# Weak solutions for the Stokes system for compressible fluids with general
pressure

Maja Szlenk

###### Abstract

We prove existence and uniqueness of global in time weak solutions for the
Stokes system for compressible fluids with a general, non-monotone pressure.
We construct the solution at the level of Lagrangian formulation and then
define the transformation to the original Eulerian coordinates. For
nonnegative and bounded initial density the solution is also nonnegative for
all $t$ and belongs to $L^{\infty}([0,\infty)\times\mathbb{T}^{d})$. A key
point of our considerations is the uniqueness of such transformation. Since
the velocity might not be Lipschitz continuous, we develop a method which
relies on the results of Crippa & De Lellis, concerning regular Lagriangian
flows. The uniqueness is obtained thanks to the application of a certain
weighted flow and detail analysis based on the properties of the $BMO$ space.

Keywords: compressible Stokes system, weak solutions, regular Lagrangian
flows, $BMO$ space, non-monotone pressure laws

##  1 Introduction

The Stokes system is an approximation of the Navier-Stokes equations for small
Reynolds number. In such cases the advective intertial forces are relatively
small and explicit dependence on time and convective term can be omitted. This
is a typical situation for highly viscous fluids or when the velocities are
very small. The flows satisfying these conditions are called Stokes or
creeping, and they occur in numerous biological and physical problems, e.g. to
describe dynamics of the blood in a process of sedimentation [31], or to model
swimming of microorganisms [11, 17, 20]. Other applications include also
engineering, where the Stokes flow is used in the process of designing
microfluids and microdevices [18, 36]. The Stokes model is also connected to
the Darcy law, which describes the flow of a fluid through porous media. Such
phenomena are observed in biological tissues [2, 10] and have many
applications in petroleum engineering [21, 27]. The other situation, where the
fluid motion is governed by the Stokes equation is a laminar flow. In this
case the fluid particles move in adjacent layers, with little mixing between
them.

In this article we consider the compressible Stokes flow on the
$d$-dimensional torus $\mathbb{T}^{d}$

$\left\\{\begin{aligned} &\varrho_{t}+\text{div}(\varrho u)=0,\\\ &-\mu\Delta
u-\nabla(\lambda+\mu)\mathrm{div}\,u+\nabla p(\varrho)=0,\end{aligned}\right.$
(1.1)

where $\varrho\colon[0,T]\times\mathbb{T}^{d}\to\mathbb{R}$ and
$u\colon[0,T]\times\mathbb{T}^{d}\to\mathbb{R}^{d}$ are the sought fluid
density and velocity field. The function $p(\varrho)$ denotes the pressure
term, and the parameters $\mu,\lambda$ represent the first and the second
viscosity.

We further assume that the flow is irrotational, namely $\mathrm{rot}u=0$. It
is equivalent to velocity having the structure of a gradient flow, namely
$u(t,x)=\nabla\phi(t,x)$ for some
$\phi\colon[0,T]\times\mathbb{T}^{d}\to\mathbb{R}$. In consequence, we obtain
the condition $\displaystyle\int_{\mathbb{T}^{d}}u(t,x)\mathrm{d}x=0$. The
second equation of (1.1) can be then rewritten in terms of effective viscous
flux, which turns out to be constant. Therefore instead of the second equation
of (1.1) we obtain

$(\lambda+2\mu)\mathrm{div}\,u=p(\varrho)-\\{p(\varrho)\\},$

where
$\displaystyle\\{f\\}=\frac{1}{\mathbb{T}^{d}}\int_{\mathbb{T}^{d}}f(x)\mathrm{d}x$.
As the qualitative properties of solutions do not depend of the values of
$\lambda$ and $\mu$, without loss of generality we take $\lambda+2\mu=1$.
Under these assumptions the system (1.1) can be transformed into

$\displaystyle\varrho_{t}$ $\displaystyle+\text{div}(\varrho u)=0,$ (1.2)
$\displaystyle\mathrm{div}\,u$ $\displaystyle=p(\varrho)-\\{p(\varrho)\\}.$

Equation (1.2) is coupled with the initial condition on the density, which is
assumed to be bounded and nonnegative, namely

$\varrho_{|_{t=0}}:=\varrho_{0}\in
L^{\infty}(\mathbb{T}^{d}),\quad\varrho_{0}\geq 0.$

In particular, we do not require the density to be strictly positive, hence
for example $\varrho_{0}$ can be equal $\mathbbm{1}_{A}$ for some
$A\subset\mathbb{T}^{d}$.

Our method allows the pressure to be in a quite general form. We require
$p(\varrho)$ to be of class $C^{1}$ and unbounded, so that in particular we
can choose a sequence $\varrho_{n}\to\infty$ such that
$p(\varrho_{n})\to\infty$. Moreover, we assume that there exist constants
$C,C_{1},C_{2},\bar{\varrho}$ such that $p$ satisfies the inequality

$-C\leq p(\varrho)\leq
CP(\varrho):=C\left(\varrho\int_{\bar{\varrho}}^{\varrho}\frac{p(s)}{s^{2}}\,\mathrm{d}s+C_{1}\varrho+C_{2}\right).$
(1.3)

Observe that as $p(\varrho)$ is bounded from below, we can choose
$\bar{\varrho},C_{1},C_{2}$ such that $P(\varrho)\geq 0$ for all $\varrho\geq
0$, hence from now on we will assume that $P(\varrho)\geq 0$. The properties
of functions satisfying (1.3) are discussed in subsection 1.2.

###  1.1 Statement of the main theorem.

The mathematical theory of weak solutions to compressible fluid equations has
been widely developing in the last twenty years, since the groundbreaking
results of Lions in 1998 [22] and Feireisl [12, 16] in 2001. They proved the
existence of weak solutions to compressible Navier-Stokes equations, provided
that the pressure term is of the form $p(\varrho)=\varrho^{\gamma}$ with
$\gamma>\frac{9}{5}$ and $\gamma>\frac{3}{2}$ respectively. In [13] and [37],
this method was also adjusted to the pressure which is non-monotone on some
finite interval. In particular, it allows to deal with the pressures expressed
via equations of state, which are of more complex form than ideal gas, the
model example being the van der Waals’ equation of state. The Lions & Feireisl
approach can be also adjusted to more complex systems, for example Navier-
Stokes-Fourier system [15, 14] and other including entropy transport [23] or
heat conductivity [3].

The important results concerning non-monotone pressure laws are due to D.
Bresch and P.-E. Jabin [4, 5]. Their method, based on the Kolmogorov
compactness criterion, allows to deal with the pressure satisfying

$C^{-1}\varrho^{\gamma}-C\leq p(\varrho)\leq C\varrho^{\gamma}+C\quad\text{
and }\quad|p^{\prime}(\varrho)|\leq\bar{p}\varrho^{\gamma-1}$ (1.4)

for $\gamma>\frac{9}{5}$. In context of our work, we refer the reader
especially to [5], where they presented their approach also on a modification
of the Stokes system. In this case they proved the existence of global weak
solutions for any $\gamma>1$ with the same regularity as in the isentropic
case, namely $\varrho\in L^{\infty}(0,T;L^{\gamma})$.

Besides the relaxed conditions on the pressure, the Bresch & Jabin compactness
criterion can be also applied to various classes of equations, where the Lions
and Feireisl method was insufficient. One of the examples are the systems with
the additional term in the continuity equation. Such models can be obtained
from the multi-fluid systems [6] or appear in the mathematical modelling of
tumor growth [39]. In this case the additional term results in lack of
compensated compactness between $\textrm{div}u$ and the pressure and therefore
the classical method fails. However, the Bresch & Jabin criterion allows to
omit this problem. The problem with the convergence of effective viscous flux
arise also in the anisotropic case. The recent result [1], concerning
anisotropic compressible Stokes system, omits this problem by controlling a
certain defect measure associated to the pressure.

The other related topic are steady flows, where the behaviour of the fluid
does not depend on time. The equations describing such flows are the classical
equations of fluid mechanics with the time derivative set as zero, for example
compressible Navier-Stokes [33, 34] or Navier-Stokes-Fourier system [25, 19].
Again, in the case of the more general pressure laws, the classical Lions-
Feireisl method cannot be directly applied to obtain a weak solution. One of
the ways to cope with that problem was introducing the variational entropy
solutions [28, 29], where the total energy balance is replaced with entropy
inequality. Another analyzed system is the steady Oseen flow [32], which is a
linearization of the Navier-Stokes system with partial consideration of the
convective forces. Note that in our case the explicit dependence on time is
removed only in the momentum equation, therefore this system can be consider
as an intermediate step between steady and unsteady flows.

As the Stokes system has a simpler structure than the Navier-Stokes, the
analysis can be carried out in the more general setting. If $\varrho_{0}$ has
higher regularity, then the solutions to the Stokes problem exist and are
unique for the general $p$ (see [22], Remark 8.14). However, in case of
$\varrho_{0}\in L^{\infty}(\mathbb{T}^{d})$, the classical method requires the
monotonicity condition on the pressure and the uniqueness was not established.
In this paper we obtain the uniqueness of solutions to the Stokes system in
case of the low regularity of the initial density and under very general
pressure laws. Our main theorem states as follows:

######  Theorem 1.1.

Let $\varrho_{0}\in L^{\infty}(\mathbb{T}^{d})$, $\varrho_{0}\geq 0$ and the
pressure satisfy (1.3). Then there exists a unique global in time weak
solution to (1.2), satisfying

$\displaystyle\varrho$ $\displaystyle\in
L^{\infty}([0,\infty)\times\mathbb{T}^{d}),$ $\displaystyle u$
$\displaystyle\in L^{\infty}([0,\infty)\times\mathbb{T}^{d}),$
$\displaystyle\nabla u$ $\displaystyle\in
L^{\infty}([0,\infty);BMO),\;\mathrm{div}\,u\in
L^{\infty}([0,\infty)\times\mathbb{T}^{d}).$

Our approach is based on the Lagrangian reformulation of the system, which
allows us to obtain a global $L^{\infty}$ estimate on the density. Having that
estimate, we can straightforwardly apply Bresch and Jabin method to obtain
compactness, and in consequence existence of solutions besides the relaxed
conditions on the pressure. However, using the results from the theory of
transport equations and classical harmonic analysis, we were able to establish
also uniqueness of solutions.

The first step is to show the existence of a unique solution in the Lagrangian
formulation. Then, it suffices to define a transformation back to the original
Eulerian coordinates. However, there appear some difficulties in the
construction, concerning low regularity of the solution of the Lagrangian
reformulation. The obtained regularity of $u$ provides that $\nabla u\in
L^{\infty}(0,T;BMO)$. In particular, $\nabla u$ may not be bounded in $x$ and
in consequence the flow $x(t,y)$ generated by $u$ may not be invertible on the
whole torus. Nevertheless, the divergence of $u$ remains bounded, therefore
$x(t,y)$ is a regular Lagrangian flow and we can treat it using the properties
from [7] and [8]. The key tool needed in our analysis is the recent result of
Crippa and De Lellis [8] concerning the stability of regular Lagrangian flows
in $L^{1}$, which allows us to pass to the limit with the smooth approximation
of the system.

The above reasoning, however, does not preserve uniqueness. We need to prove
the latter by taking two solutions $u_{0},u_{1}$, which coincide in Lagrangian
coordinates, and show that $u_{0}=u_{1}$. For this purpose we introduce a
certain weighted flow between $u_{0}$ and $u_{1}$ satisfying

$\dot{x}_{s}=su_{1}(t,x_{s})+(1-s)u_{0}(t,x_{s}),\quad x_{s}(0,y)=y.$

Having the crucial $BMO$ regularity of $\nabla u_{i}$, $i=1,2$, we are able to
use the John-Nirenberg inequality and a certain integral inequality for
functions of bounded mean oscillation, proved in [26, 24]. Then, carefully
combining the estimates we show that $x_{s}$ does not depend on $s$ and in
consequence $u_{0}=u_{1}$.

