|h|^{N-Np+p},\end{array}$
(8.4)

since the last integral is convergent for $p<\frac{N}{N-1}$. ∎

######  Theorem 8.2.

Let $(f_{n},E_{n})$ be a sequence of Lagrangian solutions to the Vlasov-
Poisson system satisfying

$f_{n}^{0}\to f^{0}\text{ in
}L^{1}(\mathbb{R}^{N}_{x}\times\mathbb{R}^{N}_{v}),$ (8.5)

and

$\iint|v|^{2}f_{n}(t,x,v)dxdv+\int|E_{n}(t,x)|^{2}dx\leq C,\qquad\mbox{for all
}t\in[0,T].$ (8.6)

Then, up to a subsequence $f_{n}$ converges strongly in
$C([0,T];L^{1}(\mathbb{R}^{N}_{x}\times\mathbb{R}^{N}_{v}))$ to $f$, $E_{n}$
converges in $C([0,T];L^{1}_{loc}(\mathbb{R}^{N}))$ to $E$, and $(f,E)$ is a
Lagrangian solution to the Vlasov-Poisson system with initial datum $f^{0}$.
Moreover, the regular forward-backward Lagrangian flow $Z_{n}(s,t,x,v)$
converges to $Z(s,t,x,v)$ locally in measure in
$\mathbb{R}^{N}\times\mathbb{R}^{N}$, uniformly in $s,t\in[0,T]$.

###### Proof.

Step 1.) (Equi-integrability)
Because of (7.2) and (8.5) we have

$\|f_{n}(t,\cdot,\cdot)\|_{L^{1}(\mathbb{R}^{N}\times\mathbb{R}^{N})}=\|f_{n}^{0}\|_{L^{1}(\mathbb{R}^{N}\times\mathbb{R}^{N})}\leq
M.$ (8.7)

Then because of the bounds (8.7), (8.6), and applying Propositions 6.2, 6.3,
one has for any $r>0$

$\mathscr{L}^{2N}\\{(x,v)\in B_{r}:\sup_{0\leq s\leq
T}|Z_{n}(s,t,x,v)|>\gamma\\}\rightarrow 0,\mbox{ as
}\gamma\rightarrow\infty,\mbox{uniformly in }t,n.$ (8.8)

Since the sequence $f_{n}^{0}$ is uniformly equi-integrable, and since $Z_{n}$
is measure-preserving, we have by (7.2) and (8.8) that $f_{n}(t,\cdot)$ is
equi-integrable, uniformly in $t,n$. Consequently, $\rho_{n}(t,\cdot)$ is also
equi-integrable, uniformly in $t,n$. Using the bound (8.6), $vf_{n}(t,\cdot)$
is also equi-integrable, and therefore ${J}_{n}(t,\cdot)$ is also equi-
integrable, uniformly in $t,n$.

Step 2.) (Compactness of the field)
In order to prove that $E_{n}(t,x)\to E(t,x)\textrm{ in
}L^{1}_{\rm{loc}}((0,T)\times\mathbb{R}^{N}_{x})$, we first look at the
compactness in $x$. Denote by $\tau_{h}E_{n}(t,x)=E_{n}(t,x+h)$. Then using
(1.6),

$\begin{array}[]{l}\displaystyle\hphantom{\leq}||\tau_{h}E_{n}(t,\cdot)-E_{n}(t,\cdot)||_{L^{p}(\mathbb{R}^{N})}\\\
\displaystyle\leq
c\left\|\tau_{h}\frac{x}{|x|^{N}}-\frac{x}{|x|^{N}}\right\|_{L^{p}(\mathbb{R}^{N})}||\rho_{n}(t,\cdot)-\rho_{b}(\cdot)||_{L^{1}(\mathbb{R}^{N})}.\end{array}$
(8.9)

Thus according to Lemma 8.1, we get for any $1<p<\frac{N}{N-1}$ that

$||E_{n}(t,x+h)-E_{n}(t,x)||_{L^{p}(\mathbb{R}^{N})}\to 0,\qquad\mbox{as }h\to
0,\mbox{ uniformly in }t,n.$ (8.10)

Then, because of (2.6), we have that $\partial_{t}E_{n}$ is bounded in
$L^{\infty}((0,T);\mathscr{S}^{\prime}(\mathbb{R}^{N}))$. Applying Aubin’s
lemma, we conclude that $E_{n}$ is compact in
$L^{1}_{\rm{loc}}([0,T]\times\mathbb{R}^{N}_{x})$. Thus after extraction of a
subsequence, $E_{n}(t,x)\to E(t,x)$ strongly in
$L^{1}((0,T);L^{1}_{\rm{loc}}(\mathbb{R}^{N}_{x}))$.

Step 3.) (Convergence of the flow)
Because of the bound (8.6), one has $E\in
L^{\infty}((0,T),L^{2}(\mathbb{R}^{N}))$. Also, using the uniform bounds on
$\rho_{n}$, ${J}_{n}$ in $L^{\infty}((0,T);L^{1}(\mathbb{R}^{N}))$ and the
uniform equi-integrability obtained in Step 1, one has up to a subsequence
$\rho_{n}\rightarrow\rho$, ${J}_{n}\rightarrow{J}$ in the sense of
distributions, with $\rho,{J}\in L^{\infty}((0,T);L^{1}(\mathbb{R}^{N}))$. We
can pass to the limit in (1.6) and (2.6). Therefore, $b=(v,E)$ satisfies the
assumptions (R1a), (R2a), (R3) and Definition 7.2 applies. According to [10,
Lemma 6.3], since (8.8) holds, and $\rho_{n}$ are equi-integrable, we deduce
the convergence of $Z_{n}$ to $Z$ locally in measure in
$\mathbb{R}^{N}_{x}\times\mathbb{R}^{N}_{v}$, uniformly with respect to
$s,t\in[0,T]$, where $Z$ is the regular forward-backward Lagrangian flow
associated to $b$.

