# On the Well-Posedness of a Fractional Stokes-Transport System

Dimitri Cobb
Universität Bonn
Mathematisches Institut
Endenicher Allee 60, 53115 Bonn, Germany
<EMAIL_ADDRESS>
#### Abstract

The purpose of this paper is to study the existence, uniqueness and lifespan
of solutions for a fractional Stokes-Transport system. This problem should be
understood as a model for sedimentation in a fluid where the viscosity law is
given by a fractional Lapalce operator $(-\Delta)^{\alpha/2}$, with $\alpha=2$
corresponding to the case of a normal viscous fluid, and $\alpha=0$ reducing
the problem to the Inviscid Incompressible Porous Media equation. For each
value of $\alpha\in[0,d]$, we prove various results related to well-posedness
in critical function spaces, such as the existence of global weak solutions
(for $\alpha>0$), local existence and uniqueness (for $\alpha\geq 0$), global
existence and uniqueness (for $\alpha\geq 1$), as well as study the lifespan
of local solutions (for $0\leq\alpha<1$). In particular, we show that gravity
stratification leads to a directional blow-up criterion for local solutions
(for $\alpha\in[0,1[$) and find a lower bound for the lifespan of solutions
which depends on the value of the dissipation parameter $\alpha\in[0,1[$.

##### 2020 Mathematics Subject Classification:

35Q35 (primary); 76D03, 35Q49, 35S10, 76D50 (secondary).

##### Keywords:

Stokes-Transport System, Fractional Laplacian, Incompressible Fluids, Critical
Regularity, Lifespan of Solutions.

##  1 Introduction

In this article, we study the existence, uniqueness and lifespan of solutions
for the initial value problem related to the following active scalar equation:

(1) $\begin{cases}\big{(}\partial_{t}+u\cdot\nabla\big{)}\rho=0\\\
(-\Delta)^{\alpha/2}u+\nabla\pi=\rho g\\\
\operatorname*{\rm{div}}(u)=0.\end{cases}$

Here, the equations are set in the whole space $\mathbb{R}^{d}$ with $d\geq
2$, although our results could certainly be extended to the torus
$\mathbb{T}^{d}$ with minimal adaptations. The unknowns are the scalar
function $\rho(t,x)\in\mathbb{R}$ which represents a particle density, the
vector field $u(t,x)\in\mathbb{R}^{d}$ which is a fluid velocity, and a scalar
pressure function $\pi(t,x)\in\mathbb{R}$. In the second equation,
$g=e_{d}=(0,...,0,1)\in\mathbb{R}^{d}$ is a constant vector and $\alpha\geq 0$
a real number. The fractional Laplace operator $(-\Delta)^{\alpha/2}$ is is
defined by its Fourier transform:

$\forall
f\in\mathcal{S},\qquad\mathcal{F}[(-\Delta)^{\alpha/2}f](\xi)=|\xi|^{s}\widehat{f}(\xi).$

The physical background for the equations of (1) is the study of the
sedimentation of a cloud of particles under the effect of gravity and lying in
a fluid with a generalized –fractionary– viscosity law. As a matter of fact,
the special cases $\alpha=2$ and $\alpha=0$, where the second equation then
reduces to the Stokes equation and Darcy’s law,

$-\Delta u+\nabla\pi=\rho g\qquad\text{and}\qquad u+\nabla\pi=\rho g,$

correspond, respectively, to the Stokes-transport equation derived by Höfer
[25] as the homogenization limit of particle system in a Stokes fluid, and the
Inviscid Incompressible Porous Medium equation, studied in e.g. [20].

The main motive for considering a fractional viscosity law is to better
understand the mechanisms that provide existence and uniqueness of solutions
in the special cases: when $\alpha=2$, the initial value problem is known to
have global solutions (this will be further discussed below), whereas even the
existence of global weak solutions is challenging in the extreme, in the case
$\alpha=0$. Although the study of well-posedness for problem (1) is
interesting in itself and already presents a number of issues, it is also
meant to contribute to a better understanding the “classical” problems
$\alpha\in\\{0,2\\}$. In particular, investigating the questions related to
well-posedness for all intermediate values $\alpha\in[0,2]$ will allow us to
better isolate the difficulties when dealing with well-posedness when
$\alpha=0$.

To be precise, we will investigate the existence of global weak solutions (for
$\alpha>0$), local well-posedness (for $\alpha\geq 0$), global well-posedness
(for $\alpha\geq 1$), as well as properties of the lifespan of the local
solutions (for $0\leq\alpha<1$) such as lower bounds and continuation
criteria.

###  1.1 Broader Context: Active Scalar Equations

The fractional Stokes-Transport equation is a part of a larger family of PDE
problems which have been studied extensively in the past years, namely active
scalar equations. These take the following form: the unknown is a scalar
quantity $\rho(t,x)$ that is advected by a vector field $u(t,x)$, thus solving
the transport equation

$\partial_{t}\rho+u\cdot\nabla\rho=0,$

and the vector field is given as a linear, and possibly non-local, function of
$\rho$. In our case, it is fairly easy to see how $u$ can be expressed as a
function of $\rho$: define the Leray projection operator $\mathbb{P}={\rm
Id}+\nabla(-\Delta)^{-1}\operatorname*{\rm{div}}$, which is the
$L^{2}$-orthogonal projection on the space of divergence-free vector fields.
Applying this operator to the second equation in (1) gives

(2) $u=(-\Delta)^{-\alpha/2}\mathbb{P}(\rho g).$

Many other equations fall into this general form. The best known example is
probably the 2D Euler equations in their vorticity form, where the velocity
field is given by

$u=-\nabla^{\perp}(-\Delta)^{-1}\rho$

and $\rho$ represents the (scalar) vorticity of the fluid. Similarly, one may
think of the Surface Quasi-Geostrophic (SQG) equation, where $u$ is given by
$u=-\nabla^{\perp}(-\Delta)^{-1/2}\rho$. Both the 2D Euler equation and the
SQG problem are instances of the so-called generalized SQG equations (gSGQ in
the literature), where the velocity law is given by

$u=\nabla^{\perp}(-\Delta)^{-\beta/2}m(D)\rho,$

and $\beta\in\mathbb{R}$ and $m(D)$ is a positive Fourier multiplication
operator, such as $m(D)=\log({\rm Id}-\Delta)^{\mu}$ with $\mu\in\mathbb{R}$
(see [8] for an example of such a velocity law, or [9] in the case of a
generalized Boussinesq equation). Many such variants exists, but the unifying
feature of gSQG equations is the presence of the orthogonal gradient
$\nabla^{\perp}$ in the velocity law, so that $u$ is the image of $\rho$ by a
skew-symmetric operator $\nabla^{\perp}(-\Delta)^{-\beta/2}m(D)$. This endows
the non-linearity $u\cdot\nabla\rho$ with a commutator structure and allows
for solutions to exist even when the velocity should be more singular than the
unknown $0\leq\beta<1$ (see [29] or [8] for explanations). Because the Leray
projection operator $\mathbb{P}$ is symmetric, this is not true in the case of
(2), even in 2D when $\mathbb{P}=\nabla^{\perp}(-\Delta)^{-1}{\rm curl}\,$.

