# Classical flows of vector fields with exponential or sub-exponential
summability

Luigi Ambrosio  Scuola Normale Superiore, Pisa, Italy<EMAIL_ADDRESS>,  Sebastiano Nicolussi Golo  Department of Mathematics and Statistics,
University of Jyväskylä, Finland<EMAIL_ADDRESS>and
Francesco Serra Cassano  Dipartimento di Matematica, Università di Trento,
Italy<EMAIL_ADDRESS>
###### Abstract.

We show that vector fields $b$ whose spatial derivative $D_{x}b$ satisfies a
Orlicz summability condition have a spatially continuous representative and
are well-posed. For the case of sub-exponential summability, their flows
satisfy a Lusin (N) condition in a quantitative form, too. Furthermore, we
prove that if $D_{x}b$ satisfies a suitable exponential summability condition
then the flow associated to $b$ has Sobolev regularity, without assuming
boundedness of ${\rm div}_{x}b$. We then apply these results to the
representation and Sobolev regularity of weak solutions of the Cauchy problem
for the transport and continuity equations.

###### Key words and phrases:

Vector Fields, Flow, Sobolev–Orlicz Spaces, Transport Equation, Continuity
Equation

###### 2010 Mathematics Subject Classification:

35F10 35A01 35A02

Corresponding author: S. Nicolussi Golo.

###### Contents

  1. 1 Introduction
    1. 1.1 Well-posedness
    2. 1.2 Regularity
    3. 1.3 Transport and continuity equations
  2. 2 Preliminaries on homeomorphisms
    1. 2.1 Weak derivatives of homeomorphisms
    2. 2.2 Mappings of finite distortion
  3. 3 Preliminaries on flows of vector fields
    1. 3.1 Well-posedness of vector fields
    2. 3.2 The flow of a well-posed vector field
  4. 4 Well-posedness with Orlicz condition
    1. 4.1 A class of subexponential summability types
  5. 5 Regularity of the flow with subexponential summability
  6. 6 Regularity of the flow with exponential summability
  7. 7 Applications to PDEs
  8. 8 Examples
    1. 8.1 Well-posedness does not imply absolute continuity of the flow
    2. 8.2 Sub-exponential condition does not imply high Sobolev regularity
    3. 8.3 Exponential summability does not imply the divergence in $BMO$ and it is only sufficient for the Sobolev regularity

##  1\. Introduction

In this paper we are concerned with the study of the existence and uniqueness
of classical solutions of the Cauchy problem for the ODE system

(1) $\left\\{\begin{array}[]{r@{\,=\,}l}\dot{\gamma}(t)&b(t,\gamma(t))\\\
\gamma(s)&x\,,\end{array}\right.$

with $x\in\Omega$, an open domain in $\mathbb{R}^{n}$, $s\in I$, an open
interval in $\mathbb{R}$, and $b:\,I\times\Omega\to\mathbb{R}^{n}$ a
continuous, possibly non-autonomous vector field Even though we will mostly
deal with the case when $b$ is continuous, we will point out which proofs can
easily be adapted to the case when $b$ is only measurable with respect to $t$.
If solutions to (1) exist and are unique for every $s$ and $x$, we say that
the vector field $b$ is _well-posed_ in $I\times\Omega$ (or in
$\overline{I}\times\Omega$, see Definition 3.1 for a more precise statement).
For every well posed vector field $b:\,I\times\Omega\to\mathbb{R}^{n}$ we have
a _flow_ , that is, a map $X:I\times I\times\Omega\to\Omega$, defined as
$X(t,s,x):=\gamma(t)$ where $\gamma$ is the unique absolutely continuous
solution of (1). More precisely, for each $t,\,s\in I$ we denote by
$\Omega_{(t,s)}\subset\Omega$ the open set of all $x\in\Omega$ such that the
path starting at $x$ at time $s$ can be extended until time $t$ (see Section
3.2 and Remark 3.3). Then $X(t,s,\cdot)$ is a well defined homeomorphism
$\Omega_{(t,s)}\to\Omega_{(s,t)}$ (see Remark 3.3).

Di Perna–Lions [17] carried out a pioneering and far-reaching theory by
introducing a generalized notion of flow for vector fields $b\in
L^{1}_{\mathrm{loc}}((0,T);W^{1,1}_{\mathrm{loc}}(\mathbb{R}^{n},\mathbb{R}^{n}))$
with important applications to the existence and uniqueness of weak solutions
for the Cauchy problem of the _transport equation_ associated to a weakly
differentiable vector field $b$, that is,

(2) $\begin{cases}\partial_{t}u+b\cdot D_{x}u=0&\text{ in
$(0,T)\times\mathbb{R}^{n}$ }\\\ u(0,\cdot)=\bar{u}\,.\end{cases}$

The theory was later remarkably extended by the first author [2] to vector
fields $b(t,\cdot)$ with $BV$ regularity. In these works, the regularity of
$b$ is paired with the boundedness of its spatial divergence, that is

(3) ${\rm div}_{x}b\in L^{1}((0,T);L^{\infty}(\mathbb{R}^{n})),$

which ensures the existence and uniqueness of the generalized flow of $b$. If
(3) does not hold, then uniqueness of the flow may fail, as it was already
shown in [17, Section IV.1]. The existence and uniqueness of a generalized
flow associated to a weakly regular vector field $b$ has been the object of an
intensive study with applications to the Cauchy problem for the transport
equation as well as for the _continuity equation_ associated to $b$, that is,

(4) $\begin{cases}\partial_{t}\rho+{\rm div}_{x}(b\rho)=0&\text{ in
$(0,T)\times\mathbb{R}^{n}$ }\\\ \rho(0,\cdot)=\bar{\rho}&\text{ in
$\mathbb{R}^{n}$\ . }\end{cases}$

Existence, uniqueness and regularity of solutions of these three problems (1),
(2) and (4) are connected with each other. In particular, the existence of a
unique flow $X$ with enough regularity implies existence and uniqueness of
solutions to both the transport equation and the continuity equation. A fairly
complete account of the development in this topic can be found in [5] and
references therein. A sample of the literature on this subject is [4, 8, 9,
10, 13, 11, 12, 14, 15, 16, 27, 29, 28].

Our contribution focuses on two problems. First, we want to weaken the
boundedness assumption on the divergence (3). We will show in Theorem A that
sub-exponential summability of $\|D_{x}b\|$ guarantees the existence of a
unique _classical_ flow (in the Di Perna–Lions–Ambrosio theory, flows have a
weaker definition).

Second, we want to find conditions on $b$ for the flow to have Sobolev
regularity, instead of just $L^{p}$ integrability. It is well-known that high
$L^{p}$ integrability of matrix Jacobian $D_{x}b$, even coupled with (3), is
not enough in order to provide Sobolev regularity of the flow $X$ (see, for
instance, [26]). A strategy used in the recent papers [13, 8] was to
strengthen the hypotheses by requiring exponential summability of
$\|D_{x}b\|$. We refer in particular to the recent paper [8], where it has
been shown that that $b$ has a unique flow with Sobolev regularity under the
condition

(5)
$\sup_{t\in\mathbb{R}}\int_{\mathbb{T}^{n}}\exp(\beta\|D_{x}b(t,x)\|)\,\mathrm{d}x<\infty\quad\text{and}\quad{\rm
div}_{x}b\in\mathrm{L}^{\infty}_{\mathrm{loc}}(\mathbb{R}\times\mathbb{T}^{n}),$

for some $\beta>0$, where $\mathbb{T}^{n}$ is the $n$-dimensional torus. We
prove analogous results without conditions on the divergence of $b$ in Theorem
D, see also Remark 6.3.

Our results are of three types. We first provide integral conditions of sub-
exponential type on $Db$ that ensure well-posedness. Then, we study the
Sobolev regularity of the homeomorphisms $X(t,s,\cdot)$. Finally, we apply
these results to both the transport equation and the continuity equation.

###  1.1. Well-posedness

Let us focus, first, on the well-posedness. If $\left\|D_{x}b\right\|$
satisfies an _exponential summability_ , that is, conditions of the form

(6)
$\int_{I}\int_{\Omega}\exp\left(\beta\|D_{x}b(t,x)\|\right)\,\mathrm{d}x\,\mathrm{d}t<+\infty$

for some $\beta>\,0$, then it is well-known that $b$ is well-posed. Indeed, in
this case, the vector field $b(t,\cdot)$ satisfies a so-called Log-Lipschitz
condition; see, for instance, [8, 32]. However, reformulating the condition of
exponential summability in a Orlicz-like form, we extend the result to some
sub-exponential cases.

######  Theorem A.

Let $\Theta:[0,+\infty)\to(0,+\infty)$ be a non decreasing locally Lipschitz
function. Assume that

  1. (A.I)

if $n>\,1$, there exists $\alpha\in(1,\frac{n}{n-1})$ such that
$\Theta^{\frac{\alpha-1}{\alpha}}$ is convex, while, if $n=1$, $\Theta$ is
convex;

  2. (A.II)

there exists $C_{\Theta}\geq 1$ such that
$\Theta:\,[C_{\Theta},+\infty)\to[\Theta(C_{\theta}),+\infty)$ is bijective
and

(7) $\Theta(s_{1})\Theta(s_{2})\leq\Theta(C_{\Theta}\,s_{1}\,s_{2})\text{ for
all }s_{1},s_{2}\geq C_{\Theta}\,;$

  3. (A.III)

$\displaystyle{\int_{1}^{\infty}\frac{\Theta^{\prime}(s)}{s\Theta(s)}\,\mathrm{d}s=+\infty}$.

Let
$b\in\mathrm{L}^{1}_{\mathrm{loc}}(I;W^{1,1}_{\mathrm{loc}}(\Omega;\mathbb{R}^{n}))$
and assume that for every $o\in\Omega$ there exist $c>0$, $R>0$ such that
$B(o,2R)\subset\Omega$ and the function

(8) $t\mapsto\psi(t):=\,\int_{B(o,2R)}\Theta(c\|D_{x}b(t,z)\|)\,\mathrm{d}z$

belongs to $\mathrm{L}^{1}_{\mathrm{loc}}(I)$. Then
$b(t,\cdot):\Omega\to\mathbb{R}^{n}$ has a continuous representative
$\tilde{b}(t,\cdot)$ for a.e. $t\in I$ and $\tilde{b}$ is well-posed in
$I\times\Omega$. Moreover, if there exists $m\in L^{1}(I)$ such that

(9) $|b(t,x)|\leq\,m(t)\quad\text{ for a.e.\leavevmode\nobreak\ }t\in I,\text{
for a.e.\leavevmode\nobreak\ }x\in B(o,R)\,,$

then $\tilde{b}$ is also well-posed in $\overline{I}\times\Omega$.

Notice that a byproduct of the proof of Theorem A is that the Sobolev-Orlicz
space $W^{1}L_{\Theta}(\Omega)$ embeds in $C^{0}(\Omega)$, with modulus of
continuity that depends only on $\Theta$. See [7, Section 2.6] for the
definition of $W^{1}L_{\Theta}(\Omega)$.

The proof is inspired from [32]. In fact, we shall prove that a continuous
representative $\tilde{b}(t,\cdot)$ of $b(t,\cdot)$ satisfies the Osgood’s
criterion (see [23, Chap. III, Corollary 6.2] or Proposition 3.2 below). The
proof of this result is given in Section 4.

We note in Proposition 4.4 that if an increasing function $\Theta$ satisfies
condition (A.III) of Theorem A, then it does not have polynomial growth.
Examples of functions $\Theta$ satisfying the properties (A.I)-(A.III) are of
the form

(10)
$\mathscr{E}_{k,\beta}(s)=\exp\left(\frac{s}{\log(s)\log\log(s)\dots(\underbrace{\log\dots\log}_{\text{
$k$-times}}s)^{\beta}}\right)\quad\text{ for }s\geq\bar{s},$

with $\mathscr{E}_{k,\beta}(s)=\mathscr{E}_{k,\beta}(\bar{s})$ for
$s<\bar{s}$, where $\bar{s}$ is large enough, $k\geq\,1$ is an integer and
$0\leq\beta\leq 1$, see Proposition 4.6. Notice that the asymptotic behaviour
of $\mathscr{E}_{k,\beta}$ as $s\to\infty$ is almost sharp in order that
assumption (A.III) holds, see Remark 4.7. If
$\mathscr{E}_{k,1}(c\left\|D_{x}b\right\|)\in\mathrm{L}_{\mathrm{loc}}^{1}(\Omega)$
for some $c>0$, we say that $\left\|D_{x}b\right\|$ satisfies a
_subexponential summability_ of order $k$. Therefore, Theorem A shows that, if
$\left\|D_{x}b\right\|$ has _subexponential summability_ , then $b$ has a
_classical_ unique flow. However, we stress that, under the hypothesis of
Theorem A, $\left\|D_{x}b\right\|$ does not need not be in $L^{p}_{\rm
loc}(I\times\Omega)$ for each $p>1$ (see Remark 4.5).

###  1.2. Regularity

Moving on to the regularity of the flow, we can prove that, if $D_{x}b$
satisfies a subexponential summability of order $1$, the associated flow
$X(t,s,\cdot)$ satisfies a weak regularity property, namely it maps the
Lebesgue measure into absolutely continuous measures. Notice that, in this
case, $\left\|D_{x}b(t,\cdot)\right\|$ does belong to $L^{p}_{\rm
loc}(\Omega)$ for every $p>1$, for almost every $t\in I$. A quantitative
version, that we obtain adapting [11], is the following.

######  Theorem B.

Let
$b\in\mathrm{L}^{1}_{\mathrm{loc}}(I;W^{1,1}_{\mathrm{loc}}(\mathbb{R}^{n};\mathbb{R}^{n}))$.
Suppose that

(11) $\frac{|b(t,x)|}{1+|x|\,\log^{+}|x|}\in
L^{1}(I;L^{\infty}(\mathbb{R}^{n}))\,,$

and

(12)
$\int_{I}\int_{\mathbb{R}^{n}}\exp\left(\frac{\|D_{x}b\|}{\log^{+}\|D_{x}b\|}\right)(t,x)\,d\gamma_{n}(x)dt<\,+\infty,$

where $\gamma_{n}$ is the Gaussian measure on $\mathbb{R}^{n}$, namely

$\gamma_{n}:=\,\frac{1}{(2\pi)^{n/2}}\exp\left(-\frac{|x|^{2}}{2}\right)\mathcal{L}^{n}\,$

with $\mathcal{L}^{n}$ the Lebesgue measure in $\mathbb{R}^{n}$.

Then the space continuous representative $\tilde{b}$, granted by the Sobolev
embedding, is well-posed in $\overline{I}\times\Omega$ and the associated flow
$X$ of $\tilde{b}$ is globally defined, that is,
$X:\overline{I}\times\overline{I}\times\mathbb{R}^{n}\to\mathbb{R}^{n}$.
Moreover, for every $t,\,s\in\overline{I}$, the image measure
$X(t,s,\cdot)_{\\#}\mathcal{L}^{n}$ is absolutely continuous with respect to
$\mathcal{L}^{n}$ and there exists a positive constant $\alpha_{0}(t,s)>0$
such that

$\frac{\,\mathrm{d}}{\,\mathrm{d}\mathcal{L}^{n}}(X(t,s,\cdot)_{\\#}\mathcal{L}^{n})\in\mathrm{L}^{\Phi_{\alpha}}_{\mathrm{loc}}(\mathbb{R}^{n})$

for each $0<\alpha<\alpha_{0}(s,t)$, where
$\mathrm{L}^{\Phi_{\alpha}}_{\mathrm{loc}}(\mathbb{R}^{n})$ is the Orlicz
space with

$\Phi_{\alpha}:[0,+\infty)\to[0,+\infty),\qquad\Phi_{\alpha}(w):=w\exp((\log^{+}w)^{\alpha}),$

and $\alpha_{0}:I\times I\to\mathbb{R}$ is continuous with $\alpha_{0}(t,t)=1$
for every $t\in I$.

See Section 5 for the proof. By trivial considerations, in the one-dimensional
case, we can improve Theorem B to absolutely continuity of the flow.

######  Theorem C.

If $n=1$ and $b$ satisfies the conditions of Theorem B, then, for every
$t,\,s\in I$, the map $X(t,s,\cdot)$ is an absolutely continuous homeomorphism
between intervals of $\mathbb{R}$.

Sobolev regularity stated in Theorem C is sharp, as we show in Example 8.2. We
don’t know whether Theorem C can be extended to the case of sub-exponential
summability of order $k>1$.

In higher dimensions, for subexponential summability, we have a partial
negative result with an example in Section 8.2, or the example constructed in
[8], see also Remark 7.2: there are vector fields satisfying a subexponential
summability of order $1$ whose flow is not in $W^{1,p}_{\mathrm{loc}}$ for any
$p>n$. However, it remains open whether vector fields satisfying a
subexponential summability of order $1$ can fail to have the flow in
$W^{1,1}_{\mathrm{loc}}$.

On the other hand, in higher dimensions, for exponential summability we have a
positive result:

######  Theorem D.

Let $I\subset\mathbb{R}$ and $\Omega\subset\mathbb{R}^{n}$ be bounded open
sets and let
$b\in\mathrm{L}^{1}_{\mathrm{loc}}(I;W^{1,1}_{\mathrm{loc}}(\Omega;\mathbb{R}^{n}))$
be bounded. Assume that for some $p>2n$ the vector field $b$ satisfies the
global geometric condition

(13)
$\Lambda_{p}:=\int_{\Omega}\int_{I}\max\left\\{\ell^{\frac{n}{n-p}},\frac{({\rm
dist}(x,\partial\Omega))^{\frac{n}{n-p}}}{(\sup|b|)^{\frac{n}{n-p}}}\right\\}\exp\left(\frac{\ell
p^{2}}{p-n}\|D_{x}b(s,x)\|\right)\,\mathrm{d}s\,\mathrm{d}x<+\infty,$

with $\ell$ equal to the length of $I$. Then the space-continuous
representative $\tilde{b}$ is well-posed in $\overline{I}\times\Omega$ thanks
to Theorem A and, in addition, for every $t\in{I}$ and for almost every $s\in
I$, one has

(14) $\text{for a.e.\leavevmode\nobreak\ }s\in I,\quad X(t,s,\cdot)\in
W^{1,p}(\Omega_{(t,s)};\mathbb{R}^{n})\quad\text{and}\quad X(s,t,\cdot)\in
W^{1,p}(\Omega_{(s,t)};\mathbb{R}^{n}),$

with

(15)
$\int_{I}\int_{\Omega_{(t,s)}}\|D_{x}X(t,s,x)\|^{p}\,\mathrm{d}x\,\mathrm{d}s\leq\ell^{\frac{n}{p-n}}\Lambda_{p}.$

With regards to (14), a lower Sobolev regularity for the flow can be proved
for all pairs of times $s,t\in I$, see Corollary 6.9. For $b\in
C^{0}(I;C^{1}(\Omega;\mathbb{R}^{n}))$, we have a more detailed statement in
Theorem 6.1. We may also consider the case when
$b:I\times\mathbb{R}^{n}\to\mathbb{R}^{n}$ and the support of $b(t,\cdot)$ is
contained in a compact set independent of $t$. See also Remarks 6.3 and 6.2.

######  Theorem E.

Let $I\subset\mathbb{R}$ be a bounded open interval and let
$b\in\mathrm{L}^{1}_{\mathrm{loc}}(I;W^{1,1}_{\mathrm{loc}}(\mathbb{R}^{n};\mathbb{R}^{n}))$
be a bounded vector field. Assume that there exist a bounded open set
$\Omega\subset\mathbb{R}^{n}$ and $p>n$ such that

(16) $\mathtt{spt}(b(t,\cdot))\subset\Omega\quad\text{for each $t\in I$}$

and (6) holds with $\beta=\,{\ell p^{2}}/{(p-n)}$ and $\ell$ equal to the
length of $I$.

Then the space-continuous representative $\tilde{b}$, understood as vector
field in $I\times\Omega$, is well-posed in $\overline{I}\times\Omega$ thanks
to Theorem A, $\Omega_{(t,s)}=\,\Omega$ and for every $t,\,s\in\overline{I}$
one has

(17) $\begin{split}&X(t,s,\cdot)\in
W^{1,p}(\Omega;\mathbb{R}^{n})\quad\text{with}\\\
&\int_{\Omega}\|D_{x}X(t,s,x)\|^{p}\,\mathrm{d}x\leq\frac{1}{\ell}\int_{I}\int_{\Omega}\exp\left(\frac{\ell
p^{2}}{p-n}\|D_{x}b(v,y)\|\right)\,\mathrm{d}y\,\mathrm{d}v.\end{split}$

Notice also that, in the previous theorem, we obviously have that the flow map
$X:\,\overline{I}\times\overline{I}\times\mathbb{R}^{n}\to\mathbb{R}^{n}$ of
$b$ is identically equal to the identity on
$\overline{I}\times\overline{I}\times(\mathbb{R}^{n}\setminus\overline{\Omega})$.

######  Remark 1.1.

(1) Since ${\rm dist}(x,\partial\Omega)$ is bounded, (13) can be stated in the
equivalent form

$\int_{\Omega}\int_{I}({\rm
dist}(x,\partial\Omega))^{n/(n-p)}\exp\left(\frac{\ell
p^{2}}{p-n}\|D_{x}b(s,x)\|\right)\,\mathrm{d}s\,\mathrm{d}x<+\infty.$

We used that specific form for the purpose of the estimate (14).