Step 4.) (Convergence of $f$)
Using the convergence (8.5), we can apply [10, Proposition 7.3], and we
conclude that $f_{n}\rightarrow f$ in
$C([0,T];L^{1}(\mathbb{R}^{N}_{x}\times\mathbb{R}^{N}_{v}))$, where $f$ is the
Lagrangian solution to the Vlasov equation with coefficient $b$ and initial
datum $f^{0}$. It follows that $\rho_{n}\to\rho=\int fdv$ in
$C([0,T];L^{1}(\mathbb{R}^{N}_{x}))$. By lower semi-continuity, we get from
(8.6) that

$\iint|v|^{2}f(t,x,v)dxdv+\int|E(t,x)|^{2}dx\leq C,\qquad\mbox{for all
}t\in[0,T].$ (8.11)

The bound (8.6) gives also that ${J}_{n}\rightarrow J=\int vfdv$ in
$C([0,T];L^{1}(\mathbb{R}^{N}_{x}))$. Therefore, $(f,E)$ is a Lagrangian
solution to the Vlasov-Poisson system. Using (1.6), we get that
$E_{n}\rightarrow E$ in $C([0,T];L^{1}_{loc}(\mathbb{R}^{N}))$, which
concludes the proof. ∎

######  Theorem 8.3.

Let $(f_{n},E_{n})$ be a sequence of Lagrangian solutions to the Vlasov-
Poisson system satisfying

$f_{n}^{0}\rightharpoonup f^{0}\text{ weakly in
}L^{1}(\mathbb{R}^{N}_{x}\times\mathbb{R}^{N}_{v}),$ (8.12)

and the bound (8.6). Then, up to a subsequence $f_{n}$ converges in
$C([0,T];\mathrm{weak}-L^{1}(\mathbb{R}^{N}_{x}\times\mathbb{R}^{N}_{v}))$ to
$f$, $E_{n}$ converges in $C([0,T];L^{1}_{loc}(\mathbb{R}^{N}))$ to $E$, and
$(f,E)$ is a Lagrangian solution to the Vlasov-Poisson system with initial
datum $f^{0}$. Moreover, the regular forward-backward Lagrangian flow
$Z_{n}(s,t,x,v)$ converges to $Z(s,t,x,v)$ locally in measure in
$\mathbb{R}^{N}\times\mathbb{R}^{N}$, uniformly in $s,t\in[0,T]$.

###### Proof.

It is the same as that of Theorem 8.2, except the last step 4. Instead we
apply [10, Proposition 7.7] and conclude that $f_{n}\rightharpoonup f$ in
$C([0,T];\mathrm{weak}-L^{1}(\mathbb{R}^{N}_{x}\times\mathbb{R}^{N}_{v}))$,
where $f$ is the Lagrangian solution to the Vlasov equation with coefficient
$b$ and initial datum $f^{0}$. It follows that
$\rho_{n}\rightharpoonup\rho=\int fdv$ in
$C([0,T];\mathrm{weak}-L^{1}(\mathbb{R}^{N}_{x}))$. By lower semi-continuity,
we get again from (8.6) the energy bound (8.11). The bound (8.6) also enables
to conclude that ${J}_{n}\rightharpoonup J=\int vfdv$ in
$C([0,T];\mathrm{weak}-L^{1}(\mathbb{R}^{N}_{x}))$. Therefore, $(f,E)$ is a
Lagrangian solution to the Vlasov-Poisson system. Using (1.6) and the
compactness estimate (8.10), we get that $E_{n}\rightarrow E$ in
$C([0,T];L^{1}_{loc}(\mathbb{R}^{N}))$, which concludes the proof. ∎

###  8.2. Existence

We conclude this section by the existence of Lagrangian solutions to the
Vlasov-Poisson system for initial datum in $L^{1}$ with finite energy, in the
repulsive case.

######  Theorem 8.4.

Let $N=1,2$ or $3$, and let $f^{0}\in
L^{1}(\mathbb{R}^{N}_{x}\times\mathbb{R}^{N}_{v})$, $f^{0}\geq 0$. Define
$\rho^{0}$ and $E^{0}$ by

$\rho^{0}(x)=\int f^{0}(x,v)dv,\qquad
E^{0}(x)=\frac{\omega}{|S^{N-1}|}\frac{x}{|x|^{N}}*(\rho^{0}(x)-\rho_{b}(x)),$
(8.13)

with $\omega=+1$ (repulsive case), $\rho_{b}\in L^{1}(\mathbb{R}^{N})$,
$\rho_{b}\geq 0$, and in the case $N=3$ $\rho_{b}\in L^{p}(\mathbb{R}^{3})$
for some $p>3/2$. Assume that the initial energy is finite,

$\iint|v|^{2}f^{0}(x,v)dxdv+\int|E^{0}(x)|^{2}dx<\infty.$ (8.14)

Then there exists a Lagrangian solution $(f,E)$ to the Vlasov-Poisson system
defined for all time, having $f^{0}$ as initial datum, and satisfying for all
$t\geq 0$

$\iint|v|^{2}f(t,x,v)dxdv+\int|E(t,x)|^{2}dx\leq\iint|v|^{2}f^{0}(x,v)dxdv+\int|E^{0}(x)|^{2}dx.$
(8.15)

###### Proof.

We use the classical way of getting global weak solutions to the Vlasov-
Poisson system, i.e. we approximate the initial datum $f^{0}$ by a sequence of
smooth data $f_{n}^{0}\geq 0$ with compact support. We approximate also
$\rho_{b}$ by smooth $\rho_{b}^{n}\geq 0$ with compact support (with
$\int(\rho^{0}_{n}-\rho_{b}^{n})dx=0$ if $N=1,2$). It is possible to do that
with the upper bounds

$\begin{array}[]{l}\displaystyle\mathop{\mathrm{limsup}}_{n\to\infty}\iint|v|^{2}f^{0}_{n}(x,v)dxdv\leq\iint|v|^{2}f^{0}(x,v)dxdv,\\\
\displaystyle\mathop{\mathrm{limsup}}_{n\to\infty}\int|E^{0}_{n}(x)|^{2}dx\leq\int|E^{0}(x)|^{2}dx.\end{array}$
(8.16)

Then, for each $n$, there exists a smooth classical solution $(f_{n},E_{n})$
with initial datum $f^{0}_{n}$, to the Vlasov-Poisson system, defined for all
time $t\geq 0$. Note that we can alternatively consider a regularized Vlasov-
Poisson system. Since $\omega=+1$, the conservation of energy (2.8) gives for
all $t\geq 0$,

$\iint|v|^{2}f_{n}(t,x,v)dxdv+\int|E_{n}(t,x)|^{2}dx=\iint|v|^{2}f_{n}^{0}(x,v)dxdv+\int|E_{n}^{0}(x)|^{2}dx.$
(8.17)