The rest of the paper is divided into sections, which contain the main steps
of the proof of Theorem 1.1. The structure of the proof is as follows:

  * •

In section 2 we present the a priori estimates and results at the level of
Lagrangian coordinates, namely the $L^{\infty}$ bounds and unique existence of
a solution in the Lagrangian reformulation.

  * •

Section 3 contains the necessary tools and definitions, together with the
proof of the uniqueness of the solutions to equation (1.2).

  * •

In section 4 we define the transformation from Lagrangian to Eulerian
coordinates using the construction from [6], and therefore prove the existence
of solutions to (1.2). Note that the estimates obtained in section 2 provide
the $BMO$ regularity of the gradient, necessary to obtain uniqueness, and the
existence of solutions can be done independently, using for example standard
Lions method and the Bresch & Jabin compactness criterion. However, we present
here an alternative approach, which is set in the framework of Lagrangian
regular flows and therefore is more consistent with the rest of this paper.

#### Notation remarks:

For the notational simplicity, we omit the subscript while integrating over
the torus, namely

$\int\mathrm{d}x\ :=\ \int_{\mathbb{T}^{d}}\mathrm{d}x.$

By $\\{\cdot\\}_{Q}$ we denote the mean integral over $Q$, while in the case
of the whole torus we again omit the subscript.
Moreover, to distinguish between the norms in $L^{\infty}(\mathbb{T}^{d})$ and
$L^{\infty}([0,T]\times\mathbb{T}^{d})$, we denote

$\|\cdot\|_{\infty}:=\|\cdot\|_{L^{\infty}(\mathbb{T}^{d})}\ \text{ and }\
\|\cdot\|_{\infty,T}:=\|\cdot\|_{L^{\infty}([0,T]\times\mathbb{T}^{d})}.$

###  1.2 Discussion on the pressure

Observe that our condition in particular includes the assumptions from [5]: if
$p$ satisfies (1.4), then $P(\varrho)\geq c\varrho^{\gamma}-c\varrho$ and for
a sufficiently large $C$ we obtain (1.3). However, our assumptions also allow
the pressure to drop to $0$ for arbitrary large $\varrho$ and we do not
require the bounds on the derivative.

Such class of possible pressures includes many physical situations. It
contains the cases which were covered before, for example van der Waals’
fluid. Moreover, the admissible pressures also can be expressed via virial
expansion, namely defined as a power series of the density:

$p(\varrho)=\sum_{k=1}^{\infty}B_{k}\varrho^{k},$

where coefficients $B_{k}$ depend on the temperature and are derived from
statistical mechanics. The virial equation of state was also considered in
[4], but our result allows wider range of pressures of this type. The other
case, where our result may be applicable, is the use in biological models,
where the pressure term is responsible for interactions between different
biological agents and therefore can take form other than resulting from
physical constitutive laws.

Let us present some further properties of $p$ satisfying (1.3):

  * •

Condition (1.3) implies that in particular
$p(\varrho)\leq\tilde{C}\varrho^{\gamma}+\tilde{C}$ for
$\varrho\geq\bar{\varrho}$ and some $\gamma>1$:

Let $p$ satisfy (1.3) and define
$\alpha(\varrho)=\int_{\bar{\varrho}}^{\varrho}\frac{p(s)}{s^{2}}\mathrm{d}s$,
$\alpha(\bar{\varrho})=0$. Then

$\alpha^{\prime}(\varrho)=\frac{p(\varrho)}{\varrho^{2}}\leq\frac{C\alpha(\varrho)}{\varrho}+\frac{C}{\varrho}\left(1+\frac{1}{\varrho}\right)$

and by the comparison criterion
$\alpha(\varrho)\leq\bar{C}\varrho^{C}-\frac{C}{C+1}\frac{1}{\varrho}-1$ for
$\varrho\geq\bar{\varrho}$, where $\bar{C}$ depends on $\bar{\varrho}$.
Therefore

$p(\varrho)\leq
C(\varrho\alpha(\varrho)+\varrho+1)\leq\bar{C}C\varrho^{C+1}+\frac{C^{2}}{C+1}\leq\tilde{C}\varrho^{\gamma}+\tilde{C}$

for $\gamma=C+1$ and a suitable $\tilde{C}$.

  * •

On any finite interval we can estimate $p$ by sufficiently large constant,
hence in particular (1.3) is fulfilled. Therefore to check if indeed $p$
satisfies (1.3) for all $\varrho\geq 0$, it suffices to analyse the behaviour
of $P$ when $\varrho\to\infty$. It is immediate to check that if $P$ satisfies

$\liminf_{\varrho\to\infty}\frac{P(\varrho)}{\varrho^{\gamma}}=\liminf_{\varrho\to\infty}\frac{1}{\varrho^{\gamma-1}}\int_{\bar{\varrho}}^{\varrho}\frac{p(s)}{s^{2}}\mathrm{d}s\geq
c>0$

for some $\gamma>1$, then $p$ satisfies (1.3), however these conditions are
not equivalent.

  * •

The most significant difference between our class of admissible pressures and
the cases considered before is that we allow the pressure to drop to $0$ even
for large $\varrho$. Moreover, the derivative of $p$ may grow arbitrarily
fast. For example, let $f\colon[0,\infty)\to[0,\infty)$ be a smooth,
increasing function such that $f^{\prime}$ is also increasing and define

$p(\varrho)=\varrho^{2}(1+\cos(f(\varrho))).$

By the alternating series test, the integral

$\int_{\bar{\varrho}}^{\infty}\cos(f(x))\mathrm{d}x=\int_{f(\bar{\varrho})}^{\infty}\frac{\cos(y)}{f^{\prime}(f^{-1}(y))}\mathrm{d}y$

is convergent in the sense of Riemann. Therefore we have

$\lim_{\varrho\to\infty}\frac{1}{\varrho}\int_{\bar{\varrho}}^{\varrho}\frac{p(s)}{s^{2}}\mathrm{d}s=\lim_{\varrho\to\infty}\frac{1}{\varrho}\int_{\bar{\varrho}}^{\varrho}1+\cos(f(s))\mathrm{d}s=1$

and $p$ satisfies (1.3). Moreover, it periodically drops to $0$ and the
derivative of $p$ depends on $f^{\prime}$, which can be arbitrarily large.

  * •

The condition $p(\varrho)\leq C\varrho^{\gamma}+C$ is not sufficient to obtain
(1.3). For example, let $\eta$ be a smooth function supported in $[-1,1]$ such
that $0\leq\eta\leq 1$ and $\eta(0)=1$. Define

$p(\varrho)=\left\\{\begin{aligned} &\varrho^{2}\eta(2^{k}(\varrho-k))\;\text{
for }k-2^{-k}\leq\varrho\leq k+2^{-k},k=1,2,...\\\ &0\;\text{
otherwise}.\end{aligned}\right.$

Then

$\displaystyle\int_{0}^{k}\frac{p(s)}{s^{2}}\mathrm{d}s$
$\displaystyle=\sum_{i=1}^{k-1}\int_{i-2^{-i}}^{i+2^{-i}}\eta(2^{i}(s-i))\mathrm{d}s+\int_{k-2^{-k}}^{k}\eta(2^{k}(s-k))\mathrm{d}s$
$\displaystyle=\sum_{i=1}^{k-1}2^{-i}\int_{-1}^{1}\eta\mathrm{d}x+2^{-k}\int_{-1}^{0}\eta\mathrm{d}x=(1-2^{-k+1})\int_{-1}^{1}\eta\mathrm{d}x+2^{-k}\int_{-1}^{0}\eta\mathrm{d}x$

and therefore for any $C$ we can choose sufficiently large $k$ such that

$p(k)=k^{2}\geq
C\left(k\int_{0}^{k}\frac{p(s)}{s^{2}}\mathrm{d}s+k+1\right)\approx\tilde{C}k.$

##  2 The $L^{\infty}$ bound on the density

###  2.1 Energy estimates

First, we obtain the a priori estimates for our solutions.

######  Lemma 2.1.

Let $(\varrho,u)$ be sufficiently smooth solutions to (1.2). Then they satisfy
the estimate

$\int_{0}^{T}\int(\mathrm{div}\,u)^{2}\;\mathrm{d}x\mathrm{d}t+\sup_{t\in[0,T]}\int
P(\varrho)\;\mathrm{d}x\leq C\int P(\varrho_{0})\;\mathrm{d}x.$

Proof: By multiplying (1.2)2 by $\mathrm{div}\,u$ and integrating on torus, we
get

$\int(\mathrm{div}\,u)^{2}\;\mathrm{d}x-\int
p(\varrho)\mathrm{div}\,u\;\mathrm{d}x=0.$

Moreover,

$\displaystyle-\int p(\varrho)\mathrm{div}\,u\;\mathrm{d}x$
$\displaystyle=\int\nabla p\cdot
u\;\mathrm{d}x=\int\frac{p^{\prime}(\varrho)}{\varrho}\nabla\varrho\cdot(\varrho
u)\;\mathrm{d}x=-\int P^{\prime}(\varrho)\cdot\text{div}(\varrho
u)\;\mathrm{d}x$ $\displaystyle=\int
P^{\prime}(\varrho)\varrho_{t}\;\mathrm{d}x=\frac{\mathrm{d}}{\mathrm{d}t}\int
P(\varrho)\;\mathrm{d}x.$

After integration over time, we get the desired estimate. ∎

The above a priori bounds also provide that $p(\varrho)\in
L^{\infty}(0,T;L^{1})$. Indeed, from assumption (1.3) we have

$\sup_{t\in[0,T]}\int p(\varrho)\mathrm{d}x\leq C\sup_{t\in[0,T]}\int
P(\varrho)\mathrm{d}x\leq C.$ (2.1)

###  2.2 The Lagrangian formulation

To prove the global $L^{\infty}$ estimate on the density, we need to rewrite
the equation (1.2) in Lagrangian coordinates, that is, we carry out a certain
change of variables, which allows us to reduce the continuity equation to a
simple ordinary differential equation.

If $u$ is a velocity field, then the trajectory of a single fluid parcel moves
along the integral curve of $\dot{x}=u(t,x)$. Therefore, if at the starting
time the particle was at a point $y$, then after the time $t$ it would be at
the point $x(t,y)$, where $x(t,y)$ is the solution to the Cauchy problem

$\displaystyle\dot{x}(t,y)$ $\displaystyle=u(t,x(t,y)),$ $\displaystyle
x(0,y)$ $\displaystyle=y.$

Moreover, differentiating this equation with respect to $y$ and using
Liouville formula, we obtain the equation for the Jacobian $J$ of
$\frac{\mathrm{d}x}{\mathrm{d}y}$:

$\displaystyle\dot{J}(t,y)$ $\displaystyle=\mathrm{div}\,u(t,x(t,y))J(t,y),$
$\displaystyle J(0,y)$ $\displaystyle=1,$

and in consequence $\displaystyle
J(t,y)=\exp\left(\int_{0}^{t}\mathrm{div}\,u(s,x(s,y))\mathrm{d}s\right)$.