Another type of velocity law consists in abandoning the world of
incompressible fluids and using velocity fields that are not divergence free.
For example, one may take $u=-\nabla(-\Delta)^{-\beta/2}\rho$, as in [7],
where the problem has its own particularities. Specially, in the case
$\beta=0$, it reduces to the Hamilton-Jacobi equation

$\partial_{t}\rho+|\nabla\rho|^{2}=0.$

Other examples would include Burgers equation $u=\rho$ (when $d=1$),
magnetogeostrophic equations, or agregation equations (see [2] or [3] for
examples).

Many of these equations have their own physical relevance, but a number of
them are also studied as toy-models for more complex equations: thus the case
$u=H\rho$ in dimention $d=1$, where $H$ is the Hilbert transform, is sometimes
understood as a model for the 3D Euler equations. On a different note, we
point out the proximity between active scalar equations and the 2D non-
resistive incompressible magnetohydrodynamics system, which reads

$\begin{cases}\partial_{t}a+u\cdot\nabla a=0\\\
\big{(}\partial_{t}+u\cdot\nabla-\Delta\big{)}u+\nabla P=(b\cdot\nabla)b\\\
\operatorname*{\rm{div}}(u)=0\\\ b=\nabla^{\perp}a,\end{cases}$

where $u(t,x)\in\mathbb{R}^{d}$ is the velocity of the fluid, $b(t,x)$ is the
magnetic field, $P(t,x)$ the MHD pressure and $a(t,x)\in\mathbb{R}$ is the
(scalar) magnetic potential. Here, the velocity is no longer given as a linear
function of the advected scalar quantity, but as the solution of a (non-
linear) Navier-Stokes problem. This last system concentrates huge
difficulties, and even existence of global weak solutions remains unknown,
despite progess on the front of local well-posedness (see [21] or [28] for
instance). The study of the simpler active scalar equations can also be seen
as a first step to understand these more sophisticated models.

###  1.2 Presentation of the Main Results

In this paragraph, we give an overview of our main results as well as some of
the main ideas involved in the proofs. At the same time, we attempt a
comparision with other existing results in the literature.

####  1.2.1 Global Weak Solutions: $\alpha>0$

Our first result concerns the existence of global weak solutions in Lebesgue
spaces when $\alpha>0$ (see Theorem 3.1).

######  Theorem 1.1.

Consider $\alpha\in]0,d[$ and an exponent $p$ such that

(3) $\frac{2}{1+\frac{\alpha}{d}}<p<\frac{d}{\alpha}.$

For all $\rho_{0}\in L^{p}(\mathbb{R}^{d})$, there exists a global weak
solution $\rho_{0}\in L^{\infty}(\mathbb{R}_{+};L^{p}(\mathbb{R}^{d}))$ of (1)
associated to the initial datum $\rho_{0}$.

######  Remark 1.2.

The reader will note that an important case is not covered by Theorem 1.1,
that is $\alpha=d=2$, where (1) becomes the 2D Stokes-Transport system for
which global well-posedness is known in bounded domains and in the infinite
strip $\mathbb{R}\times]0,1[$, [27], or for compactly supported solutions
[23]. But this does not really affect much the significance of our result:
when $\alpha=d$, we will see that the system in fact has global and unique
solutions (see Theorem 1.6 below). The same remark applies for the statements
pertaining to local solutions, Theorems 1.3, 1.4 and 1.7 below.

This result is based on one of the essential distinctive features of our
problem (1), that the velocity field is divergence free
$\operatorname*{\rm{div}}(u)=0$. This implies that we have at our disposal the
conservation of all $L^{p}$ norms, namely, provided the velocity field is
regular enough,

$\|\rho(t)\|_{L^{p}}=\|\rho_{0}\|_{L^{p}}.$

Theorem 1.1 combines the conservation of $L^{p}$ norms with the velocity law
of (2) to provide global estimates on $u$ in spaces of functions that are
compactly embedded in $L^{p}_{\rm loc}$.

However, while this is the main principle of the proof, we should point out
the two main difficulties. The first one is that the conservation of $L^{p}$
norms mentioned above can only be taken for granted if the velocity field
possesses sufficient regularity. For example, $u$ must be at least
$W^{1,1}_{\rm loc}$ if Di Perna-Lions theory is to be applied, whereas $u$
will have at best an $\alpha$ index of regularity by (2). Our way around that
will be to first start by constructing $L^{2}$ solutions by means of a
Friedrichs scheme, and then proceeding to general $L^{p}$ solutions through
regularization of the velocity field. But because we are below the regularity
level where Di Perna-Lions theory applies, the weak solutions may display some
decay of $\|\rho(t)\|_{L^{p}}$, instead of the conservation we expect from
strong solutions.

Secondly, when handling the nonlinear term
$u\cdot\nabla\rho=\operatorname*{\rm{div}}(\rho u)$, caution should be taken
with the product $\rho u$ when $p<2$, as it is not obvious it is properly
defined. Condition (3) is written so that the $s=\alpha$ regularity of $u$ and
embeddings of well-chosen function spaces will make the product $\rho u$ well
defined.

We mention that results of this kind already exist for SQG and gSQG equations.
Resnick proved existence of global $L^{2}$ solutions for SQG in his PhD
dissertation [32], while a result in $L^{p}$ with $p>4/3$ is due to Marchand
[29]. This has been extended to gSQG by Lazar and Xue, who construct global
weak $L^{1}\cap L^{2}$ solutions in [26]. These results rely on the commutator
structure the non-linear terms in gSQG equations present, and hence are
capable of covering more singular velocities that we can with the fractional
Stokes-Transport equation: we must have $\alpha>0$.

On the other hand, our methods do not use the structure of the velocity law in
any special way: only that $u$ is a divergence free function and that
$(-\Delta)^{-\alpha/2}\mathbb{P}g$ is a Fourier multiplier of order $-\alpha$.
Our results should also hold with no modification for gSQG equations with
velocity law

$u=\nabla^{\perp}(-\Delta)^{-\frac{1}{2}(\alpha+1)}\rho$

and $\alpha>0$ (so that the velocity is not more singular than the density),
thus extending, in the case $\alpha>0$, the $L^{1}\cap L^{2}$ existence of
[26] to $L^{p}$ solutions with (3).

####  1.2.2 Local well-posedness: $\alpha\geq 0$

We now turn to the question of uniqueness of solutions. Theorem 1.3 below
states the existence of solutions that are regular enough to be unique.