The couple $(f_{n},E_{n})$ is in particular a Lagrangian solution to the
Vlasov-Poisson system, for all intervals $[0,T]$. We can therefore apply
Theorem 8.2. Extracting a diagonal subsequence, we get the convergence of
$(f_{n},E_{n})$ to $(f,E)$ as stated in Theorem 8.2, where $(f,E)$ is a
Lagrangian solution to the Vlasov-Poisson system defined for all time, with
$f^{0}$ as initial datum. The bound (8.11), together with (8.16), gives
(8.15). ∎

Let us end with a remark on measure densities. When considering a sequence of
solutions to the Vlasov-Poisson system, the vector fields $b_{n}=(v,E_{n})$
have a gradient in $(x,v)$ of the form

$Db_{n}=\left(\begin{matrix}D_{1}b_{n}^{1}&D_{2}b_{n}^{1}\\\
D_{1}b_{n}^{2}&D_{2}b_{n}^{2}\end{matrix}\right)=\left(\begin{matrix}0&\mathrm{Id}\\\
S(\rho_{n}-\rho_{b})&0\end{matrix}\right),$ (8.18)

where the index $1$ stands for $x$, $2$ for $v$, and where $S$ is a singular
integral operator. If we require only that $D_{1}b_{n}^{2}$ converges in the
sense of distributions to $D_{1}b^{2}=S(\rho-\rho_{b})$, for some measure
$\rho\in\mathscr{M}(\mathbb{R}^{N})$, then we are in the setting of [8]. If
$\rho_{n}$ is uniformly bounded in $L^{1}((0,T);\mathscr{M}(\mathbb{R}^{N}))$,
and $b_{n}\to b$ strongly in
$L^{1}((0,T);L^{1}_{\rm{loc}}(\mathbb{R}^{N}_{x}\times\mathbb{R}^{N}_{v}))$
with $b$ satisfying (8.18), we conclude that $Z_{n}\to Z$ strongly, where $Z$
is the regular Lagrangian flow associated to $b$. However, we are not able to
define the push forward (7.2) of a measure $f^{0}$. This prevents from
applying the context of [8] to the Vlasov-Poisson system with measure data.

## References

  * [1] L. Ambrosio, _Transport equation and Cauchy problem for BV vector fields_ , Invent. Math. 158, 227-260, 2004.
  * [2] L. Ambrosio, M. Colombo, A. Figalli, _Existence and uniqueness of maximal regular flows for non-smooth vector fields_ , preprint 2014.
  * [3] L. Ambrosio, M. Colombo, A. Figalli, _On the Lagrangian structure of transport equations: the Vlasov-Poisson system_ , preprint 2014.
  * [4] L. Ambrosio, G. Crippa, _Existence, uniqueness, stability and differentiability properties of the flow associated to weakly differentiable vector fields_ , Transport equations and multi-D hyperbolic conservation laws, 3-57, Lect. Notes Unione Mat. Ital., 5, Springer, Berlin, 2008.
  * [5] L. Ambrosio, G. Crippa, _Continuity equations and ODE flows with non-smooth velocity_ , Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 144, 1191-1244, 2014.
  * [6] A.A. Arsen’ev, _Existence in the large of a weak solution of Vlasov’s system of equations (Russian)_ , Z. Vycisl. Mat. i Mat. Fiz. 15, 136-147, 276, 1975.
  * [7] J. Batt, _Global symmetric solutions of the initial value problem of stellar dynamics_ , J. Differential Equations 25, 342-364, 1977.
  * [8] A. Bohun, F. Bouchut, G. Crippa, _Lagrangian flows for vector fields with anisotropic regularity_ , preprint 2014.
  * [9] F. Bouchut, G. Crippa, _Equations de transport à coefficient dont le gradient est donné par une intégrale singulière_ , In séminaire EDP, Ecole Polytechnique, 2007-2008, exposé no 1.
  * [10] F. Bouchut, G. Crippa, _Lagrangian flows for vector fields with gradient given by a singular integral_ , J. Hyp. Diff. Eq. 10, 235-282, 2013.
  * [11] G. Crippa, C. De Lellis, _Estimates and regularity results for the DiPerna-Lions flow_ , J. Reine Angew. Math. 616, 15-46, 2008.
  * [12] R. DiPerna, P.-L. Lions, _Solutions globales d’équations du type Vlasov-Poisson_ , C. R. Acad. Sci. Paris Sér. I Math. 307, 655-658, 1988.
  * [13] R. DiPerna, P.-L. Lions, _Ordinary differential equations, transport theory and Sobolev spaces_ , Invent. Math. 98, 511-547, 1989.
  * [14] R. DiPerna, P.-L. Lions, _Global weak solutions of kinetic equations_ , Rend. Sem. Mat. Univ. Politec. Torino 46, 259-288, 1990.
  * [15] E. Horst, _On the classical solutions of the initial value problem for the unmodified non-linear Vlasov equation, Parts I and II_ , Math. Methods. Appl. Sci. 3, 229-248, 1981, Math. Methods Appl. Sci. 4,19-32, 1982.
  * [16] P.-E. Jabin, _The Vlasov-Poisson system with infinite mass and energy_ , J. Statist. Phys. 103, 1107-1123, 2001.
  * [17] P.-L. Lions, B. Perthame, _Propagation of moments and regularity for the 3-dimensional Vlasov-Poisson system_ , Invent. Math. 105, 415-430, 1991.
  * [18] A.J. Majda, G. Majda, Y.X. Zheng, _Concentrations in the one-dimensional Vlasov-Poisson equations. I. Temporal development and non-unique weak solutions in the single component case_ , Phys. D 74, 268-300, 1994.
  * [19] A.J. Majda, G. Majda, Y.X. Zheng, _Concentrations in the one-dimensional Vlasov-Poisson equations. II. Screening and the necessity for measure-valued solutions in the two component case_ , Phys. D 79, 41-76, 1994.
  * [20] B. Perthame, _Time decay, propagation of low moments and dispersive effects for kinetic equations_ , Comm. Partial Differential Equations 21, 659-686,1996.
  * [21] K. Pfaffelmoser, _Global classical solutions of the Vlasov-Poisson system in three dimensions for general initial data_ , J. Diff. Eq. 95, 281-303, 1992.
  * [22] J. Schaeffer, _Global existence of smooth solutions to the Vlasov-Poisson system in three dimensions_ , Commun. Part. Diff. Eqns. 16, 1313-1335, 1991.
  * [23] J. Schaeffer, _Asymptotic growth bounds for the Vlasov-Poisson system_ , Math. Methods Appl. Sci. 34, 262-277, 2011.
  * [24] X. Zhang, J. Wei, _The Vlasov-Poisson system with infinite kinetic energy and initial data in $L^{p}({\mathbf{R}}^{6})$_, J. Math. Anal. Appl. 341, 548-558, 2008.
  * [25] Y.X. Zheng, A.J. Majda, _Existence of global weak solutions to one-component Vlasov-Poisson and Fokker-Planck-Poisson systems in one space dimension with measures as initial data_ , Comm. Pure Appl. Math. 47, 1365-1401, 1994.