We rewrite the system (1.2) in terms of $y$ coordinates instead of $x$, so at
a time $t$ the space variable is the position of a parcel starting from $y$.
Let

$\eta(t,y)=\varrho(t,x(t,y))\;\;\;\text{ and
}\;\;\;\sigma(t,y)=\mathrm{div}\,u(t,x(t,y)).$

Then the equation (1.2) has the form

$\displaystyle\partial_{t}\eta$ $\displaystyle+\eta\sigma=0,$ (2.2)
$\displaystyle\sigma$ $\displaystyle=p(\eta)-\\{p(\eta)\\}_{\sigma},$

where $\\{\cdot\\}_{\sigma}$ is the mean integral in the new coordinates given
by

$\\{f\\}_{\sigma}=\frac{1}{|\mathbb{T}^{d}|}\int
f(t,y)\exp\left(\int_{0}^{t}\sigma(s,y)\mathrm{d}s\right)\mathrm{d}y.$

Now for the system (2.2) we obtain the following result:

######  Theorem 2.2.

For $\varrho_{0}\in L^{\infty}(\mathbb{T}^{d})$ and any $T>0$ there exists a
unique solution

$(\eta,\sigma)\in L^{\infty}([0,T]\times\mathbb{T}^{d})\times
L^{\infty}([0,T]\times\mathbb{T}^{d})$

to the equation (2.2) with the initial condition $\eta(0,y)=\varrho_{0}(y)$.
Moreover, there exists a constant $r$, independent of $T$, such that

$\|\eta\|_{\infty,T}\leq r.$

The immediate consequence of Theorem 2.2 is the similar uniform bound on
$\sigma$. As $T$ is arbitrary, we hence obtain the existence of a solution on
the whole real line.

The proof of the existence of a unique solution is a standard application of
the Banach fixed point theorem and is presented in the Appendix A. Here we
show the second part of Theorem 2.2, namely the $L^{\infty}$ bound on $\eta$.

######  Proposition 2.3.

If $(\eta,\sigma)\in L^{\infty}([0,T]\times\mathbb{T}^{d})\times
L^{\infty}([0,T]\times\mathbb{T}^{d})$ is a solution to (2.2), then

$\|\eta\|_{\infty,T}\leq r,$

where the constant $r$ does not depend on $T$.

Proof: It is useful to solve the first equation of (2.2). We get the identity

$\eta(t,y)=\varrho_{0}(y)\exp\left(-\int_{0}^{t}\sigma(s,y)\mathrm{d}s\right).$
(2.3)

Note that this explicit formula for $\eta$ provides that in particular if
$\varrho_{0}$ is strictly positive, then for any $t$ the density is strictly
positive as well.

Using (2.3), we see that $\eta$ is also continuous with respect to time. For a
fixed $y$, we have

$\displaystyle|\eta(t+\varepsilon,y)-\eta(t,y)|$
$\displaystyle=\varrho_{0}(y)\exp\left(-\int_{0}^{t}\sigma(s,y)\mathrm{d}s\right)\left|\exp\left(-\int_{t}^{t+\varepsilon}\sigma(s,y)\mathrm{d}s\right)-1\right|$
$\displaystyle\leq\varrho_{0}(y)e^{t\|\sigma\|_{\infty,T}}\left|e^{\varepsilon\|\sigma\|_{\infty,T}}-1\right|.$

Hence $|\eta(t+\varepsilon,y)-\eta(t,y)|$ goes to $0$ as $\varepsilon\to 0$
and indeed $\eta(\cdot,y)$ is continuous.

The continuity of $\eta$ allows us to show global boundedness. Recall that by
virtue of energy estimates, the mean value $\\{p(\eta)\\}_{\sigma}$ is
bounded. Let

$\sup_{t\in[0,T]}\\{p(\eta)\\}_{\sigma}=M.$

From (2.1) $M$ is finite and does not depend on $T$. As $p$ is unbounded, we
can choose $r>\|\varrho_{0}\|_{L^{\infty}(\mathbb{T}^{d})}$ such that
$p(r)>M$. Then at the point $\eta=r$ we get

$\partial_{t}\eta|_{\eta=r}=-r(p(r)-\\{p(\eta)\\}_{\sigma})<-r(p(r)-M)<0.$

However, as for a fixed $y$ $\eta(\cdot,y)$ is continuous and
$\eta(0,y)=\varrho_{0}(y)<r$, if it exceeds the value $r$, it must have a
nonnegative derivative at that point, which gives a contradiction. Hence for
any $y\in\mathbb{T}^{d}$ the function $\eta(\cdot,y)$ is also bounded by $r$.
∎

##  3 Uniqueness of solutions

Using Lagrangian coordinates introduced in the previous section, we are able
to show that the solutions to (1.2) are unique.

######  Theorem 3.1.

If $(\varrho_{i},u_{i}),\;i=1,2$ are solutions to (1.2) with the regularity
from Theorem 1.1, then $(\varrho_{1},u_{1})=(\varrho_{2},u_{2})$.

First, let us show that if $(\varrho,u)$, is a solution to (1.2) satisfying

$\varrho,u,\mathrm{div}\,u\in
L^{\infty}([0,T]\times\mathbb{T}^{d}),\quad\nabla u\in L^{\infty}(0,T;BMO),$

then in the Lagrangian coordinates it satisfies (2.2). From the classical
results from transport theory of DiPerna & Lions [9] it follows that for such
$u$ there exists unique flow $x(t,y)$, such that $x\in C(0,T;L^{p})$ for any
$1<p<\infty$ and

$\dot{x}=u(t,x),\quad x(0,y)=y.$

Moreover, if $\varrho$ is a solution to the continuity equation

$\varrho_{t}+\mathrm{div}\,(\varrho u)=0,\quad\varrho(0,\cdot)=\varrho_{0},$
(3.1)

then the function $\varrho(t,x(t,y))$ is equal to

$\varrho(t,x(t,y))=\varrho_{0}(y)\exp\left(-\int_{0}^{t}\text{div}u(s,x(s,y))\mathrm{d}s\right)=\varrho_{0}(y)\exp\left(-\int_{0}^{t}\sigma(s,y)\mathrm{d}s\right)$

and therefore $\eta(t,y)=\varrho(t,x(t,y))$ satisfies the first equation of
(2.2) with $\sigma(t,y)=\mathrm{div}\,u(t,x(t,y))$. Furthermore, taking the
second equation of (2.2) at a point $x(t,y)$, we obtain

$\sigma(t,y)=p(\eta(t,y))-\frac{1}{\mathbb{T}^{d}}\int
p(\varrho(t,x))\mathrm{d}x.$

However, by Lemma 3.1. from [7], for any $f\in L^{1}(\mathbb{T}^{d})$ we have

$\int f(x)\mathrm{d}x=\int
f(x(t,y))e^{\int_{0}^{t}\mathrm{div}\,u(s,x(s,y))\mathrm{d}s}\mathrm{d}y=\int
f(x(t,y))e^{\int_{0}^{t}\sigma(s,y)\mathrm{d}s}\mathrm{d}y.$

Thus

$\int p(\varrho(t,x))\mathrm{d}x=\int
p(\varrho(t,x(t,y))e^{\int_{0}^{t}\sigma(s,y)\mathrm{d}s}\mathrm{d}y=\int
p(\eta(t,y))e^{\int_{0}^{t}\sigma(s,y)\mathrm{d}s}\mathrm{d}y$

and $\sigma$ satisfies the second equation of (2.2).

From the uniqueness of solutions in the Lagrangian formulation, we conclude
that if $(\varrho_{i},u_{i})$, $i=1,2$ are solutions to (1.2), then they are
equal at the level of Lagrangian coordinates. In particular,

$\mathrm{div}\,u_{1}(t,x_{1}(t,y))=\mathrm{div}\,u_{2}(t,x_{2}(t,y))=\sigma(t,y).$

Therefore the uniqueness of the solutions to (1.2) is equivalent to uniqueness
of solutions to the equation

$\mathrm{div}\,u(t,x(t,y))=\sigma(t,y),$ (3.2)

where $\sigma\in L^{\infty}([0,T]\times\mathbb{T}^{d})$ is given and $x(t,y)$
satisfies an ODE

$\displaystyle\dot{x}(t,y)$ $\displaystyle=u(t,x),$ (3.3) $\displaystyle
x(0,y)$ $\displaystyle=y.$

Having the unique $u$, the uniqueness of $\varrho$ follows then again from the
classical results from transport theory. The regularity of $u$ provides that
in particular $u\in L^{1}(0,T;W^{1,p})$ for some $p\geq 1$ and
$\mathrm{div}\,u\in L^{1}(0,T;L^{\infty})$. Therefore there exists a unique
$\varrho\in L^{\infty}([0,T]\times\mathbb{T}^{d})$, which is a solution to the
continuity equation (3.1).

###  3.1 Definition of the flow $x_{s}$

For $u_{1}$ and $u_{2}$ being the solutions to (3.2)-(3.3), we introduce the
weighted flow between $u_{1}$ and $u_{2}$, that is a function
$x_{s}(t,y),\;s\in[0,1]$ such that $x_{s}$ for $s=0$ is the flow generated by
$u_{2}$ and for $s=1$ the flow generated by $u_{1}$. Such $x_{s}$ is defined
by the ordinary differential equation

$\displaystyle\dot{x_{s}}(t,y)$
$\displaystyle=su_{1}(t,x_{s})+(1-s)u_{2}(t,x_{s})$ (3.4) $\displaystyle
x_{s}(0,y)$ $\displaystyle=y$

for $s\in[0,1]$. Note that the Jacobian $J_{s}$ of $x_{s}$ satisfies the same
bounds as the Jacobian $J$ of $x_{1}$ and $x_{2}$, namely

$e^{-L}\leq J_{s}(t,y)\leq e^{L}$ (3.5)

with $\displaystyle
L=\int_{0}^{T}\|s\mathrm{div}\,u_{1}(t,x_{s})+(1-s)\mathrm{div}\,u_{2}(t,x_{s})\|_{\infty}\mathrm{d}t\leq\int_{0}^{T}\|\sigma\|_{\infty}\mathrm{d}t$.

The first step to show uniqueness is to obtain certain $L^{p}$ estimates for
the derivative of $x_{s}$ with respect to $s$.

######  Lemma 3.2.

If $x_{s}$ is defined by (3.4) for $u_{1},u_{2}\in
L^{\infty}([0,T]\times\mathbb{T}^{d})$ such that $\nabla u_{i}\in
L^{\infty}(0,T;BMO)$, $i=1,2$, then for sufficiently small $t$,
$\frac{\mathrm{d}x_{s}}{\mathrm{d}s}\left(t,\cdot\right)\in L^{p}$ for some
$p>4$.