######  Theorem 1.3.

Consider $\alpha\in[0,d[$ as well as indices $s>0$ and $p\in[1,+\infty]$ such
that

$p<\frac{d}{\alpha},\qquad\text{ and }\qquad s\geq 1-\alpha.$

For all initial datum $\rho_{0}\in B^{s}_{\infty,1}(\mathbb{R}^{d})\cap
L^{p}(\mathbb{R}^{d})$, there exists a time $T>0$ such that the system (1) has
a unique solution in that space $\rho\in
C^{0}([0,T[;B^{s}_{\infty,1}(\mathbb{R}^{d})\cap L^{p}(\mathbb{R}^{d}))$.

In addition to the local existence of a unique solution, we also prove a lower
bound for the lifespan of solutions and a continuation criterion. These are
contained in Theorem 4.1 below.

The essence of Theorem 1.3 is finding a priori estimates at a level where the
velocity field is Lipschitz, thus enabling us to perform stability estimates
and bound the difference of two solutions. The condition $s\geq 1-\alpha$ is
designed so that the velocity law (2) will bring $u$ to a
$B^{1}_{\infty,1}\subset W^{1,\infty}$ level of regularity.

Local well-posedness result already exists for many active scalar equations
with fractional velocity laws, such as [8] for gSQG, [11] for fractional
porous medium flows or [7] for two different types of problems, including (1).
As we have already noted above, the velocity law
$u=(-\Delta)^{-\alpha/2}\mathbb{P}g$ is somewhat different from gSQG problems
because the latter possess a commutator structure that provides well-posedness
even when the velocity is more singular that the unknown. Consequently,
existing results on (1) only hold for $\alpha\geq 0$, as mentioned in [7], and
ill-posedness has even been proved for $\alpha<0$, see [22].

As far as we are aware, all well-posedness results existing in the literature
concern initial data that are in $H^{s}$ Sobolev spaces, with appropriate $s$.

At this point, we should note that Theorem 1.3 is not optimal when $\alpha\geq
1$. In that case, as we will see, solutions are global and we will dedicate a
whole Section to global well-posedness with initial data in critical spaces
(see below).

####  1.2.3 A Directional Continuation Criterion

So far, none of our results have used any specific feature of the velocity law
(2), other than incompressibility $\operatorname*{\rm{div}}(u)=0$ and the
order $\alpha$ of the operator. But this does not mean that (2) has no
interesting underlying structure. Here, we use the way gravity stratification
affects the velocity to prove a directional continuation criterion.

######  Theorem 1.4.

Assume that $0\leq\alpha<1$. Under the assumptions and notations of Theorem
1.3, the unique solution $\rho\in
C^{0}([0,T[;B^{1-\alpha}_{\infty,1}(\mathbb{R}^{d})\cap
L^{p}(\mathbb{R}^{d}))$ may be continued beyond time $T$ if and only if

(4)
$\int_{0}^{T}\big{\|}\partial_{d}\rho\big{\|}_{B^{-\alpha}_{\infty,1}}\,{\rm
d}t<+\infty$

Understanding (4) is quite simple: by writing explicitly the Leray projection
operator $\mathbb{P}={\rm Id}+\nabla(-\Delta)^{-1}\operatorname*{\rm{div}}$
and remembering that the gravity vector is $g=e_{d}$, we may express the
nonlinearity in (1) as

$u\cdot\nabla\rho=(-\Delta)^{-\alpha/2}\rho\,\partial_{d}\rho+\nabla\rho\cdot\nabla(-\Delta)^{-\alpha/2}(-\Delta)^{-1}\partial_{d}\rho.$

Both products in the above involve a factor that is linearly dependent on the
derivative $\partial_{d}\rho$. This implies that blow-up of the solution is
prevented provided $\partial_{d}\rho$ remains bounded in an appropriate space.

####  1.2.4 Global well-posedness: $\alpha\geq 1$

We now turn our attention to the range $\alpha\geq 1$, in which case solutions
are global. Roughly, the idea is that conservation of $L^{q}$ norms yielded by
incompressibility automatically grant $B^{\alpha}_{q,\infty}$ regularity to
the velocity field through (2), and the embeddings

$B^{\alpha}_{q,\infty}\subset B^{1}_{\infty,1}\subset W^{1,\infty}$

will hold as long as $q>d/(1-\alpha)$, hence global Besov-Lipschitz estimates
on the velocity field. However, with this in mind, the limit case
$q=d/(1-\alpha)$ must be seen as critical. As a matter of fact, if $\rho\in
L^{d/(1-\alpha)}$, then $(-\Delta)^{-\alpha/2}\mathbb{P}(\rho g)$ will not be
in general Lipschitz, but $\log$-Lipschitz.

Therefore, if we are to prove uniqueness of global $L^{d/(1-\alpha)}$
solutions, we must turn to the theory of transport equations with non-
Lipschitz velocity fields, as in [18], which provides estimates at the cost of
a loss of regularity.

######  Theorem 1.5.

Consider $d\geq 2$ and $\alpha\in]1,d[$. Let $p\in[1,d/\alpha[$ and set
$q=d/(\alpha-1)$. Then, for any initial datum $\rho_{0}\in
L^{p}(\mathbb{R}^{d})\cap L^{q}(\mathbb{R}^{d})$, problem (1) has a unique
global solution $\rho\in L^{\infty}(\mathbb{R}_{+};L^{p}(\mathbb{R}^{d})\cap
L^{q}(\mathbb{R}^{d}))$.

In the recent preprint [30], Mecherbet and Sueur obtain essentially the same
result in the special case $\alpha=2$ and $d=3$, albeit by very different
methods. While we rely on stability estimates, the authors of [30] prove well-
posedness in $L^{1}\cap L^{3}$ using estimates in the Wasserstein distance
$W_{1}$ to prove uniqueness.

Theorem 1.5 has two limitations: two settings of special interest are excluded
from its scope. The first one is the case of the 2D Stokes-Transport problem
$\alpha=d=2$. The second is the case $\alpha=1$, which has the same scaling as
the 2D Euler system.

In those endpoint cases, we may obtain global well-posedness, but at the cost
of working in smaller (but still critical) function spaces.

######  Theorem 1.6.

Consider $d\geq 2$ and $\alpha\in[1,d]$. Let $p=d/\alpha$ and
$q=d/(\alpha-1)$. Then, for any initial datum
$\rho_{0}\in\dot{B}^{0}_{p,1}(\mathbb{R}^{d})\cap
B^{0}_{q,1}(\mathbb{R}^{d})$, problem (1) has a unique global solution
$\rho\in C^{0}(\mathbb{R}_{+};\dot{B}^{0}_{p,1}(\mathbb{R}^{d})\cap
B^{0}_{q,1}(\mathbb{R}^{d}))$.

The assumption $\rho_{0}\in\dot{B}^{0}_{p,1}$, stronger than simple $L^{p}$
integrability, serves to insure that the velocity field is well-defined. The
operator $(-\Delta)^{-\alpha}\mathbb{P}g$ has a symbol which is singular at
low frequencies, so we require the Fourier transform of $\rho$ to be regular
enough that $\widehat{u}$ be properly defined. When $\alpha=d$, even
$\rho_{0}\in L^{1}$ is not enough, and we must resort to the stronger
$\rho\in\dot{B}^{0}_{1,1}\subsetneq L^{1}$ condition. In previous works for
the 2D Stokes Transport system, i.e. when $\alpha=d=2$, global well-posedness
has been obtained under other integrability conditions, namely a compact
support assumption on the initial data [23], or in the framework of solutions
on a bounded domain $\Omega\subset\mathbb{R}^{2}$ or the infinite strip
$\Omega=\mathbb{R}\times]0,1[$, see [27].