# High Precision Indoor Localization with Dummy Antennas - An Experimental
Study

Kaixuan Huang†, Chenlu Xiang†, Shunqing Zhang†, Shugong Xu†, Xianfeng Ma‡,
Qinglong Xian‡, and Hua Yang‡
†Shanghai Institute for Advanced Communication and Data Science, Shanghai
University, Shanghai, China
‡Guangzhou Guangzhong Enterprise Group Corporation, Guangdong, China
Email: {kaixuanhuang, xcl, shunqing,
shugong}<EMAIL_ADDRESS>
###### Abstract

With the rising demand for indoor localization, high precision technique-based
fingerprints became increasingly important nowadays. The newest advanced
localization system makes effort to improve localization accuracy in the time
or frequency domain, for example, the UWB localization technique can achieve
centimeter-level accuracy but have a high cost. Therefore, we present a
spatial domain extension-based scheme with low cost and verify the
effectiveness of antennas extension in localization accuracy. In this paper,
we achieve sub-meter level localization accuracy using a single AP by
extending three radio links of the modified laptops to more antennas.
Moreover, the experimental results show that the localization performance is
superior as the number of antennas increases with the help of spatial domain
extension and angular domain assisted.

###### Index Terms:

localization, spatial domain extension, angular domain assisted, channel state
information

##  I Introduction

Location-based services (LBS) become a key element of modern mobile internet
applications, which utilize real-time geometric information from smartphones
to provide navigation, entertainment, or security [1]. Although satellite
positioning systems, such as global positioning system (GPS) [2] or Beidou
[3], can achieve centimeter-level localization accuracy in the outdoor
environment, they can hardly achieve the same level in the indoor scenario,
due to satellite signal occlusions. To address this issue, more diversified
wireless signals, including wireless fidelity (WiFi)[4], Bluetooth low energy
(BLE)[5], and increasingly popular 3GPP long term evolution (LTE) / new radio
(NR) technology [6], have been exploited to provide indoor localization
services, and accurate navigation and seamless tracking capability in both
outdoor and indoor scenarios have been specified in 3GPP Release 16 standard
[7].

In order to provide a better indoor localization accuracy, the most direct
approach is to collect more observation samples, either in the frequency or
time domain [8]. For example, the ultra wide band (UWB) localization technique
adopted in the recent iPhone product, is able to achieve centimeter-level
localization accuracy with 500 MHz bands. Fingerprint-based localization
solutions can also achieve a similar accuracy when the observation window is
sufficiently long [9]. However, improving the localization accuracy via
spatial domain extension is never straightforward due to the following
reasons. First, the existing spatial domain schemes [10, 11] rely on
calculating the geometric location through the measured time of flight (TOF)
and angle of arrival (AoA), and the corresponding resolution does not scale
linearly with respect to the number of available radio frequency (RF) chains
[12]. Second, deploying more RF chains with expensive components, including
low-noise amplifiers, analog-to-digital, or digital-to-analog converters, are
cost-prohibitive in general [13]. Third, the processing complexity increases
exponentially with respect to the number of RF chains as well [14]. As
illustrated in [15], the baseband processing of a 12-antenna software defined
radio (SDR) system has to be moved to ‘BigStation’ with 15 personal computers
for real-time signal processing.

Last but not least, to address the aforementioned issues of the spatial domain
extension for high accuracy indoor localization, we present in this paper a
low-cost WiFi based prototype localization system with dummy antennas to
improve AoA resolutions as well as localization accuracy. By regularly
rotating 12 antennas with only 3 RF chains, the proposed system is able to
achieve less than 1 meter localization accuracy in a $40m^{2}$ corridor
environment, and the main contributions of our work are listed below.

  * •

Low-Cost Spatial Domain Extension Conventional spatial domain extension
schemes require special hardware support. As reported in [16], Phaser builds a
9-antenna WiFi based AoA estimation system by combining multiple WiFi network
interface cards (NICs) and the associated RF chains together, and the hardware
cost is thus significant. In this paper, we control the number of RF chains
and extend the number of antennas by introducing a four-port switcher to
reduce the hardware cost, which can enjoy the benefit of AoA resolution
improvement and maintains the hardware cost simultaneously.

  * •

AoA Assisted Localization Different from the conventional approach to directly
minimize mean square error (MSE) in [17], we propose a two-stage localization
strategy, which contains an AoA estimation and a location prediction. Through
this approach, the localization accuracy can be improved to less than 1 meter
with more available dummy antennas.

  * •

Low Cost Hardware Implementation We use three low-cost off-the-shelf single
pole four throws (SP4T) RF switchers to connect 12 dummy antennas on top of
only one 3-port WiFi NIC. By applying a time division based activation scheme,
we are able to collect the channel behaviors of all the available antennas,
which eventually pave a way for low-cost high precision indoor localization
with dummy antennas.

The remainder of this paper is organized as follows. In Section II, we
introduce the system model and some preliminary information, and the problem
formulation is discussed in Section III. The problem analysis, as well as the
corresponding solution, are provided in Section IV. In Section V, we present
our experimental results and the concluding remarks are given in Section VI.

##  II System Model

In this section, we briefly introduce some preliminary knowledge about channel
state information (CSI), followed by the angular based localization
procedures.