Proof: differentiating $\dot{x}_{s}$ with respect to $s$, we get

$\frac{\mathrm{d}\dot{x}_{s}}{\mathrm{d}s}=u_{1}(t,x_{s})-u_{2}(t,x_{s})+(s\nabla
u_{1}(t,x_{s})+(1-s)\nabla
u_{2}(t,x_{s}))\frac{\mathrm{d}x_{s}}{\mathrm{d}s},$ (3.6)

hence from the Gronwall’s lemma

$\left|\frac{\mathrm{d}x_{s}}{\mathrm{d}s}\right|\leq\exp\left(\int_{0}^{t}|s\nabla
u_{1}(\tau,x_{s})+(1-s)\nabla
u_{2}(\tau,x_{s})|\mathrm{d}\tau\right)\int_{0}^{t}|u_{1}(\tau,x_{s})-u_{2}(\tau,x_{s})|\mathrm{d}\tau.$

Let $\nabla v=s\nabla u_{1}+(1-s)\nabla u_{2}$. From the Hölder inequality,

$\displaystyle\int\left|\frac{\mathrm{d}x_{s}}{\mathrm{d}s}\right|^{p}\mathrm{d}y\leq$
$\displaystyle\left(\int\exp\left(2p\int_{0}^{t}|\nabla
v(\tau,x_{s})|\mathrm{d}\tau\right)\mathrm{d}y\right)^{1/2}$
$\displaystyle\times\left(\int_{0}^{t}\int|u_{1}(\tau,x_{s})-u_{2}(\tau,x_{s})|^{2p}\mathrm{d}y\mathrm{d}\tau\right)^{1/2}.$

As $u_{1},u_{2}\in L^{\infty}([0,T]\times\mathbb{T}^{d})$, the term
$\displaystyle\left(\int_{0}^{t}\int|u_{1}(\tau,x_{s})-u_{2}(\tau,x_{s})|^{2p}\mathrm{d}y\mathrm{d}\tau\right)^{1/2}$
is bounded. We will now estimate

$I=\int\exp\left(2p\int_{0}^{t}|\nabla
v(\tau,x_{s}(\tau,y))\mathrm{d}\tau\right)\mathrm{d}y.$

By Jensen’s inequality and the bounds on Jacobian $J_{s}$, we have

$\displaystyle\int e^{2p\int_{0}^{t}|\nabla
v(\tau,x_{s}(\tau,y))|\mathrm{d}\tau}\mathrm{d}y$
$\displaystyle\leq\int\frac{1}{t}\int_{0}^{t}e^{2pt|\nabla
v(\tau,x_{s}(\tau,y))|}\;\mathrm{d}\tau\mathrm{d}y$
$\displaystyle\leq\left\|\frac{1}{J_{s}}\right\|_{\infty,T}\int\frac{1}{t}\int_{0}^{t}e^{2pt|\nabla
v(\tau,x_{s}(\tau,y))|}J_{s}(\tau,y)\;\mathrm{d}\tau\mathrm{d}y$
$\displaystyle\leq e^{L}\frac{1}{t}\int_{0}^{t}\int e^{2pt|\nabla
v(\tau,x)|}\;\mathrm{d}x\mathrm{d}\tau.$

From the fact that $\sup_{t\in[0,T]}\|\nabla u_{i}\|_{BMO}\leq C$ for $i=0,1$,
we know that $\nabla v\in L^{\infty}(0,T;BMO)$ and from the Corollary C.4

$\int e^{2pt|\nabla v(\tau,x)|}\mathrm{d}x\leq C\;\;\text{ for all
}\;\;p\leq\frac{C}{2t\|\nabla v\|_{L^{\infty}(0,T;BMO)}}.$

In particular, for sufficiently small $t$ we have
$\frac{\mathrm{d}x_{s}}{\mathrm{d}s}\in L^{p}(\mathbb{T}^{d})$ for some $p>4$.
∎

######  Lemma 3.3.

Let $x_{s}$ be as in Lemma 3.2 and let $T_{1}$ be such that
$\frac{\mathrm{d}x_{s}}{\mathrm{d}s}\in L^{p}$ for $p>4$ and $t\in[0,T_{1}]$.
Then
$\left\|\frac{\mathrm{d}x_{s}}{\mathrm{d}s}\left(t,\cdot\right)\right\|_{2}^{2}$
satisfies the inequality

$\frac{\mathrm{d}}{\mathrm{d}t}\left\|\frac{\mathrm{d}x_{s}}{\mathrm{d}s}\right\|_{2}^{2}\leq
C\left\|\frac{\mathrm{d}x_{s}}{\mathrm{d}s}\right\|_{2}^{2}\left(1+\left|\ln\left\|\frac{\mathrm{d}x_{s}}{\mathrm{d}s}\right\|_{2}^{2}\right|\right)+C\|u_{1}(t,\cdot)-u_{2}(t,\cdot)\|_{2}^{2}$
(3.7)

for $t\in[0,T_{1}]$.

Proof: multiplying both sides of (3.6) by
$\frac{\mathrm{d}x_{s}}{\mathrm{d}s}$, we obtain

$\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\left|\frac{\mathrm{d}x_{s}}{\mathrm{d}s}\right|^{2}=\frac{\mathrm{d}x_{s}}{\mathrm{d}s}\nabla
v(t,x_{s})\frac{\mathrm{d}x_{s}}{\mathrm{d}s}+\big{(}u_{1}(t,x_{s})-u_{2}(t,x_{s})\big{)}\frac{\mathrm{d}x_{s}}{\mathrm{d}s}.$
(3.8)

Integrating (3.8) over torus, we get

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\int\left|\frac{\mathrm{d}x_{s}}{\mathrm{d}s}\right|^{2}\mathrm{d}y$
$\displaystyle\leq 2\left|\int\frac{\mathrm{d}x_{s}}{\mathrm{d}s}\nabla
v(t,x_{s})\frac{\mathrm{d}x_{s}}{\mathrm{d}s}\mathrm{d}y\right|+2\int|u_{1}(t,x_{s})-u_{2}(t,x_{s})|\left|\frac{\mathrm{d}x_{s}}{\mathrm{d}s}\right|\mathrm{d}y$
$\displaystyle\leq 2\left|\int\frac{\mathrm{d}x_{s}}{\mathrm{d}s}\nabla
v(t,x_{s})\frac{\mathrm{d}x_{s}}{\mathrm{d}s}\mathrm{d}y\right|+\int|u_{1}(t,x_{s})-u_{2}(t,x_{s})|^{2}\mathrm{d}y+\int\left|\frac{\mathrm{d}x_{s}}{\mathrm{d}s}\right|^{2}\mathrm{d}y.$

From the bound (3.5) on $J_{s}$, we have

$\int|u_{1}(t,x_{s})-u_{2}(t,x_{s})|^{2}\mathrm{d}y\leq
C\|u_{1}(t,\cdot)-u_{2}(t,\cdot)\|_{2}^{2}.$

By the regularity of $\frac{\mathrm{d}x_{s}}{\mathrm{d}s}$ from Lemma 3.2,
$\left|\frac{\mathrm{d}x_{s}}{\mathrm{d}s}\right|^{2}\in L^{q}$ for some
$q>2$. Therefore we can apply Corollary C.7 to estimate
$\displaystyle\left|\int\frac{\mathrm{d}x_{s}}{\mathrm{d}s}\nabla
v(t,x_{s})\frac{\mathrm{d}x_{s}}{\mathrm{d}s}\mathrm{d}y\right|$. In
consequence we obtain the inequality (3.7), where $C$ depends on $\|\nabla
v\|_{BMO}$, $\left\|\frac{\mathrm{d}x_{s}}{\mathrm{d}s}\right\|_{q}$ and
$\|J\|_{\infty}$. ∎

###  3.2 The final argument

Having the results from the previous subsection, we can now prove the
uniqueness:

######  Theorem 3.4.

The solution to system (3.2)-(3.3) with regularity from Theorem 1.1 is unique.

Proof: Let $u_{1},u_{2}$ be the solutions to (3.2)-(3.3),
$u_{i}=\nabla\phi_{i}$ and $x_{1},x_{2}$ are the flows generated by
$u_{1},u_{2}$ respectively. We will show that $\|u_{1}-u_{2}\|_{2}=0$ for all
$t\in[0,T]$. By the weak formulation of
$\mathrm{div}\,u(t,x(t,y))=\sigma(t,y)$, for any $\xi\in
C^{\infty}([0,T]\times\mathbb{T}^{d})$ we have

$\displaystyle\int(u_{1}(t,x)-u_{2}(t,x))\nabla\xi(t,x)\mathrm{d}x$
$\displaystyle=-\int(\mathrm{div}\,u_{1}(t,x)-\mathrm{div}\,u_{2}(t,x))\xi(t,x)\mathrm{d}x$
$\displaystyle=-\int\sigma(t,y)J(t,y)(\xi(t,x_{1}(t,y))-\xi(t,x_{2}(t,y)))\mathrm{d}y.$

Using definition of the flow $x_{s}$, we can rewrite the last integral as

$\int\sigma(t,y)J(t,y)\int_{0}^{1}\frac{\mathrm{d}}{\mathrm{d}s}\xi(t,x_{s}(t,y))\mathrm{d}s\mathrm{d}y=\int\sigma(t,y)J(t,y)\int_{0}^{1}\nabla\xi(t,x_{s}(t,y))\frac{\mathrm{d}x_{s}(t,y)}{\mathrm{d}s}\mathrm{d}s\mathrm{d}y.$

By the density of smooth functions in $W^{1,2}$, we can choose
$\xi=\phi_{1}-\phi_{2}$. Then $\nabla\xi=u_{1}-u_{2}$ and we obtain

$\displaystyle\int|u_{1}-u_{2}|^{2}\mathrm{d}x$
$\displaystyle=-\int\sigma(t,y)J(t,y)\int_{0}^{1}(u_{1}(t,x_{s}(t,y))-u_{2}(t,x_{s}(t,y)))\frac{\mathrm{d}x_{s}}{\mathrm{d}s}(t,y)\mathrm{d}s\mathrm{d}y$
$\displaystyle\leq\|\sigma\|_{\infty}\|J\|_{\infty}\|u_{1}-u_{2}\|_{2}\int_{0}^{1}\left\|\frac{\mathrm{d}x_{s}}{\mathrm{d}s}\right\|_{2}\mathrm{d}s.$

Hence

$\|u_{1}-u_{2}\|_{2}\leq
C\int_{0}^{1}\left\|\frac{\mathrm{d}x_{s}}{\mathrm{d}s}\right\|_{2}\mathrm{d}s.$
(3.9)

Substituting (3.9) into (3.7) and integrating over $s$, we get

$\frac{\mathrm{d}}{\mathrm{d}t}\int_{0}^{1}\left\|\frac{\mathrm{d}x_{s}}{\mathrm{d}s}\right\|_{2}^{2}\;\mathrm{d}s\leq
C\int_{0}^{1}\left\|\frac{\mathrm{d}x_{s}}{\mathrm{d}s}\right\|_{2}^{2}\left(1+\left|\ln\left\|\frac{\mathrm{d}x_{s}}{\mathrm{d}s}\right\|_{2}^{2}\right|\right)\;\mathrm{d}s+C\left(\int_{0}^{1}\left\|\frac{\mathrm{d}x_{s}}{\mathrm{d}s}\right\|_{2}\mathrm{d}s\right)^{2}$

for $t\in[0,T_{1}]$ for some $T_{1}\leq T$.
Let
$\alpha(t)=\int_{0}^{1}\left\|\frac{\mathrm{d}x_{s}}{\mathrm{d}s}\right\|_{2}^{2}\mathrm{d}s$.
As the function $x(1+|\ln x|)$ is concave for $x<1$ and $x^{2}$ is convex, we
can estimate both terms in the right hand side from Jensen’s inequality and
obtain

$\dot{\alpha}\leq C\alpha(1+|\ln\alpha|)+C\alpha.$

From Osgood’s lemma, the problem

$\dot{z}=Cz(1+|\ln z|),\quad z(0)=0$

has a unique solution $z\equiv 0$. Therefore, as
$\frac{\mathrm{d}x_{s}}{\mathrm{d}s}_{|_{t=0}}=0$, we have

$\alpha(t)\leq 0\textrm{ for all }t\in[0,T_{1}].$

In conclusion, $\frac{\mathrm{d}x_{s}}{\mathrm{d}s}\equiv 0$ for all
$t\in[0,T_{1}]$ and so is $\|u_{1}-u_{2}\|_{2}$. Having that, we can perform
analogous reasoning on the consecutive intervals $[nT_{1},(n+1)T_{1}]$ to get
$u_{1}=u_{2}$ for all $t\in[0,T]$. ∎

##  4 The existence of solutions to (3.2)-(3.3)

In this section we prove that the transformation from equation (3.2) together
with (3.3) to equation (1.2) is well defined, which will end the proof of
Theorem 1.1. Having the solution $(\eta,\sigma)$ in the Lagrangian coordinates
by Theorem 2.2, we define the transformation to Eulerian coordinates
$(\varrho,u)$. In other words, we need to find $u$ such that
$\displaystyle\int u(t,x)\mathrm{d}x=0$ and $u$ satisfies (3.2)-(3.3). By
virtue of the discussion at the beginning of section 3, such $u$ provides us
also existence of the density $\varrho$.