Concerning the $B^{0}_{q,1}$ assumption, it has two uses. The first one is
that for $\dot{B}^{0}_{p,1}$ regularity to be propagated by the transport
equation, we need a Lipschitz vector field. Therefore, working with initial
data in $L^{d/(\alpha-1)}$ will not do, as it will only give a
$\log$-Lipschitz vector field. Secondly, in the case where $\alpha=1$, finding
uniqueness in $L^{d/(\alpha-1)}=L^{\infty}$ seems fraught with extreme
difficulties.

This may seem surprising, given the closeness of the case $\alpha=1$ to the 2D
Euler problem, where uniqueness with $L^{\infty}$ vorticity is well-known.
However, all $L^{\infty}$ uniqueness results we are aware of for 2D Euler
(such as the method of Yudovich [37] or that of Serfati [35], [1]) rely the
fact that the vorticity equation is the curl of a lower order equation: the
original Euler system. This “integration” of the vorticity equation is not
available in our case.

As we present Theorem 6.5 below, we give additional details and comments on
why the proof used for Theorem 1.5 does not work in the case $\alpha=1$. We
refer to the discussion below.

####  1.2.5 Lifespan of Solutions for $\alpha\rightarrow 1^{-}$

To conclude this article, we study the way the lifespan of the solution
depends on the value of $\alpha<1$. The idea is quite simple: since the
equation is globally well-posed for $\alpha=1$, then continuous dependency on
the parameters should imply that the lifespan becomes arbitrarily large as
$\alpha\rightarrow 1^{-}$. This is Theorem 1.7 below.

######  Theorem 1.7.

Assume that $d\geq 2$ and consider an exponent $p\in[1,d[$. For any
$\frac{1}{2}<\alpha<1$ and initial datum $\rho_{0}\in
X:=L^{p}(\mathbb{R}^{d})\cap B^{1-\alpha}_{\infty,1}(\mathbb{R}^{d})$, the
lifespan $T^{*}$ of the unique associated solution of (1) is at least

$T^{*}\geq\frac{C}{\|\rho_{0}\|_{X}}\log\left(1+\frac{C}{1-\alpha}\right).$

In fact, in Theorem 7.1 below is even more general, as we study the lifespan
around an equilibrium solution $R(x_{d})$. This amounts to adding an extra
linear term to the equation and changes the inequality above to account for
the norm of derivatives of $R(x_{d})$.

Theorem 1.7 is not the first of its kind: several authors have proved similar
“asymptotically global well-posedness” results. For instance, Danchin and
Fanelli [19] study the 2D non-homogeneous incompressible Euler equations and
find the lifespan to be arbitrarily large in the limit of constant densities,
with a similar logarithmic lower bound (see also [34] by Sbaiz for the case of
the 2D non-homogeneous Euler equations with fast rotation), while Fanelli and
Liao do so in [17] for a zero Mach number limit system.

Similarly, Fanelli and the present author examine the case of 2D ideal
magnetohydrodynamics in [14] and [15] in the limit of weak magnetic fields,
but because of the complexity of the system, the method only yields a triply
iterated logarithmic bound.

The proof of Theorem 1.7 is a bit different from the lower bounds from the
other works mentioned here. These usually rely on estimates for the transport
equation, in Besov spaces of regularity $s=0$, that are linear with respect to
the transport field, as in the proof of global well-posedness for the 2D Euler
problem in $B^{1}_{\infty,1}$ (see the articles of Vishik [36] and Hmidi-
Keraani [24]). In our setting, we have to adapt these estimates to a
regularity level of $\epsilon=1-\alpha\rightarrow 0^{+}$.

####  1.2.6 Outline of the paper

We start by defining precisely the notion of weak solution of (1) in Section
2. This requires some clarification on the Stokes problem, and we explain how
$u$ can be uniquely determined by $\rho$.

In Sections 3 to 7, we state comment and prove all of our results in the order
they were mentioned in this Introduction.

The paper ends with an Appendix containing definitions related to Littlewood-
Paley decompositions, Besov spaces and various results on transport equations.

####  1.2.7 Notation

We adopt the following notation throughout the paper.

  * •

The notation $C$ will be used to denote constants whose precise value are
irrelevant to the computations, and which may differ from one line to the
next. When needed, we may indicate dependency on a given parameter by writing
the constant as $C(\,.\,)$.

  * •

We note $f\lesssim g$ for $f\leq Cg$. The notation $f\approx g$ will mean
$\frac{1}{C}f\leq g\leq Cf$.

  * •

Unless otherwise mentioned, all derivatives or pseudo-differential operators
refer to the space variables only. For example, if
$f:(t,x)\in\mathbb{R}\times\mathbb{R}^{d}\longmapsto f(t,x)\in\mathbb{R}$, the
Laplacian $\Delta f$ is given by $\Delta
f(t,x)=\partial_{1}^{2}f(t,x)+\cdots+\partial_{d}^{2}f(t,x)$.

  * •

If $X$ is a Banach space and $T>0$, we note $C^{0}_{T}(X):=C^{0}([0,T[;X)$, or
$C^{0}(X):=C^{0}(\mathbb{R}_{+};X)$ when $T=+\infty$. We define the spaces
$L^{p}_{T}(X)$, $L^{p}(X)$, $W^{1,\infty}_{T}(X)$, $W^{1,\infty}(X)$, etc. in
the same way.

  * •

In order to simplify notations, we will omit the reference to the
$\mathbb{R}^{d}$ when using function spaces, so we write for example
$L^{p}:=L^{p}(\mathbb{R}^{d})$, $H^{s}:=H^{s}(\mathbb{R}^{d})$ and so on. For
instance, we have

$C^{0}_{T}(L^{p})=C^{0}([0,T[;L^{p}(\mathbb{R}^{d})).$

  * •

We note $D=-i\nabla$. More generally, for
$\phi:\mathbb{R}^{d}\operatorname*{\longrightarrow}\mathbb{R}^{d}$, the
Fourier multiplication operator $\phi(D)$ is defined by the Fourier transform

$\forall
f\in\mathcal{S},\qquad\widehat{\phi(D)f}(\xi)=\phi(\xi)\widehat{f}(\xi).$

  * •

We note $C^{0}_{b}(X)$ the space of continuous and bounded functions on
$\mathbb{R}_{+}$ with values in a Banach space $X$, while $C^{0}(X)$ without
the subscript refers to the space of continuous functions
$\mathbb{R}_{+}\operatorname*{\longrightarrow}X$ that are not necessarily
bounded.