(2) If $\alpha=n/(p-n)$ is smaller than 1 (i.e., $p>2n$) and $\partial\Omega$
is regular, then the geometric condition (13), thanks to the Hölder
inequality, is implied by the simpler condition

(18)
$\int_{\Omega}\int_{I}\exp\left(c\|D_{x}b(s,x)\|\right)\,\mathrm{d}s\,\mathrm{d}x<+\infty$

provided $c>(1/\alpha)^{\prime}p^{2}/(p-n)$, where
$(1/\alpha)^{\prime}=(1-\alpha)^{-1}$ is the dual exponent.

We also point out that the Sobolev exponent $p$, related to the constant in
the exponential integrability condition in (13), may not be sharp. In other
words, we can have $\Lambda_{p}=\infty$ but $X(t,s,\cdot)\in
W^{1,p}(\Omega_{(t,s)};\mathbb{R}^{n})$, see Example 8.3.

Several regularity properties of $X$ follow from Theorem D, see Corollaries
6.7, 6.8 and 6.10. In particular we show that, as in Theorem B,
$X(t,s,\cdot)_{\\#}\mathcal{L}^{n}\ll\mathcal{L}^{n}$, for each
$t,\,s\in\overline{I}$.

######  Remark 1.2.

Suppose that
$b\in\mathrm{L}^{1}_{\mathrm{loc}}(I;W^{1,1}_{\mathrm{loc}}(\mathbb{R}^{n};\mathbb{R}^{n}))$
has compact support and (18) holds for some $c>0$. For every $p>2n$, we get
(17) on subintervals of $I$ of length $\ell$ such that $\frac{\ell
p^{2}}{p-n}=c$. Since $\frac{p^{2}}{p-n}>4n$ for $p>2n$, then
$\ell<\frac{c}{4n}$.

###  1.3. Transport and continuity equations

Finally, we apply Theorem B and Theorem D to the transport equation (2) and
the continuity equation (4), respectively. By standard methods, we provide
existence, uniqueness, representation and regularity of solutions.

In [12, Theorem 1], uniqueness of weak solutions $u\in
L^{\infty}((0,T);L^{\infty}(\mathbb{R}^{n}))$ has been proved, provided that
the spatial derivatives of $b$ satisfies a sub-exponential summability and
$\bar{u}\in L^{\infty}(\mathbb{R}^{n})$. Our next result provides even in this
context the classical representation of this unique solution in terms of the
flow map, granted by Theorem B.

######  Theorem F (Representation of the solutions for the transport
equation).

Let $b$ be a vector field as in Theorem B, and let $X$ be the flow associated
to the continuous representative $b$. Then, for each $\bar{u}\in
L^{\infty}(\mathbb{R}^{n})$, for each $t\in[0,T]$, the function

(19) $v(t,x):=\,\bar{u}\left(X(t,0,\cdot)^{-1}(x)\right)\text{ for
a.e.\leavevmode\nobreak\ }x\in\mathbb{R}^{n}\,,$

is the unique weak solution in $L^{\infty}((0,T);L^{\infty}(\mathbb{R}^{n}))$
of the Cauchy problem for the transport equation (2), understood in the sense
of distributions.

By means of the theory of maps with finite distortion, see for instance [25],
we can also show Sobolev regularity of the solution $v$ to the transport
equation, see Corollary 7.1.

By applying the well-posedness and representation results proved in [3, 14]
(see Theorem 7.4 and Remark 7.5), we make more explicit the representation of
the weak solutions of (4), under the assumptions of Theorem B and Theorem E.
It is useful to deal with the case where $\rho(t,\cdot)$ belongs to the space
of signed Borel measures on $\mathbb{R}^{n}$ with finite total variation,
which we will denote by $\mathcal{M}(\mathbb{R}^{n})$.

######  Theorem G (Representation of the solutions for the continuity
equation).

Let $I=(0,T)$, let $b:I\times\mathbb{R}^{n}\to\mathbb{R}^{n}$ be a vector
field with $b(t,\cdot)$ continuous for a.e. $t\in I$.

  * (i)

Suppose that $b$ satisfies the assumptions of Theorem B and let $X$ be the
flow associated to $b$. Then, for each signed measure
$\bar{\rho}\in\mathcal{M}(\mathbb{R}^{n})$,

(20) $\rho_{t}=\,\rho(t,\cdot)=\,X(t,0,\cdot)_{\\#}\bar{\rho}\quad t\in[0,T]$

is the unique weak solution for the Cauchy problem (4) in
$L^{\infty}((0,T);\mathcal{M}(\mathbb{R}^{n}))$.

Moreover, if $\bar{\rho}\in L^{1}(\mathbb{R}^{n})$, $\rho_{t}$ can be
represented as

(21) $\rho_{t}:=(\bar{\rho}J_{X,t})\circ X(0,t,\cdot)\,,$

where
$J_{X,t}(y)=\frac{\,\mathrm{d}X(t,0,\cdot)_{\\#}\mathcal{L}^{n}}{\,\mathrm{d}\mathcal{L}^{n}}(X(t,0,y))$.
In particular we have $\rho\in L^{\infty}((0,T);L^{1}(\mathbb{R}^{n}))$.

  * (ii)

If $b$ satisfies the stronger hypotheses of Theorem E and $\bar{\rho}\in
L^{1}(\mathbb{R}^{n})$, then the unique weak solution for the Cauchy problem
(4) can also be represented as

(22) $\rho_{t}=\,\frac{\bar{\rho}}{J_{X(t,0,\cdot)}}\circ X(0,t,\cdot)\,,$

where $J_{X(t,0,\cdot)}$ denotes the Jacobian of homeomorphism
$X(t,0,\cdot):\,\mathbb{R}^{n}\to\mathbb{R}^{n}$ (see (24)).

Notice that existence and uniqueness of distributional solutions of continuity
equation (4) has been obtained in [15] by weakening assumption (3). More
precisely, the authors showed that if $\bar{\rho}\in L^{\infty}(\mathbb{R})$
and

(23) ${\rm div}_{x}b\in(BMO\cap L^{1})(\mathbb{R}^{n}),$

then there exists a unique distributional solution $\rho\in
L^{\infty}((0,T)\times\mathbb{R}^{n})$ of the Cauchy problem (4) However, we
point out that condition (6) does not imply (23), see Example 8.3.

### Structure of the paper

We start in Sections 2 and 3 by listing known results about homeomorphisms of
finite distortion and classical flows of vector fields. In Section 4 we prove
Theorem A. Theorems B and C are proven in Section 5. The proof of Theorem D is
given in Section 6. Theorems F and G are shown in Section 7. Finally, Section
8 is devoted to the illustration of a few examples.

### Acknowledgements

We thank A. Clop for useful suggestions and discussions on this topic. We are
also grateful to R. Serapioni for several discussions about Section 4. We want
to thank also the anonymous referee for their useful comments and corrections.

L.A. and F.S.C. have been supported by the PRIN 2017 project “Gradient flows,
Optimal Transport and Metric Measure Structures.” S.N.G. has been supported by
University of Padova STARS Project “Sub-Riemannian Geometry and Geometric
Measure Theory Issues: Old and New” and by the Academy of Finland (grant
328846 “Singular integrals, harmonic functions, and boundary regularity in
Heisenberg groups”, grant 322898 “Sub-Riemannian Geometry via Metric-geometry
and Lie-group Theory”, grant 314172 “Quantitative rectifiability in Euclidean
and non-Euclidean spaces”). S.N.G. and F.S.C. have been supported by the INdAM
– GNAMPA Project 2019 “Rectifiability in Carnot groups”.

##  2\. Preliminaries on homeomorphisms

In this section we present some results about homeomorphisms that we will need
later. Most of the material comes from the book [25].

###  2.1. Weak derivatives of homeomorphisms

If $\Omega\subset\mathbb{R}^{n}$ is open and $\Psi:\Omega\to\mathbb{R}^{n}$,
we denote by $D\Psi$ the weak differential of $\Psi$. When a point-wise
analysis is needed, we assume that $D\Psi(x)$ is equal to the classical
differential of $\Psi$ at every $x\in\Omega$ where $\Psi$ is
differentiable.111Notice that the distributional derivative of a Sobolev
function $\Psi$ agrees with the classical derivative at a.e. $x$ where $\Psi$
is differentiable. If the reader needs an argument, let us point out that the
approximate differential of $\Psi$ agrees with the distributional derivative
almost everywhere by [18, Theorem 6.1.4, page 233], and that the classical
derivative is equal to the approximate differential wherever it exists. The
_Jacobian_ of $\Psi$ is

(24) $J_{\Psi}(x)=\det(D\Psi(x))\,.$

A map $\Psi:\Omega\to\mathbb{R}^{n}$ satisfies the _Lusin (N) condition_ if
for all $E\subset\Omega$ the vanishing of $\mathcal{L}^{n}(E)$ implies
$\mathcal{L}^{n}(\Psi(E))=0$. An _embedding_ is a homeomorphism onto its
image.

######  Lemma 2.1 ([25, Lemma A.29]).

Let $\Omega\subset\mathbb{R}^{n}$ be open and let
$\Psi:\Omega\to\mathbb{R}^{n}$ be an embedding that is differentiable at $x$.
The map $\Psi^{-1}$ is differentiable at $\Psi(x)$ if and only if
$\det(D\Psi(x))\neq 0$. In this case, we have

$D(\Psi^{-1})(\Psi(x))=(D\Psi(x))^{-1}.$

######  Lemma 2.2 ([25, Theorem A.35]: Area Formula).

If $\Omega\subset\mathbb{R}^{n}$ is open, $\Psi\in
W^{1,1}_{\mathrm{loc}}(\Omega;\mathbb{R}^{n})$ and
$\eta:\mathbb{R}^{n}\to[0,+\infty]$ is a Borel function, then

(25)
$\int_{\Omega}\eta(\Psi(x))|J_{\Psi}(x)|\,\mathrm{d}x\leq\int_{\mathbb{R}^{n}}\eta(y)N(\Psi,\Omega,y)\,\mathrm{d}y,$

where $N(\Psi,\Omega,y)=\\#(\Psi^{-1}(y)\cap\Omega)$ is the cardinality of the
set $\Psi^{-1}(y)\cap\Omega$. If $\Psi$ satisfies also the Lusin (N) condition
then equality holds in (25).

######  Remark 2.3.

Observe that the Lusin (N) condition for a homeomorphism
$\Psi:\Omega\to\Omega^{\prime}$ is equivalent to assume that the push-forward
measure $(\Psi^{-1})_{\\#}(\mathcal{L}^{n}\mathop{\hbox{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt\vrule
height=0.5pt,width=6.0pt,depth=0.0pt}}\nolimits\Omega^{\prime})$ is absolutely
continuous with respect to $\mathcal{L}^{n}$. An abstract area-type formula
with respect to a general measure $\mu$ is also introduced in Lemma 7.6.

######  Lemma 2.4 ([25, Theorem 4.2 and Theorem 6.2.1]).

If $\Omega\subset\mathbb{R}^{n}$ is open, $\Psi\in
W^{1,p}_{\mathrm{loc}}(\Omega;\mathbb{R}^{n})$ and $p>n$, then the continuous
representative $\tilde{\Psi}$ is differentiable at a.e. $x\in\Omega$ and
$\tilde{\Psi}$ satisfies the Lusin (N) condition.

###  2.2. Mappings of finite distortion

We denote by $\|M\|$ the operator norm of a linear map
$\mathbb{R}^{n}\to\mathbb{R}^{n}$ or $m\times n$-matrix, that is,

$\|M\|:=\sup\left\\{|Mv|:v\in\mathbb{R}^{n},\ |v|=1\right\\},$

where $|\cdot|$ is the Euclidean norm. With this choice of the norm, one can
easily show the following inequalities (the first one is called _Hadamard’s
inequality_):

(26) $|\det M|\leq\Pi_{i=1}^{n}|Me_{i}|\leq\|M\|^{n}.$

######  Definition 2.5.

A _map of finite distortion_ on an open set $\Omega\subset\mathbb{R}^{n}$ is a
function $\Psi\in W^{1,1}_{\mathrm{loc}}(\Omega;\mathbb{R}^{n})$ such that
there exists $K:\Omega\to[1,+\infty)$ with

(27) $\|D\Psi(x)\|^{n}\leq K(x)J_{\Psi}(x)\qquad\text{for
a.e.\leavevmode\nobreak\ }x\in\Omega.$

For $q\geq 1$, the _$q$ -distortion function_ of a map of finite distortion
$\Psi$ is

$K_{q}^{\Psi}(x):=\begin{cases}\displaystyle{\frac{\|D\Psi(x)\|^{q}}{J_{\Psi}(x)}}&\text{
if }J_{\Psi}(x)\neq 0,\\\ 1&\text{ otherwise }.\end{cases}$

Our main reference on maps of finite distortion is [25]. Notice that
$K_{n}^{\Psi}$ is the optimal distortion function for the inequality (27) to
hold.

######  Lemma 2.6.

If $\Omega\subset\mathbb{R}^{n}$ is open and $\Psi\in
W^{1,1}_{\mathrm{loc}}(\Omega;\mathbb{R}^{n})$ is an embedding with
$J_{\Psi}>0$ almost everywhere in $\Omega$, then $\Psi$ has finite distortion.

###### Proof.

The function $K^{\Psi}_{n}$ is finite almost everywhere and (27) holds
whenever $J_{\Psi}(x)>0$. Since $J_{\Psi}>0$ almost everywhere in $\Omega$,
then $\Psi$ has finite distortion. ∎

######  Lemma 2.7.

Let $\Psi:\Omega_{1}\to\Omega_{2}$ be a homeomorphism between open subsets of
$\mathbb{R}^{n}$, $q>0$ and $p>n$. Suppose that

$\Psi\in W^{1,p}_{\mathrm{loc}}(\Omega;\mathbb{R}^{n}),\quad\Psi^{-1}\in
W^{1,p}_{\mathrm{loc}}(\Psi(\Omega);\mathbb{R}^{n})\quad\text{and
}J_{\Psi}(x)>0\text{ for a.e.\leavevmode\nobreak\ }x\in\Omega.$

Then $\Psi$ is a homeomorphism of finite distortion and

$K^{\Psi}_{q}\in\mathrm{L}^{r}_{\mathrm{loc}}(\Omega)$

with $r=\left(\frac{q}{p}+\frac{n}{p-n}\right)^{-1}$, which may be smaller
than 1.

###### Proof.

The map $\Psi$ has finite distortion by Lemma 2.6. Let $r>0$ and
$U\Subset\Omega$. By Hölder inequality, we have

(28)
$\int_{U}|K^{\Psi}_{q}|^{r}\,\mathrm{d}x\leq\left(\int_{U}\|D\Psi\|^{rq\alpha}\,\mathrm{d}x\right)^{1/\alpha}\left(\int_{U}|J\Psi|^{-r\beta}\,\mathrm{d}x\right)^{1/\beta}$

for $\alpha,\beta\geq 1$ with $\frac{1}{\alpha}+\frac{1}{\beta}=1$. If
$rq\alpha=p$, then the first term of the right-hand side is finite.

By Lemma 2.4, both $\Psi$ and $\Psi^{-1}$ are differentiable almost
everywhere. Therefore, using Lemma 2.1, it follows that, for almost every
$x\in U$,

$|J\Psi(x)|^{-1}=|J\Psi^{-1}(\Psi(x))|.$

Using the area inequality (25), we have for all $\gamma>1$

$\displaystyle\int_{U}|J\Psi(x)|^{-\gamma}\,\mathrm{d}x$
$\displaystyle=\int_{U}|J\Psi^{-1}(\Psi(x))|^{\gamma+1}|J\Psi(x)|\,\mathrm{d}x$
$\displaystyle\leq\int_{\Psi(U)}|J\Psi^{-1}(y)|^{\gamma+1}\,dy\leq\int_{\Psi(U)}\|D\Psi^{-1}(y)\|^{n(\gamma+1)}\,\mathrm{d}x,$

where we used (26) in the last step. Now, if $\gamma=r\beta$, then

$\int_{U}|J\Psi|^{-r\beta}\,\mathrm{d}x\leq\int_{\Psi(U)}\|D\Psi^{-1}(y)\|^{n(r\beta+1)}\,\mathrm{d}x,$

which is finite when $n(r\beta+1)=p$.

Finally, solving the three equations $\frac{1}{\alpha}+\frac{1}{\beta}=1$,
$rq\alpha=p$ and $n(r\beta+1)=p$ in $\alpha,\beta$ and $r$, we get

$r=\left(\frac{q}{p}+\frac{n}{p-n}\right)^{-1}$

and $\alpha=\frac{p}{rq}=1+\frac{pn}{q(p-n)}>1$, by the assumption $p>n$.
Therefore, we get from (28) that $\int_{U}|K_{\Psi}|^{r}\,\mathrm{d}x<\infty$.
∎

######  Proposition 2.8 ([25, Theorem 5.13 and equation (5.18)]: Regularity of
composition).

Let $\Psi:\Omega_{1}\to\Omega_{2}$ be a homeomorphism between open subsets of
$\mathbb{R}^{n}$ and let $1\leq\,p<\,q<\,\infty$. Define the composition
operator $T_{\Psi}u=u\circ\Psi$ for $u:\Omega_{2}\to\mathbb{R}$. Suppose that
$\Psi$ has finite distortion and

(29) $K^{\Psi}_{q}\in\mathrm{L}^{\frac{p}{q-p}}_{\mathrm{loc}}(\Omega_{1}).$

Then $T_{\Psi}$ is continuous from $W_{\mathrm{loc}}^{1,q}(\Omega_{2})$ to
$W_{\mathrm{loc}}^{1,p}(\Omega_{1})$ for every $1\leq q\leq\infty$, where the
operator norm on $T_{\Psi}$ is controlled by
$\|K^{\Psi}_{q}\|_{\mathrm{L}^{\frac{p}{q-p}}}^{1/q}$ on balls.

##  3\. Preliminaries on flows of vector fields

In this section we present a part of the classical theory of flows of vector
fields that we will need later.

###  3.1. Well-posedness of vector fields

We denote by $B(x,r)$ the Euclidean ball of radius $r$ and center $x$. Recall
that a vector field $b(t,x)$ is said to be _autonomous_ if it does not depend
on $t$, _non-autonomous_ if it may depend on $t$.

######  Definition 3.1 (Well posedness).

Let $\Omega\subset\mathbb{R}^{n}$ open, $I\subset\mathbb{R}$ an open interval
and $b:I\times\Omega\to\mathbb{R}^{n}$ a function. We say that $b$ is _well-
posed in $I\times\Omega$_ if for every $(s,x)\in I\times\Omega$ there exist
$\epsilon>0$ and, in the interval $(s-\epsilon,s+\epsilon)\subset I$, a unique
absolutely continuous solution $\gamma:\,(s-\epsilon,s+\epsilon)\to\Omega$ of
(1), that is, $\dot{\gamma}(t)=\,b(t,\gamma(t))$ for a.e.
$t\in(s-\epsilon,s+\epsilon)$ and $\gamma(s)=x$. Analogously, we say that $b$
is _well-posed in $\overline{I}\times\Omega$_ if for every
$(s,x)\in\overline{I}\times\Omega$ there exist $\epsilon>0$ and, in the
interval $I\cap(s-\epsilon,s+\epsilon)$, a unique absolutely continuous
solution $\gamma:\,I\cap(s-\epsilon,s+\epsilon)\to\Omega$ of (1).

The following classical result provides a sufficient condition for well-
posedness, in local and global form with respect to the time variable. The
proof is rather classical and well known, see for instance [19, 23].

######  Proposition 3.2 (Well-posedness with Osgood Condition).

Let $\omega:[0,+\infty)\to[0,+\infty)$ be non-decreasing with $\omega(0)=0$,
$\omega(t)>0$ for $t>0$ and

(30) $\int_{0}^{1}\frac{1}{\omega(t)}\,\mathrm{d}t=+\infty.$

Let $I\subset\mathbb{R}$ be an interval, and $\Omega\subset\mathbb{R}^{n}$ an
open set. Let $b:I\times\Omega\to\mathbb{R}^{n}$ be a function such that:

  1. (a)

there exists a positive function $\phi\in L^{1}_{\mathrm{loc}}(I)$ such that

(31) $|b(t,x)-b(t,y)|\leq\phi(t)\omega(|x-y|)$

for all $t\in I$ and $x,\,y\in\Omega$;

  2. (b)

for every $x\in\Omega$, the function $t\mapsto b(t,x)$ is measurable;

  3. (c)

there exists $m\in L^{1}_{\mathrm{loc}}(I)$ such that $|b(t,x)|\leq m(t)$ for
all $(t,x)\in I\times\Omega$.

Then the following hold:

  1. (i)

The vector field $b$ is well posed in $I\times\Omega$. For $(s,x)\in
I\times\Omega$, denote by $\gamma_{(s,x)}$ the maximal solution of (1) with
$\gamma_{(s,x)}(s)=x$ and by $I_{(s,x)}$ its maximal domain of existence.
Define also $\alpha(s,t)=\inf I_{(s,x)}$ and $\beta(s,t)=\sup I_{(s,x)}$.

  2. (ii)

If $\beta(s,x)\neq\sup I$, then $\lim_{t\to\beta(s,x)}\gamma_{(s,x)}$ exists
and belongs to $\partial\Omega$. Similarly for $\alpha(s,x)\neq\inf I$.