######  Theorem 4.1.

Let $\sigma\in L^{\infty}([0,T]\times\mathbb{T}^{d})$. There exists $u\in
L^{\infty}([0,T]\times\mathbb{T}^{d})$ such that $u$ is a solution to system
(3.2)-(3.3) and

$\mathrm{div}\,u\in L^{\infty}([0,T]\times\mathbb{T}^{d}),\;\nabla u\in
L^{\infty}(0,T;BMO).$

First, we prove the existence for smoothened $\sigma$, by putting

$\sigma_{\delta}=\sigma\ast\kappa_{\delta},$

where $\kappa_{\delta}$ is a standard mollifier.

######  Lemma 4.2.

There exists a unique $u_{\delta}\in C(0,T;W^{1,\infty})$ satisfying

$\mathrm{div}\,u_{\delta}(t,x_{\delta}(t,y))=\sigma_{\delta}(t,y),$

where $x_{\delta}(t,y)$ is given by (3.3).

Proof: We define the suitable map, and then apply the Banach fixed point
theorem. Let

$\Phi\colon C(0,T;W^{1,\infty})\to C(0,T;W^{1,\infty})$

be defined in the following way:

  1. 1.

If $\bar{u}\in C(0,T;W^{1,\infty})$, then $\bar{u}$ is Lipschitz, so there
exists a unique solution to system

$\dot{x}=\bar{u}(t,x),\quad x(0)=y.$ (4.1)

  2. 2.

We can now invert $x(t,y)$ to get $y(t,x)$ instead. After differentiation of
(4.1) with respect to $y$, we get an ODE for the matrix $H(t,y)=\frac{\partial
x}{\partial y}(t,y)$:

$\partial_{t}H=\nabla_{x}\bar{u}(t,x(t,y))H,\;H(0,y)=I.$ (4.2)

Moreover, the equation for $J(t,y)=\det H(t,y)$ yields

$\partial_{t}J(t,y)=\text{div}_{x}\bar{u}(t,x(t,y))J(t,y).$ (4.3)

Therefore we have the estimates

$\exp\left(-\int_{0}^{T}\|\nabla\bar{u}\|_{\infty}\mathrm{d}s\right)\leq\|H\|_{\infty}\leq\exp\left(\int_{0}^{T}\|\nabla\bar{u}\|_{\infty}\mathrm{d}s\right),$

and

$\exp\left(-\int_{0}^{T}\|\text{div}\bar{u}\|_{\infty}\mathrm{d}s\right)\leq\|J\|_{\infty}\leq\exp\left(\int_{0}^{T}\|\text{div}\bar{u}\|_{\infty}\mathrm{d}s\right),$

and $H$ is invertible, which allows us to treat $y$ as a function of $x$.

  3. 3.

Finally, we put $\Phi(\bar{u})=u$, where $u$ is a unique solution to the
system

$\displaystyle u(t,x)$ $\displaystyle=\nabla\phi(t,x),$ (4.4)
$\displaystyle\Delta\phi(t,x)$ $\displaystyle=\sigma_{\delta}(t,y(t,x))$

for $y(t,x)$ being the inverse flow associated with $\bar{u}$.

The function $\sigma_{\delta}(t,y(t,\cdot))$ is Lipschitz, as

$\displaystyle|\sigma_{\delta}(t,y(t,x_{1}))-\sigma_{\delta}(t,y(t,x_{2}))|$
$\displaystyle\leq\|\nabla\sigma_{\delta}\|_{\infty}|y(t,x_{1})-y(t,x_{2})|$
$\displaystyle\leq\|\nabla\sigma_{\delta}\|_{\infty}\left\|\frac{\mathrm{d}y}{\mathrm{d}x}\right\|_{\infty}|x_{1}-x_{2}|.$

Therefore $\phi(t,\cdot)\in C^{2,1}(\mathbb{T}^{d})$ and in consequence
$u(t,\cdot)\in C^{1,1}(\mathbb{T}^{d})$ and we have the estimates

$\displaystyle\sup_{0\leq t\leq T}\|u\|_{\infty}$ $\displaystyle\leq
C\|\sigma_{\delta}\|_{\infty,T},$ (4.5) $\displaystyle\sup_{0\leq t\leq
T}\|\nabla u\|_{C^{0,1}}$ $\displaystyle\leq C\sup_{0\leq t\leq
T}\|\sigma_{\delta}(t,y(t,\cdot))\|_{C^{0,1}}.$

Moreover, as $\sigma\in L^{\infty}([0,T]\times\mathbb{T}^{d})$, by the
classical results for $L^{p}$ regularity of strong solutions to Poisson
equation (see e.g. [40]), for any $p\in(1,\infty)$ we have the estimate

$\sup_{0\leq t\leq T}\|u\|_{\infty}+\sup_{0\leq t\leq
T}\|u\|_{W^{1,p}}+\sup_{0\leq t\leq T}\|\nabla u\|_{BMO}\leq
C\|\sigma\|_{\infty,T},$ (4.6)

which are uniform with respect to $\delta$.

###  4.1 Additional regularity of $u$

We will now show that the fixed points of $\Phi$ are uniformly bounded in
$L^{\infty}(0,T;W^{2,p})$. By use of an appropriate logarithmic inequality, it
implies that $\Phi(K)\subseteq K$ for some bounded and closed $K\subset
C(0,T;W^{1,\infty})$.

######  Proposition 4.3.

If $u_{\delta}$ is a fixed point of $\Phi$, then

$\sup_{0\leq t\leq T}\|\nabla u_{\delta}\|_{W^{1,p}}\leq C(p,\delta,T).$ (4.7)

Proof: After differentiation of (4.4) with respect to $x_{i}$, we get

$\Delta_{x}\frac{\partial\phi}{\partial
x_{i}}=\nabla_{y}\sigma_{\delta}\cdot\frac{\partial y}{\partial x_{i}}.$

Hence for any $1<p<\infty$ the standard elliptic estimate gives

$\|\nabla\phi\|_{W^{2,p}}\leq
C(p)\|\nabla\sigma_{\delta}\|_{\infty}\left\|\frac{\mathrm{d}y}{\mathrm{d}x}\right\|_{\infty},$

which leads to

$\sup_{0\leq t\leq T}\|\nabla u\|_{W^{1,p}}\leq
C(p)\|\nabla\sigma_{\delta}\|_{\infty,T}e^{T\|\nabla\bar{u}\|_{\infty,T}}.$
(4.8)

Now take $\bar{u}=u=u_{\delta}$ in (4.8) and apply the following inequality
for $p>d$:

$\|\nabla f\|_{\infty}\leq C(p)\left(1+\|\nabla
f\|_{BMO}\left(\ln^{+}(\|\nabla
f\|_{W^{1,p}}+\|f\|_{\infty})\right)^{1/2}\right).$ (4.9)

The proof of the above estimate one can find in [30], Corollary 2.4. By the
estimates (4.6) and Cauchy inequality, we have

$\displaystyle\|\nabla u_{\delta}\|_{L^{\infty}(0,T;W^{1,p})}$
$\displaystyle\leq
C(p)\|\nabla\sigma_{\delta}\|_{\infty,T}\exp\left(T\sup_{0\leq t\leq
T}\|\nabla u_{\delta}\|_{\infty}\right)$
$\displaystyle\leq\exp\left(CT\sup_{0\leq t\leq T}\|\nabla
u_{\delta}\|_{BMO}\left(\ln^{+}(\|\nabla
u_{\delta}\|_{L^{\infty}(0,T;W^{1,p})}+\|u_{\delta}\|_{\infty,T})\right)^{1/2}\right)$
$\displaystyle\leq\exp\left(CT^{2}+\frac{1}{2}\ln^{+}\left(\|\nabla
u_{\delta}\|_{L^{\infty}(0,T;W^{1,p})}+\|u_{\delta}\|_{\infty,T}\right)\right)$
$\displaystyle\leq C\left(\|\nabla
u_{\delta}\|_{L^{\infty}(0,T;W^{1,p})}+\|u_{\delta}\|_{\infty,T}\right)^{1/2}.$

Therefore

$\|\nabla u_{\delta}\|_{L^{\infty}(0,T;W^{1,p})}^{2}-C\|\nabla
u_{\delta}\|_{L^{\infty}(0,T;W^{1,p})}-C\leq 0$

and in consequence

$\|\nabla u_{\delta}\|_{L^{\infty}(0,T;W^{1,p})}\leq C$

for some $C$ depending on $p$, $\delta$ and $T$. ∎

The analogous reasoning also provides that $\Phi(K_{T_{1}})\subseteq
K_{T_{1}}$ for

$K_{T_{1}}=\\{u\in C(0,T_{1};W^{1,\infty}):\ \|u\|_{\infty,T_{1}},\|\nabla
u\|_{L^{\infty}(0,T_{1};BMO)}\leq C_{1}\ \text{ and }\ \|\nabla
u\|_{\infty,T_{1}}\leq C_{2}\\},$

where $C_{1}$ is the constant from estimates (4.6) and
$C_{2}=C(p)\left(1+C_{1}\left(\ln^{+}(C_{p}+C_{1})\right)^{1/2}\right)$ is the
right hand side of (4.9) for $C_{p}$ being the constant from (4.7).

###  4.2 The fixed point argument.

#### Local existence.

We show that $\Phi\colon K\to K$ is a contraction for sufficiently small
$T_{1}$. From the elliptic estimates, we have

$\displaystyle\|u_{1}(t,\cdot)-u_{2}(t,\cdot)\|_{\infty}$ $\displaystyle\leq
C\|\sigma_{\delta}(t,y_{1}(t,\cdot))-\sigma_{\delta}(t,y_{2}(t,\cdot))\|_{\infty}$
$\displaystyle\leq
C\|\nabla\sigma_{\delta}\|_{\infty,T}\|y_{1}(t,\cdot)-y_{2}(t,\cdot)\|_{\infty}.$

By Lemma B.1 from the Appendix, for small $t$ we have

$\|y_{1}(t,\cdot)-y_{2}(t,\cdot)\|_{\infty}\leq
Ct\|\bar{u}_{1}-\bar{u}_{2}\|_{\infty,T}$

and due to the uniform bound on $\|\nabla u\|_{\infty}$ for $u\in K$ the
constant $C$ does not depend on $\bar{u}_{1}$ and $\bar{u}_{2}$. Hence

$\|u_{1}-u_{2}\|_{\infty,T_{1}}\leq
C\|\nabla\sigma_{\delta}\|_{\infty,T}(T_{1}+o(T_{1}))\|\bar{u}_{1}-\bar{u}_{2}\|_{\infty,T_{1}}.$
(4.10)

Choosing $T_{1}$ such that $C\|\nabla\sigma_{\delta}\|_{\infty,T}T_{1}<1$, we
get that $\Phi$ is a contraction on $K$. Therefore there exists a unique fixed
point $u_{\delta}$ of $\Phi$ on the interval $[0,T_{1}]$.