###  II-A CSI Modeling

Consider an orthogonal frequency division multiplexing (OFDM) based modern
WiFi communication system with $N_{SC}$ subcarriers. For a given location
$\mathcal{L}$, the received signal vector,
$\mathbf{y}(\mathcal{L})=\left[y_{1}(\mathcal{L}),\cdots,y_{N_{R}}(\mathcal{L})\right]^{T}\in\mathbb{C}^{N_{R}\times
1}$, can be modeled through,

$\displaystyle\mathbf{y}(\mathcal{L})=\mathbf{H}(\mathcal{L})\cdot\mathbf{x}+\mathbf{n}(\mathcal{L}),$
(1)

where $\mathbf{H}(\mathcal{L})\in\mathbb{C}^{N_{R}\times N_{SC}}$ denotes the
channel frequency responses, $\mathbf{x}\in\mathbb{C}^{N_{SC}\times 1}$
denotes the transmitted symbols, and
$\mathbf{n}(\mathcal{L})\in\mathbb{C}^{N_{R}\times 1}$ denotes the additive
white Gaussian noise (AWGN), respectively. If we further choose
$h_{i}(\mathcal{L},n)$ to be the collected channel state information of the
$i^{th}$ subcarrier at the $n^{th}$ received antenna for a given location
$\mathcal{L}$, $\mathbf{H}(\mathcal{L})$ can be rewritten as,

$\displaystyle\mathbf{H}(\mathcal{L})=\left[\begin{array}[]{ccc}h_{1}(\mathcal{L},1)&\cdots&h_{N_{SC}}(\mathcal{L},1)\\\
\vdots&\ddots&\vdots\\\
h_{1}(\mathcal{L},N_{R})&\cdots&h_{N_{SC}}(\mathcal{L},N_{R})\end{array}\right].$
(5)

Without loss of generality, we denote $a_{i}(\mathcal{L},n)$ and
$p_{i}(\mathcal{L},n)$ to be the amplitude and phase information of the
corresponding channel responses, $h_{i}(\mathcal{L},n)$ can be modeled as,

$\displaystyle h_{i}(\mathcal{L},n)=a_{i}(\mathcal{L},n)\cdot e^{j\cdot
p_{i}(\mathcal{L},n)}.$ (6)
$\displaystyle\mathbf{P}(\mathcal{L})=\left[\begin{array}[]{ccc}p_{1}(\mathcal{L},1)&\cdots&p_{N_{SC}}(\mathcal{L},1)\\\
\vdots&\ddots&\vdots\\\
p_{1}(\mathcal{L},N_{R})&\cdots&p_{N_{SC}}(\mathcal{L},N_{R})\end{array}\right].$
(10)

and thus we obtain phase matrix $\mathbf{P}(\mathcal{L})$ extracted by
$\mathbf{H}(\mathcal{L})$.

###  II-B AoA Assisted Localization

As illustrated in Fig. 1, the geometric distances between two adjacent
antennas are selected to be half of the wavelength, and therefore, the AoA of
uniform linear array (ULA) configuration can be estimated via equation (11),

$\displaystyle\bar{\theta}=\frac{1}{N_{SC}}\sum_{i=1}^{N_{SC}}\arccos\left(\frac{p_{i}(\mathcal{L},n+1)-p_{i}(\mathcal{L},n)}{\pi}\right).$
(11)

However, the measured CSI usually contains information of the non-line-of-
sight (NLOS) path, in practical, which makes the AoA estimation inaccurate,
equation (11) does not work directly. In order to mitigate the effect of NLOS,
we need to extend the number of the antenna array to estimate AoA more
accurately. When the number of antennas is greater than the number of NLOS
paths, the MUSIC algorithm could be employed to estimate the AOA of the multi-
path signal.

Figure 1: Illustration diagram for AoA estimation with multiple receiving
antennas. With extended dummy antennas, the AoA estimation becomes inaccurate.

Considering a ULA with $N_{R}$ antennas, the correlation matrix of received
signals vector $\mathbf{R}_{\mathbf{yy}}$ can be expressed by,

$\displaystyle\mathbf{R}_{\mathbf{yy}}=\mathbb{E}[\mathbf{yy}^{H}]$ (12)

where $\mathbb{E}(\cdot)$ and $(\cdot)^{H}$ denote conjugate transpose and
expectation, respectively. The singular value decomposition of
$\mathbf{R}_{\mathbf{yy}}$ is performed, and obtain $N_{R}$ eigenvalues
corresponding to $N_{R}$ vectors $\mathbf{U}=[u_{1},u_{2},\cdots,u_{N_{R}}]$,
which is divided to two parts given as,

$\displaystyle\mathbf{U}=[\mathbf{U_{S}},\mathbf{U_{N}}]=[\underbrace{u_{1},\cdots,u_{p}},\underbrace{u_{p+1},\cdots,u_{N_{R}}}]$
(13)

where $\mathbf{U_{S}}$ and $\mathbf{U_{N}}$ are the corresponding vectors of
signal and noise parts. The signal steering vector
$a(\theta_{x})=[1,e^{-j\cdot w(2,\theta_{x})},\cdots,e^{-j\cdot
w(N_{R},\theta_{x})}]^{T}$ and noise are uncorrelated, we have,

$\displaystyle a^{H}(\theta_{x})\mathbf{U_{N}}=0$ (14)

Based on the orthogonality based signal and noise, the AoA spectrum of MUSIC
can be expressed as,

$\displaystyle\mathbf{B}(\theta)=\frac{1}{a^{H}(\theta)\mathbf{U_{N}}\mathbf{U_{N}^{H}}a(\theta)}$
(15)

We could obtain the closed expression of steering vector $a(\theta)$ by
minimizing its spectrum function, and then AoA of incident signal $\theta_{x}$
can be obtained by spectrum peaking search. We define $\mathcal{M}(\cdot)$ to
represent the peak search process of MUSIC algorithm as follows,

$\displaystyle\theta_{x}=\mathcal{M}\left(\mathbf{B}(\theta)\right)$ (16)

Once we obtain the AoA, $\theta_{x}$, the location information could be
further derived by combining the distance information.

##  III Problem Formulation

In this section, we formulate the localization problem using a generalized
optimization framework. Denote $\hat{\mathcal{L}}^{m}$ and $\mathcal{L}^{m}$
to be the $m^{th}$ predicted and true location, respectively, and we choose
MSE of them to be the performance measure, where111$\|\cdot\|_{2}$ denotes the
$l_{2}$ norm of the inner vector.
$\overline{\Delta\mathcal{L}}=\frac{1}{M}\sum_{m=1}^{M}\|\hat{\mathcal{L}}^{m}-\mathcal{L}^{m}\|_{2}$.
Mathematically, the location error minimization problem can be described as
follows.

######  Problem 1 (MSE Minimization).

The original location error minimization problem is given as follows.