  3. (iii)

If $(s_{j},x_{j})\in I\times\Omega$ is a sequence converging to
$(s_{\infty},x_{\infty})\in I\times\Omega$, with
$\gamma_{j}=\gamma_{(s_{j},x_{j})}:I_{(s_{j},x_{j})}\to\Omega$ the
corresponding maximal solutions, $j\in\mathbb{N}\cup\\{\infty\\}$, and if
$s_{\infty}\in J\Subset I_{(s_{\infty},x_{\infty})}$, then $J\subset
I_{(s_{j},x_{j})}$ for $j$ large enough and $\gamma_{j}$ converge to
$\gamma_{\infty}$ uniformly on $J$.

In particular, $\alpha$ is upper semicontinuous and $\beta$ is lower
semicontinuous.

  4. (iv)

If $m\in L^{1}(I)$, then $\gamma_{(s,x)}$ is well defined also for
$(s,x)\in\partial I\times\Omega$ and the limits

$\lim_{t\to\beta(s,x)}\gamma_{(s,x)}(t)\quad\text{and}\quad\lim_{t\to\alpha(s,x)}\gamma_{(s,x)}(t)$

exist in $\bar{\Omega}$. In particular, $b$ is well posed in
$\overline{I}\times\Omega$.

###  3.2. The flow of a well-posed vector field

Fix an open set $\Omega\subset\mathbb{R}^{n}$, an open interval
$I\subset\mathbb{R}$ and a well-posed vector field
$b:I\times\Omega\to\mathbb{R}^{n}$. For every $(s,x)\in I\times\Omega$, let
$I_{(s,x)}=\,(\alpha(s,x),\beta(s,x))\subset I$ be the maximal open interval
of existence of a solution $\gamma_{(s,x)}:I_{(s,x)}\to\Omega$ to the Cauchy
problem (1) with initial datum $\gamma_{(s,x)}(s)=x$. Define

$\mathscr{D}_{b}:=\left\\{(t,s,x):(s,x)\in I\times\Omega,\ t\in
I_{(s,x)}\right\\}\subset I\times I\times\Omega,$

and the _flow_ $X:\mathscr{D}_{b}\to\Omega$ of $b$ as
$X(t,s,x)=\gamma_{(s,x)}(t)$. For every $(t,s)\in I\times I$, we define
$\Omega_{(t,s)}=\\{x\in\Omega:(t,s,x)\in\mathscr{D}_{b}\\}=\\{x\in\Omega:t\in
I_{(s,x)}\\}$ (which is possibly empty) and
$X_{(t,s)}:\Omega_{(t,s)}\to\Omega$ as $X_{(t,s)}(x)=X(t,s,x)$.

An analogous notation can be used for the case of vector fields well posed in
$\overline{I}\times\Omega$, just allowing $s$ and $t$ to belong also to
$\partial I$.

We recall several well-known facts about flows of vector fields, referring to
fix the ideas to the weaker case of well posedness in $I\times\Omega$. See for
instance [23] for reference.

######  Remark 3.3.

If $b:I\times\Omega\to\mathbb{R}^{n}$ satisfies the hypothesis of Proposition
3.2, then the sets $\mathscr{D}_{b}$ and $\Omega_{(t,s)}$ are open and $X$ is
continuous, thanks to Proposition 3.2(iii).

Uniqueness gives the semigroup-type property

(32) $X(t_{3},t_{2},X(t_{2},t_{1},x))=X(t_{3},t_{1},x)\ ,$

whenever the expressions make sense. Moreover, the map $X_{(t,s)}$ is a
homeomorphism $\Omega_{(t,s)}\to\Omega_{(s,t)}$ with inverse $X_{(s,t)}$.
Finally, if $k\geq 1$ and $b\in C^{k}(I\times\Omega;\mathbb{R}^{n})$, then
$X\in C^{k+1}(\mathscr{D}_{b};\Omega)$.

######  Lemma 3.4 ([23, Theorem 3.1, Chap. V]).

Any $b\in C^{0}(I;C^{1}(\Omega;\mathbb{R}^{n}))$ is well posed in
$I\times\Omega$ and the homeomorphisms
$X_{(t,s)}:\Omega_{(t,s)}\to\Omega_{(s,t)}$ are of class $C^{1}$.

Fix $(s,x)\in I\times\Omega$ and set the following functions defined on
$I_{(s,x)}$

$\displaystyle y(t)$ $\displaystyle:=\,D_{x}X(t,s,x),$ $\displaystyle B(t)$
$\displaystyle:=\,(D_{x}b)(t,X(t,s,x)),$ $\displaystyle J(t)$
$\displaystyle:=\det(D_{x}X(t,s,x)),$ $\displaystyle\beta(t)$
$\displaystyle:={\rm div}_{x}(b)(t,X(t,s,x))=\mathrm{trace}(B(t)).$

Then $y$ and $J$ are the unique $C^{1}$ solutions of the following initial
value problems:

$\begin{cases}\dot{y}(t)&=B(t)y(t),\\\
y(s)&=\mathrm{Id},\end{cases}\qquad\text{and}\qquad\begin{cases}\dot{J}(t)&=\beta(t)J(t),\\\
J(s)&=1.\end{cases}$

When considering smooth approximations of vector fields, we need to control
the convergence of the corresponding flows, as in Lemma 3.5 below. The
following results are fairly well-known, but in absence of a reference
precisely adapted to our purposes we provide sketch of proofs for the reader’s
convenience. An analogous statement holds for vector fields well posed in
$\overline{I}$, considering the additional points
$(s_{\infty},x_{\infty})\in\partial I\times\Omega$ and the stronger
convergence in $L^{1}(I;C(\Omega^{\prime};\mathbb{R}^{n}))$ for any
$\Omega^{\prime}\Subset\Omega$.

######  Lemma 3.5.

Let $\\{b_{h}\\}_{h\in\mathbb{N}\cup\\{\infty\\}}$ be well posed vector fields
in $I$ and denote by $I^{h}_{(s,x)}$ their maximal existence times and by
$X^{h}$ their flows. If $b_{h}\to b_{\infty}$ in
$L^{1}_{\mathrm{loc}}(I;C(\Omega^{\prime};\mathbb{R}^{n}))$ for any
$\Omega^{\prime}\Subset\Omega$ and $(s_{h},x_{h})\to(s_{\infty},x_{\infty})\in
I\times\Omega$ as $h\to\infty$, then for any compact interval $J\subset
I^{\infty}_{(s_{\infty},x_{\infty})}$ one has

$J\subset I^{h}_{(s_{h},x_{h})}\qquad\text{for $h$ large enough}$

and $X^{h}(\cdot,s_{h},x_{h})$ converge uniformly to
$X^{\infty}(\cdot,s_{\infty},x_{\infty})$ on $J$.

###### Proof.

Let us consider the compact curve
$\Gamma=\\{(v,X^{\infty}(v,s_{\infty},x_{\infty})):\ v\in J\\}$ and let $U$ be
the open set $J^{\prime}\times\Omega^{\prime}$, with $J\Subset
J^{\prime}\Subset I$ and $\Omega^{\prime}\Subset\Omega$ chosen in such a way
that $\Gamma\subset U$. By assumption,
$\int_{J^{\prime}}\sup_{\Omega^{\prime}}|b_{h}(t,\cdot)-b(t,\cdot)|\,\mathrm{d}t$
is infinitesimal as $h\to\infty$. Let us consider the maximal solutions
$\gamma_{h}(v)$ for the ODE relative to $b_{h}$, starting from $x_{h}$ at
$s_{h}$ and with graph remaining in $U$ (in particular, restrictions of
$X^{h}(\cdot,s_{h},x_{h})$), and let $J_{h}\subset I^{h}_{(s_{h},x_{h})}$
their maximal existence intervals. Since the curves $\gamma_{h}$ are
equicontinuous, it is clear that limit points of the graph of these curves
exist, and that any limit point is the graph of a solution $\gamma_{\infty}$
to the ODE relative to $b_{\infty}$, with
$\gamma_{\infty}(s_{\infty})=x_{\infty}$, defined on an interval $J_{\infty}$
and with the property that, as $v$ tends to one, if any, of the extreme points
of $J_{\infty}$ different from $\partial I$, $(v,\gamma_{\infty}(v))$ tends to
$\partial U$. Since $U\Subset I\times\Omega$, the well-posedness of
$b_{\infty}$ yields that the curve $\gamma_{\infty}$ is not only a restriction
of the maximal curve $X^{\infty}(v,s_{\infty},x_{\infty})$, $v\in
I^{\infty}_{(s_{\infty},x_{\infty})}$, but also contains the curve
$X^{\infty}(v,s_{\infty},x_{\infty})$, $v\in J$, since this restricted curve
does not touch the boundary of $U$. ∎

The previous lemma grants, in particular, the lower semicontinuity of
$(s,x)\mapsto{\rm length}(I_{(s,x)})$ in $I\times\Omega$. Moreover, a simple
contradiction argument gives

(33) $\begin{split}\text{ for any compact interval $J\subset
I^{\infty}_{(s_{\infty},x_{\infty})}$ and any open set $A\Subset\Omega$, one
has}\\\ \exists\,\bar{h}\text{ such that }J\subset
I^{h}_{(s,x)}\qquad\text{for each $h>\bar{h}$ and $(s,x)\in J\times
A$}\,;\end{split}$ (34)
$A\Subset\Omega^{\infty}_{(t,s)}\quad\text{implies}\quad
A\Subset\Omega^{h}_{(t,s)}\quad\text{for $h$ large enough}$

and the uniform convergence of $X_{h}(t,s,\cdot)$ to $X_{\infty}(t,s,\cdot)$
on $A$.

##  4\. Well-posedness with Orlicz condition

In this section we are going to prove Theorem A. We fix some dimensional
constants for $n\geq 1$. Let $\omega_{n}=|B(0,1)|$ be the volume of the unit
Euclidean ball in $\mathbb{R}^{n}$ and $\sigma_{n-1}=n\omega_{n}$ the
perimeter of $B(0,1)$; let $\tau_{n}=|B(0,1)\cap B(q,1)|$ for any
$q\in\partial B(0,1)$; finally, $C_{n}$ is the constant from Lemma 4.1 and
$\kappa_{n}=2\tau_{n}^{-1}\omega_{n}C_{n}\sigma_{n-1}$ will appear in Lemma
4.3.

######  Lemma 4.1.

Let $b\in W^{1,1}(B(x,r);\mathbb{R}^{n})$ and assume that $x$ is a Lebesgue
point of $b$. Then, for some dimensional constant $C_{n}$, one has

(35)
$\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{B(x,r)}|b(x)-b(y)|\,\mathrm{d}y\leq
C_{n}\int_{B(x,r)}\frac{\|Db(y)\|}{|x-y|^{n-1}}\,\mathrm{d}y.$

###### Proof.

We assume for simplicity $n\geq 2$. From the same argument of [18, Lemma 1,
Sec.4.5.2], based on a radial integration, for all $x\in\mathbb{R}^{n}$,
$r>0$, $\epsilon\in(0,1)$ and $f\in C^{1}(B(x,r))$ we have

(36)
$\begin{split}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{B(x,r)\setminus
B(x,\epsilon r)}|f(x)-f(y)|\,\mathrm{d}y&\leq\int_{\epsilon
r}^{r}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{B(x,s)}|y-x||Df(y)|\,\mathrm{d}y\,\mathrm{d}s\\\
&\leq\int_{\epsilon
r}^{r}s\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{B(x,s)}|Df(y)|\,\mathrm{d}y\,\mathrm{d}s\,.\end{split}$

Assume now that $f\in L^{1}_{\rm loc}(\Omega)$, $x$ is a Lebesgue point of
$f$, $B(x,r)\Subset\Omega$ and $f\in W^{1,1}(B(x,r))$. Let $\delta:=\,{\rm
dist}(B(x,r),\mathbb{R}^{n}\setminus\Omega)$,
$\Omega_{\delta}:=\,\\{x\in\Omega:\,{\rm
dist}(x,\mathbb{R}^{n}\setminus\Omega)>\,\delta/2\\}$ and $(\rho_{h})_{h}$ be
a sequence of mollifiers in $\mathbb{R}^{n}$. Then it is well-defined the
sequence of regularized functions
$f_{h}:=\,f*\rho_{h}:\,\Omega_{\delta}\to\mathbb{R}$ if $h>\,1/\delta$ and it
satisfies the following properties: $f_{h}\in C^{\infty}(\Omega_{\delta})$,

(37) $f_{h}(z)\to f(z)\text{, for each $z\in\Omega_{\delta}$ Lebegue point of
$f$}\text{, as }h\to\infty,$ (38) $f_{h}\to f\text{ in }W^{1,1}(B(x,r))\,.$

Applying (36) with $f\equiv f_{h}$ and let $h\to\infty$, by (37) and (38), we
get the same inequality when $f\in W^{1,1}(B(x,r))$ and $x$ is a Lebesgue
point of $f$. Now we can let $\epsilon\to 0$ and use Fubini’s theorem in the
right hand side to get

(39)
$\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{B(x,r)}|f(x)-f(y)|\,\mathrm{d}y\leq\frac{1}{n}\int_{B(x,r)}\frac{|Df(y)|}{|x-y|^{n-1}}\,\mathrm{d}y\,.$

Finally, notice that there is a constant $c_{n}$ such that, if $M$ is a
$n\times n$ matrix with rows $M_{j}$, then $\sum_{j}|M_{j}|\leq c_{n}\|M\|$.
Therefore,

$\displaystyle\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{B(x,r)}|b(x)-b(y)|\,\mathrm{d}y$
$\displaystyle\leq\sum_{j=1}^{n}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{B(x,r)}|b_{j}(x)-b_{j}(y)|\,\mathrm{d}y$
$\displaystyle\leq\frac{1}{n}\sum_{j=1}^{n}\int_{B(x,r)}\frac{|Db_{j}(y)|}{|x-y|^{n-1}}\,\mathrm{d}y\leq\frac{c_{n}}{n}\
\int_{B(x,r)}\frac{\|Db(y)\|}{|x-y|^{n-1}}\,\mathrm{d}y,$

which proves (35) with $C_{n}=c_{n}/n$. ∎

Recall Jensen’s inequality

(40)
$\Phi\left(\int_{X}\chi(x)\,\mathrm{d}\mu(x)\right)\leq\int_{X}\Phi(\chi(x))\,\mathrm{d}\mu(x),$

where $\Phi:[0,+\infty)\to[0,+\infty)$ is a convex function, $\mu$ is a
probability measure on $X$ and $\chi:X\to\mathbb{R}$ is a $\mu$-measurable
non-negative function. The following lemma is a direct application of Jensen’s
inequality.

######  Lemma 4.2.

Let $\Phi:[0,+\infty)\to[0,+\infty)$ be a convex function. If $n\geq 1$,
$x\in\mathbb{R}^{n}$, $r>0$,
$\chi\in\mathrm{L}^{1}_{\mathrm{loc}}(\mathbb{R}^{n})$ is non-negative, then

(41)
$\Phi\left(\frac{1}{r}\int_{B(x,r)}\frac{\chi(z)}{|z-x|^{n-1}}\,\mathrm{d}z\right)\leq\frac{1}{r\sigma_{n-1}}\int_{B(x,r)}\frac{\Phi(\sigma_{n-1}\chi(z))}{|x-z|^{n-1}}\,\mathrm{d}z.$

###### Proof.

Using polar coordinates, one easily checks that

$\int_{B(0,r)}\frac{1}{|z|^{n-1}}\,\mathrm{d}z=r\sigma_{n-1}.$

Therefore, the measure
$\,\mathrm{d}\mu:=\frac{\chi_{B(x,r)}(z)}{r\sigma_{n-1}}\frac{\,\mathrm{d}z}{|x-z|^{n-1}}$
is a probability measure on $\mathbb{R}^{n}$ and Jensen’s inequality (40)
applies. ∎

######  Lemma 4.3.

Let $\Phi:[0,+\infty)\to[0,+\infty)$ be a convex non-decreasing function. Let
$o\in\mathbb{R}^{n}$, $R>0$ and $b\in W^{1,1}(B(o,2R);\mathbb{R}^{n})$. Then,
for all $\alpha\in(1,\frac{n}{n-1})$, and all $x,\,y\in B(o,R)$ distinct
Lebesgue points of $b$, one has

(42)
$\displaystyle\Phi\left(\frac{|b(x)-b(y)|}{|x-y|}\right)\leq\frac{1}{\sigma_{n-1}}\left(\int_{B(o,2R)}\frac{1}{|z|^{\alpha(n-1)}}\,\mathrm{d}z\right)^{1/\alpha}\times$
$\displaystyle\hskip
113.81102pt\times\frac{1}{|x-y|}\left(\int_{B(o,2R)}\Phi\left(\kappa_{n}\|Db(z)\|\right)^{\frac{\alpha}{\alpha-1}}\,\mathrm{d}z\right)^{\frac{\alpha-1}{\alpha}}.$

In the case $n=1$, the estimate (42) is understood to hold for
$\alpha\in(1,+\infty)$ and we also have

(43)
$\Phi\left(\frac{|b(x)-b(y)|}{|x-y|}\right)\leq\frac{1}{2|x-y|}\int_{B(o,2R)}\Phi\left(\kappa_{n}\|Db(z)\|\right)\,\mathrm{d}z.$

###### Proof.

Set $r:=|x-y|>0$ and $W=B(x,r)\cap B(y,r)$. Notice that $|W|=\tau_{n}r^{n}$.
Then

$\displaystyle|b(x)-b(y)|$
$\displaystyle\leq\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{W}|b(x)-b(z)|\,\mathrm{d}z+\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{W}|b(y)-b(z)|\,\mathrm{d}z$
$\displaystyle\leq\frac{\omega_{n}}{\tau_{n}}\left(\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{B(x,r)}|b(x)-b(z)|\,\mathrm{d}z+\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{B(y,r)}|b(y)-b(z)|\,\mathrm{d}z\right)$
$\displaystyle\leq\frac{2\omega_{n}}{\tau_{n}}\sup_{w\in
B(o,R)}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{B(w,r)}|b(w)-b(z)|\,\mathrm{d}z.$

Since $\Phi$ is non-decreasing, we have:

$\displaystyle\Phi\left(\frac{|b(x)-b(y)|}{|x-y|}\right)$
$\displaystyle\leq\sup_{w\in
B(o,R)}\Phi\left(\frac{2\omega_{n}}{r\tau_{n}}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{B(w,r)}|b(w)-b(z)|\,\mathrm{d}z\right)$
$\displaystyle\overset{\\!\\!\\!\text{by }\eqref{eq03201820}}{\leq}\sup_{w\in
B(o,R)}\Phi\left(\frac{2\omega_{n}C_{n}}{r\tau_{n}}\int_{B(w,r)}\frac{\|Db(z)\|}{|w-z|^{n-1}}\,\mathrm{d}z\right)$
$\displaystyle\overset{\\!\\!\\!\text{by
}\eqref{eq03202019}}{\leq}\frac{1}{r\sigma_{n-1}}\sup_{w\in
B(w,r)}\int_{B(w,r)}\Phi\left(\frac{2\omega_{n}C_{n}\sigma_{n-1}}{\tau_{n}}\,\|Db(z)\|\right)\frac{1}{|w-z|^{n-1}}\,\mathrm{d}z\,.$

If $n=1$, we have already obtained (43). However, if $\alpha>1$, we apply the
Hölder inequality to obtain

$\int_{B(w,r)}\Phi\left(\kappa_{n}\|Db(z)\|\right)\frac{1}{|w-z|^{n-1}}\,\mathrm{d}z\\\
\leq\left(\int_{B(w,r)}\Phi\left(\kappa_{n}\|Db(z)\|\right)^{\frac{\alpha}{\alpha-1}}\,\mathrm{d}z\right)^{\frac{\alpha-1}{\alpha}}\times\left(\int_{B(w,r)}\frac{1}{|w-z|^{\alpha(n-1)}}\,\mathrm{d}z\right)^{1/\alpha}\\\
\leq\left(\int_{B(o,2R)}\Phi\left(\kappa_{n}\|Db(z)\|\right)^{\frac{\alpha}{\alpha-1}}\,\mathrm{d}z\right)^{\frac{\alpha-1}{\alpha}}\times\left(\int_{B(0,2R)}\frac{1}{|z|^{\alpha(n-1)}}\,\mathrm{d}z\right)^{1/\alpha}.$

Notice that $\int_{B(0,2R)}\frac{1}{|z|^{\alpha(n-1)}}\,\mathrm{d}z<\infty$ if
and only if $\alpha(n-1)<n$, i.e., $\alpha<\frac{n}{n-1}$ when $n>1$ or
$\alpha<\infty$ when $n=1$. Applying this estimate to the former inequality,
we obtain (42). ∎

###### Proof of Theorem A.

Notice that assumption (A.I) implies that $\Theta$ is also convex, and
condition (7) is equivalent to

(44) $\Theta^{-1}(s_{1}s_{2})\leq
C_{\Theta}\,\Theta^{-1}(s_{1})\Theta^{-1}(s_{2})\qquad\forall
s_{1},\,s_{2}\geq\Theta(C_{\Theta}).$

Let $n>\,1$, fix $\alpha\in(1,\frac{n}{n-1})$ as in the assumption (A.I), so
that $\Phi(s):=\Theta(s/\kappa_{n})^{\frac{\alpha-1}{\alpha}}$ is convex,
where $\kappa_{n}$ is the constant defined at the beginning of the section.