#### Extension to $[0,T]$.

Having the uniqueness on $[0,T_{1}]$, we are able to perform the same
reasoning on $[T_{1},2T_{1}]$. Note that the estimate (4.10) on
$[T_{1},2T_{1}]$ would still depend only on the length of the interval.
Therefore again $\Phi$ is a contraction on $[T_{1},2T_{1}]$, which gives us
the unique fixed point on $[0,2T_{1}]$. Performing this procedure on the
consecutive intervals $[nT_{1},(n+1)T_{1}]$, we obtain the existence of a
unique fixed point on the whole interval $[0,T]$, which completes the proof of
Lemma 4.2. ∎

###  4.3 Letting $\delta\to 0$.

In Lemma 4.2 we obtained the unique $u_{\delta}=\nabla\phi_{\delta}$, which
satisfies the equation

$\Delta\phi_{\delta}(t,x)=\sigma_{\delta}(t,y_{\delta}(t,x))$ (4.11)

and the uniform estimates (4.6). Moreover, it turns out that
$\partial_{t}u_{\delta}$ is uniformly bounded in $L^{2}$. Indeed, from the
weak formulation of (4.11), for any $\pi\in C^{\infty}(\mathbb{T}^{d})$ we
have

$\int\Delta\phi_{\delta}(t,x)\pi(x)\;\mathrm{d}x=\int\sigma_{\delta}(t,y_{\delta}(t,x))\pi(x)\;\mathrm{d}x=\int\sigma_{\delta}(t,y)\pi(x_{\delta}(t,y))J_{\delta}(t,y)\;\mathrm{d}y,$

where
$J_{\delta}(t,y)=\exp\left(\int_{0}^{t}\sigma_{\delta}(s,y)\mathrm{d}s\right)$
is the Jacobian of $x_{\delta}$. Differentiating this equality with respect to
time, we obtain

$\displaystyle\int\Delta\partial_{t}\phi_{\delta}(t,x)\pi(x)\;\mathrm{d}x=$
$\displaystyle\int\partial_{t}\sigma_{\delta}(t,y)\pi(x_{\delta}(t,y))J_{\delta}(t,y)\;\mathrm{d}y$
(4.12)
$\displaystyle+\int\sigma_{\delta}(t,y)\partial_{t}\pi(x_{\delta}(t,y))J_{\delta}(t,y)\;\mathrm{d}y$
$\displaystyle+\int\sigma_{\delta}(t,y)\pi(x_{\delta}(t,y))\partial_{t}J_{\delta}(t,y)\;\mathrm{d}y.$

Let us now estimate the terms on the right hand side of (4.12). First, observe
that from equation (2.2) $\partial_{t}\sigma$ is bounded, and therefore

$|\partial_{t}\sigma_{\delta}|=|(\partial_{t}\sigma)\ast\kappa_{\delta}|\leq|\partial_{t}\sigma|\in
L^{\infty}([0,T]\times\mathbb{T}^{d}).$

For the second term, we have

$\partial_{t}\pi(x_{\delta}(t,y))=\nabla\pi(x_{\delta}(t,y))\dot{x_{\delta}}(t,y)=\nabla\pi(x_{\delta}(t,y))u(t,x_{\delta}(t,y))$

and then using uniform estimates on $u_{\delta}$, we get

$\int|\nabla\pi(x_{\delta}(t,y))|^{2}|u_{\delta}(t,x_{\delta}(t,y))|^{2}\;\mathrm{d}y\leq\left\|\frac{1}{J_{\delta}}\right\|_{\infty}\int|\nabla\pi(t,x)|^{2}|u_{\delta}(t,x)|^{2}\;\mathrm{d}x\leq
C,$

therefore $\partial_{t}\pi(x_{\delta}(t,y))\in L^{\infty}(0,T;L^{2})$. The
third term is bounded as well, as
$\partial_{t}J_{\delta}=\sigma_{\delta}J_{\delta}$ and both $\sigma_{\delta}$
and $J_{\delta}$ are bounded by some $C(\|\sigma\|_{\infty,T})$.
The above estimates imply that $\Delta\partial_{t}\phi_{\delta}$ is bounded in
$L^{\infty}(0,T;W^{-1,2})$ uniformly in $\delta$ and therefore

$\|\partial_{t}u_{\delta}\|_{L^{\infty}(0,T;L^{2})}=\|\nabla\partial_{t}\phi_{\delta}\|_{L^{\infty}(0,T;L^{2})}\leq
C.$ (4.13)

We now let $\delta\to 0^{+}$ and therefore obtain the solution to equation
(3.2). The estimates (4.6) give

$\|\phi_{\delta}\|_{L^{\infty}(0,T;W^{2,p})}\leq C,$

so $\phi_{\delta}\rightharpoonup^{*}\phi$ in $L^{\infty}(0,T;W^{2,p})$ up to a
subsequence. From the uniform estimates (4.6) and (4.13), Aubin-Lions Lemma
implies that in particular $u_{\delta}$ is compact in
$L^{1}([0,T]\times\mathbb{T}^{d})$. Moreover, using Theorem 2.9 from [8], we
get

$\sup_{0\leq t\leq T}\|x(t,y)-x_{\delta}(t,y)\|_{L^{1}(\mathbb{T}^{d})}\leq
C\left|\ln\left(\|u-u_{\delta}\|_{L^{1}([0,T]\times\mathbb{T}^{d})}\right)\right|^{-1},$

where $x(t,y)$ is the flow generated by this weak* limit $u=\nabla\phi$.
Therefore $x_{\delta}\to x$ in $L^{\infty}(0,T;L^{1})$.

The above convergence allows us to pass to the limit with $\delta\to 0$ in a
weak formulation of (4.4). For any $\xi\in
C^{\infty}([0,T]\times\mathbb{T}^{d})$, we have

$\displaystyle\int_{0}^{T}\int\Delta\phi_{\delta}(t,x)\xi(t,x)\mathrm{d}x\mathrm{d}t$
$\displaystyle=\int_{0}^{T}\int(\sigma\ast\kappa_{\delta})(t,y_{\delta}(t,x))\xi(t,x)\mathrm{d}x\mathrm{d}t$
$\displaystyle=\int_{0}^{T}\int(\sigma\ast\kappa_{\delta})(t,y)\xi(t,x_{\delta}(t,y))J_{\delta}(t,y)\mathrm{d}y\mathrm{d}t.$

Letting $\delta\to 0$, we get

$\int_{0}^{T}\int\Delta\phi(t,x)\xi(t,x)\mathrm{d}x\mathrm{d}t=\int_{0}^{T}\int\sigma(t,y)\xi(t,x(t,y))J(t,y)\mathrm{d}y\mathrm{d}t,$
(4.14)

where $J=\exp\left(\int_{0}^{t}\sigma(s,y)\mathrm{d}s\right)$ is the Jacobian
of the limit flow $x(t,y)$.

To deduce that indeed we have $\mathrm{div}\,u(t,x(t,y))=\sigma(t,y)$, we need
to change the variables in one of the sides in (4.14). Despite the fact that
$x(t,\cdot)$ is not a diffeomorphism, Lemma 3.1 from [7] allows us to perform
the change of variables in the left hand side of (4.14) and obtain

$\int_{0}^{T}\int\mathrm{div}\,u(t,x(t,y))\xi(t,x(t,y))J(t,y)\mathrm{d}y\mathrm{d}t.$

Therefore the equality (4.14) is transformed into

$\int_{0}^{T}\int\big{[}\mathrm{div}\,u(t,x(t,y))-\sigma(t,y)\big{]}\xi(t,x(t,y))J(t,y)\mathrm{d}y\mathrm{d}t=0.$

As $\mathrm{div}\,u\in L^{\infty}([0,T]\times\mathbb{T}^{d})$, the Jacobian
$J(t,y)$ is strictly positive. Hence, from the arbitrary choice of $\xi$, we
have $\mathrm{div}\,u(t,x(t,y))=\sigma(t,y)$ in the sense of distributions,
which ends the proof of Theorem 4.1. ∎

As by Theorem 2.2 the norms $\|\eta\|_{\infty,T}$ and $\|\sigma\|_{\infty,T}$
does not depend on $T$, so are $\|\varrho\|_{\infty,T}$,
$\|\mathrm{div}\,u\|_{\infty,T}$ and the estimates given by (4.6). Then again
from arbitrary choice of $T$ we obtain the unique existence on the whole real
line and hence the proof of Theorem 1.1 is completed.

#### Acknowledgments.

The author has been supported by National Science Centre grant
2018/29/B/ST1/00339 (Opus) and would like to thank Piotr Mucha and Tomasz
Piasecki for their invaluable help and remarks through the process of creating
the paper.

##  Appendix A Existence of solutions to (2.2)

We here prove that there exists a unique solution to (2.2), which completes
the proof of Theorem 2.2.

######  Lemma A.1.

For $\varrho_{0}\in L^{\infty}(\mathbb{T}^{d})$ and any $T>0$ there exists a
unique global in time solution

$(\eta,\sigma)\in L^{\infty}([0,T]\times\mathbb{T}^{d})\times
L^{\infty}([0,T]\times\mathbb{T}^{d})$

to the equation (2.2) with the initial condition $\eta(0,y)=\varrho_{0}(y)$.

Proof: The proof relies on double application of the Banach fixed point
theorem. First, observe that for a fixed $\eta\in
L^{\infty}([0,T]\times\mathbb{T}^{d})$, there exists a unique $\sigma$,
satisfying

$\sigma=p(\eta)-\frac{1}{|\mathbb{T}^{d}|}\int
p(\eta(t,y))\exp\left(\int_{0}^{t}\sigma(s,y)\mathrm{d}s\right)\mathrm{d}y.$
(A.1)

To see that, take $\Psi\colon L^{\infty}([0,T]\times\mathbb{T}^{d})\to
L^{\infty}([0,T]\times\mathbb{T}^{d})$ such that

$\Psi(\sigma)=p(\eta)-\frac{1}{|\mathbb{T}^{d}|}\int
p(\eta(t,y))\exp\left(\int_{0}^{t}\sigma(s,y)\mathrm{d}s\right)\mathrm{d}y.$

Then

$\displaystyle|\Psi(\sigma_{1})-\Psi(\sigma_{2})|$
$\displaystyle\leq\frac{1}{|\mathbb{T}^{d}|}\int|p(\eta(t,y))|\left|\exp\left(\int_{0}^{t}\sigma_{1}(s,y)\mathrm{d}s\right)-\exp\left(\int_{0}^{t}\sigma_{2}(s,y)\mathrm{d}s\right)\right|\mathrm{d}y.$
$\displaystyle\leq\frac{1}{|\mathbb{T}^{d}|}\int|p(\eta(t,y)|\int_{0}^{t}|\sigma_{1}(s,y)-\sigma_{2}(s,y)|\mathrm{d}s\mathrm{d}y$
$\displaystyle\leq Ct\|\sigma_{1}-\sigma_{2}\|_{\infty,\tau}.$

Hence taking $\tau$ such that $C\tau<1$, from the Banach fixed point theorem
we get the existence of a unique solution on the interval $[0,\tau]$. However,
as $\sigma$ does not blow up on an interval $[0,T]$, we are able to extend the
solution to (A.1) to the whole interval.