#### Acknowledgements

The author is grateful to Francesco Fanelli for many long discussions on this
problem and his constant help throughout the project. Some of the ideas behind
Theorem 3.1 were kindly suggested by Rafael Granero-Belinchón.

This work has been partially supported by the Deutsche Forschungsgemeinschaft
(DFG, German Research Foundation) Project ID 211504053 - SFB 1060, and by the
project CRISIS (ANR-20-CE40-0020-01), operated by the French National Research
Agency (ANR).

##  2 A Definition of Weak solution

In this paragraph, we define the notion of weak solution of (1) and give the
precise meaning of the fractional Laplace operator we use throughout the
article.

###  2.1 The Fractional Stokes Equation

We focus for a while on the fractional Stokes equation:

(5) $\begin{cases}(-\Delta)^{\alpha/2}u+\nabla\pi=\rho g\\\
\operatorname*{\rm{div}}(u)=0.\end{cases}$

Here, $\rho$ is considered as a datum for the system while the unknowns are
$u$ and $\pi$. The fractional Laplace operator is defined by its Fourier
transform (see for example Proposition A.8).

The first remark we make is that solutions of this system are in general
highly non-unique. For example, in the case of the Stokes system $\alpha=2$,
the velocity $u$ is only given up to the addition of a (divergence free)
harmonic function, and $\pi$ up to the addition of a constant. But on top of
that, there is a further ambiguity that cannot be reduced to harmonic or
constant summands: for example, if $\rho=1$, then the pairs

$\begin{split}&\pi(x)=x\cdot g\qquad u(x)=0\\\ &\pi(x)=0,\qquad
u(x)=-\frac{1}{2d}|x|^{2}g\end{split}$

are both solutions of (5). This phenomenon is due to the lack of a far-field
condition, such as decay at $|x|\rightarrow+\infty$, and is not specific to
the Stokes system. We refer to [12] and the references therein for a
discussion of a similar problem in the Euler equations set on
$\mathbb{R}^{d}$.

In order to map the density $\rho$ to a unique velocity field that is solution
of (5), we impose sufficient decay at $|x|\rightarrow+\infty$. Our goal in
this paragraph is to prove the following result, which involves the space
$\mathcal{S}^{\prime}_{h}$ defined in the Appendix (see Definition A.2).

######  Proposition 2.1.

Consider $d\geq 2$. The following assertions hold:

  1. (i)

Assume that $\rho\in L^{p}$ with $p\in[1,d/p[$. Then there is a unique
$u\in\mathcal{S}^{\prime}_{h}$ such that (5) is satisfied for some pressure
function $\pi$. In addition, $\pi$ is uniquely determined up to a constant.

  2. (ii)

The same holds when $\rho\in\dot{B}^{0}_{p,1}$ with $p=\frac{d}{\alpha}$.

###### Proof.

Formally, applying the Leray projection operator $\mathbb{P}={\rm
Id}+\nabla(-\Delta)^{-1}\operatorname*{\rm{div}}$ to the first equation in (5)
gives the velocity as a function of the density:

$u=(-\Delta)^{-\alpha/2}\mathbb{P}(\rho g).$

The obvious problem is that the operator $(-\Delta)^{-\alpha/2}\mathbb{P}g$
may not be well defined, as it is a Fourier multiplier whose symbol is
singular at $\xi=0$. We set

$v:=\sum_{j\in\mathbb{Z}}\dot{\Delta}_{j}(-\Delta)^{-\alpha/2}\mathbb{P}(\rho
g).$

Now, thanks to the embedding $L^{p}\subset\dot{B}^{0}_{p,\infty}$, which holds
because $p<+\infty$, Lemma A.5 shows that this sum defines an element of the
homogeneous Besov space $\dot{B}^{\alpha}_{p,\infty}$. In particular, we have
$v\in\mathcal{S}^{\prime}_{h}$.

Similarly, note that the pressure force $F=\nabla\pi$ can be defined as the
sum of its homogeneous Littlewood-Paley decomposition:

$F=\sum_{j\in\mathbb{Z}}\dot{\Delta}_{j}\nabla(-\Delta)^{-1}\operatorname*{\rm{div}}(\rho
g),$

and this sum defines an element of the homogeneous Besov space
$F\in\dot{B}^{0}_{p,\infty}$. The existence of a pressure function
$\pi\in\mathcal{S}^{\prime}$ such that $F=\nabla\pi$ follows from, for
example, Lemma 3.1 in [14].

Now consider another solution $(v^{\prime},\pi^{\prime})$ of the problem such
that $v^{\prime}\in\mathcal{S}^{\prime}_{h}$ and let
$F^{\prime}=\nabla\pi^{\prime}$. Because the symbol of the operator
$(-\Delta)^{-\alpha/2}\mathbb{P}g$ is smooth outside of $\xi=0$, we see that
the Fourier transforms of $v$ and $v^{\prime}$ must agree on
$\mathbb{R}^{d}\setminus{0}$, namely

$\widehat{v}(\xi)=\widehat{v^{\prime}}(\xi)\qquad\text{in
}\mathcal{D}^{\prime}(\mathbb{R}^{d}\setminus{0}),$

so the difference $v-v^{\prime}$ must be reduced to a polynomial. As there is
no polynomial function in $\mathcal{S}^{\prime}_{h}$, we conclude that
$v=v^{\prime}$, and hence $F=F^{\prime}$ so the pressure functions $\pi$ and
$\pi^{\prime}$ are equal up to a constant summand. ∎

The consequence the computations in the proof of Proposition 2.1 is that we
have an explicit representation of the velocity field. This makes it possible
to bound it in appropriate norms.

######  Proposition 2.2.

Consider $0\leq\alpha\leq d$ and assume that
$\rho:\mathbb{R}^{d}\operatorname*{\longrightarrow}\mathbb{R}$ is a density
such that one of the assumptions of Proposition 2.1 holds. Then, the uniquely
determined velocity field $u\in\mathcal{S}^{\prime}_{h}$ satisfies the
following properties:

  1. (i)

if $\rho\in L^{p}$ for some $p<d/\alpha$ then
$\|\Delta_{-1}u\|_{L^{\infty}}\lesssim\|\rho\|_{L^{p}}$. Moreover, if $\rho\in
L^{p}\cap B^{s}_{\infty,1}$ for some $s\in\mathbb{R}$ then

$\|u\|_{B^{s+\alpha}_{\infty,1}}\lesssim\|\rho\|_{L^{p}}+\|\rho\|_{B^{s}_{\infty,1}}\,;$

  2. (ii)

if $\rho\in\dot{B}^{0}_{p,1}$ for $p=d/p$ then
$\|\Delta_{-1}u\|_{L^{\infty}}\lesssim\|\rho\|_{\dot{B}^{0}_{p,1}}$. Moreover,
if $\rho\in\dot{B}^{0}_{p,1}\cap B^{s}_{\infty,1}$ for some $s\in\mathbb{R}$
then