Fix $o\in\Omega$ and the corresponding $R>0$ and $c>0$ as in (8). We claim
that the space continuous representative $\tilde{b}$ is well posed in $I\times
B(o,R)$. Since $b:\,I\times B(o,R)\to\mathbb{R}^{n}$ is well posed and
satisfies (8) if and only if $c\,b$ does, in the proof of the claim we can
assume with no loss of generality $c=1$. Applying Lemma 4.3 with the above
$\alpha\in(1,\frac{n}{n-1})$ and $\Phi$, we have, for almost every $t$ and all
$x,\,y\in B(o,R)$ distinct Lebegue points of $b(t,\cdot)$, one has

$\Theta\left(\frac{|b(t,x)-b(t,y)|}{\kappa_{n}|x-y|}\right)^{\frac{\alpha-1}{\alpha}}\leq
C(n,R,\alpha)\frac{1}{|x-y|}\left(\int_{B(o,2R)}\Theta\left(\|Db(t,z)\|\right)\,\mathrm{d}z\right)^{\frac{\alpha-1}{\alpha}}.$

We write $s_{1}\vee s_{2}$ for $\max\\{s_{1},s_{2}\\}$ and $s_{1}\wedge s_{2}$
for $\min\\{s_{1},s_{2}\\}$. Then, by applying (44) twice we obtain that

$\frac{|b(t,x)-b(t,y)|}{\kappa_{n}|x-y|}\leq
C_{\Theta}^{2}\Theta^{-1}(\Theta(C_{\Theta})\vee
C(n,R,\alpha)^{\frac{\alpha}{\alpha-1}})\times\\\
\times\Theta^{-1}\left(\Theta(C_{\Theta})\vee\int_{B(o,2R)}\Theta\left(\|Db(t,z)\|\right)\,\mathrm{d}z\right)\times\\\
\times\Theta^{-1}\left(\Theta(C_{\Theta})\vee\left(\frac{1}{|x-y|}\right)^{\frac{\alpha}{\alpha-1}}\right).$

If we set

(45)
$\omega(\delta):=\delta\Theta^{-1}\left(\Theta(C_{\Theta})\vee\left(\frac{1}{\delta}\right)^{\frac{\alpha}{\alpha-1}}\right)$

and

(46) $\varphi(t):=\kappa_{n}C_{\Theta}^{2}\Theta^{-1}(\Theta(C_{\Theta})\vee
C(n,R,\alpha)^{\frac{\alpha}{\alpha-1}})\,\Theta^{-1}\left(\Theta(C_{\Theta})\vee\int_{B(o,2R)}\Theta\left(\|Db(t,z)\|\right)\,\mathrm{d}z\right)\,,$

we obtain that $\omega\in C^{0}((0,\infty))$ and $b$ satisfies

(47) $|b(t,x)-b(t,y)|\leq\,\varphi(t)\,\omega(|x-y|)$

for each $x,\,y\in B(o,R)$ Lebesgue points of $b(t,\cdot)$. From (47), it
follows that $b(t,\cdot)$ is uniformly continuous on the set of Lebesgue
points of $b(t,\cdot)$ contained in $B(o,R)$. Since this set is dense in
$B(o,R)$, it follows that there exists a unique continuous extension
$\tilde{b}(t,\cdot):\,B(o,R)\to\mathbb{R}^{n}$ still satisfying (47) on the
whole $B(o,R)$. Therefore we can conclude that
$\tilde{b}(t,\cdot):\,\mathbb{R}^{n}\to\mathbb{R}^{n}$ is continuous,
condition holds (47) on $B(o,R)$ and

(48) $\tilde{b}=\,b\text{ a.e.\leavevmode\nobreak\ on }I\times B(o,R)\,.$

Moreover, it is clear that
$\varphi\in\mathrm{L}^{1}_{\mathrm{loc}}(I;\mathbb{R})$, because $\Theta^{-1}$
is concave and the function in (8) belongs to
$\mathrm{L}^{1}_{\mathrm{loc}}(I;\mathbb{R})$ by assumption.

We claim that condition (A.III) implies
$\int_{0}^{1}\frac{1}{\omega(\delta)}\,\mathrm{d}\delta=\infty$ and
$\lim_{\delta\to 0}\omega(\delta)=0$. Indeed, on the one hand, using the
monotonicity of $\Theta^{-1}$ and the change of variables
$s=\Theta^{-1}((1/\delta)^{\frac{\alpha}{\alpha-1}})$, we obtain, if
$\bar{\delta}:=\Theta(C_{\Theta})^{\frac{1-\alpha}{\alpha}}\wedge 1$ and
$\bar{s}=\Theta^{-1}((1/\bar{\delta})^{\frac{\alpha}{\alpha-1}})$,

$\begin{split}\int_{0}^{1}\frac{1}{\omega(\delta)}\,\mathrm{d}\delta\geq\int_{0}^{\bar{\delta}}\frac{1}{\delta\Theta^{-1}((1/\delta)^{\frac{\alpha}{\alpha-1}})}\,\mathrm{d}\delta=\frac{\alpha-1}{\alpha}\int_{\bar{s}}^{\infty}\frac{\Theta^{\prime}(s)}{s\Theta(s)}\,\mathrm{d}s=+\infty.\end{split}$

On the other hand, to prove ${\displaystyle\lim_{\delta\to
0}}\omega(\delta)={\displaystyle\lim_{\delta\to
0}}\delta\Theta^{-1}((1/\delta)^{\frac{\alpha}{\alpha-1}})=0$, we only need to
show that the function
$\delta\mapsto\delta\Theta^{-1}((1/\delta)^{\frac{\alpha}{\alpha-1}})$ is
monotone for $\delta$ small enough, because the above integral is not bounded.
Again with the change of variables
$s=\Theta^{-1}((1/\delta)^{\frac{\alpha}{\alpha-1}})$, we see that this
monotonicity is equivalent to the monotonicity of the function
$\displaystyle{\phi:s\mapsto s^{-1}\Theta(s)^{-\frac{\alpha-1}{\alpha}}}$ for
$s$ large. Inspecting the derivative of $\phi$, we see that

$\phi^{\prime}(s)=\frac{\Theta(s)^{-\frac{\alpha-1}{\alpha}}}{s^{2}}\left(-\frac{\alpha-1}{\alpha}\frac{\Theta^{\prime}(s)}{s\Theta(s)}-1\right)<0,$

and so $\phi$ is monotone for $s$ sufficiently large. We conclude that, for
each $o\in\Omega$ and $t$ such that $\varphi(t)<+\infty$ there exists $R>0$
such that $B(o,R)\subset\Omega$ and
$\tilde{b}(t,\cdot):\,B(o,R)\to\mathbb{R}^{n}$ satisfying Osgood’s condition

(49) $|\tilde{b}(t,x)-\tilde{b}(t,y)|\leq\,\varphi(t)\,\omega(|x-y|)\text{ for
each }x,y\in B(o,R)\,.$

(50)
$\begin{split}|\tilde{b}(t,x)|&\leq\,|\tilde{b}(t,x)-\tilde{b}(t,x_{0})|+\,|\tilde{b}(t,x_{0})|\leq\,\varphi(t)\,\omega(|x-x_{0}|)+\,|\tilde{b}(t,x_{0})|\\\
&\leq\,\omega(2R)\,\varphi(t)+|\tilde{b}(t,x_{0})|\text{ for
a.e.\leavevmode\nobreak\ }t\in I\text{ and for each }x,\,x_{0}\in
B(o,R)\,.\end{split}$

Now observe that, since $b\in L^{1}_{\rm loc}(I\times\Omega;\mathbb{R}^{n})$,
by (48), $\tilde{b}\in L^{1}_{\rm loc}(I\times B(o,R);\mathbb{R}^{n})$. Thus
there exists $x_{0}\in B(o,R)$ such that $|\tilde{b}(t,x_{0})|\in L^{1}_{\rm
loc}(I)$. By (49), (50) and Proposition 3.2 (i), we obtain that
$\tilde{b}:\,I\times B(o,R)\to\mathbb{R}^{n}$ is well-posed. As a consequence
also the vector field $\tilde{b}:\,I\times\Omega\to\mathbb{R}^{n}$ is well-
posed.

Moreover, if (9) holds, it also follows that

$|\tilde{b}(t,x)|\leq\,m(t)\text{ for a.e.\leavevmode\nobreak\ }t\in
I,\text{for a.e.\leavevmode\nobreak\ }x\in B(o,R)\,.$

Thus, by applying now Proposition 3.2 (iv), we can conclude that the vector
field $\tilde{b}:\,\overline{I}\times B(o,R)\to\mathbb{R}^{n}$ is well-posed,
which implies that $\tilde{b}:\,\overline{I}\times\Omega\to\mathbb{R}^{n}$ is
well-posed, too.

When $n=1$, let $\Phi(s):=\Theta(s/\kappa_{n})$. Then we can repeat the same
arguments of the previous case, showing that the space continuous
representativre of $b$ satisfies Osgood’s condition. ∎

The next proposition shows why we could not use a standard Sobolev embedding
in the proof of Theorem A to obtain a stronger Sobolev regularity for $b$. See
also Remark 4.5 below.

######  Proposition 4.4.

Condition (A.III) in Theorem A implies that $\Theta$ cannot have polynomial
growth, that is,

(51) $\forall
m\in\mathbb{N}\qquad\limsup_{s\to\infty}\frac{\Theta(s)}{s^{m}}=+\infty,$

but it does not imply that $\Theta$ has more than polynomial growth.

###### Proof.

On the one hand, we have

$\displaystyle\infty$
$\displaystyle=\int_{1}^{\infty}\frac{\Theta^{\prime}(s)}{s\Theta(s)}\,\mathrm{d}s=\sum_{k=1}^{\infty}\int_{k}^{k+1}\frac{\Theta^{\prime}(s)}{s\Theta(s)}\,\mathrm{d}s$
$\displaystyle\leq\sum_{k=1}^{\infty}\frac{1}{k}\int_{k}^{k+1}\frac{\Theta^{\prime}(s)}{\Theta(s)}\,\mathrm{d}s=\sum_{k=1}^{\infty}\frac{1}{k}\left(\log(\Theta(k+1))-\log(\Theta(k))\right)$
$\displaystyle=-\log(\Theta(1))+\sum_{k=2}^{\infty}\frac{1}{k^{2}-k}\log(\Theta(k))+\limsup_{k\to\infty}\frac{\log(\Theta(k))}{k}$

If the series in the last row is infinite, then for every $0<\alpha<1$ there
exists a sequence $k_{j}\to\infty$ such that

$\frac{1}{k_{j}^{2}-k_{j}}\log(\Theta(k_{j}))\geq\frac{1}{k_{j}^{1+\alpha}},$

that is, for all $j$ with $k_{j}\geq 2$,

$\Theta(k_{j})\geq\exp\left(\frac{1}{2}k_{j}^{1-\alpha}\right).$

If instead the series in the last row is finite, then
$\limsup_{k\to\infty}\frac{\log(\Theta(k))}{k}=\infty$, that is, there exists
a sequence $k_{j}\to\infty$ such that

$\Theta(k_{j})\geq\exp\left(k_{j}\right).$

On the other hand, we cannot improve the $\limsup$ in (51) with a $\liminf$
(or a $\lim$). Indeed, with a similar estimate as above, we obtain

$\int_{1}^{\infty}\frac{\Theta^{\prime}(s)}{s\Theta(s)}\,\mathrm{d}s\geq-\frac{1}{2}\log(\Theta(1))+\sum_{k=2}^{\infty}\frac{1}{k^{2}+k}\log(\Theta(k)).$

Thus, taking a sequence $\\{k_{j}\\}_{j}$ sparse enough, one can construct a
piece-wise linear increasing convex function
$\Theta:[0,+\infty)\to[0,+\infty)$ such that
$\Theta(k_{j})=\exp(k_{j}^{2}+k_{j})$ (thus
$\int_{1}^{\infty}\frac{\Theta^{\prime}(s)}{s\Theta(s)}\,\mathrm{d}s=\infty$)
and $\Theta(x_{j})=x_{j}^{2}$ for some $k_{j}<x_{j}<k_{j+1}$ (hence
$\liminf_{s\to\infty}\frac{\Theta(s)}{s^{2}}<\infty$).

Let us give more details about such construction. As a start, define
$k_{1}=1$, $\Theta(1)=\exp(2)$ and $\alpha_{1}=\exp(2)$. Then define
$\Theta(t)=\alpha_{1}t$ for $t\in[0,x_{1}]$, where $x_{1}$ is so that
$\Theta(x_{1})=x_{1}^{2}$, that is, $x_{1}=\alpha_{1}$. Iteratively, given
$x_{j}$ and $\alpha_{j}$, and the function $\Theta$ defined in $[0,x_{j}]$
with $\Theta(x_{j})=x_{j}^{2}$, define

$\displaystyle k_{j+1}$ $\displaystyle=\lceil 2x_{j}\rceil$
$\displaystyle\alpha_{j+1}$
$\displaystyle=\frac{\exp(k_{j+1}^{2}+k_{j+1})-x_{j}^{2}}{k_{j+1}-x_{j}}$
$\displaystyle\Theta(t)$
$\displaystyle=\Theta(x_{j})+\alpha_{j+1}(t-x_{j})\text{ for
}t\in[x_{j},x_{j+1}]$ $\displaystyle\text{ where $x_{j+1}$ is so that
}\Theta(x_{j})+\alpha_{j+1}(x_{j+1}-x_{j})=x_{j+1}^{2}$

By construction, we have $\Theta(k_{j+1})=\exp(k_{j+1}^{2}+k_{j+1})$ and
$\Theta(x_{j+1})=x_{j+1}^{2}$. Moreover, since $\exp(x^{2}+x)>x^{2}$ for
$x\geq 0$, we have $\alpha_{j+1}>\alpha_{j}$ and thus the resulting function
$\Theta$ is convex; in fact, $\Theta$ is the sup of the linear functions
$t\mapsto\Theta(x_{j})+\alpha_{j+1}(t-x_{j})$, $j=1,\dots$, and thus it is
convex. ∎

######  Remark 4.5.

Using the function $\Theta$ constructed in the previous Proposition 4.4, we
can give an example of a continuous vector field $b:\mathbb{R}\to\mathbb{R}$
that is well posed, even though $b\notin W^{1,p}_{\mathrm{loc}}(\mathbb{R})$
for all $p>2$.

Indeed, define $b(t)=\int_{0}^{t}\chi(s)\,\mathrm{d}s$, where $\chi\in
L^{1}(\mathbb{R})$ is the function

$\chi(s)=\sum_{j=1}^{\infty}\chi_{I_{j}}(s)\cdot x_{j}\text{ where
}I_{j}:=\left(\sum_{k=1}^{j-1}\frac{1}{k^{2}x_{k}^{2}},\sum_{k=1}^{j}\frac{1}{k^{2}x_{k}^{2}}\right).$

Notice that, by construction,

$\exp(4x_{j}^{2}+x_{j})\leq\exp(k_{j}^{2}+k_{j})=\Theta(k_{j+1})<\Theta(x_{j+1})=x_{j+1}^{2},$

and thus $x_{j+1}>\exp(2x_{j}^{2}+x_{j})>\exp(x_{j})>\exp(j)$ for all $j\geq
1$. It follows that $\bigcup_{j=1}^{\infty}I_{j}$ is bounded and that

$\int_{\mathbb{R}}\chi(s)^{p}\,\mathrm{d}s=\sum_{j=1}^{\infty}\frac{x_{j}^{p}}{j^{2}x_{j}^{2}}=\sum_{j=1}^{\infty}\frac{x_{j}^{p-2}}{j^{2}}$

is finite if and only if $p\leq 2$.

###  4.1. A class of subexponential summability types

Examples of functions $\Theta$ satisfying the properties listed in Theorem A
are of the form $\mathscr{E}_{k,\beta}(s)$ as in (10), as we will show in this
section. It is clear that the subexponential summability of type
$\mathscr{E}_{k,\beta}$ implies the subexponential summability of type
$\mathscr{E}_{k^{\prime},\beta^{\prime}}$ for all $\beta\leq\beta^{\prime}\leq
1$ and $k^{\prime}\geq k$.

If $\left\|D_{x}b\right\|$ satisfies an exponential summability, then it is
well-known that $b$ is well-posed. Indeed, in this case, the vector field $b$
satisfies a so-called Log-Lipschitz condition (see, for instance, [8, 32]).
Theorem A extends the well-posedness to subexponential summability order
$k\geq 1$. We will show in Section 8.2 that the upper bound on $\beta$ for the
subexponential summability order $1$ is in fact necessary; see also [12,
Section 6], [11, p. 1240].

Let us now show that $\mathscr{E}_{k,\beta}$ satisfies the assumptions of
Theorem A. Let us introduce some notation in order to better represent
$\mathscr{E}_{k,\beta}$. Let us denote by $E_{k}:\,\mathbb{R}\to\mathbb{R}$
the _$k$ th-iterated exponential function_, that is, by induction on $k$,

$E_{1}(s):=\exp(s),\quad E_{k+1}(s):=\exp\left(E_{k}(s)\right)\text{ if
}s\in\mathbb{R},\,k\geq\,1.$

Since $\lim_{s\to-\infty}E_{1}(s)=0$, notice that
$\lim_{s\to-\infty}E_{k}(s)=E_{k-1}(0)=:s_{k}$ for all $k>1$, and that
$\lim_{k\to\infty}s_{k}=\infty$. Denote by
$L_{k}:E_{k}(\mathbb{R})\to\mathbb{R}$ the _$k$ th-iterated logarithm
function_ as the inverse of $E_{k}$, that is

$L_{k}(s):=E_{k}^{-1}(s)\text{ if }s\in E_{k}(\mathbb{R})=(s_{k},+\infty)\,.$

Define then $P_{k}(s):=\prod_{j=1}^{k}L_{j}(s)$, so that one can easily check
that

$L_{k+1}^{\prime}(s)=\frac{1}{sP_{k}(s)},\text{ so that
}P^{\prime}_{k}(s)=\frac{P_{k}(s)}{s}\sum_{j=1}^{k}\frac{1}{P_{j}(s)}.$

With this notation, we have

$\mathscr{E}_{k,\beta}(s)=\exp\left(\frac{s}{P_{k-1}(s)L_{k}^{\beta}(s)}\right).$

A direct computation shows that

(52)
$\mathscr{E}_{k,\beta}^{\prime}=\mathscr{E}_{k,\beta}H_{k,\beta}\quad\text{
where }\quad
H_{k,\beta}=\frac{1}{P_{k-1}L_{k}^{\beta}}\left(1-\sum_{j=1}^{k-1}\frac{1}{P_{j}}-\frac{\beta}{P_{k}}\right),$

and that

(53)
$H_{k,\beta}^{\prime}=\frac{1}{sP_{k-1}L_{k}^{\beta}}\bigg{(}-\Big{(}\sum_{j=1}^{k-1}\frac{1}{P_{j}}+\frac{\beta}{P_{k}}\Big{)}\Big{(}1-\sum_{j=1}^{k-1}\frac{1}{P_{j}}-\frac{\beta}{P_{k}}\Big{)}+\sum_{j=1}^{k-1}\frac{1}{P_{j}}\Big{(}\sum_{i=1}^{j}\frac{1}{P_{i}}+\frac{\beta}{P_{k}}\Big{)}\bigg{)}.$

######  Proposition 4.6.

The function $\mathscr{E}_{k,\beta}:\,\mathbb{R}\to\mathbb{R}$ satisfies the
assumptions of Theorem A for each integer $k\geq\,1$ and
$0\leq\,\beta\leq\,1$.

###### Proof.

Monotonicity of $\mathscr{E}_{k,\beta}$: From (52) it is evident that
$\mathscr{E}_{k,\beta}^{\prime}$ is positive for $s$ large enough and thus
$\mathscr{E}_{k,\beta}$ is strictly increasing in $[\alpha,+\infty)$ for
$\alpha$ large enough.

Verification of (A.I): If $\gamma>0$, then

$\frac{\,\mathrm{d}^{2}}{\,\mathrm{d}s^{2}}\mathscr{E}_{k,\beta}(s)^{\gamma}=\gamma\mathscr{E}_{k,\beta}(s)^{\gamma}\left(\gamma
H_{k,\beta}^{2}+H_{k,\beta}^{\prime}\right).$

Because of the presence of the factor $1/s$ in (53), one can see that
$\lim_{s\to\infty}\frac{H_{k,\beta}^{\prime}}{H_{k,\beta}^{2}}=0$ and thus
$\frac{\,\mathrm{d}^{2}}{\,\mathrm{d}s^{2}}\mathscr{E}_{k,\beta}(s)^{\gamma}$
is positive for $s$ large enough. Thus we obtain that
$\mathscr{E}_{k,\beta}(s)^{\gamma}$ is convex for $s$ large.

Verification of (A.II): Firstly, notice that
$\log(st)=\log(s)+\log(t)\leq\log(s)\log(t)$ for all $s,t>e$. Therefore, by
induction on $k$, we have $L_{k}(st)\leq L_{k}(s)L_{k}(t)$ for all
$s,t>E_{n}(1)$. Hence,
$P_{k}(st)=\prod_{j=1}^{k}L_{j}(st)\leq\prod_{j=1}^{k}L_{j}(s)L_{j}(t)=P_{k}(s)P_{k}(t)$,
or all $s,t>E_{n}(1)$.