Now define $\Phi\colon L^{\infty}([0,T]\times\mathbb{T}^{d})\to
L^{\infty}([0,T]\times\mathbb{T}^{d})$ as

$\Phi(\eta)=\varrho_{0}(y)-\int_{0}^{t}\eta(s,y)\sigma(s,y)\mathrm{d}s,$

where $\sigma$ is given by (A.1). Similarly as before, we want to obtain the
estimate

$\|\Phi(\eta_{1})-\Phi(\eta_{2})\|_{\infty,\tau}\leq
C\tau\|\eta_{1}-\eta_{2}\|_{\infty,\tau}$

and choose $\tau$ such that $C\tau<1$ and
$\Phi\colon\\{\eta:\|\eta\|_{\infty,\tau}<r\\}\to\\{\eta:\|\eta\|_{\infty,\tau}<r\\}$
is a contraction. We have

$|\Phi(\eta_{1})-\Phi(\eta_{2})|=|\int_{0}^{t}\eta_{1}\sigma_{1}-\eta_{2}\sigma_{2}\mathrm{d}s|\leq\int_{0}^{t}|\eta_{1}-\eta_{2}\|\sigma_{1}|\mathrm{d}s+\int_{0}^{t}\eta_{2}|\sigma_{1}-\sigma_{2}|\mathrm{d}s.$

The first integral can be bounded by $Ct\|\eta_{1}-\eta_{2}\|_{\infty,\tau}$,
so to complete the desired estimate we need to estimate the difference of
$\sigma_{1}$ and $\sigma_{2}$. We have

$|\sigma_{1}-\sigma_{2}|\leq|p(\eta_{1})-p(\eta_{2})|+|\\{p(\eta_{1})\\}_{\sigma_{1}}-\\{p(\eta_{2})\\}_{\sigma_{2}}|.$

As the derivative of $p$ is bounded on an interval $[0,r]$, we can estimate
the first element by $C\|\eta_{1}-\eta_{2}\|_{\infty}$. Moreover,

$\displaystyle|\mathbb{T^{d}}|$
$\displaystyle|\\{p(\eta_{1})\\}_{\sigma_{1}}-\\{p(\eta_{2})\\}_{\sigma_{2}}|=\left|\int
p(\eta_{1})\exp\left(\int_{0}^{t}\sigma_{1}\mathrm{d}s\right)-p(\eta_{2})\exp\left(\int_{0}^{t}\sigma_{2}\mathrm{d}s\right)\mathrm{d}y\right|$
$\displaystyle\leq\int\exp\left(\int_{0}^{t}\sigma_{1}\mathrm{d}s\right)|p(\eta_{1})-p(\eta_{2})|\mathrm{d}y+\int
p(\eta_{2})\left|\exp\left(\int_{0}^{t}\sigma_{1}\mathrm{d}s\right)-\exp\left(\int_{0}^{t}\sigma_{2}\mathrm{d}s\right)\right|\mathrm{d}y$
$\displaystyle\leq
C\|\eta_{1}-\eta_{2}\|_{\infty}+C\int\left|\int_{0}^{t}(\sigma_{1}-\sigma_{2})\mathrm{d}s\right|\mathrm{d}y$
$\displaystyle\leq C\|\eta_{1}-\eta_{2}\|_{\infty}+Ct\sup_{0\leq s\leq
t}|\\{p(\eta_{1})\\}_{\sigma_{1}}-\\{p(\eta_{2})\\}_{\sigma_{2}}|.$

Hence for $\tau$ small enough, we get

$\sup_{0\leq
t\leq\tau}|\\{p(\eta_{1})\\}_{\sigma_{1}}-\\{p(\eta_{2})\\}_{\sigma_{2}}|\leq
C\sup_{0\leq t\leq\tau}\|\eta_{1}-\eta_{2}\|_{\infty},$

which gives us the desired estimate and ends the proof. ∎

##  Appendix B Estimate for the inverse flows.

In this section we prove the useful lemma for estimating the difference of the
inverse flows $y_{i}=x_{i}^{-1}(t,\cdot)$ by the difference of vector fields
generating $x_{1},x_{2}$.

######  Lemma B.1.

Consider two ordinary differential equations with the same initial value:

$\displaystyle\dot{x}_{1}$ $\displaystyle=u_{1}(t,x_{1}),$
$\displaystyle\dot{x}_{2}$ $\displaystyle=u_{2}(t,x_{2}),$ $\displaystyle
x_{1}(0)$ $\displaystyle=x_{2}(0)=y,$

where $u_{1},u_{2}\in C(0,T;W^{1,\infty})$. Let $y_{1}(t,x)$ and $y_{2}(t,x)$
be the inversions of $x_{1}$ and $x_{2}$ with respect to $y$. Then for
sufficiently small $t$

$\|y_{1}(t,\cdot)-y_{2}(t,\cdot)\|_{\infty}\leq Ct\|u_{1}-u_{2}\|_{\infty,T},$

where $C=C\left(||\nabla u_{1}\|_{\infty,T}\right)$.

Proof: Let $M=\|\nabla u_{1}\|_{\infty,T}$. We have

$\displaystyle|u_{1}(t,x_{1})-u_{2}(t,x_{2})|$
$\displaystyle\leq|u_{1}(t,x_{1})-u_{1}(t,x_{2})|+|u_{1}(t,x_{2})-u_{2}(t,x_{2})|$
$\displaystyle\leq\|\nabla
u_{1}\|_{\infty}|x_{1}-x_{2}|+\|u_{1}-u_{2}\|_{\infty}.$

Substituting it into the difference of $x_{1}$ and $x_{2}$, we get

$\displaystyle|x_{1}-x_{2}|$
$\displaystyle\leq\int_{0}^{t}|u_{1}(s,x_{1})-u_{2}(s,x_{2})|\mathrm{d}s$
$\displaystyle\leq\int_{0}^{t}\|\nabla
u_{1}\|_{\infty}|x_{1}-x_{2}|\;\mathrm{d}s+\int_{0}^{t}\|u_{1}-u_{2}\|_{\infty}\mathrm{d}s.$

Hence from the Gronwall’s lemma,

$\displaystyle\|x_{1}-x_{2}\|_{\infty}$
$\displaystyle\leq\int_{0}^{t}\|u_{1}-u_{2}\|_{\infty}\mathrm{d}s+\int_{0}^{t}\|\nabla
u_{1}\|_{\infty}\exp\left(\int_{s}^{t}\|\nabla
u_{1}\|_{\infty}\mathrm{d}\tau\right)\int_{0}^{s}\|u_{1}-u_{2}\|_{\infty}\mathrm{d}\tau\mathrm{d}s$
$\displaystyle\leq\|u_{1}-u_{2}\|_{\infty,T}\int_{0}^{t}1+Me^{M(t-s)}s\;\mathrm{d}s$
$\displaystyle\leq Ct\|u_{1}-u_{2}\|_{\infty,T}$

Analogously, we obtain the estimate

$\displaystyle|x_{1}(t,y_{1})-x_{2}(t,y_{2})|$
$\displaystyle\leq|x_{1}(t,y_{1})-x_{1}(t,y_{2})|+|x_{1}(t,y_{2})-x_{2}(t,y_{2})|$
$\displaystyle\leq\left\|\frac{\partial x_{1}}{\partial
y}\right\|_{\infty}|y_{1}-y_{2}|+\|x_{1}-x_{2}\|_{\infty}$
$\displaystyle\leq\exp\left(\int_{0}^{t}\|\nabla
u_{1}\|_{\infty}\mathrm{d}s\right)|y_{1}-y_{2}|+Ct\|u_{1}-u_{2}\|_{\infty,T}.$

Combining the above estimates, we get

$\displaystyle|y_{1}(t,x)-y_{2}(t,x)|$
$\displaystyle\leq\int_{0}^{t}|u_{1}(s,x_{1}(s,y_{1}(t,x)))-u_{2}(s,x_{2}(s,y_{2}(t,x)))|\mathrm{d}s$
$\displaystyle\leq\int_{0}^{t}\|\nabla
u_{1}\|_{\infty}|x_{1}(s,y_{1}(t,x))-x_{2}(s,y_{2}(t,x))|+\|u_{1}-u_{2}\|_{\infty}\mathrm{d}s$
$\displaystyle\leq\int_{0}^{t}\|\nabla
u_{1}\|_{\infty}\exp\left(\int_{0}^{s}\|\nabla
u_{1}\|_{\infty}\mathrm{d}\tau\right)|y_{1}(t,x)-y_{2}(t,x)|\mathrm{d}s$
$\displaystyle\ \ \ +\int_{0}^{t}\|\nabla
u_{1}\|_{\infty}Cs\|u_{1}-u_{2}\|_{\infty,T}\mathrm{d}s+\int_{0}^{t}\|u_{1}-u_{2}\|_{\infty}\mathrm{d}s$
$\displaystyle\leq\left(e^{Mt}-1\right)|y_{1}(t,x)-y_{2}(t,x)|+(t+Ct^{2})\|u_{1}-u_{2}\|_{L^{\infty}([0,T]\times\mathbb{T}^{d})}.$

For small $t$ we have $e^{Mt}<2$ and therefore

$|y_{1}(t,x)-y_{2}(t,x)|\leq\frac{t+Ct^{2}}{2-e^{Mt}}\|u_{1}-u_{2}\|_{\infty,T}\leq
Ct\|u_{1}-u_{2}\|_{\infty,T}$

##  Appendix C Properties of the $BMO$ space

We present here the useful properties of the $BMO$ functions, which can be
found for example in [35] and [38].

######  Definition C.1.

A function $f\in L^{1}(\Omega)$ belongs to space of bounded mean oscillation
$BMO(\Omega)$ iff

$\|f\|_{BMO}=\sup_{Q\subset\Omega}\frac{1}{|Q|}\int_{Q}|f-\\{f\\}_{Q}|\mathrm{d}x<\infty,$

where the supremum is taken over all cubes in $\Omega$.

Note that $\|\cdot\|_{BMO}$ is not a norm, as $\|f\|_{BMO}=0$ for $f$
constant. However, we can equip the space $BMO(\Omega)$ with the norm

$\|\cdot\|_{L^{1}}+\|\cdot\|_{BMO}$

and then it becomes the Banach space.

It is straightforward from the definition that the standard mollification is
bounded in $BMO$:

######  Proposition C.2.

For $f\in BMO$ and $\kappa_{\delta}$ the standard mollifier we have

$\|f\ast\kappa_{\delta}\|_{BMO}\leq\|f\|_{BMO}.$

One of the important tools concerning the $BMO$ spaces is the John-Nirenberg
inequality:

######  Lemma C.3 (John-Nirenberg).

There exist constants $c_{1},c_{2}>0$ such that for any cube $Q\subset\Omega$
and $f\in BMO(\Omega)$

$|\\{x\in Q:|f-\\{f\\}_{Q}|>\lambda\\}|\leq
c_{1}\exp\left(-\frac{c_{2}\lambda}{\|f\|_{BMO}}\right)|Q|.$

The useful applications of the John-Nirenberg inequality are the following:

######  Corollary C.4.

Let $f\in BMO(\Omega)$. Then

  1. 1.

$f\in L_{loc}^{p}(\Omega)$ for any $1\leq p<\infty$.

  2. 2.

$\displaystyle\sup_{Q\subset\Omega}\int_{Q}\exp\left(\frac{|f-\\{f\\}_{Q}|}{\|f\|_{BMO}}\right)\mathrm{d}x<\infty.$

###  C.1 The logarithmic inequality

We recall here the inequality from [26]:

######  Lemma C.5.

Let $f\in BMO(\mathbb{R}^{d})$ with compact support and $g\in
L^{1}(\mathbb{R}^{d})\cap L^{\infty}(\mathbb{R}^{d})$. Then

$\left|\int_{\mathbb{R}^{d}}fg\mathrm{d}x\right|\leq
C\|f\|_{BMO}\|g\|_{L^{1}}\big{(}|\ln\|g\|_{L^{1}}|+\ln(e+\|g\|_{L^{\infty}})\big{)}.$

It turns out that after slight modifications, the similar inequality holds for
$g\in L^{q}(\mathbb{T}^{d})$ for sufficiently large $q$.