$\displaystyle\underset{g(\cdot)}{\textrm{minimize}}$
$\displaystyle\frac{1}{M}\sum_{m=1}^{M}\|\hat{\mathcal{L}}^{m}-\mathcal{L}^{m}\|_{2},$
(17) subject to
$\displaystyle\hat{\mathcal{L}}^{m}=g\left({\widehat{\mathbf{H}}}(\mathcal{L}^{m}),\bar{\theta}^{m}_{x}\right),$
(18)
$\displaystyle\bar{\theta}^{m}_{x}=\mathcal{M}\left(\mathbf{B}(\theta)\right),$
$\displaystyle\hat{\mathcal{L}}^{m},\mathcal{L}^{m}\in\mathcal{A},$

where $\mathcal{A}$ denotes all the feasible localization areas.

In the indoor localization, the accuracy of AoA estimation based on MUSIC
algorithm is in general limited, due to the complicated NLOS fading
environments. Meanwhile, the associated computational complexity could be
significant, specially when the received SNR is small [18]. To overcome this
obstacle, we minimize the mean absolute error (MAE) of AoA rather than
directly calculate the AoA value using MUSIC. Denote $\hat{\theta}^{m}$ and
$\theta^{m}$ to be predicted and true AoA values, we can define the AoA MAE
minimization problem as follows.

######  Problem 2 (AoA MAE Minimization).

The AoA MAE minimization problem can be formulated as follows.

$\displaystyle\underset{f(\cdot)}{\textrm{minimize}}$
$\displaystyle\frac{1}{M}\sum_{m=1}^{M}\|\hat{\theta}^{m}-\theta^{m}\|,$ (19)
subject to
$\displaystyle\hat{\theta}^{m}=f\left({\widehat{\mathbf{P}}}(\mathcal{L}^{m})\right),$
(20)

where $\|\cdot\|$ denotes the $l_{1}$ norm and $\theta^{m}$ is the ground
truth value that can be measured from $\mathcal{L}^{m}$ and the location of
AP, $\mathcal{L}^{AP}$.

By combining the above two optimization problems together, we propose the
following angular domain assisted localization scheme, where we first estimate
the AoA value from Problem 2 and then minimize the location error of Problem 1
with the estimated AoA value, $\hat{\theta}^{m}$. Mathematically, we can
directly model the regression function $g^{\star}(\cdot)$ described as
follows.

######  Problem 3 (Angular Domain Assisted Localization).

The angular domain assisted localization problem can be formulated as,

$\displaystyle g^{\star}(\cdot)=\underset{g(\cdot)}{\textrm{minimize}}$
$\displaystyle\frac{1}{M}\sum_{m=1}^{M}\|\hat{\mathcal{L}}^{m}-\mathcal{L}^{m}\|_{2}.$
(21) subject to
$\displaystyle\hat{\mathcal{L}}^{m}=g\left({\widehat{\mathbf{H}}}(\mathcal{L}^{m}),\hat{\theta}^{m}\right),$
(22)
$\displaystyle\hat{\theta}^{m}=f^{\star}\left({\widehat{\mathbf{P}}}(\mathcal{L}^{m})\right),$
$\displaystyle\hat{\mathcal{L}}^{m},\mathcal{L}^{m}\in\mathcal{A},$

where $f^{\star}(\cdot)$ is the optimal solution of Problem 2.

Through the above formulation, we can approximate $f^{\star}(\cdot)$ using the
classical logistic regression model in the offline training rather than
running MUSIC algorithms in the online stage, which greatly reduces the
computational complexity in the localization. Meanwhile, it is also worth
mentioning that the localization accuracy and the computational complexities
can trade-off with each other in the theoretical sense, when the numbers of
training reference points or receive antennas increase, e.g., the localization
accuracy can be improved if we increase the experimental cost and algorithm
complexity in the offline stage.

Figure 2: The architecture of the system. In the offline stage, we collected
CSI of different antennas at Reference Point with modified PC by receiving
packets from WiFi AP and the amplitude and phase extracted from collected CSI
is the input data of DNN. In offline stage, we collect the CSI at Test Point
as test data.

##  IV Deep Learning based Localization Solution

In this section, we propose to apply the deep neural network (DNN)
architectures to solve the AoA MAE minimization and the angular domain
assisted localization as defined in Problem 2 and Problem 3.

###  IV-A Phase Alignment

One of the key factors for applying deep learning based solution is to
generate high quality measured CSI. However, the measured phase information
usually contains random jitters and noises, due to the imperfect hardware
components that used in the practical systems. To solve this issue and obtain
high accurate localization results, we model the main phase errors from
carrier frequency offset (CFO) and sampling frequency offset (SFO), which is
given below.

$\displaystyle\widehat{p_{i}}=p_{i}+2\pi\frac{i}{N_{SC}}\Delta{t}+\beta+Z,$
(23)

where $\widehat{\angle{p_{i}}}$ and $\angle{p_{i}}$ denote the measured and
genuine phase of subcarrier $i$, respectively. $\Delta{t}$ is the SFO time
lag, $\beta$ is the CFO, and $Z$ is the random measured noise with zero mean
and unit variance.

In order to eliminate these random phase offsets, we perform the linear
regression [19] on the raw phase information across the entire frequency band,
where the slope $k$ and the y-intercept of the regression line $b$ can be
obtained from,

$\displaystyle p_{i}$ $\displaystyle=$ $\displaystyle\widehat{p_{i}}-ki-b,$
(24) $\displaystyle k$ $\displaystyle=$
$\displaystyle\frac{\widehat{p_{N_{sc}}}-\widehat{p_{1}}}{N_{sc}-1},$ (25)
$\displaystyle b$ $\displaystyle=$
$\displaystyle\frac{1}{N_{sc}}\sum_{i=1}^{N_{sc}}\widehat{p_{i}}.$ (26)

With the above linear regression, the calibrated phase information can be
obtained by subtracting the linear phase offset from the measured phase
according to formula (24).

###  IV-B Neural Network Configuration

The full connection structure of DNN networks can extract the features from
CSI across the spatial domains. To improve the ability to extract features, we
implement AoA estimation and localization using two DNNs with four hidden
layers, respectively. For two DNNs, we adopt rectified linear unit (ReLU) as
the non-linear activation functions in each hidden layer to avoid the
vanishing gradient problem and adopt the linear function as the activation
function of the last output layer. The architecture of DNN is shown in Fig. 3.