Secondly, condition (7), i.e.,
$\mathscr{E}_{k,\beta}(ts)\leq\mathscr{E}_{k,\beta}(s)\mathscr{E}_{k,\beta}(t)$
for $s,t$ large enough, is equivalent to

$\frac{s}{P_{k-1}(s)L_{k}(s)^{\beta}}+\frac{t}{P_{k-1}(t)L_{k}(t)^{\beta}}\leq\frac{st}{P_{k-1}(st)L_{k}(st)^{\beta}},$

that is

(54)
$\frac{P_{k-1}(t)L_{k}(t)^{\beta}}{t}+\frac{P_{k-1}(s)L_{k}(s)^{\beta}}{s}\leq\frac{P_{k-1}(s)L_{k}(s)^{\beta}P_{k-1}(t)L_{k}(t)^{\beta}}{P_{k-1}(st)L_{k}(st)^{\beta}}.$

Thirdly, on the one hand we have
$\lim_{t\to\infty}\frac{P_{k-1}(t)L_{k}(t)^{\beta}}{t}=0$, and thus the left-
hand side of (54) smaller than 1 for $s$ and $t$ large. On the other hand, by
the initial observation,

$\frac{P_{k-1}(s)L_{k}(s)^{\beta}P_{k-1}(t)L_{k}(t)^{\beta}}{P_{k-1}(st)L_{k}(st)^{\beta}}\geq\frac{P_{k-1}(s)L_{k}(s)^{\beta}P_{k-1}(t)L_{k}(t)^{\beta}}{P_{k-1}(s)P_{k-1}(t)L_{k}(s)^{\beta}L_{k}(t)^{\beta}}=1.$

So, inequality (54) holds true and so condition (7).

Verification of (A.III): Notice that, by (52), for $s>E_{k}(1)$,

$\frac{\mathscr{E}_{k,\beta}^{\prime}(s)}{s\mathscr{E}_{k,\beta}(s)}\sim_{s\to\infty}\frac{1}{sP_{k-1}(s)L_{k}(s)^{\beta}}.$

Thus, since
$\int_{E_{k}(1)}^{\infty}\frac{1}{sP_{k-1}(s)L_{k}(s)^{\beta}}\,\mathrm{d}s=\infty$
if and only if $\beta\leq 1$, we also have
$\int_{1}^{\infty}\frac{\mathscr{E}_{k,\beta}^{\prime}(s)}{s\mathscr{E}_{k,\beta}(s)}\,\mathrm{d}s=+\infty$
if and only if $\beta\leq 1$. ∎

######  Remark 4.7.

Observe that, if (A.III) holds, then

(55)
$\limsup_{s\to\infty}\frac{P_{k-1}(s)L_{k}(s)^{1+\alpha}\Theta^{\prime}(s)}{\Theta(s)}=\infty\text{
for every }\alpha>0\text{ and }k\geq 1\,.$

##  5\. Regularity of the flow with subexponential summability

This section is devoted to the proof of Theorem B and its consequence in
dimension 1 as written in Theorem C. Let us recall that, if
$\Phi:\,[0,+\infty)\to[0,+\infty)$ is an increasing homeomorphism, so that
$\Phi(0)=\,0$ and $\lim_{t\to+\infty}\Phi(t)=+\infty$, the Orlicz space
$\mathrm{L}^{\Phi}(\mathbb{R}^{n})$ is the space of measurable functions
$f:\mathbb{R}^{n}\to\mathbb{R}$ for which the _Luxembourg norm_

$\|f\|_{\mathrm{L}^{\Phi}}:=\,\inf\left\\{\lambda>0:\,\int_{\mathbb{R}^{n}}\Phi\left(\frac{f(x)}{\lambda}\right)\,\mathrm{d}x\leq\,1\right\\}$

is finite. $L_{\mathrm{loc}}^{\Phi}(\mathbb{R}^{n})$ will denote, as usual,
the space of measurable functions $f$ such that
$f\,\chi_{K}\in\mathrm{L}^{\Phi}(\mathbb{R}^{n})$ for each compact set
$K\subset\mathbb{R}^{n}$. If

(56) $\Phi(t)=\exp\left(\frac{t}{\log^{+}t}\right)-1,$

then we will denote the obtained $\mathrm{L}^{\Phi}(\mathbb{R}^{n})$
(respectively $\mathrm{L}_{\mathrm{loc}}^{\Phi}(\mathbb{R}^{n})$) by ${\rm
Exp}\bigl{(}{L}/{\log L}\bigr{)}$ (respectively ${\rm
Exp}_{\mathrm{loc}}\bigl{(}{L}/{\log L}\bigr{)}$). When changing the reference
measure from the Lebegues measure to a Radon measure $\mu$ on
$\mathbb{R}^{n}$, we will denote $\mathrm{L}^{\Phi}(\mathbb{R}^{n},\mu)$
(respectively $L_{\mathrm{loc}}^{\Phi}(\mathbb{R}^{n},\mu)$) the associated
Orlicz space and, if $\Phi$ is as in (56), by ${\rm
Exp}_{\mu}\bigl{(}{L}/{\log L}\bigr{)}$ (respectively ${\rm
Exp}_{\mu,\mathrm{loc}}\bigl{(}{L}/{\log L}\bigr{)}$).

###### Proof of Theorem B.

We assume with no loss of generality that $b$ coincides with its space
continuous representative. It is easy to see that, from assumptions (11) and
(12), it follows that, for a given $R>\,0$ and

$m(t):=\,(1+R\log^{+}R)\,\sup_{x\in
B(0,R)}\frac{|b(t,x)|}{1+|x|\log^{+}|x|}\text{ for a.e.\leavevmode\nobreak\
}t\in I\,,$

then

(57) $m\in L^{1}(I)\text{ and }|b(t,x)|\leq\,m(t)\text{ for
a.e.\leavevmode\nobreak\ }t\in I,\text{ for each }x\in B(0,R)$

and

(58) $\exp\left(\frac{\|D_{x}b\|}{\log^{+}\|D_{x}b\|}\right)\in
L^{1}(I;L^{1}_{\rm loc}(\mathbb{R}^{n}))\,.$

Thus, by (57) and (58), the vector field $b$ is well-posed in
$\overline{I}\times\mathbb{R}^{n}$ by Theorem A and Proposition 4.6. Moreover
the flow $X$ associated to $b$ is globally defined thanks to the growth
condition (11), so that $X\in
C^{0}(\overline{I}\times\overline{I}\times\mathbb{R}^{n};\mathbb{R}^{n})$
(see, for instance, [23, Theorem 5.1, Chap. III]). In particular, $b$
satisfies the assumptions of [11, Main Theorem]. Therefore the push-forward
measure $X(t,s,\cdot)_{\\#}\gamma_{n}$ is absolutely continuous with respect
to the Gaussian measure $\gamma_{n}$ with density $w(t,s)$ belonging to
$\mathrm{L}^{\Phi_{\alpha}}(\mathbb{R}^{n},\gamma_{n})$ for each
$0<\alpha<\alpha_{0}(s,t)$, where

$\alpha_{0}(t,s)=\exp\left(-16\,e^{2}\int_{s}^{t}\|{\rm
div}_{\gamma_{n}}b(v,\cdot)\|_{{\rm Exp}_{\gamma_{n}}\left(\frac{L}{\log
L}\right)}\,\mathrm{d}v\right),$

where

${\rm div}_{\gamma_{n}}b(v)={\rm div}b(v)-v\cdot b(v),$

that is, ${\rm div}_{\gamma_{n}}$ is the adjoint of the gradient operator with
respect to the measure $\gamma_{n}$. Notice that $\alpha_{0}$ is a continuous
function with $\alpha_{0}(t,t)=1$ for every $t\in I$.

To complete the proof, it is enough to show that

(59)
$\frac{\,\mathrm{d}}{\,\mathrm{d}\mathcal{L}^{n}}(X(t,s,\cdot)_{\\#}\mathcal{L}^{n})(x)=D(x)w(t,s)(x)\quad\text{for
a.e.\leavevmode\nobreak\ }x\in\mathbb{R}^{n}\,,$

where $D(x)=\exp\left({\frac{|X(s,t,x)|^{2}-|x|^{2}}{2}}\right)$ and
${\frac{d}{d\mathcal{L}^{n}}}$ denotes the Radon-Nikodym derivative with
respect to $\mathcal{L}^{n}$. Indeed, since the map
$D:\mathbb{R}^{n}\to[0,\infty)$ is continuous and since
$\mathrm{L}^{\Phi_{\alpha}}(\mathbb{R}^{n},\mu)\subset\mathrm{L}^{\Phi_{\alpha}}_{\mathrm{loc}}(\mathbb{R}^{n})$,
we can conclude (B).

In order to prove (59), fix $t,\,s\in I$ and set $\psi:=X(s,t,\cdot)$. By the
differentiation theorem for Radon measures (see, for instance, [6]), we have

$\frac{\,\mathrm{d}}{\,\mathrm{d}\mathcal{L}^{n}}\psi_{\\#}\mathcal{L}^{n}=\lim_{r\to
0^{+}}\frac{\mathcal{L}^{n}(\psi(B(x,r)))}{\mathcal{L}^{n}(B(x,r))},$

for almost every $x\in\mathbb{R}^{n}$. Notice that

$\frac{\mathcal{L}^{n}(\psi(B(x,r)))}{\mathcal{L}^{n}(B(x,r))}=\frac{\mathcal{L}^{n}(\psi(B(x,r)))}{\gamma_{n}(\psi(B(x,r)))}\frac{\gamma_{n}(\psi(B(x,r)))}{\gamma_{n}(B(x,r))}\frac{\gamma_{n}(B(x,r)))}{\mathcal{L}^{n}(B(x,r))},$

where, again by differentiation theorem for Radon measures,

$\displaystyle\lim_{r\to
0^{+}}\frac{\gamma_{n}(B(x,r)))}{\mathcal{L}^{n}(B(x,r))}$
$\displaystyle=\frac{1}{(2\pi)^{n/2}}\exp\left(-\frac{|x|^{2}}{2}\right),$
$\displaystyle\lim_{r\to
0^{+}}\frac{\gamma_{n}(\psi(B(x,r)))}{\gamma_{n}(B(x,r))}$
$\displaystyle=w(t,s)(x),$ $\displaystyle\lim_{r\to
0^{+}}\frac{\gamma_{n}(\psi(B(x,r)))}{\mathcal{L}^{n}(\psi(B(x,r)))}$
$\displaystyle=\frac{1}{(2\pi)^{n/2}}\exp\left(-\frac{|\psi(x)|^{2}}{2}\right).$

A short computation leads to (59). ∎

When the spatial dimension $n$ equals 1, we have stronger results.

###### Proof of Theorem C.

By Theorem B, $b$ is well-posed and both maps
$X_{(t,s)}:\Omega_{(t,s)}\to\Omega_{(s,t)}$ and
$X_{(t,s)}^{-1}=X_{(s,t)}:\Omega_{(s,t)}\to\Omega_{(t,s)}$ satisfy the Lusin
(N) condition. Thus, by a well-known result of real analysis (see, for
instance, [20, Theorem 7.45]) they must be locally absolutely continuous. ∎

######  Remark 5.1.

We do not know whether, if $n\geq 2$, there exists a flow $X$ associated to a
vector field $b$ satisfying (12), but $X(t,s,\cdot)\notin
W^{1,1}_{\mathrm{loc}}(\mathbb{R}^{n};\mathbb{R}^{n})$ for some
$t,s\in\mathbb{R}$. Recall it has been proven in [21] that there exists an
almost everywhere approximately differentiable, orientation and measure
preserving homeomorphism $\Psi:\mathbb{R}^{n}\to\mathbb{R}^{n}$, whose
Jacobian is equal to $-1$ almost everywhere on the unit $n$-dimensional cube
$Q$, $\Psi(x)=x$ if $x\in\mathbb{R}^{n}\setminus Q$ and $\Psi\notin
W^{1,1}(Q;\mathbb{R}^{n})$. In particular it holds that

$\Psi_{\\#}\mathcal{L}^{n}\ll\mathcal{L}^{n}\text{ and
}\frac{\,\mathrm{d}}{\,\mathrm{d}\mathcal{L}^{n}}(\Psi_{\\#}\mathcal{L}^{n})(x)=1\text{
for a.e.\leavevmode\nobreak\ }x\in\mathbb{R}^{n}.$

We are not aware whether such a homeomorphism $\Psi$ could be induced by the
flow of a suitable vector field.

######  Remark 5.2.

Sobolev regularity stated in Theorem C is sharp, as we show in Example 8.2
below. Moreover it can be compared with the result in [27], which proves
existence and uniqueness of a unique flow $X$, with $D_{x}X(t,s,\cdot)\in
A_{\infty}(\mathbb{R})$, provided that $D_{x}b\in
L^{1}((0,T);BMO(\mathbb{R}))$ (see [22, Chap. 7] for the definition of the
class of functions $BMO(\mathbb{R}^{n})$). Notice that a function in $BMO$
admits exponential summability, as defined in Section 4.1. In particular, (12)
holds, too. Whereas there are vector fields $b$ with $D_{x}b$ exponentially
summable, but $D_{x}b\notin\mathrm{L}^{1}((0,T);BMO(\mathbb{R}))$ (see Example
8.3 below). Notice also that there are flows associated to well-posed vector
fields, which are not absolutely continuous (see Example 8.1 below).

##  6\. Regularity of the flow with exponential summability

This section is devoted to the proof of Theorem D. Under the stronger
regularity assumption

(60) $b\in C^{0}(I;C^{1}(\Omega;\mathbb{R}^{n}))$

we are able to derive a more precise a priori estimate, involving the quantity
$\ell(s,x)$, namely the length of the maximal interval $I_{(s,x)}$. Notice
that we clearly have $\ell(s,x)\leq\ell$, with $\ell$ the length of $I$.
Moreover, when $b$ is a well posed vector field, the maximal integral curves
of $b$ stop either at the boundary of $I$ or of $\Omega$ and thus, for all
$s\in I$,

(61) $\ell(s,x)\geq\min\\{\ell,{\rm dist}(x,\partial\Omega)/\sup|b|\\}.$

From this follows that, $\Lambda_{p}^{\prime}\leq\Lambda_{p}$, where
$\Lambda_{p}^{\prime}$ is defined in (65) below and $\Lambda_{p}$ in (13). In
particular, the finiteness condition (62) implies the finiteness of
$\Lambda^{\prime}_{p}$ as stated in (65).

######  Theorem 6.1.

Let $I\subset\mathbb{R}$ and $\Omega\subset\mathbb{R}^{n}$ be bounded open
sets and let $b\in C^{0}(I;C^{1}(\Omega;\mathbb{R}^{n}))$ be a bounded vector
field. Assume that for some $p>2n$ one has

(62) $\int_{I}\int_{\Omega}{\rm
dist}(x,\partial\Omega)^{\frac{n}{n-p}}\exp\left(\frac{\ell
p^{2}}{p-n}\|D_{x}b(s,x)\|\right)\,\mathrm{d}x\,\mathrm{d}s<+\infty.$

Then $b$ is a well-posed vector field in $\overline{I}\times\Omega$ thanks to
Theorem A.

For all $t\in\overline{I}$

(63) $\text{for almost every $s\in{I}$,\ }X(t,s,\cdot)\in
W^{1,p}(\Omega_{(t,s)};\mathbb{R}^{n})\quad\text{and}\quad X(s,t,\cdot)\in
W^{1,p}(\Omega_{(s,t)};\mathbb{R}^{n}).$

Moreover, we have that, for all $t\in\overline{I}$,

(64)
$\int_{I}\int_{\Omega_{(t,s)}}\|D_{x}X(t,s,x)\|^{p}\,\mathrm{d}x\,\mathrm{d}s\leq\Lambda^{\prime}_{p},$

where

(65)
$\Lambda_{p}^{\prime}:=\int_{I}\int_{\Omega}\left(\frac{\ell}{\ell(s,x)}\right)^{\frac{n}{p-n}}\exp\biggl{(}\frac{\ell
p^{2}}{p-n}\|D_{x}b(s,x)\|\biggr{)}\,\mathrm{d}x\,\mathrm{d}s<\infty.$

If we also assume that there exists $\hat{\ell}>0$ such that

(66) $\ell(t,x)\geq\hat{\ell}\text{ for all $t\in I$ and $x\in\Omega$},$

then, for every $s,t\in I$,

(67)
$\int_{\Omega_{(t,s)}}\|D_{x}X(t,s,x)\|^{p}\,\mathrm{d}x\leq\frac{1}{\hat{\ell}}\left(\frac{\ell}{\hat{\ell}}\right)^{\frac{n^{2}}{p(p-n)}}\Lambda^{\prime\prime}_{p}$

where

$\Lambda^{\prime\prime}_{p}:=\int_{I}\int_{\Omega}\exp\left(\frac{\ell
p^{2}}{p-n}\|D_{x}b(v,y)\|\right)\,\mathrm{d}y\,\mathrm{d}v.$

Moreover, if there exists an open set $\Omega^{\prime}\Subset\Omega$ such that

(68) $\mathtt{spt}(b(t,\cdot))\subset\Omega^{\prime}\text{ for each }t\in
I\,,$

then (62) implies (67) also for $p>n$.

######  Remark 6.2.

Typical cases where where (66) holds are the following. First, as we already
observed in the statement above, if there exists an open set
$\Omega^{\prime}\Subset\Omega$ such that (68) holds then it turns out that
$\Omega_{(t,s)}=\,\Omega$ and $\ell(t,x)=\ell$ for every
$t,\,s\in\overline{I}$ and $x\in\Omega$.

Second, if $\partial\Omega$ is smooth and compact and $b$ is tangent to
$\partial\Omega$, then we have again $\Omega_{(t,s)}=\,\Omega$ and
$\ell(t,x)=\ell$ for every $t,\,s\in\overline{I}$ and $x\in\Omega$.

Third, one can easily have $\hat{\ell}<\ell$: for example, if on the plane
$\mathbb{R}^{2}$ with coordinates $(x,y)$, we take $b(t,(x,y))=\partial_{x}$
and $\Omega=\\{(x,y):-2+y<x<2-y,\ 0<y<1\\}$. In this case we have
$\ell(t,(x,y))=4-2y$.

The proof of Theorem 6.1 is based on the well-known estimate (69) below and a
bootstrap argument in three steps. The proof gives also an intermediate
estimate (74) on subdomains $A\Subset\Omega$ that involves the function
$\ell(s,x)$ in place of ${\rm dist}(x,\partial\Omega)$.

######  Remark 6.3.

The proof of Sobolev regularity of the map $X(t,s,\cdot)$ becomes even simpler
when the spatial domain $\Omega$ is replaced by a compact Riemannian manifold
$M$ without boundary, as for instance the $n$-dimensional torus
$\mathbb{T}^{n}$ considered in [8] (see also [13, 11, 12, 14, 10]). Indeed, in
this case the quantity $\ell(s,x)$ equals the length of $I$. The extension to
the case of Sobolev regularity with respect to the space variable, along the
lines of Theorem D (i.e., dropping assumption (60)), simply requires a global
approximation of $b$ by more regular vector fields. In the case
$M=\mathbb{T}^{n}$, one may argue by convolution with respect to the space
variable, viewing the vector field as a spatially periodic one defined in
$I\times\mathbb{R}^{n}$.

The first lemma is one of the many variants of Gronwall’s lemma, whose simple
proof is omitted.

######  Lemma 6.4.

Let $f:[s,t]\to[0,+\infty)$ be an absolutely continuous function and $\beta\in
L^{1}(s,t)$ non-negative. If $f^{\prime}\leq\beta f$ a.e. in $(s,t)$ and
$f(s)=1$, then $f(t)\leq\exp(\int_{s}^{t}\beta(v)\,\mathrm{d}v)$.