$\|u\|_{B^{s+\alpha}_{\infty,1}}\lesssim\|\rho\|_{\dot{B}^{0}_{p,1}}+\|\rho\|_{B^{s}_{\infty,1}}.$

###### Proof.

The proof is fairly straightforward. Start by assuming that $\rho\in L^{p}$
for some $p<d/\alpha$. Then, as $u$ belongs to the Besov space
$\dot{B}^{\alpha}_{p,\infty}$ the series

(6) $\Delta_{-1}u=\sum_{j\leq
1}\dot{\Delta}_{j}(-\Delta)^{-\alpha/2}\mathbb{P}(\rho g)$

is normally convergent in $L^{\infty}$, and we deduce that
$\|\Delta_{-1}u\|_{L^{\infty}}\lesssim\|\rho\|_{L^{p}}$. On the other hand, if
we also have $\rho\in B^{s}_{\infty,1}$ for some $s\in\mathbb{R}$, then it
follows from Lemma A.5 that

$\begin{split}\|u\|_{B^{s}_{\infty,1}}&\lesssim\|\Delta_{-1}u\|_{L^{\infty}}+\sum_{j\geq
0}2^{j(s+\alpha)}\big{\|}\Delta_{j}(-\Delta)^{-\alpha/2}\mathbb{P}(\rho
g)\big{\|}_{L^{\infty}}\\\ &\lesssim\|\rho\|_{L^{p}}+\sum_{j\geq
0}2^{js}\|\Delta_{j}\rho\|_{L^{\infty}}\\\
&\lesssim\|\rho\|_{L^{p}}+\|\rho\|_{B^{s}_{\infty,1}}.\end{split}$

This proves point (i) of the Proposition. Point (ii) follows from nearly
identical computations. ∎

###  2.2 Defining Weak Solutions

In this paragraph, we define the notion of weak solution of problem (1).
Throughout this article, we will work with solutions which do not necessarily
have locally integrable derivatives, so we will always understand the non-
linear convective term in the sense of the divergence form:

$u\cdot\nabla\rho=\operatorname*{\rm{div}}(\rho u),$

which is possible thanks to incompressibility $\operatorname*{\rm{div}}(u)=0$.
Even then, we will have to take care that the product $\rho u$ is always
properly defined.

######  Definition 2.3.

Consider $\alpha\in[0,d]$ and $q\in[1,+\infty]$ such that
$q>\frac{2}{1+\frac{\alpha}{d}}$ and $T>0$. A function $\rho\in L^{2}_{\rm
loc}([0,T[;L^{q})$ is said to be a weak solution of the fractional Stokes-
Transport problem (1) with initial datum $\rho_{0}\in L^{q}$ if

  1. (i)

we have $\rho\in L^{2}_{\rm loc}([0,T[;L^{p})$ for some $p\in[1,d/\alpha[$ or
$\rho\in L^{2}_{\rm loc}([0,T[;\dot{B}^{0}_{p,1})$ for $p=d/\alpha$ ;

  2. (ii)

the density $\rho$ solves the weak form of the transport equation

$\forall\phi\in\mathcal{D}([0,T[\times\mathbb{R}^{d};\mathbb{R}),\qquad\iint\Big{\\{}\rho\partial_{t}\phi+\rho
u\cdot\nabla\phi\Big{\\}}\,{\rm d}x\,{\rm d}t+\int\rho_{0}(x)\phi(0,x)\,{\rm
d}x=0$

with the velocity field given, for almost all times, as the unique
$\mathcal{S}^{\prime}_{h}$ solution of the Stokes system (5), as specified by
Proposition 2.1 above. In that case, it follows from Lemma 3.6 below (whose
proof is independent from this definition) that the product $\rho u$ is well
defined as an element of the space $\rho u\in L^{1}_{\rm
loc}([0,T[;B^{\alpha-d/q}_{q,\infty})$.

##  3 Global Weak Solutions for $\alpha>0$

######  Theorem 3.1.

Consider $\alpha\in]0,d[$ and an exponent $p$ such that

(7) $\frac{2}{1+\frac{\alpha}{d}}<p<\frac{d}{\alpha}.$

For all $\rho_{0}\in L^{p}$, there exists a global weak solution $\rho\in
L^{\infty}(L^{p})$ of (1) associated to the initial datum $\rho_{0}$. In
addition, for any $q\in[1,+\infty]$, if $\rho_{0}\in L^{q}$ then this solution
satisfies

(8) $\|\rho\|_{L^{\infty}(L^{q})}\leq\|\rho_{0}\|_{L^{q}}.$

######  Remark 3.2.

We remark that condition (7) is not void because $0<\alpha<d$. In addition, as
long as $\alpha<d/2$, the value $p=2$ always works. It should also be noted
that the bound (8) remains valid in the case where $q=1$.

######  Remark 3.3.

The case of a plane problem $d=2$ could seem to be a major limitation of our
result, as is will not then apply to $\alpha=2$, which is a value of special
interest. However, we will have other means of proving global well-posedness
whenever $\alpha\geq 1$ (see Theorem 6.5 below), so we should not be too
worried about this issue.

Before giving the full proof, let us sketch ou argument. The main difficulty
is the propagation of $L^{p}$ norms in the transport equation at such a low
level of regularity for $u$. For instance, Di Perna-Lions theory requires the
velocity field to be at least $W^{1,1}_{\rm loc}$, and such a bound is
unavailable here, as $u$ will have a regularity level of at most $\alpha$.

In order to replace Di Perna-Lions theory, we will Sarto by using energy
estimates and a Friedrichs scheme to construct global $L^{\infty}(L^{2})$
solutions. Incidentally, it should be noted that fixed-point arguments (a.k.a.
iterative schemes) do not work at this level of regularity, as they require
stability estimates to hold. This will be the first part of the proof.

Secondly, in order to propagate $L^{p}$ norms, we work with a regularization
of the velocity field and construct approximate solutions that are uniformly
bounded in $L^{\infty}(L^{p})$. The proof will then be a matter of finding the
estimates we need to obtain convergence of the approximate solutions.

We therefore start by the case where $p=2$. In order to prepare the way for
further estimates, we already introduce a regularization of the velocity
field.

######  Proposition 3.4.

Consider a Fourier multiplier $T=\phi(D)$ whose symbol $\phi(\xi)$ is a
$C^{\infty}$ function with compact support bounded away from $\xi=0$ such that
$|\phi|\leq 1$ and let $\rho_{0}\in L^{2}$. Then the PDE system

(9) $\begin{cases}\partial_{t}\rho+\operatorname*{\rm{div}}(\rho u)=0\\\
u=T\mathbb{P}(-\Delta)^{-\alpha/2}(\rho g)\end{cases}$

has a global weak solution $\rho\in L^{\infty}(L^{2})$ associated to the
initial datum $\rho_{0}$.

######  Remark 3.5.

Although the proof of this Proposition is by no means challenging, the PDE
problem (9) still is a fully non-linear set of equations, so that the
existence of solutions cannot be considered as totally trivial.