In DNN1 used for AoA estimation, the input size of the input layer is
$30\times N$, which contains processed phase of 30 subcarriers at $N$ received
antennas, the output of DNN1 is AoA. And we select the loss function,
$\mathbb{L}_{AoA}$, to be the original definition of $f^{*}(\cdot)$ expressed
as,

$\displaystyle\mathbb{L}_{AoA}=\frac{1}{N_{RP}}\sum_{m=1}^{N_{RP}}\|\hat{\theta}^{m}-\theta^{m}\|,$
(27)

In DNN2 for used localization, every antenna has an additional angle provided
by DNN1 and amplitude information of 30 subcarriers, and thus the size of
input layer is 61. The output of DNN2 is the coordinate of every RP.
Therefore, we select the loss function $\mathbb{L}_{Loc}$ expressed as,

$\displaystyle\mathbb{L}_{Loc}=\frac{1}{N_{RP}}\sum_{m=1}^{N_{RP}}\|\hat{\mathcal{L}}^{m}-\mathcal{L}^{m}\|_{2}.$
(28)

Figure 3: The architecture of the DNN, which is consisted of four full connect
layers as hidden layer.

However, in order to avoid the biasing effects of some unusual samples, we
adopt the max-pooling technique to get rid of unimportant features and apply
the dropout technique to further reduce the unimportant connections in both
DNNs. The logistic regression based approach can gradually converge to the
non-convex function $\mathbb{L}_{AoA}$ and $\mathbb{L}_{Loc}$ with satisfied
performance via respective DNN networks. The detail of the configuration is
shown in Table I.

Table I: An Overview Of Network Configuration and Parameters. Layers | DNN1 | DNN2
---|---|---
Input Layers | $30\times N$ | $61\times N$
Hidden Layer 1 | FC 256 + ReLU | FC 256 + ReLU
Hidden Layer 2 | FC 256 + ReLU | FC 256 + ReLU
Hidden Layer 3 | FC 256 + ReLU | FC 256 + ReLU
Hidden Layer 4 | FC 256 + ReLU | FC 256 + ReLU
| \+ Dropout 0.3 | \+ Dropout 0.3
Output Layer | FC N + Linear | FC 2 + Linear

##  V Experimental Results

In this section, we provide some experimental results to validate the proposed
high precision indoor localization scheme with 12 dummy antennas. With more
receiving antennas installed at the localization terminals, we offer a low
cost implementation strategy on top of an off-the-shelf laptop, and compare
AoA estimation and localization results with baselines in what follows.

###  V-A Low Cost Implementation

The laptop used in this experiment consists of three receiving antennas. In
order to facilitate more receiving antennas, we connect the original three
radio chains with 12 antennas via three SP4T RF switches as shown in Fig. 4.
Through this approach, each radio chain can serve four antennas through SP4T
RF switches via a time division multiplexing mode, and an external micro-
controller unit is applied to control the active durations for each antenna.

Figure 4: The hardware design of extended antenna array. PE42442 is an
integrated board of three SP4Ts, which is used to extend 3 radio chains to
12-antenna-array.

To implement localization service, we select a $1.2\times 12m^{2}$ corridor
environment with 20 reference points (RPs) and a single WiFi AP. The entire
layout of the experimental environment is shown in Fig. 5, where we uniformly
select 9 additional testing points to obtain more reliable results. The
12-antenna modified laptop communicates with WiFi AP to obtain the CSI
information through Linux-802.11n-CSI-Tool [20] and WiFi network interface
cards (NICs). Since we can not concurrently measure all 12 antennas, we divide
them into four groups, and collect the corresponding CSIs by fast switching
and matching.

During each receiving period, we obtain one CSI packet per antenna, which
contains a total of 30 subcarriers. In the testing stage, we collect $20000$
CSI packets per RP for each antenna to construct the training data set, while
in the validation stage, $1000$ CSI packets per testing point (TP) for each
antenna is applied.

Figure 5: A sketch map of the experiment environment. Blue, red and yellow
points represent WiFi AP, reference point, and additional test point,
respectively.

###  V-B AoA Estimation Results

In the first experiment, we compare the accuracy of AoA estimation for
different antenna configurations. The cumulative distribution functions (CDFs)
of absolute AoA errors for 3, 6, 9, and 12 receiving antennas are plotted in
Fig. 6 and the AoA MAE values are summarized in Table II.

From the above results, we find that the AoA MAE value decreases when the
number of receiving antennas increases from 3 to 9 antennas. The physical
interpretation is that the performance of AoA estimation can be improved, when
the number of receiving antennas increases. However, when the number of
receiving antennas is greater than 12, the AoA estimation error may become
worse. This is due to the fact that the estimated AoA value will be inaccurate
when the size of receiving antennas becomes significant as illustrated in Fig.
1.

Figure 6: The CDF of AoA errors for different number of antennas.

###  V-C Localization Results

In the second experiment, we further compare the localization accuracy in
terms of RMSE for different antenna configurations. The CDF of absolute
localization errors for 3, 6, 9, and 12 antennas are plotted in Fig. 7 and the
overall RMSEs of localization schemes are listed in Table II.

As illustrated in Fig. 7, the localization accuracy in terms of RMSE improves
when the number of receiving antennas grows, e.g. from 1.747 m for 3 antennas
to 0.918 m for 12 antennas, which is equivalent to 47.5% improvement. From
Table II, we can also conclude that the absolute localization errors improves
when the number of receiving antennas grows as well.

It is worth noting that, different from AoA estimation, the localization
accuracy for 12 antennas is still better than that of other antenna
configurations. This is because the localization estimation utilizes the
amplitude information of CSI, which compensates the AoA estimation errors in
the previous process.

Table II: The MAE of AoA estimation and RMSE of Localization. Numbers of Antennas | AoA errors(°) | Localization errors(m)
---|---|---
3 antennas | 5.633 | 1.747
6 antennas | 4.234 | 1.445
9 antennas | 2.893 | 1.172
12 antennas | 3.635 | 0.918

##  VI Conclusion

In this paper, we propose an angular domain assisted localization scheme for
multi-antennas configuration. With more spatial domain diversity, we propose a
logistic regression based AoA estimation scheme, and substitute the AoA
estimated results in the later location. Through this approach, we can achieve
better localization accuracy. With the increased number of receiving antennas,
e.g. from 1.746 meters with 3 antennas to 0.918 meters with 12 antennas.