######  Lemma 6.5.

Let $b\in C^{0}(I;C^{1}(\Omega;\mathbb{R}^{n}))$ and let
$X:\mathscr{D}_{b}\to\Omega$ the corresponding flow. Then, for every
$(t,s,x)\in\mathscr{D}_{b}$ one has

(69)
$\|D_{x}X(t,s,x)\|\leq\exp\left|\int_{s}^{t}\|D_{x}b(v,X(v,s,x))\|\,\mathrm{d}v\right|.$

###### Proof.

Set $y(v)=D_{x}X(v,s,x)$, $B(v)=(D_{x}b)(v,X(v,s,x))$ and assume, to fix the
ideas, $s\leq t$ (the proof in the other case is similar). Since, thanks to
Lemma 3.4, $y$ is of class $C^{1}$, the function $[s,t]\ni v\mapsto|y(v)|$ is
absolutely continuous. Using once more Lemma 3.4, we have

$\frac{\,\mathrm{d}}{\,\mathrm{d}v}|y(v)|=\langle\frac{y(v)}{|y(v)|},\dot{y}(v)\rangle=\langle\frac{y(v)}{|y(v)|},B(v)y(v)\rangle\leq\|B(v)\||y(v)|\text{
for a.e.\leavevmode\nobreak\ }v\in(s,t)\,.$

Lemma 6.4 implies (69), where the absolute value in the argument of the
exponential is necessary when $t<s$. ∎

###### Proof of Theorem 6.1.

Fix $t\in I$ and $A\subset\Omega$ open; first, we use Lemma 6.5 and Jensen’s
inequality (40) to get

(70)
$\displaystyle\int_{I}\int_{A_{(t,s)}}\|D_{x}X(t,s,x)\|^{p}\,\mathrm{d}x\,\mathrm{d}s$
$\displaystyle\leq\int_{I}\int_{A_{(t,s)}}\exp\left(p\left|\int_{s}^{t}\|D_{x}b(v,X(v,s,x))\|\,\mathrm{d}v\right|\right)\,\mathrm{d}x\,\mathrm{d}s$
$\displaystyle\leq\int_{I}\int_{A}\int_{I}\frac{\chi_{A(v,s)}(x)}{\ell_{A}(s,x)}\exp\left(\ell
p\|D_{x}b(v,X(v,s,x))\|\right)\,\mathrm{d}v\,\mathrm{d}x\,\mathrm{d}s,$

where we write $\ell_{A}(s,x)$ for the length of the interval of the maximal
integral curve of $b$ in $A$ that starts from $x$ at time $s$. Setting
$J(s,v,\cdot)=J_{X}(s,v,\cdot)=\det D_{x}X(s,v,\cdot)$, we apply the change of
variable $y=X(v,s,x)$, i.e., $X(s,v,y)=x$ and
$\chi_{A(v,s)}(x)\,\mathrm{d}x=J(s,v,y)\chi_{A(s,v)}(y)\,\mathrm{d}y$; notice
in particular that $x\in A(v,s)$ if and only if $y=X(v,s,x)\in A(s,v)$.
Together with Hadamard’s inequality (26) and the Hölder inequality with
exponents $q=p/n$ and $q^{\prime}=p/(p-n)$, we get

(71)
$\displaystyle\int_{I}\int_{I}\int_{A}\frac{\chi_{A(v,s)}(x)}{\ell_{A}(s,x)}\exp\left(\ell
p\|D_{x}b(v,X(v,s,x))\|\right)\,\mathrm{d}x\,\mathrm{d}v\,\mathrm{d}s$
$\displaystyle\leq\int_{I}\int_{I}\int_{A}\frac{\chi_{A(s,v)}(y)}{\ell_{A}(s,X(s,v,y))}\exp\left(\ell
p\|D_{x}b(v,y)\|\right)\|D_{y}X(s,v,y)\|^{n}\,\mathrm{d}y\,\mathrm{d}v\,\mathrm{d}s$
$\displaystyle\leq\left(\int_{I}\int_{I}\int_{A}\frac{\chi_{A(s,v)}(y)}{\ell_{A}(s,X(s,v,y))^{q^{\prime}}}\exp\left(\ell
pq^{\prime}\|D_{x}b(v,y)\|\right)\,\mathrm{d}y\,\mathrm{d}v\,\mathrm{d}s\right)^{1/q^{\prime}}\times$
$\displaystyle\phantom{AAAAAAAAAAAAAAAAAAAAA}\times\left(\int_{I}\int_{I}\int_{A}\|D_{y}X(s,v,y)\|^{nq}\,\mathrm{d}y\,\mathrm{d}v\,\mathrm{d}s\right)^{1/q}.$

The identity $\ell_{A}(s,X(s,v,y))=\ell_{A}(v,y)$ gives us

$\int_{I}\frac{\chi_{A(s,v)}(y)}{\ell_{A}(s,X(s,v,y))^{q^{\prime}}}\,\mathrm{d}s=\ell_{A(v,y)}(y)^{1-q^{\prime}},$

and we conclude that

(72)
$\displaystyle\int_{I}\int_{A_{(t,s)}}\|D_{x}X(t,s,x)\|^{p}\,\mathrm{d}x\,\mathrm{d}s$
$\displaystyle\leq\left(\int_{I}\int_{A}\ell_{A}^{1-q^{\prime}}(v,y)\exp\left(\ell
pq^{\prime}\|D_{x}b(v,y)\|\right)\,\mathrm{d}y\,\mathrm{d}v\right)^{1/q^{\prime}}\times$
$\displaystyle\phantom{AAAAAAAAAAAAAAAAAAAAA}\times\left(\int_{I}\int_{I}\int_{A}\|D_{y}X(s,v,y)\|^{nq}\,\mathrm{d}y\,\mathrm{d}v\,\mathrm{d}s\right)^{1/q}.$

If we take $A\Subset\Omega$, the latter triple integral is finite. By
integrating this inequality with respect to $t$, and then rearranging the
terms, we obtain that

(73)
$\int_{I}\int_{I}\int_{A_{(t,s)}}\|D_{x}X(t,s,x)\|^{p}\,\mathrm{d}x\,\mathrm{d}s\,\mathrm{d}t\leq\ell^{q^{\prime}}\int_{I}\int_{A}\ell_{A}^{1-q^{\prime}}(v,y)\exp\left(\ell
pq^{\prime}\|D_{x}b(v,y)\|\right)\,\mathrm{d}y\,\mathrm{d}v$

If we plug the estimate (73) into (72), we obtain

(74)
$\int_{I}\int_{A_{(t,s)}}\|D_{x}X(t,s,x)\|^{p}\,\mathrm{d}x\,\mathrm{d}s\leq\ell^{\frac{n}{p-n}}\int_{I}\int_{A}\ell_{A}^{\frac{n}{n-p}}(v,y)\exp\left(\frac{\ell
p^{2}}{p-n}\|D_{x}b(v,y)\|\right)\,\mathrm{d}y\,\mathrm{d}v.$

To prove (64), we need to take a limit in (74) along a sequence of sets
$A\Subset\Omega$ that fills $\Omega$. To this aim, we define

$A^{\epsilon}:=\\{x\in\Omega:{\rm dist}(x,\partial\Omega)>\epsilon\\}$

for $\epsilon>0$, plug $A^{\epsilon}$ in (74) and take $\epsilon\to 0$.

For the left-hand side of (74), we observe that
$A^{\epsilon_{1}}_{(t,s)}\subset A^{\epsilon_{2}}_{(t,s)}$ whenever
$\epsilon_{1}>\epsilon_{2}$. Thus, we have

$\lim_{\epsilon\to
0}\int_{I}\int_{A^{\epsilon}_{(t,s)}}\|D_{x}X(t,s,x)\|^{p}\,\mathrm{d}x\,\mathrm{d}s=\int_{I}\int_{\Omega_{(t,s)}}\|D_{x}X(t,s,x)\|^{p}\,\mathrm{d}x\,\mathrm{d}s.$

The right-hand side of (74) is more tricky. Define $f(v,y):=\exp\left(\ell
pq^{\prime}\|D_{x}b(v,y)\|\right)$, $\delta(x):={\rm dist}(x,\partial\Omega)$,
and $\alpha=\frac{n}{n-p}$. Notice that, since $p\geq 2n$, we have
$\alpha\in(-1,0)$.

We claim that

(75) $\liminf_{\epsilon\to
0}\int_{A^{\epsilon}}\int_{I}\ell_{A^{\epsilon}}^{\alpha}(v,y)f(v,y)\,\mathrm{d}v\,\mathrm{d}y\leq\int_{\Omega}\int_{I}\ell_{\Omega}^{\alpha}(v,y)f(v,y)\,\mathrm{d}v\,\mathrm{d}y.$

If $K\Subset\Omega$, the monotone convergence theorem implies that

$\lim_{\epsilon\to
0}\int_{K}\int_{I}\ell_{A^{\epsilon}}^{\alpha}(v,y)f(v,y)\,\mathrm{d}v\,\mathrm{d}y=\int_{K}\int_{I}\ell_{\Omega}^{\alpha}(v,y)f(v,y)\,\mathrm{d}v\,\mathrm{d}y.$

Thus, (75) is shown if we can prove that

(76) $\inf_{K\Subset\Omega}\liminf_{\epsilon\to 0}\int_{A^{\epsilon}\setminus
K}\int_{I}\ell_{A^{\epsilon}}^{\alpha}(v,y)f(v,y)\,\mathrm{d}v\,\mathrm{d}y=0.$

Indeed, thanks to (65), for every $K\Subset\Omega$ we have

$\left|\liminf_{\epsilon\to
0}\int_{A^{\epsilon}}\int_{I}\ell_{A^{\epsilon}}^{\alpha}(v,y)f(v,y)\,\mathrm{d}v\,\mathrm{d}y-\int_{\Omega}\int_{I}\ell_{\Omega}^{\alpha}(v,y)f(v,y)\,\mathrm{d}v\,\mathrm{d}y\right|\\\
\leq\liminf_{\epsilon\to 0}\int_{A^{\epsilon}\setminus
K}\int_{I}\ell_{A^{\epsilon}}^{\alpha}(v,y)f(v,y)\,\mathrm{d}v\,\mathrm{d}y+\int_{\Omega\setminus
K}\int_{I}\ell_{\Omega}^{\alpha}(v,y)f(v,y)\,\mathrm{d}v\,\mathrm{d}y.$

If there is a sequence $K_{j}\Subset\Omega$ such that

$\lim_{j\to\infty}\liminf_{\epsilon\to 0}\int_{A^{\epsilon}\setminus
K_{j}}\int_{I}\ell_{A^{\epsilon}}^{\alpha}(v,y)f(v,y)\,\mathrm{d}v\,\mathrm{d}y=0,$

then $\lim_{j\to\infty}|\Omega\setminus K_{j}|=0$, and
$\lim_{j\to\infty}\int_{\Omega\setminus
K_{j}}\int_{I}\ell_{\Omega}^{\alpha}(v,y)f(v,y)\,\mathrm{d}v\,\mathrm{d}y=0$,
and so (75) holds.

To prove (76) we use the inequality (61). If $K$ is large enough, so that
${\rm dist}(y,\partial A^{\epsilon})<\ell$ for all $y\in A_{\epsilon}\setminus
K$, the inequality (61) simplifies to

$\ell_{A^{\epsilon}}(v,y)\|b\|_{L^{\infty}}\geq{\rm dist}(y,\partial
A^{\epsilon}).$

We use the latter inequality to compute the following averaged integral

$\displaystyle\frac{1}{\eta}\int_{0}^{\eta}$
$\displaystyle\int_{A^{\epsilon}\setminus
K}\int_{I}\ell_{A^{\epsilon}}^{\alpha}(v,y)f(v,y)\,\mathrm{d}v\,\mathrm{d}y\,\mathrm{d}\epsilon$
$\displaystyle\leq\frac{1}{\eta}\int_{0}^{\eta}\int_{A^{\epsilon}\setminus
K}\int_{I}\frac{{\rm dist}(y,\partial
A^{\epsilon})^{\alpha}}{\|b\|_{L^{\infty}}^{\alpha}}f(v,y)\,\mathrm{d}v\,\mathrm{d}y\,\mathrm{d}\epsilon$
$\displaystyle=\frac{1}{\|b\|_{L^{\infty}}^{\alpha}}\int_{\Omega\setminus
K}\int_{I}\left(\frac{1}{\eta}\int_{0}^{\eta}\chi_{A^{\epsilon}}(y){\rm
dist}(y,\partial
A^{\epsilon})^{\alpha}\,\mathrm{d}\epsilon\right)f(v,y)\,\mathrm{d}v\,\mathrm{d}y.$

Since $\delta(y)\leq{\rm dist}(y,\partial A^{\epsilon})+\epsilon$, i.e., ${\rm
dist}(y,\partial A^{\epsilon})\geq\delta(y)-\epsilon$, we compute

$\displaystyle\int_{0}^{\eta}\chi_{A^{\epsilon}}(y){\rm dist}(y,\partial
A^{\epsilon})^{\alpha}\,\mathrm{d}\epsilon$
$\displaystyle\leq\int_{0}^{\eta}\chi_{A^{\epsilon}}(y)(\delta(y)-\epsilon)^{\alpha}\,\mathrm{d}\epsilon$
$\displaystyle=\int_{0}^{\min\\{\eta,\delta(y)\\}}(\delta(y)-\epsilon)^{\alpha}\,\mathrm{d}\epsilon$
$\displaystyle\overset{(*)}{\leq}\min\left\\{2^{-\alpha},\,\frac{2}{\alpha+1}\right\\}\eta\delta(y)^{\alpha}.$

In $(*)$, we considered two cases: first, when $\eta<\delta(y)/2$, the
integral is bounded by $2^{-\alpha}\delta(y)^{\alpha}\eta$; second, when
$\eta\geq\delta(y)/2$, using the fact that $\alpha>-1$, the integral is
bounded by $\frac{2}{\alpha+1}\eta\delta(y)^{\alpha}$.

Therefore, there exists $\eta_{0}>0$ such that for all $\eta\in(0,\eta_{0})$,
we can estimate the averaged integral with

$\frac{1}{\eta}\int_{0}^{\eta}\int_{A^{\epsilon}\setminus K}\int_{I}{\rm
dist}(y,\partial
A^{\epsilon})^{\alpha}f(v,y)\,\mathrm{d}v\,\mathrm{d}y\,\mathrm{d}\epsilon\leq
2^{-\alpha}\int_{\Omega\setminus
K}\int_{I}\delta(y)^{\alpha}f(v,y)\,\mathrm{d}v\,\mathrm{d}y.$

It follows that

$\liminf_{\epsilon\to 0}\int_{A^{\epsilon}\setminus K}\int_{I}{\rm
dist}(y,\partial A^{\epsilon})^{\alpha}f(v,y)\,\mathrm{d}v\,\mathrm{d}y\leq
2^{-\alpha}\int_{\Omega\setminus
K}\int_{I}\delta(y)^{\alpha}f(v,y)\,\mathrm{d}v\,\mathrm{d}y.$

Then, since we assumed
$\int_{\Omega}\int_{I}\delta(y)^{\alpha}f(v,y)\,\mathrm{d}v\,\mathrm{d}y<\infty$
in (62), we obtain the estimate (76) and thus our claim (75).

We have thus completed the proof of (64), and thus (63). Next we work on the
second part of the theorem. If (66) holds, i.e.,
$\hat{\ell}\leq\ell(t,x)\leq\ell$ for all $t\in I$ and $x\in\Omega$, then

(77)
$\Lambda^{\prime\prime}_{p}\leq\Lambda^{\prime}_{p}\leq\left(\frac{\ell}{\hat{\ell}}\right)^{\frac{n}{p-n}}\Lambda^{\prime\prime}_{p},$

since $p\geq 2n$. We can repeat the steps in (70) and (71) without the
integral in $s$, and then apply (64) and (77) to obtain

$\displaystyle\int_{\Omega_{(t,s)}}\|D_{x}X(t,s,x)\|^{p}\,\mathrm{d}x$
$\displaystyle\leq\left(\int_{I}\int_{\Omega}\frac{\chi_{\Omega(s,v)}(y)}{\ell_{\Omega}(s,X(s,v,y))^{q^{\prime}}}\exp\left(\ell
pq^{\prime}\|D_{x}b(v,y)\|\right)\,\mathrm{d}y\,\mathrm{d}v\right)^{1/q^{\prime}}\times$
$\displaystyle\phantom{AAAAAAAAAAAAAAAAAAAAA}\times\left(\int_{I}\int_{\Omega}\|D_{y}X(s,v,y)\|^{nq}\,\mathrm{d}y\,\mathrm{d}v\right)^{1/q}\leq\frac{1}{\hat{\ell}}\left(\frac{\ell}{\hat{\ell}}\right)^{\frac{n^{2}}{p(p-n)}}\Lambda^{\prime\prime}_{p},$

and this proves (67).

If (68) holds, then the triple integral in (72) is finite also for $A=\Omega$
and $p>n$. In this case, we can obtain directly (74) for $A=\Omega$ without
passing through the approximation $A^{\epsilon}$, where we used the stronger
hypothesis $p>2n$. ∎

The extension of Theorem 6.1 to the case of Sobolev spatial regularity
requires a global approximation of $b$ by more regular vector fields that
seems to be not trivial, also because of the weight function $\ell(s,x)$
depending on the vector field itself. In addition, the non-doubling property
of the exponential function is source of extra difficulties, when performing
standard convolution arguments. In the case of a weight independent of $t$, we
addressed this problem in the note [7], from which we extract the following
result.

######  Theorem 6.6 ([7, Theorem 3 and Remark 25]).

Let $\Omega$ be a bounded open set and let $w:\Omega\to(0,+\infty)$, with
$w+w^{-1}\in L^{\infty}_{\mathrm{loc}}(\Omega)$.

  * (i)

If $b\in\mathrm{L}^{1}(I;W^{1,1}_{\mathrm{loc}}(\Omega;\mathbb{R}^{n}))$
satisfies

(78)
$\int_{I}\int_{\Omega}w(x)\exp(c\|D_{x}b(s,x)\|)\,\mathrm{d}x\,\mathrm{d}s<+\infty,$

for some $c>0$, then there exist $b_{h}\in
C^{\infty}(I\times\Omega;\mathbb{R}^{n})$ satisfying, whenever
$\Omega^{\prime}\Subset\Omega$,

(79) $\text{$b_{h}$ converge to $\tilde{b}$ in
$L^{1}(I;C(\Omega^{\prime};\mathbb{R}^{n}))$ as $h\to\infty$}\,,$

where $\tilde{b}$ is the space continuous representative of $b$,

(80) $D_{x}b_{h}\to D_{x}b\text{ in
}L^{1}(I;L^{1}(\Omega^{\prime};\mathbb{R}^{n})),\text{ as $h\to\infty$}\,,$

and

(81)
$\lim_{h\to\infty}\int_{I}\int_{\Omega}w(x)\exp(c\|D_{x}b_{h}(s,x)\|)\,\mathrm{d}x\,\mathrm{d}s=\int_{I}\int_{\Omega}w(x)\exp(c\|D_{x}b(s,x)\|)\,\mathrm{d}x\,\mathrm{d}s.$

  * (ii)

If $b\in L^{1}(I;W^{1,1}_{\rm loc}(\mathbb{R}^{n};\mathbb{R}^{n}))$ and there
exists a bounded open set $\Omega$ such that

(82) ${\rm spt}(b(t,\cdot))\subset\Omega\text{ for each }t\in I\,,$

and $b$ satisfies (78) on $\Omega$, then there exist $b_{h}\in
C^{\infty}(I\times\mathbb{R}^{n};\mathbb{R}^{n})$ still satisfying (79), (80),
(81) and also

(83) ${\rm spt}(b_{h}(t,\cdot))\subset\Omega\text{ for each }t\in I\text{ and
}h\in\mathbb{N}\,.$

###### Proof of Theorem D.

Let

$w(x):=\max\left\\{\ell^{n/(n-p)},\frac{({\rm
dist}(x,\partial\Omega))^{n/(n-p)}}{(\sup|b|)^{n/(n-p)}}\right\\},$

so that, with $c=\,\ell p^{2}/(p-n)$, one has

$\int_{I}\int_{\Omega}w(x)\exp(c\|Db(s,x)\|)\,\mathrm{d}x\,\mathrm{d}s<+\infty$

and we may apply Theorem 6.6 (i) to $b$. Since $b$ is bounded, we can also
assume, by a truncation argument, that $\sup|b_{h}|\leq\sup|b|$. In addition,
the quantity $\Lambda_{p}^{\prime}$ in (65) is finite and satisfies
$\Lambda_{p}^{\prime}\leq\Lambda_{p}$, using the inequality (61).

It follows from these considerations that

$\limsup_{h\to\infty}\Lambda_{p,h}^{\prime}\leq\limsup_{h\to\infty}\Lambda_{p,h}\leq\Lambda_{p}<+\infty\,,$

where

(84)
$\Lambda_{p,h}^{\prime}:=\int_{I}\int_{\Omega}(\ell_{h}(s,x))^{n/(n-p)}\exp\biggl{(}\frac{\ell
p^{2}}{p-n}\|D_{x}b_{h}(s,x)\|\biggr{)}\,\mathrm{d}x\,\mathrm{d}s\,,$

and similarly $\Lambda_{p,h}$ is defined with $b_{h}$.

Now, in order to apply (64) from Theorem 6.1 to $b_{h}$ and the pass to the
limit $h\to\infty$, it suffices by Fatou’s lemma to prove that

(85)
$\int_{\Omega_{(t,s)}}\|D_{x}X(t,s,x)\|^{p}\,\mathrm{d}x\leq\liminf_{h\to\infty}\int_{\Omega^{h}_{(t,s)}}\|D_{x}X^{h}(t,s,x)\|^{p}\,\mathrm{d}x\,,$

where $\Omega^{h}_{(t,s)}$ and $X^{h}$ are relative to the vector fields
$b_{h}$. Now, (34), derived from Lemma 3.5, yields that for any open domain
$A\Subset\Omega_{(t,s)}$ one has $A\Subset\Omega^{h}_{(t,s)}$ for $h$ large
enough, and that $X^{h}(t,s,\cdot)$ converge to $X(t,s,\cdot)$ uniformly on
$A$. Hence the lower semicontinuity of
$w\mapsto\int_{A}|D_{x}w|^{p}\,\mathrm{d}x$ yields

$\int_{A}\|D_{x}X(t,s,x)\|^{p}\,\mathrm{d}x\leq\liminf_{h\to\infty}\int_{A}\|D_{x}X^{h}(t,s,x)\|^{p}\,\mathrm{d}x\leq\liminf_{h\to\infty}\int_{\Omega^{h}_{(t,s)}}\|D_{x}X^{h}(t,s,x)\|^{p}\,\mathrm{d}x\,.$

Letting $A\uparrow\Omega_{(t,s)}$ we obtain the claimed semicontinuity
property (85). ∎

###### Proof of Theorem E.

First assume that $b$ is understood as vector field
$b:\,I\times\Omega\to\mathbb{R}^{n}$. Let $w\equiv 1$ so that, with
$c=\,\ell\,p^{2}/(p-n)$, one has

$\int_{I}\int_{\Omega}\exp(c\|Db(s,x)\|)\,\mathrm{d}x\,\mathrm{d}s<+\infty\,.$

From Theorem 6.6 (ii), let $b_{h}:\,I\times\Omega\to\mathbb{R}^{n}$ be the
regular sequence of vector fields satisfying (79), (80), (81) and (83).

It follows from (82) that $\ell_{h}(t,x)=\ell$ and thus

$\limsup_{h\to\infty}\Lambda_{p,h}^{\prime}\leq\frac{1}{\ell}\int_{I}\int_{\Omega}\exp(c\|Db(s,x)\|)\,\mathrm{d}x\,\mathrm{d}s<+\infty\,,$

where $\Lambda_{p,h}^{\prime}$ is the quantity in (84).