###### Proof (of the Proposition).

We start by noting that, by assumption on the function $\phi(\xi)$, the
operator $T\mathbb{P}(-\Delta)^{-\alpha/2}g$ has a smooth, bounded and
compactly supported symbol. It therefore is
$L^{2}\operatorname*{\longrightarrow}B^{k}_{\infty,1}$ continuous for all
$k>0$. This fact will be of constant use throughout the proof.

STEP 1: approximate system. In order to implement a Friedrichs scheme, we
introduce the following set of approximate equations: for all $n\geq 1$,
consider

(10)
$\begin{cases}\partial_{t}\rho_{n}+A_{n}\operatorname*{\rm{div}}(\rho_{n}u_{n})=0\\\
u_{n}=T\mathbb{P}(-\Delta)^{-\alpha/2}(\rho_{n}g),\end{cases}\qquad\text{with
initial data }\rho_{n}(0)=A_{n}\rho_{0}.$

Here, $A_{n}$ is the spectral projector on the ball $|\xi|\leq n$. More
precisely, for any $f\in L^{2}$, the function $A_{n}f\in L^{2}$ has Fourier
transform $\mathds{1}_{|\xi|\leq n}\widehat{f}(\xi)$. We wish to make use of
the Cauchy-Lipschitz theorem to construct solutions of (10) in the Banach
space $L^{2}$. We must therefore prove that
$\partial_{t}\rho_{n}=-A_{n}\operatorname*{\rm{div}}(\rho_{n}u_{n})$ is a
locally Lipschitz function of $\rho_{n}$ in $L^{2}$.

Firstly, since the function $A_{n}\operatorname*{\rm{div}}(\rho_{n}u_{n})$ has
a compactly supported Fourier transform, the first Bernstein inequality gives

$\big{\|}A_{n}\operatorname*{\rm{div}}(\rho_{n}u_{n})\big{\|}_{L^{2}}\leq
C(n)\|\rho_{n}u_{n}\|_{L^{2}}\leq
C(n)\|u_{n}\|_{L^{\infty}}\|\rho_{n}\|_{L^{2}}.$

Next, thanks to the boundedness properties of the operator
$T\mathbb{P}(-\Delta)^{-\alpha/2}g$ mentioned at the beginning of the proof,
we get

$\big{\|}A_{n}\operatorname*{\rm{div}}(\rho_{n}u_{n})\big{\|}_{L^{2}}\leq
C(n)\|\rho_{n}\|_{L^{2}}^{2},$

so that the quantity $A_{n}\operatorname*{\rm{div}}(\rho_{n}u_{n})$ is indeed
locally Lipschitz in $L^{2}$ with respect to $\rho_{n}$.

Consequently, the Cauchy-Lipschitz theorem provides, for every $n\geq 1$, a
solution of (10) $\rho_{n}\in C^{0}([0,T_{n}[;L^{2})$ associated to the
initial datum $\rho_{0}$, with positive lifespan $T_{n}>0$.

STEP 2: uniform estimates. Because the velocity fields $u_{n}$ are divergence-
free and smooth, we may test the evolution equation in (10) against $\rho_{n}$
and obtain conservation of the $L^{2}$ norms of the approximate solutions:

$\forall
t\in[0,T_{n}[,\qquad\|\rho_{n}(t)\|_{L^{2}}=\|\rho_{n}(0)\|_{L^{2}}\leq\|\rho_{0}\|_{L^{2}}.$

As a consequence, the approximate solutions $\rho_{n}$ provided by the Cauchy-
Lipschitz theorem do not blow-up in $L^{2}$, and so they must be global
$T_{n}=+\infty$.

We deduce from the continuity of the operator
$T\mathbb{P}(-\Delta)^{-\alpha/2}g:L^{2}\operatorname*{\longrightarrow}B^{k}_{\infty,1}$
that the sequence of velocity fields $(u_{n})_{n}$ is uniformly bounded in
$L^{\infty}(B^{k}_{\infty,1})$ for every $k>0$. In order to also provide time
compactness, we use the equation and bound the time derivative
$\partial_{t}u_{n}$. First, we see that

$\|\partial_{t}\rho_{n}\|_{H^{-1}}=\big{\|}A_{n}\operatorname*{\rm{div}}(\rho_{n}u_{n})\big{\|}_{H^{-1}}\leq\|\rho_{n}\|_{L^{2}}\|u_{n}\|_{L^{\infty}}\lesssim\|\rho_{0}\|_{L^{2}}^{2},$

so that, since the operator $T\mathbb{P}(-\Delta)^{-\alpha/2}g$ has a smooth
and compactly supported symbol, the Bernstein inequalities provide, for any
$k>0$,

$\|\partial_{t}u_{n}\|_{B^{k}_{\infty,1}}\lesssim\|\partial_{t}\rho_{n}\|_{H^{-1}}\lesssim\|\rho_{0}\|_{L^{2}}^{2},$

and the sequence $(u_{n})_{n}$ is therefore uniformly bounded in the space
$W^{1,\infty}_{T}(B^{k}_{\infty,1})$ for every $T>0$.

STEP 3: convergence to a solution. Thanks to the uniform bounds we have
obtained above, namely

$\|\rho_{n}\|_{L^{\infty}(L^{2})}\leq\|\rho_{0}\|_{L^{2}}\qquad\text{and}\qquad\|u_{n}\|_{W^{1,\infty}_{T}(B^{k}_{\infty,1})}\leq\|\rho_{0}\|_{L^{2}}+\|\rho_{0}\|_{L^{2}}^{2}$

for every $T>0$ and $k>0$, we have convergence of the sequences
$(\rho_{n})_{n}$ and $(u_{n})_{n}$ up to an extraction: there is a $\rho\in
L^{\infty}(L^{2})$ and a $u\in L^{\infty}_{\rm loc}(L^{\infty})$ such that, as
$n\rightarrow+\infty$,

$\begin{split}&\rho_{n}\operatorname*{\,\,\rightharpoonup\,\,}^{*}\rho\qquad\text{in
}L^{\infty}(L^{2}),\\\ &\rho_{n}(0)\rightarrow\rho_{0}\qquad\text{in
}L^{2},\\\ &u_{n}\rightarrow u\qquad\text{in }L^{\infty}_{T}(L^{\infty}_{\rm
loc}).\end{split}$

for every $T>0$. We now study the convergence of all the terms in the PDE to
show that the limit $(\rho,u)$ is in fact a solution of the system. Firstly,
we have,

$\partial_{t}\rho_{n}\operatorname*{\longrightarrow}\partial_{t}\rho\qquad\text{in
}\mathcal{D}^{\prime}(\mathbb{R}_{+}^{*}\times\mathbb{R}^{d}).$

Next, we study the convergence of the nonlinear term $\rho_{n}u_{n}$. Let
$\psi\in\mathcal{D}(]0,T[\times\mathbb{R}^{d})$. Then, if
$K\subset\mathbb{R}^{d}$ is a compact set such that ${\rm
supp}\big{(}\psi(t)\big{)}\subset K$ for all times $t\in]0,T[$,