Figure 7: The CDF of localization errors for different number of antennas.

## Acknowledgement

This work was supported by the National Natural Science Foundation of China
(NSFC) under Grants 62071284, 61871262, 61901251 and 61904101, the National
Key Research and Development Program of China under Grants 2019YFE0196600, the
Innovation Program of Shanghai Municipal Science and Technology Commission
under Grant 20JC1416400, Pudong New Area Science & Technology Development
Fund, Key-Area Research and Development Program of Guangdong Province under
Grants 2020B0101130012, Foshan Science and Technology Innovation Team Project
under Grants FS0AA-KJ919-4402-0060 and research funds from Shanghai Institute
for Advanced Communication and Data Science (SICS).

## References

  * [1] C. De Lima, D. Belot, R. Berkvens, A. Bourdoux, D. Dardari, M. Guillaud, M. Isomursu, E.-S. Lohan, Y. Miao, A. N. Barreto, M. R. K. Aziz, J. Saloranta, T. Sanguanpuak, H. Sarieddeen, G. Seco-Granados, J. Suutala, T. Svensson, M. Valkama, B. Van Liempd, and H. Wymeersch, “Convergent communication, sensing and localization in 6g systems: An overview of technologies, opportunities and challenges,” _IEEE Access_ , vol. 9, pp. 26 902–26 925, Jan. 2021.
  * [2] L. Yen, C. Yan, S. Renu, A. Belay, H. Lin, and Y. Ye, “A modified wknn indoor wi-fi localization method with differential coordinates,” in _IEEE Int. Conf. Appl. Syst. Innov. (ICASI)_ , May. 2017, pp. 1822–1824.
  * [3] W. Jiang, S. Chen, B. Cai, J. Wang, W. ShangGuan, and C. Rizos, “A multi-sensor positioning method-based train localization system for low density line,” _IEEE Trans. Veh. Technol._ , vol. 67, no. 11, pp. 10 425–10 437, 2018.
  * [4] W. Jaworski, P. Wilk, P. Zborowski, W. Chmielowiec, A. Y. Lee, and A. Kumar, “Real-time 3d indoor localization,” in _IEEE Proc. Int. Conf. Indoor Posit. Indoor Navi._ , 2017, pp. 1–8.
  * [5] C. Xiao, D. Yang, Z. Chen, and G. Tan, “3-d ble indoor localization based on denoising autoencoder,” _IEEE Access_ , vol. 5, pp. 12 751–12 760, Jun. 2017.
  * [6] H. Zhang, Z. Zhang, S. Zhang, S. Xu, and S. Cao, “Fingerprint-based localization using commercial lte signals: A field-trial study,” in _IEEE Veh. Technol. Conf._ , 2019, pp. 1–5.
  * [7] TR38.855, “Study on nr positioning support.” in _3GPP (Release 16)_ , 2019\.
  * [8] D. Vasisht, S. Kumar, and D. Katabi, “Decimeter-level localization with a single WiFi access point.” in _USENIX Symp. Netw. Sys. Design & Implem._, vol. 16, 2016, pp. 165–178.
  * [9] C. Laoudias, A. Moreira, S. Kim, S. Lee, L. Wirola, and C. Fischione, “A survey of enabling technologies for network localization, tracking, and navigation,” _IEEE Commun. Surv. Tutor._ , vol. 20, no. 4, pp. 3607–3644, Jul. 2018.
  * [10] C. Laoudias, C. G. Panayiotou, and P. Kemppi, “On the RBF-based positioning using WLAN signal strength fingerprints,” in _IEEE Works. Posit. Navi. Commun._ , 2010, pp. 93–98.
  * [11] A. Cazzorla, G. De Angelis, A. Moschitta, M. Dionigi, F. Alimenti, and P. Carbone, “A 5.6-ghz uwb position measurement system,” _Instrum. Meas._ , vol. 62, no. 3, pp. 675–683, Mar. 2013.
  * [12] Y. Chapre, A. Ignjatovic, A. Seneviratne, and S. Jha, “Csi-mimo: Indoor wi-fi fingerprinting system,” in _IEEE Conf. Local Comp. Netw._ , Oct. 2014, pp. 202–209.
  * [13] Y. Xie, Y. Zhang, J. C. Liando, and M. Li, “Swan: Stitched wi-fi antennas,” in _ACM Int. Conf. on Mobile Comp. & Netw._, Oct. 2018, p. 51–66.
  * [14] C. Husmann, G. Georgis, K. Nikitopoulos, and K. Jamieson, “Flexcore: Massively parallel and flexible processing for large MIMO access points,” in _USENIX Symp. Netw. Sys. Design & Implem._, Mar. 2017, pp. 197–211.
  * [15] Q. Yang, X. Li, H. Yao, J. Fang, K. Tan, W. Hu, J. Zhang, and Y. Zhang, “Bigstation: Enabling scalable real-time signal processing in large mu-mimo systems,” in _ACM Spec. Inter. Group Data Commun._ , Aug. 2013, pp. 399–410.
  * [16] J. Gjengset, J. Xiong, G. Mcphillips, and K. Jamieson, “Phaser: enabling phased array signal processing on commodity wifi access points,” in _ACM Int. Conf. on Mobile Comp. & Netw._, 2014.
  * [17] C. Xiang, S. Zhang, S. Xu, X. Chen, S. Cao, G. C. Alexandropoulos, and V. K. N. Lau, “Robust sub-meter level indoor localization with a single wifi access point—regression versus classification,” _IEEE Access_ , vol. 7, pp. 146 309–146 321, Oct. 2019.
  * [18] J. Wang, J. Xiong, H. Jiang, X. Chen, and D. Fang, “D-watch: Embracing “bad” multipaths for device-free localization with cots rfid devices,” _IEEE ACM Trans. on Network._ , vol. 25, no. 6, pp. 3559–3572, Sep. 2017\.
  * [19] X. Wang, L. Gao, and S. Mao, “Phasefi: Phase fingerprinting for indoor localization with a deep learning approach,” in _IEEE Glob. Commun. Conf._ , Dec. 2015, pp. 1–6.
  * [20] D. Halperin, W. Hu, A. Sheth, and D. Wetherall, “Predictable 802.11 packet delivery from wireless channel measurements,” _ACM SIGCOMM_ , vol. 40, no. 4, pp. 159–170, Aug. 2010.