Now, in order to apply (67) with $p>n$ from Theorem 6.1 to $b_{h}$ and then
pass to the limit $h\to\infty$, we only need to use (85) again.

Assume now that $b$ is understood as vector field
$b:\,I\times\mathbb{R}^{n}\to\mathbb{R}^{n}$. It is clear that the flow map
$X:\,I\times I\times\mathbb{R}^{n}\to\mathbb{R}^{n}$ is identically equal to
the identity on $I\times
I\times(\mathbb{R}^{n}\setminus\overline{\Omega^{\prime}})$. Since (17) holds
for any bounded open set $\Omega\Supset\Omega^{\prime}$, then $X(t,s,\cdot)\in
W^{1,p}_{\rm loc}(\mathbb{R}^{n})$. ∎

######  Corollary 6.7.

Under the assumptions of Theorem D, we have, for every $t\in I$,

(86) for almost every $s\in I$ $\displaystyle
X(t,s,\cdot)_{\\#}\mathcal{L}^{n}\mathop{\hbox{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt\vrule
height=0.5pt,width=6.0pt,depth=0.0pt}}\nolimits\Omega_{(t,s)}=J_{X}(s,t,\cdot)\mathcal{L}^{n}\mathop{\hbox{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt\vrule
height=0.5pt,width=6.0pt,depth=0.0pt}}\nolimits\Omega_{(s,t)}=\,\frac{1}{J_{X(t,s,X(s,t,\cdot))}}\mathcal{L}^{n}\mathop{\hbox{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt\vrule
height=0.5pt,width=6.0pt,depth=0.0pt}}\nolimits\Omega_{(s,t)}\,,$

where
$J_{X}(t,s,\cdot)=\det(D_{x}X(t,s,\cdot))\in\mathrm{L}^{p/n}(\Omega_{(t,s)})$
is non-zero almost everywhere. Under the assumptions of Theorem E, then (86)
holds for every $s,\,t\in I$ replacing both $\Omega_{(t,s)}$ and
$\Omega_{(s,t)}$ by $\mathbb{R}^{n}$.

In fact, $J_{X}$ is strictly positive, as we will show in the next corollary.

###### Proof.

Since $X(t,s,\cdot)^{-1}=X(s,t,\cdot)$ is also in
$W^{1,p}_{\mathrm{loc}}(\Omega_{(s,t)};\mathbb{R}^{n})$ with $p>2n$, then both
maps $X(t,s,\cdot)$ and $X(t,s,\cdot)^{-1}$ are differentiable almost
everywhere by Lemma 2.4. Therefore, we have $J_{X}(t,s,x)\neq 0$ by Lemma 2.1.
By Lemma 2.4, $X(t,s,\cdot)$ satisfies Lusin’s (N) condition, and thus, by
Lemma 2.2, the area formula holds, that is, (86) holds. Moreover, by the
Laplace expansion of the determinant, one can easily prove, by induction on
the order matrix and Hölder inequality, that
$J_{X}(t,s,\cdot)=\det(D_{x}X(t,s,\cdot))\in\mathrm{L}^{p/n}(\Omega_{(t,s)})$.
The same argument applies under the assumptions of Theorem E. ∎

######  Corollary 6.8.

Under the assumptions of Theorem D, for every $(s,x)\in I\times\Omega$, if we
set

$\displaystyle y(t)$ $\displaystyle:=D_{x}X(t,s,x),$ $\displaystyle B(t)$
$\displaystyle:=(D_{x}b)(t,X(t,s,x)),$ $\displaystyle J(t)$
$\displaystyle:=\det(D_{x}X(t,s,x)),$ $\displaystyle\beta(t)$
$\displaystyle:={\rm div}_{x}b(t,X(t,s,x))=\mathrm{trace}(B(t)),$

then $y$ and $J$ are absolutely continuous solutions to the initial value
problems

(87) $\displaystyle\begin{cases}\dot{y}(t)=B(t)y(t),\\\
y(s)=\mathrm{Id}.\end{cases}$ (88)
$\displaystyle\begin{cases}\dot{J}(t)=\beta(t)J(t),\\\ J(s)=1.\end{cases}$

Moreover, for almost every $(t,s,x)\in\mathscr{D}_{b}$ we have

(89) $\displaystyle D_{x}X(t,s,x)$
$\displaystyle=\exp\left(\int_{s}^{t}D_{x}b(v,X(v,s,x))\,\mathrm{d}v\right),$
(90) $\displaystyle J_{X}(t,s,x)$ $\displaystyle=\exp\left(\int_{s}^{t}{\rm
div}_{x}b(v,X(v,s,x))\,\mathrm{d}v\right).$

In particular, $J_{X}>0$ almost everywhere.

###### Proof.

First, we claim that, given $s$, for almost every $x$ the matrix $B$ belongs
to $\mathrm{L}^{1}(I_{(s,x)};\mathbb{R}^{n^{2}})$, where $I_{(s,x)}$ is
defined in Section 3.2. Indeed, the change of variables $z=X(t,s,x)$ and the
identity $J_{X}(s,t,X(t,s,x))=1/J_{X}(t,s,x)$ give

$\int_{I}\int_{\Omega_{(s,t)}}\|(D_{x}b)(t,X(t,s,x))\|\,\mathrm{d}x\,\mathrm{d}t=\int_{I}\int_{\Omega_{(t,s)}}\|(D_{x}b)(t,z)\|J_{X}(s,t,z)\,\mathrm{d}z\,\mathrm{d}t.$

The latter integral is finite because $D_{x}b\in\mathrm{L}^{q}$ for all $q$
and $J_{X}\in\mathrm{L}^{p/n}$. Therefore, the claim is true.

Second, using the same approximation of $b$ as in the proof of Theorem D, we
know that $D_{x}X^{h}\to D_{x}X$ weakly in $\mathrm{L}^{p}(\mathscr{D}_{b})$,
since $D_{x}X^{h}$ are uniformly bounded bounded in
$\mathrm{L}^{p}(\mathscr{D}_{b})$.

Third, we see that the distributional derivative $\partial_{t}D_{x}X$ has the
following form: for every $\phi\in
C^{\infty}_{c}(\mathscr{D}_{b};\mathbb{R}^{n})$,

$\displaystyle\partial_{t}D_{x}X[\phi]$
$\displaystyle=-\int_{\mathscr{D}_{b}}D_{x}X\partial_{t}\phi\,\mathrm{d}t\,\mathrm{d}s\,\mathrm{d}x$
$\displaystyle=-\lim_{\epsilon\to
0}\int_{\mathscr{D}_{b}}D_{x}X_{\epsilon}\partial_{t}\phi\,\mathrm{d}t\,\mathrm{d}s\,\mathrm{d}x$
$\displaystyle=\lim_{\epsilon\to
0}\int_{\mathscr{D}_{b}}D_{x}b_{\epsilon}(t,X_{\epsilon}(t,s,x))D_{x}X_{\epsilon}(t,s,x)\phi\,\mathrm{d}t\,\mathrm{d}s\,\mathrm{d}x.$

Now, $X_{\epsilon}\to X$ uniformly on compact sets and $D_{x}b_{\epsilon}\to
D_{x}b$ in $\mathrm{L}^{q}_{\mathrm{loc}}(\mathscr{D}_{b})$ for all $q$. So,
using Hölder inequality and the Lebesgue dominated convergence theorem, we
obtain that the limit above is

$\partial_{t}D_{x}X[\phi]=\int_{\mathbb{R}^{2+n}}D_{x}b(t,X(t,s,x))D_{x}X(t,s,x)\phi\,\mathrm{d}t\,\mathrm{d}s\,\mathrm{d}x.$

In other words, $\partial_{t}D_{x}X=D_{x}b(t,X(t,s,x))D_{x}X(t,s,x)$. This
shows that $y$ is solution to the Cauchy system (87). Since $B$ is integrable,
then we get (89) by integrating this Cauchy system.

Finally, the validity of (88) follows in a standard way from (87) and (88)
implies (90). ∎

######  Corollary 6.9.

Under the assumptions of Theorem D, for every $1<r<p-n$ and for every
$s,t\in\overline{I}$, we have that

(91) $X(t,s,\cdot)\in W^{1,r}(\Omega_{(t,s)};\mathbb{R}^{n}).$

###### Proof.

The proof is an improvement of (14) through an application of Corollary 6.8,
Lemma 2.7 and Proposition 2.8. Indeed, for every $1<r<p-n$ there exists
$0<q<p^{2}-\frac{p^{2}}{p-n}$ such that

$r=\frac{p^{2}(p-n)}{q(p-n)+p^{2}},\quad\text{ that is,
}\quad\frac{r}{p-r}=\left(\frac{q}{p}+\frac{n}{p-n}\right)^{-1}.$

So, given $s,t\in I$, (14) implies that there exists $u\in I$ such that

$X(s,t,\cdot)=X(s,u,X(u,t,\cdot))$

and both maps $X(s,u,\cdot)$ and $X(u,t,\cdot)$ belong to $W^{1,p}$ on their
domains, with non-zero Jacobian by Corollary 6.8. We then apply Lemma 2.7 and
Proposition 2.8 to prove that their composition is of class $W^{1,r}$ on its
domain. ∎

######  Corollary 6.10.

Under the assumptions of Theorem D, we have

$X\in W^{1,p}(\mathscr{D}_{b};\mathbb{R}^{n}).$

###### Proof.

We know that $X$ is continuous and, from (15), we have that
$D_{x}X\in\mathrm{L}^{p}(\mathscr{D}_{b};\mathbb{R}^{n^{2}})$. Moreover, by
the identity $X(t,s,x)=x+\int_{s}^{t}b(v,X(v,s,x))\,\mathrm{d}v$, we have

$\partial_{t}X(t,s,x)=b(t,X(t,s,x)),$

and thus $\partial_{t}X$ is continuous. Finally, by differentiating with
respect to $s$ the semigroup identity (32) in the form
$X(t,v,x)=X(t,s,X(s,v,x))$, we get

$\partial_{s}X(t,s,X(s,v,x))+D_{x}X(t,s,X(s,v,x))b(s,X(s,v,x))=0,$

for all $t,s,v\in\mathbb{R}$ and $x\in\mathbb{R}^{n}$ for which the expression
make sense. Therefore, $\partial_{s}X(t,s,y)=-D_{x}X(t,s,y)b(s,y)$ and so
$\partial_{s}X\in\mathrm{L}^{p}(\mathscr{D}_{b};\mathbb{R}^{n})$. ∎

######  Remark 6.11.

Corollary 6.10 improves [13, Theorem 4] and [8, Corollary 1.8], by dropping
assumption (3), that is ${\rm
div}_{x}b\in\mathrm{L}^{1}_{\mathrm{loc}}(\mathbb{R},\mathrm{L}^{\infty}(\mathbb{R}^{n}))$.
Notice that, if $n=1$, assumption (3) reduces to the classical Lipschitz
condition of $b$ with respect to $x$, uniformly in $t$. Corollary 6.10 also
applies to a non Lipschitz one-dimensional vector field $b$ (see Example 8.3
below).

######  Remark 6.12.

Corollary 6.10 looks almost sharp. Indeed, it was proved in [26] that no
Sobolev regularity can be expected for the flow, when assuming only that
$b\in\mathrm{L}^{1}(I;W^{1,p}(\mathbb{R}^{n};\mathbb{R}^{n}))$ for all finite
$p\in[1,+\infty)$, even when $b$ is compactly supported and divergence-free.
See also Remark 5.1.

##  7\. Applications to PDEs

In this section we apply the Sobolev regularity of flows for getting the
representation of weak solutions of the Cauchy problems both for the transport
and continuity equations.

###### Proof of Theorem F.

By [12, Theorem 1], we can infer the uniqueness of weak solutions $u\in
L^{\infty}((0,T);L^{\infty}(\mathbb{R}^{n}))$ for (2), provided that the
spatial derivative of $b$ satisfies a sub-exponential summability and
$\bar{u}\in L^{\infty}(\mathbb{R}^{n})$. Therefore we have only to show that
the function $v$ in (19) is a weak solution of (2), that is,

(92) $\int_{0}^{T}\int_{\mathbb{R}^{n}}v\,\left(\partial_{t}\varphi+{\rm
div}(b\,\varphi)\,\right)dtdx=\,-\int_{\mathbb{R}^{n}}\bar{u}\,\varphi(0,\cdot)\,\mathrm{d}x$

for each $\varphi\in C^{\infty}_{c}([0,T)\times\mathbb{R}^{n})$. We divide the
proof in two steps.

1st step. Let us assume that $\bar{u}\in C^{\infty}(\mathbb{R}^{n})\cap
L^{\infty}(\mathbb{R}^{n})$. Let
$b_{\epsilon}:I\times\mathbb{R}^{n}\to\mathbb{R}^{n}$ be the family of vector
fields defined as

$b_{\epsilon}(t,x):=\,(\tilde{b}_{\epsilon}(t,\cdot)*\rho_{\epsilon})(x)\text{
if }(t,x)\in I\times\mathbb{R}^{n}\,,$

where

$\tilde{b}_{\epsilon}(t,x):=\,(\tilde{b}_{\epsilon,1}(t,x),\dots,\tilde{b}_{\epsilon,n}(t,x))\,,$
$\tilde{b}_{\epsilon,i}(t,x):=\,\max\\{\\{\min\\{b_{i}(t,x),1/\epsilon\\},-1/\epsilon\\}\text{
if }(t,x)\in I\times\mathbb{R}^{n},\,i=1,\dots,n\,,$

and $(\rho_{\epsilon})_{\epsilon}$ denotes a family of mollifiers depending on
the space variable $x$. Then, standard properties of convolutions yield

(93) $|b_{\epsilon}(t,x)|\leq\,\frac{\sqrt{n}}{\epsilon}\text{ for each
}(t,x)\in I\times\mathbb{R}^{n}\,,$ (94) $b_{\epsilon}(t,\cdot)\in
C^{\infty}(\mathbb{R}^{n};\mathbb{R}^{n})\text{ and
}D_{x}b_{\epsilon}(t,x)=\,\int_{\mathbb{R}^{n}}D_{x}\rho_{\epsilon}(x-y)\,\tilde{b}_{\epsilon}(t,y)\,dy\quad\forall(t,x)\in
I\times\mathbb{R}^{n}\,,$ (95) $\text{
$b_{\epsilon}(t,\cdot):\mathbb{R}^{n}\to\mathbb{R}^{n}$ is Lipschitz
continuous, uniformly with respect to $t\in I$},$ (96) $b_{\epsilon}\to
b\quad\text{in $L^{1}(I;L^{1}(\Omega;\mathbb{R}^{n}))$ as $\epsilon\to 0$,
whenever $\Omega\Subset\mathbb{R}^{n}$}\,.$

Then, from (95) we get that $b_{\epsilon}$ is well-posed and the flow maps
$X_{\epsilon}$ associated to $b_{\epsilon}$ is globally defined, that is,
$X_{\epsilon}:\,\overline{I}\times\mathbb{R}^{n}\to\mathbb{R}^{n}$ and is
Lipschitz regular. Define

$v_{\epsilon}(t,x)=\bar{u}(X_{\epsilon}(0,t,x)).$

By (96), Lemma 3.5 and the subsequent remark, $X_{\epsilon}\to X$ uniformly on
compact subsets of $\overline{I}\times\mathbb{R}^{n}$. Since $\bar{u}$ is
assumed to be continuous, we obtain that $v_{\epsilon}\to v$ uniformly on
compact sets of $\overline{I}\times\mathbb{R}^{n}$. Moreover, since
$\bar{u}\in L^{\infty}(\mathbb{R}^{n})$, by the dominated convergence theorem,
we can also assume that $v_{\epsilon}\to v$ in $L^{2}$ on each compact set of
$\overline{I}\times\mathbb{R}^{n}$.

By the classical Cauchy-Lipschitz theory (see, for instance, [5, Section 2]),
it is well-known that $v_{\epsilon}$ is a classical solution to (2) with
$b_{\epsilon}$ in place of $b$ and, in particular, a weak solution, i.e.,

$\int_{0}^{T}\int_{\mathbb{R}^{n}}v_{\epsilon}\left(\partial_{t}\varphi+{\rm
div}(b_{\epsilon}\varphi)\right)\,\mathrm{d}x\,\mathrm{d}t=-\int_{\mathbb{R}^{n}}\bar{u}(x)\varphi(0,x)\,\mathrm{d}x\,,$

for each $\varphi\in C^{\infty}_{c}([0,T)\times\mathbb{R}^{n})$. Since
$\varphi$ has compact support in $[0,T)\times\mathbb{R}^{n}$, it is easy to
check that ${\rm div}(b_{\epsilon}\varphi)\to{\rm div}(b\varphi)$ in $L^{2}$.
Hence

$\lim_{\epsilon\to
0^{+}}\int_{0}^{T}\int_{\mathbb{R}^{n}}v_{\epsilon}\left(\partial_{t}\varphi+{\rm
div}(b_{\epsilon}\varphi)\right)\,\mathrm{d}x\,\mathrm{d}t=\int_{0}^{T}\int_{\mathbb{R}^{n}}v\left(\partial_{t}\varphi+{\rm
div}(b\varphi)\right)\,\mathrm{d}x\,\mathrm{d}t,$

that is, $v$ is a weak solution to (2).

2nd step. Let $\bar{u}\in\mathrm{L}^{\infty}(\mathbb{R}^{n})$ and, by
mollification in $\mathbb{R}^{n}$, let
$\bar{u}_{j}\in\mathrm{L}^{\infty}(\mathbb{R}^{n})\cap
C^{\infty}(\mathbb{R}^{n})$, with $j\in\mathbb{N}$, be a sequence of functions
that converges to $\bar{u}$ almost everywhere and such that
$\|\bar{u}_{j}\|_{\mathrm{L}^{\infty}}\leq\|\bar{u}\|_{\mathrm{L}^{\infty}}$
for every $j$.

Define

$v_{j}(t,x)=\bar{u}_{j}(X(0,t,x)).$

By Theorem B, for every $t$ the homeomorphism $X(0,t,\cdot)$ satisfies the
Lusin (N) condition, and therefore $v_{j}(t,\cdot)\to v(t,\cdot)$ almost
everywhere in $\mathbb{R}^{n}$. Since this is true for every $t$, we get that
$v_{j}\to v$ almost everywhere in $[0,T]\times\mathbb{R}^{n}$. Moreover, it is
clear that
$\|v_{j}\|_{\mathrm{L}^{\infty}}\leq\|\bar{u}\|_{\mathrm{L}^{\infty}}$ for all
$j$.

By the previous step, we have, for all $j$,

$\int_{0}^{T}\int_{\mathbb{R}^{n}}v_{j}\left(\partial_{t}\varphi+{\rm
div}(b\varphi)\right)\,\mathrm{d}x\,\mathrm{d}t=-\int_{\mathbb{R}^{n}}\bar{u}_{j}(x)\varphi(0,x)\,\mathrm{d}x\,,$

for each $\varphi\in C^{\infty}_{c}([0,T)\times\mathbb{R}^{n})$. We can now
apply the Dominated Convergence Theorem and pass to the limit $j\to\infty$ to
obtain that $v$ is a weak solution to (2). ∎

######  Corollary 7.1 (Sobolev regularity of the solutions of the transport
equation).

Let $b$ $\in C^{0}(I\times\mathbb{R}^{n};\mathbb{R}^{n})$ as in Theorem E and
let $X$ be the flow of $b$. Let $p>2n$ and $1\leq\tilde{q}\leq q<\infty$ be
such that

(97)
$\frac{\tilde{q}}{q-\tilde{q}}=\left(\frac{q}{p}+\frac{n}{p-n}\right)^{-1},\quad\text{i.e.,}\quad\tilde{q}=\frac{pq(p-n)}{q(p-n)+p^{2}}\,.$

If $\bar{u}\in\mathrm{L}^{\infty}(\mathbb{R}^{n})\cap
W^{1,q}_{\mathrm{loc}}(\mathbb{R}^{n})$, then the function $v$ in (19)
satisfies

$v\in L^{\infty}([0,T];W^{1,\tilde{q}}_{\mathrm{loc}}(\mathbb{R}^{n}))\,.$

###### Proof.

By Theorem E, the function $\Psi:=X(0,t,\cdot)$ and its inverse are in
$W^{1,p}_{\mathrm{loc}}$ with $p>2n$. By Lemma 2.7 and Corollary 6.8, $\Psi$
is of finite distortion and $K^{\Psi}_{q}\in\mathrm{L}^{r}_{\mathrm{loc}}$ for
$r=\left(\frac{q}{p}+\frac{n}{p-n}\right)^{-1}$. By Proposition 2.8, the
composition operator $T_{\Psi}$ is continuous from
$W_{\mathrm{loc}}^{1,q}(\Omega_{2})$ to
$W_{\mathrm{loc}}^{1,\tilde{q}}(\Omega_{1})$. Since
$v(t,\cdot)=T_{\Psi}(\bar{u})$, the proof is concluded. ∎

######  Remark 7.2.

Notice that the propagation of regularity, in the spirit of Corollary 7.1, may
fail below the exponential summability of $D_{x}b$, even though $\bar{u}\in
C^{\infty}_{c}(\mathbb{R}^{n})$. Indeed, in [8, Theorem 2.1], the authors
constructed a divergence-free vector field
$b:\mathbb{R}\times\mathbb{R}^{n}\to\mathbb{R}^{n}$ ($n\geq\,2$) satisfying
the subexponential summability condition (12), and a weak bounded, compactly
supported solution $u(t,x)$ of (2) such that $\bar{u}:=\,u(0,\cdot)\in
C^{\infty}_{c}(\mathbb{R}^{n})$ but
$u(t,\cdot)\notin\dot{W}^{s,p}(\mathbb{R}^{n})$ for all $t>0$, $s>0$ and
$p\geq 1$, where $\dot{W}^{s,p}(\mathbb{R}^{n})$ denotes the so-called
homogeneous Sobolev space. The example is based on the work [1]. Let us recall
that when $s=1$ and $1<\,p<\,\infty$, then $\dot{W}^{1,p}(\mathbb{R}^{n})\cap
L^{p}(\mathbb{R}^{n})$ coincides with the classical Sobolev space
${W}^{1,p}(\mathbb{R}^{n})$ (see [1, Section 2]).