######  Lemma C.6.

Let $f\in BMO(\mathbb{T}^{d})$ and $g\in L^{q}(\mathbb{T}^{d})$ for some
$q>2$. Then

$\displaystyle\left|\int_{\mathbb{T}^{d}}fg\mathrm{d}x\right|\leq$
$\displaystyle C\|f\|_{BMO}\|g\|_{L^{1}}$ (C.1)
$\displaystyle\times\big{(}|\ln\|g\|_{L^{1}}|+\ln(e+\|g\|_{L^{q}})+(1+|\ln\|g\|_{L^{1}}|)\|g\|_{L^{q}}^{\frac{q-2}{2}}\big{)}.$

Proof: Assume $\int g\mathrm{d}x=0$. Then $g\in\mathcal{H}^{1}$ and from
duality of $\mathcal{H}^{1}$ and $BMO$ we have

$\left|\int fg\;\mathrm{d}x\right|\leq\|f\|_{BMO}\|g\|_{\mathcal{H}^{1}}.$

By the characterization of $\mathcal{H}^{1}$ (see e.g. paragraph III.4 in
[35]) we can write $\|g\|_{\mathcal{H}^{1}}$ as

$\|g\|_{\mathcal{H}^{1}}=\|g\|_{L^{1}}+\sum_{k=1}^{d}\|R_{k}g\|_{L^{1}},$

where $R_{k}$ is the Riesz transform given as
$\mathcal{F}(R_{k}g)=-i\frac{\xi_{k}}{|\xi|}\mathcal{F}(g)$. As the Riesz
transform is an operator of weak-type $(1,1)$ and strong-type $(p,p)$, we can
apply Proposition V.3.2. from [38] and obtain

$\|R_{k}g\|_{L^{1}}\leq C+C\int|g(x)|\ln^{+}|g(x)|\;\mathrm{d}x.$ (C.2)

By scaling, we can rewrite (C.2) as

$\|R_{k}g\|_{L^{1}}\leq\lambda+C\int|g(x)|\ln^{+}(|g(x)|/\lambda)\mathrm{d}x$

for any $\lambda>0$. For $|g|\geq\lambda$, we have
$\ln^{+}(|g|/\lambda)=\ln|g|-\ln\lambda$. Then

$|\ln(|g|_{|_{|g|\geq\lambda}})|=\left|\ln\left(\frac{|g|_{|_{|g|\geq\lambda}}}{1+\|g\|_{L^{q}}}\right)+\ln(1+\|g\|_{L^{q}})\right|\leq\ln(1+\|g\|_{L^{q}})+\left|\ln\left(\frac{|g|_{|_{|g|\geq\lambda}}}{1+\|g\|_{L^{q}}}\right)\right|.$

Now assume $\lambda<1+\|g\|_{L^{q}}$ and take $x$ such that
$\lambda\leq|g(x)|\leq\frac{(1+\|g\|_{L^{q}})^{2}}{\lambda}$. Then

$\left|\ln\left(\frac{|g(x)|}{1+\|g\|_{L^{q}}}\right)\right|\leq\left|\ln\left(\frac{\lambda}{1+\|g\|_{L^{q}}}\right)\right|$

and in consequence

$|\ln|g(x)|\leq 2\ln(1+\|g\|_{L^{q}})+|\ln\lambda|.$

Choose $\lambda=\|g\|_{L^{1}}$. Then

$\displaystyle\int_{\big{\\{}|g|\leq\frac{(1+\|g\|_{L^{q}})^{2}}{\lambda}\big{\\}}}|g|\ln^{+}(|g|/\lambda)\mathrm{d}x$
$\displaystyle\leq\int_{\big{\\{}|g|\leq\frac{(1+\|g\|_{L^{q}})^{2}}{\lambda}\big{\\}}}|g|(2\ln(1+\|g\|_{L^{q}})+|\ln\|g\|_{L^{1}}|)\mathrm{d}x$
$\displaystyle\leq\|g\|_{L^{1}}(2\ln(1+\|g\|_{L^{q}})+|\ln\|g\|_{L^{1}}|)$

What is left is the case $|g(x)|>\frac{(1+\|g\|_{L^{q}})^{2}}{\lambda}$. From
the Chebyshev inequality, we have

$\left|\left\\{x:|g(x)|>\frac{(1+\|g\|_{L^{q}})^{2}}{\|g\|_{L^{1}}}\right\\}\right|\leq\left(\frac{\|g\|_{L^{1}}}{1+\|g\|_{L^{q}}}\right)^{2}.$

Therefore from the Hölder inequality

$\int_{\big{\\{}|g|>\frac{(1+\|g\|_{L^{q}})^{2}}{\lambda}\big{\\}}}|g|\ln^{+}(|g|/\lambda)\mathrm{d}x\leq\frac{\|g\|_{L^{1}}}{1+\|g\|_{L^{q}}}\left(\int|g|^{2}\ln(|g|/\lambda)^{2}\mathrm{d}x\right)^{1/2}.$

Using the fact that both $\int|g|^{2}\ln|g|\mathrm{d}x$ and
$\int|g|^{2}\ln^{2}|g|\mathrm{d}x$ are bounded by

$C\|g\|_{L^{q}}^{4-q}+\|g\|_{L^{q}}^{q}$

for any $q>2$, we obtain

$\displaystyle\int|g|^{2}\ln(|g|/\lambda)^{2}\mathrm{d}x$
$\displaystyle=\int|g|^{2}\ln^{2}|g|\mathrm{d}x-2\ln\lambda\int|g|^{2}\ln|g|\mathrm{d}x+|\ln\lambda|^{2}\int|g|^{2}\mathrm{d}x$
$\displaystyle\leq(C\|g\|_{L^{q}}^{4-q}+\|g\|_{L^{q}}^{q})(1+2|\ln\lambda|+|\ln\lambda|^{2})$
$\displaystyle=(C\|g\|_{L^{q}}^{4-q}+\|g\|_{L^{q}}^{q})(1+|\ln\lambda|)^{2}.$

As $\frac{(Cs^{4-q}+s^{q})^{1/2}}{1+s}\sim s^{\frac{q-2}{2}}$, after combining
the estimates we get

$\int_{\big{\\{}|g|>\frac{(1+\|g\|_{L^{q}})^{2}}{\lambda}\big{\\}}}|g|\ln^{+}(|g|/\lambda)\mathrm{d}x\leq
C\|g\|_{L^{1}}(1+|\ln\|g\|_{L^{1}})\|g\|_{L^{q}}^{\frac{q-2}{2}}.$

Putting all terms together,

$\|R_{k}g\|_{L^{1}}\leq
C\|g\|_{L^{1}}\left(1+\ln(1+\|g\|_{L^{q}})+|\ln\|g\|_{L^{1}}|+(1+|\ln\|g\|_{L^{1}}|)\|g\|_{L^{q}}^{\frac{q-2}{2}}\right)$

and therefore we obtain inequality (C.1).
If $\int g\;\mathrm{d}x\neq 0$, then we can apply this inequality to
$\bar{g}(x)=g(x)-\frac{1}{|\mathbb{T}^{d}|}\int g\mathrm{d}x$ and use the fact
that the $L^{p}$ norms of $\bar{g}$ are bounded by norms of $g$ up to a
constant. ∎

The same result holds if we replace $f$ by a composition of $f$ and the flow
$x(t,y)$:

######  Corollary C.7.

If $f$ and $g$ satisfy assumptions of Lemma C.6 and $x(t,y)$ is the regular
Lagrangian flow of some $u(t,x)$ with bounded divergence, then

$\displaystyle\left|\int f(x(t,y))g(y)\mathrm{d}y\right|\leq$ $\displaystyle
C\|f\|_{BMO}\|g\|_{L^{1}}$
$\displaystyle\times\big{(}|\ln\|g\|_{L^{1}}|+\ln(e+\|g\|_{L^{q}})+(1+|\ln\|g\|_{L^{1}}|)\|g\|_{L^{q}}^{\frac{q-2}{2}}\big{)}.$

Proof: We will first approximate $u$ with smooth vector fields, then make the
change of variables and apply Lemma C.6, and at the end show the convergence
to the non-smooth case.
Let $J(t,y)$ be the Jacobian of $x$. By the properties of Lagrangian flows we
have

$e^{-L}\leq J(t,y)\leq e^{L},\text{ where
}L=\int_{0}^{T}\|\textrm{div}u\|_{\infty}\mathrm{d}t.$

Now let us approximate $u$ by convolution, defining $u_{\varepsilon}$ as a
convolution with standard convolution kernels in time and space. Then if
$x_{\varepsilon}$ is a flow of $u_{\varepsilon}$, then
$x_{\varepsilon}(t,\cdot)$ is the diffeomorphism and the Jacobian
$J_{\varepsilon}$ of $x_{\varepsilon}$ still satisfies the bounds

$e^{-L}\leq J_{\varepsilon}(t,y)\leq e^{L}.$

By the change of variables, we have

$\int f(x_{\varepsilon}(t,y))g(y)\mathrm{d}y=\int
f(x)\frac{g(y_{\varepsilon}(t,x))}{J_{\varepsilon}(t,y_{\varepsilon}(t,x))}\mathrm{d}x,$

where $y_{\varepsilon}(t,\cdot)=x_{\varepsilon}(t,\cdot)^{-1}$. Applying Lemma
C.6, we obtain inequality (C.1) but with $L^{1}$ and $L^{q}$ norms of
$\frac{g(y_{\varepsilon}(t,\cdot))}{J_{\varepsilon}(t,y_{\varepsilon}(t,\cdot))}$
instead of $g$. However, changing the variables again and using the bounds on
$J_{\varepsilon}$, we obtain for any $p\geq 1$

$\int\frac{|g(y_{\varepsilon}(t,x)|^{p}}{J_{\varepsilon}(t,y_{\varepsilon}(t,x))^{p}}\mathrm{d}x=\int|g(y)|^{p}J_{\varepsilon}(t,y)^{1-p}\mathrm{d}y\leq
e^{(p-1)L}\int|g(y)|^{p}\mathrm{d}y$

and we are done.
Now we will show that indeed

$\int f(x_{\varepsilon}(t,y))g(y)\mathrm{d}y\to\int
f(x(t,y))g(y)\mathrm{d}y\;\text{ with }\;\varepsilon\to 0.$ (C.3)

By the stability of the flow, we have the pointwise convergence
$x_{\varepsilon}(t,y)\to x(t,y)$ up to a subsequence. If $f\in
C^{\infty}(\mathbb{T}^{d})$, then (C.3) holds by the dominated convergence
theorem. Let us approximate $f$ by $f_{\delta}=f\ast\kappa_{\delta}$, where
$\kappa_{\delta}$ is again the standard mollifier. As $f\in L^{p}$ for
$p=q^{\prime}$, we have

$\left|\int(f_{\delta}(x(t,y))-f(x(t,y))g(y)\mathrm{d}y\right|\leq\|f_{\delta}(x(t,\cdot))-f(x(t,\cdot))\|_{p}\|g\|_{q}$

and by the bounds on $J(t,y)$,

$\int|f_{\delta}(x(t,y))-f(x(t,y))|^{p}\;\mathrm{d}x\leq
e^{pL}\int|f_{\delta}(x)-f(x)|^{p}\;\mathrm{d}x\to 0,$

therefore we have the desired convergence. Moreover, by the Proposition C.2
the norms $\|f_{\delta}\|_{BMO}$ in the right hand side of (C.1) are bounded
by $\|f\|_{BMO}$, which ends the proof of the Corollary. ∎

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