Let us point out that, although $b$ satisfies the hypothesis of Theorem F,
$u(t,\cdot)\notin{W}^{1,p}(\mathbb{R}^{n})$ for each $t\in(0,\infty)$ and
$p\in(1,\infty)$. This implies that the flow of $b$ has not Sobolev regularity
$W^{1,p}$ for some $p>n$, otherwise the same proof of Corollary 7.1 could be
repeated. Our Example 8.2 shows the same phenomenon. Notice that we don’t know
whether $u(t,\cdot)\notin{W}^{1,1}(\mathbb{R}^{n})$.

######  Remark 7.3.

Corollary 7.1 shows that the Sobolev regularity of the flow $X$ implies
Sobolev regularity for solutions of the transport equation in the form (19).
We notice that the converse implication is almost true. Indeed, suppose that
for every $1\leq\tilde{q}\leq q<\infty$ satisfying (97), and for every
$\bar{u}\in\mathrm{L}^{\infty}(\mathbb{R}^{n})\cap
W^{1,q}_{\mathrm{loc}}(\mathbb{R}^{n})$, the function $v$ in (19) satisfies
$v\in L^{\infty}((-T,T);W^{1,\tilde{q}}_{\mathrm{loc}}(\mathbb{R}^{n}))$.
Then, if we take $\bar{u}=\phi x_{j}$, where $\phi\in
C^{\infty}_{c}(\mathbb{R}^{n})$ and $x_{j}$ a the $j$-th coordinate function,
we see that $x_{j}(X(0,t,\cdot))\in
W^{1,\tilde{q}}_{\mathrm{loc}}(\mathbb{R}^{n})$ for almost every $t$ and for
all $\tilde{q}<p$.

Before proving Theorem G, let us introduce some notation and recall some
existence and uniqueness results as concern as the Cauchy problem (4) for the
continuity equation. It is well-known that the non trivial issue in the well-
posedness turns out to be the uniqueness of weak solutions. Nonnegative
measure-valued solutions of the continuity equation are uniquely determined by
their initial condition if the characteristic ODE associated to the vector
field $b$ has a unique solution (see [5, Theorem 9]). A partial extension of
this result to signed measures was given in [3], under a quantitative two-
sided Osgood condition on $b$. For the proof of Theorem G we will use the
following more general result.

######  Theorem 7.4 ([14, Theorem 3]).

Let $b:\,(0,T)\times\mathbb{R}^{n}\to\mathbb{R}^{n}$ be such that the
following two conditions hold: There exists a continuous and nondecreasing
function $G:\,[0,\infty)\to[0,\infty)$ satisfying
$\int_{r}^{\infty}\frac{ds}{G(s)}ds=\infty$ for some $r>\,0$ such that

(98) $\sup_{x\in\mathbb{R}^{n}}\frac{|b(t,x)|}{G(|x|)}\in L^{\infty}(0,T)\\\
,$

and there exists a continuous and nondecreasing function
$\omega:\,[0,\infty)\to[0,\infty)$ satisfying
$\int_{0}^{r}\frac{ds}{\omega(s)}ds=\infty$ for some $r>\,0$ such that

(99) $\sup_{x,\,y\in B(0,R),\,x\neq y}\frac{|b(t,x)-b(t,y)|}{\omega(|x-y|)}\in
L^{\infty}(0,T)\,,$

for any radius $R>\,0$. Then for any $\rho_{0}\in\mathcal{M}(\mathbb{R}^{n})$
there exists a unique solution $\rho$ in
$L^{1}((0,T);\mathcal{M}(\mathbb{R}^{n}))$ for the Cauchy problem (4), that
is,

(100) $\int_{0}^{T}\int_{\mathbb{R}^{n}}\left(\partial_{t}\varphi(t,x)+\langle
b(t,x),D_{x}\varphi(t,x)\rangle\right)\,d\rho_{t}(x)dt=\,-\int_{\mathbb{R}^{n}}\varphi(0,x)\,d\rho_{0}(x)$

for each $\varphi\in C^{\infty}_{c}([0,T)\times\mathbb{R}^{n})$, with
$\rho_{t}:=\,\rho(t,\cdot)$. Moreover this solution representable as
$\rho_{t}=\,X(t,0,\cdot)_{\\#}\rho_{0}$, where $X(t,s,\cdot)$ denotes the flow
of the vector field $b$.

######  Remark 7.5.

As pointed out in [14, p. 49], by assuming $\rho\in
L^{\infty}((0,T);\mathcal{M}(\mathbb{R}^{n}))$, the uniqueness still holds if
one replaces $L^{\infty}(0,T)$ by $L^{1}(0,T)$ in both (98) and (99).

We will also use the following abstract area-type formula, which can be proved
in a standard way.

######  Lemma 7.6.

Let $\Psi:\,\mathbb{R}^{n}\to\mathbb{R}^{n}$,
$\mu:\mathcal{M}_{n}\to[0,\infty]$ and
$\bar{\rho}:\,\mathbb{R}^{n}\to[-\infty,\infty]$ be a homeomorphism, a Radon
measure on $\mathbb{R}^{n}$ and a function in $L^{1}(\mathbb{R}^{n},\mu)$,
respectively. Suppose that $\Psi_{\\#}\mathcal{\mu}$ is absolutely continuous
with respect to $\mu$ and denote

$w:=\frac{d\Psi_{\\#}\mu}{d\mu}\in L_{\rm loc}^{1}(\mathbb{R}^{n},\mu)\,.$

Then

$\Psi_{\\#}(\bar{\rho}\mu)=\,\bar{\rho}(\Psi^{-1})\,w\,\mu\,.$

In particular it also follows that

$\bar{\rho}(\Psi^{-1})\,w\in L^{1}(\mathbb{R}^{n},\mu)\,.$

###### Proof of Theorem G.

(i) In view of the use of Remark 7.5, let us begin to show that (98) and (99)
hold with $L^{1}((0,T))$ in place of $L^{\infty}((0,T))$. It is immediate
that, by choosing

$G(s):=\,s\,\log^{+}s\text{ if }s\geq\,0\,,$

(98) in the weaker form follows from (11). As for (99), observe that, from
(58), it follows that

(101)
$\int_{0}^{T}\int_{B(0,R)}\exp\left(\frac{\|D_{x}b\|}{\log^{+}\|D_{x}b\|}\right)\,dxdt<+\infty\text{
for each }R>\,0\,.$

Let $\theta:[0,\infty)\to[0,\infty)$ be the function defined as

$\theta(s):=\begin{cases}\exp\displaystyle{\left(\frac{\bar{s}}{\log\bar{s}}\right)}&\text{
if }0\leq\,s\leq\,\bar{s}\,;\\\ \exp\displaystyle{\left(\frac{s}{\log
s}\right)}&\text{ if }s\geq\,\bar{s}\,.\end{cases}$

Then, by Proposition 4.6, $\theta$ satisfies the assumption of Theorem A, for
$\bar{s}$ large enough. In particular, we can deduce, as in the proof of
Theorem A, the estimate

$\sup_{x,\,y\in B(0,R),\,x\neq
y}\frac{|b(t,x)-b(t,y)|}{\omega(|x-y|)}\leq\,\varphi(t)$

where $\omega$ and $\varphi$ are the modulus of continuity defined in (45) and
the function in (46), respectively. From (101), it follows that $\varphi\in
L^{1}(0,T)$. Thus (99) in the weaker form follows, too. Hence, the uniqueness
of weak solutions for (4) in $L^{\infty}((0,T);\mathcal{M}(\mathbb{R}^{n}))$
is granted by Theorem 7.4 and Remark 7.5 .

The existence of weak solutions for (4) can be proved as in [3, 14]. More
precisely, if $X:\,[0,T]\times[0,T]\times\mathbb{R}^{n}\to\mathbb{R}^{n}$
denotes the flow associated to $b$ and $\rho$ is defined as in (20), then one
can verbatim repeat the same proof, by showing that $\rho$ is a weak solution
of (4) in $L^{\infty}((0,T);\mathcal{M}(\mathbb{R}^{n}))$, that is, (100)
holds.

If now $\bar{\rho}\in L^{1}(\mathbb{R}^{n})$, we can apply the previous result
with initial value measure $\bar{\rho}\,\mathcal{L}^{n}$ and we get that the
unique weak solution for (4) is given by

$\rho_{t}=\,X(t,0,\cdot)_{\\#}(\bar{\rho}\,\mathcal{L}^{n})\text{ for each
}t\in[0,T]\,.$

From Theorem B, $\rho_{t}=\,X(t,0,\cdot)_{\\#}\mathcal{L}^{n}$ is absolutely
continuous with respect to $\mathcal{L}^{n}$. Therefore, by Lemma 7.6 with
$\Psi=\,X(t,0,\cdot)$ and $\mu=\,\mathcal{L}^{n}$, (21) follows.

(ii) Observe that in this case the assumptions of Theorem B are still
satisfied. Thus, by applying the previous claim (i), it follows that (21)
holds. We have only to prove that

(102) $J_{X,t}=\,J_{X(0,t,\cdot)}\text{ a.e.\leavevmode\nobreak\ in
}\mathbb{R}^{n}\text{, for each }t\in[0,T]\,$

in order to conclude the proof. From Theorem E, the flow $X(t,s,\cdot)\in
W^{1,p}_{\rm loc}(\mathbb{R}^{n};\mathbb{R}^{n})$ with $p>\,n$, for each
$t,\,s\in[0,T]$. In particular, from Corollary 6.7, we get that

$\,X(t,0,\cdot)_{\\#}(\mathcal{L}^{n})=\,J_{X(0,t,\cdot)}\mathcal{L}^{n}\,.$

Thus (102) follows. ∎

##  8\. Examples

Most of the examples we provide are in low dimension. However, notice that we
can always extend the examples to higher dimensions as follows. If $b(t,x)$ is
a vector field on $\mathbb{R}^{n}$ and $m>0$, then define the vector field
$h(t,(x,y)):=(b(t,x),0)$ on $\mathbb{R}^{n}\times\mathbb{R}^{m}$. If
$X(t,s,x)$ is the flow of $b$, then $Z(t,s,(x,y)):=(X(t,s,x),y)$ is the flow
of $h$. Notice that $Z(t,s)\in
W^{1,p}_{\mathrm{loc}}(\mathbb{R}^{n}\times\mathbb{R}^{m})$ if and only if
$X(t,s)\in W^{1,p}_{\mathrm{loc}}(\mathbb{R}^{n})$.

###  8.1. Well-posedness does not imply absolute continuity of the flow

Here we show that there is a vector field $b\in
C^{0}(\mathbb{R}\times\mathbb{R})$ that is well-posed, but whose flow on
$\mathbb{R}$ is not absolutely continuous. This shows that Theorems B and C
are not a direct consequence only of the well-posedness.

A first example is given as follows. It is well-known that a quasisymmetric
homeomorphism $\Psi:\mathbb{R}\to\mathbb{R}$ need not be absolutely continuous
(see, for instance, [24, p. 107]). Moreover, by [31, p. 250], for each
quasisymmetric homeomorphism $\Psi:\mathbb{R}\to\mathbb{R}$ there is a well
-posed continuous vector field $b:\mathbb{R}^{2}\to\mathbb{R}$ such that, if
$X(t,x)$ denote the (unique) flow $X(t,0,x)$ associated to $b$, then, for some
$t>0$, $X(t,\cdot)=\Psi$. We conclude that there exists a well-posed vector
field $b$ whose flow is not absolutely continuous at some time.

We provide a second more explicit example from [30, Section 8]. (Be aware that
what is denoted by $t$ in [30] is actually the spatial variable.) In the plane
$\mathbb{R}\times\mathbb{R}$ with coordinates $(t,x)$, consider the family of
parabolas of the form $x=a(s)t^{2}+s$, where $a:\mathbb{R}\to\mathbb{R}$ is
$0$ for $s\leq 0$, $1$ for $s\geq 1$ and coincides with the Cantor Staircase
Function on $[0,1]$. These parabolas foliate the plane and are the integral
curves of a vector field of the form $(1,b(t,x))$. One can show that $b\in
W^{1,2}_{\mathrm{loc}}(\mathbb{R})\cap C^{0}(\mathbb{R})\cap{\rm
Lip}_{\mathrm{loc}}(\mathbb{R}\setminus\\{0\\}))$ and that $(1,b)$ is a well-
posed non-autonomous vector field on $\mathbb{R}^{2}$.222The uniqueness of
integral curves is clear, because $b$ is locally Lipschitz outside the axis
$\mathbb{R}\times\\{0\\}$ and its unique integral curves are then the
parabolas $x=a(s)t^{2}+s$. However, the flow $X$ of $b$ is not absolutely
continuous. Indeed, for $t\neq 0$, the function $X(t,0,\cdot)$ maps the Cantor
set to a set of positive measure: with the notation of [30, page 141], if
$C\subset\mathbb{R}$ is the Cantor set, then
$\mathcal{L}^{1}(X(t,0,C))=\mathcal{L}^{1}(C_{t})=t^{2}/2$.

###  8.2. Sub-exponential condition does not imply high Sobolev regularity

Here we show that there is a vector field $b\in C^{0}(\mathbb{R})$ that
satisfies (12), but the flow of $b$ is no better that
$W^{1,1}_{\mathrm{loc}}(\mathbb{R})$. The same example allows us to show that
the upper bound on $\beta$ in Proposition 4.6 is necessary for the well-
posedness.

Given $\alpha\geq 1$, let $b:\mathbb{R}\to\mathbb{R}$ be the autonomous vector
field on $\mathbb{R}$ given by

$b(x)=\begin{cases}0&\text{ if }x\leq 0\text{ or }x\geq e^{-e},\\\ \beta
x\log\frac{1}{x}(\log\log\frac{1}{x}-1)^{\alpha}&\text{ if
}0<x<e^{-e}.\end{cases}$

We clearly have that $b\in C^{0}(\mathbb{R})\cap
W^{1,p}_{\mathrm{loc}}(\mathbb{R})$ for all $p\in[1,\infty)$, and that
$\mathtt{spt}(b)=[0,e^{-e}]$. Moreover,

$D_{x}b(x)=\begin{cases}0&\text{ if }x<0\text{ or }x>e^{-e},\\\
\beta\log\frac{1}{x}(\log\log\frac{1}{x})^{\alpha}(1+R(x))&\text{ if
}0<x<e^{-e},\end{cases}$

where $R$ is a remainder satisfying $\lim_{x\to 0^{+}}R(x)=0$. It is easy to
see that

(103)
$\exp\left(\frac{\left|D_{x}b\right|}{\left(\log^{+}\left|D_{x}b\right|\right)^{\alpha}}\right)\in\mathrm{L}^{1}_{\mathrm{loc}}(\mathbb{R}).$

Let us consider now the case $\alpha=1$. Then $b$ satisfies (12) and thus, by
Theorem C, $b$ is well-posed and the unique flow $X$ of $b$ satisfies
$X(t,s)\in W^{1,1}_{\mathrm{loc}}(\mathbb{R})$ for every $t,s\in\mathbb{R}$.
In fact, one can explicitly compute $X$ by separation of variables: for every
$t,s\in\mathbb{R}$,

$X(t,s,x)=\begin{cases}x&\text{ if }x\leq 0,\\\
\exp\left(-e\left(\frac{1}{e}\log\frac{1}{x}\right)^{k(t,s)}\right)&\text{ if
}0<x<e^{-e},\\\ x&\text{ if }x\geq e^{-e},\end{cases}$

where $k(t,s)=\exp(s-t)$. The spatial derivative of $X$ for $0<x<e^{-e}$ is

$D_{x}X(t,s,x)=\exp\left(-e\left(\frac{1}{e}\log\frac{1}{x}\right)^{k(t,s)}\right)k(t,s)\exp\left(-k(t,s)+1\right)\left(\log\frac{1}{x}\right)^{k(t,s)-1}\frac{1}{x}.$

If we now choose $s=0$ and $t>0$, then it is easy to see that, for every
$p>1$,

$\int_{0}^{e^{-e}}|D_{x}X(t,0,x)|^{p}dx=+\infty;$

in particular, $X(t,0,\cdot)\notin W^{1,p}_{\mathrm{loc}}(\mathbb{R})$
whenever $p>1$. Notice also that $X(t,0,\cdot)\notin
C^{0,\gamma}_{\mathrm{loc}}(\mathbb{R}_{x})$ for all $\gamma\in(0,1)$.

In the other case, when $\alpha>1$, the vector field $b$ still satisfies (103)
and (8) with $\Theta$ of the form as in (10) (but with $\beta>1$). However,
$b$ is not well-posed any more. For instance, it is easy to see that both
functions $\gamma_{1}\equiv 0$ and

$\gamma_{2}(t)=\begin{cases}0&\text{if }t\leq 0,\\\
\displaystyle{\exp\left(-\exp\left(-\left(1+\frac{1}{(\alpha-1)t}\right)^{\frac{1}{\alpha-1}}\right)\right)}&\text{if
}t>0,\end{cases}$

satisfy $\gamma^{\prime}=b(\gamma)$ and $\gamma(0)=0$.

###  8.3. Exponential summability does not imply the divergence in $BMO$ and
it is only sufficient for the Sobolev regularity

Here we show that there is a vector field $b$ satisfying the assumptions of
Theorem E such that ${\rm div}_{x}b\notin BMO(\mathbb{R})$. On the other hand
there exist $\ell>0$ and an exponent $p\in(1,\infty)$ such that, for each
$t,\,s\in[-\ell/2,\ell/2]$, $X(t,s,\cdot)\in W^{1,p}_{\rm loc}(\mathbb{R})$,
even though $\exp(\beta|D_{x}b|)\notin L^{1}_{\rm loc}(\mathbb{R})$ if
$\beta:=\,\frac{\ell p^{2}}{p-1}$.

Let $b:\mathbb{R}\to\mathbb{R}$ be the autonomous vector field on $\mathbb{R}$
given by

$b(x)=\begin{cases}0&\text{ if }x\leq 0\text{ or }x\geq e,\\\
x\log\displaystyle{\frac{e}{x}}\,&\text{ if }0<x<e.\end{cases}$

It is easy to see that $b\in C^{0}(\mathbb{R})\cap
W^{1,p}_{\mathrm{loc}}(\mathbb{R})$ for every $p\in[1,\infty)$, and that
$\mathtt{spt}(b)=\mathbb{R}\times[0,e]$. Moreover,

$D_{x}b(x)=\begin{cases}0&\text{ if }x<0\text{ or }x>e,\\\
\log\frac{1}{x}&\text{ if }0<x<e.\end{cases}$

Let $I:=\,(-\ell/2,\ell/2)$ and $\Omega:=\,(-1,3)$. Observe that, in this
case, condition (6) amounts to

(104) $\beta:=\,\frac{\ell\,p^{2}}{p-1}<1.$

It is also easy to check (see [22, Example 7.1.4]) that

$D_{x}b(\cdot)\notin BMO(\mathbb{R})\,.$

Using (104) and Theorem E, one easily proves that
$b:I\times\mathbb{R}\to\mathbb{R}$ is well-posed. In fact, we can integrate
$b$ by separation of variables and obtain

$X(t,s,x)=\begin{cases}x&\text{ if }x\leq 0,\\\
e\left(\frac{x}{e}\right)^{k(t,s)}&\text{ if }0<x<e,\\\ x&\text{ if }x\geq
e,\end{cases}$

where $k(t,s)=\exp(s-t)$. Moreover

$D_{x}X(t,s,x)=\begin{cases}1&\text{ if }x<0,\\\
k(t,s)\left(\frac{x}{e}\right)^{{k(t,s)-1}}&\text{ if }0<x<e,\\\ 1&\text{ if
}x>e.\end{cases}$

It follows that $X(t,s,\cdot)\in W^{1,q}_{\mathrm{loc}}(\mathbb{R})$ if and
only if

(105)
$(\exp(s-t)-1)q>-1,\quad\text{i.e.,}\quad\begin{cases}q<\frac{1}{1-\exp(s-t)}&\text{
if }s<t,\\\ q\geq 1&\text{ if }s>t.\end{cases}$

This example shows how the Sobolev regularity of $X(t,0,\cdot)$ can
deteriorate with time. It also shows that condition (6) is only sufficient:
indeed, if $\ell\geq 1/4$ then (104) is not satisfied by any $p$, but
$X(t,s,\cdot)\in W^{1,p}_{\rm loc}(\mathbb{R})$, for each
$t,\,s\in[-\ell/2,\ell/2]$, if $1<\,p<\,\frac{1}{1-\exp(-\ell)}$. Thus Theorem
E does not apply. Notice also that, by (105), if $t>\,0$, $X(t,0,\cdot)\notin
W^{1,q}_{\rm loc}(\mathbb{R})$ for any $q\geq\,\frac{1}{1-\exp(-t)}$.

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