$\begin{split}\left|\langle\rho_{n}u_{n}-\rho
u,\psi\rangle\right|&\leq\|u_{n}-u\|_{L^{\infty}(K)}\|\psi\|_{L^{2}}\|\rho_{n}\|_{L^{2}}+\left|\langle\rho_{n}-\rho,u\psi\rangle\right|\\\
&\leq\|u_{n}-u\|_{L^{\infty}(K)}\|\psi\|_{L^{2}}\|\rho_{0}\|_{L^{2}}+\left|\langle\rho_{n}-\rho,u\psi\rangle\right|\end{split}$

On the one hand, $u_{n}\operatorname*{\longrightarrow}u$ in the space
$L^{\infty}_{T}\big{(}L^{\infty}(K)\big{)}$, so the first summand tends to
zero, and on the other hand, the fact that $u\psi\in L^{\infty}_{T}(L^{2})$
insures that the bracket converges to zero. We have shown that $\rho$ solves
the equation

$\partial_{t}\rho+\operatorname*{\rm{div}}(\rho u)=0\qquad\text{in
}\mathcal{D}^{\prime}(\mathbb{R}_{+}^{*}\times\mathbb{R}^{d}),$

and the assertion concerning the initial datum follows from the convergence
$A_{n}\rho_{0}\operatorname*{\longrightarrow}\rho_{0}$ in $L^{2}$. ∎

Now that we have constructed $L^{2}$ solutions, we may now finish the proof of
Theorem 3.1.

###### Proof (of Theorem 3.1).

Our argument runs pretty much along the same lines as that of Proposition 3.4,
starting from a set of approximate solutions whose convergence must be shown,
except that this time much more care is required to find estimates in the
correct spaces.

STEP 1: approximate solutions. We use Proposition 3.4 in order to construct a
family of approximate solutions. For any $N\geq 1$, we define a sequence of
initial data $\rho_{0,N}$ by

$\rho_{0,N}(x)=\Psi(x/N)S_{N}\rho_{0}(x),$

where $\Psi\in\mathcal{D}$ is such that $0\leq\Psi\leq 1$ and $\Psi(x)=1$ for
all $|x|\leq 1$, and $S_{N}$ is the Littlewood-Paley operator defined in
Subsection A.1 of the appendix. As a consequence, $\rho_{0,N}\in L^{2}$ for
every $N\geq 1$ and $\|\rho_{0,N}\|_{L^{r}}\leq\|\rho_{0}\|_{L^{r}}$ for every
$N\geq 1$ and $r\in[1,+\infty]$. In addition, we define the smoothing operator
$T_{N}$ by

$T_{N}:=\sum_{-N}^{N}\dot{\Delta}_{j}=\chi(2^{-N}D)\big{(}{\rm
Id}-\chi(2^{N}D)\big{)},$

where $\chi(\xi)$ is the Littlewood-Paley function from Subsection A.1. Note
that the operator $T_{N}$ is uniformly bounded in the
$L^{r}\operatorname*{\longrightarrow}L^{r}$ topology, as it is a composition
of scaled versions of the Fourier multiplier $\chi(D)=\Delta_{-1}$.

We consider the following approximate system, for which the velocity field has
been regularized thanks to the operator $T_{N}$. For every $N\geq 1$, consider
the problem

(11)
$\begin{cases}\partial_{t}\rho_{N}+\operatorname*{\rm{div}}(\rho_{N}u_{N})=0\\\
u_{N}=T_{N}\mathbb{P}(-\Delta)^{-\alpha/2}(\rho_{N}g),\end{cases}\qquad\text{with
initial data }\rho_{N}(0)=\rho_{0,N}.$

The existence of global solutions $\rho_{N}\in L^{\infty}(L^{2})$ is
guaranteed by Proposition 3.4.

STEP 2: uniform estimates. First of all, the $\rho_{N}$ solve pure transport
equations with divergence-free velocity fields $u_{N}$. Since, by
construction, each $u_{N}$ is $C^{\infty}$ smooth, all the Lebesgue norms of
the solutions are conserved: for all $r\in[1,+\infty]$,

$\forall t\geq
0,\qquad\|\rho_{N}(t)\|_{L^{r}}=\|\rho_{0,N}\|_{L^{r}}\lesssim\|\rho_{0}\|_{L^{r}}.$

Now, we seek uniform estimates on the velocity field in order to provide
compactness on the sequence $(u_{N})_{N}$. Here, unlike in the proof of
Proposition 3.4 above, we have to deal with the full operator
$(-\Delta)^{-\alpha/2}\mathbb{P}g$ whose symbol involves a singularity at low
frequencies $\xi=0$. To better understand the specific low and high frequency
issues, we split $u_{N}$ into

$u_{N}=\Delta_{-1}u_{N}+({\rm Id}-\Delta_{-1})u_{N}$

and will prove compactness separately on each part. On the one hand, because
of Proposition 2.2, we may handle the low-frequency part

(12)
$\|\Delta_{-1}u_{N}\|_{L^{\infty}}=\big{\|}\Delta_{-1}T_{N}(-\Delta)^{-\alpha/2}\mathbb{P}(\rho_{N}g)\big{\|}_{L^{\infty}}\lesssim\|\rho_{N}\|_{L^{p}}\leq\|\rho_{0}\|_{L^{p}}.$

And since $\Delta_{-1}u_{N}$ has a compactly supported Fourier transform, this
means that we in fact have the uniform bound
$\|\Delta_{-1}u_{N}\|_{B^{k}_{\infty,1}}\lesssim\|\rho_{0}\|_{L^{p}}$.
Concerning the high frequency part, we have, thanks to Lemma A.5,

(13) $\big{\|}({\rm
Id}-\Delta_{-1})u_{N}\big{\|}_{B^{\alpha}_{p,\infty}}\lesssim\big{\|}({\rm
Id}-\Delta_{-1})\rho_{N}\big{\|}_{B^{0}_{p,\infty}}\lesssim\|\rho_{0}\|_{L^{p}}.$

These two estimates provide space compactness on the velocity field. In order
to achieve time compactness, we will need to estimate the derivative
$\partial_{t}\rho_{N}$. This requires a careful analysis of the product
$\rho_{N}u_{N}$. Once again, we use the low and high frequency decomposition
of the velocity field and write

(14) $\rho_{N}u_{N}=\rho_{N}\Delta_{-1}u_{N}+\rho_{N}({\rm
Id}-\Delta_{-1})u_{N}.$

Estimating the first of these two terms is straightforward: because
$\Delta_{-1}u_{N}$ is already known to be bounded in $L^{\infty}$, we have, by
(12)

$\|\rho_{N}\Delta_{-1}u_{N}\|_{L^{p}}\leq\|\Delta_{-1}u_{N}\|_{L^{\infty}}\|\rho_{N}\|_{L^{p}}\lesssim\|\rho_{0}\|_{L^{p}}^{2}.$

For the other summand, things are much more delicate: the function $({\rm
Id}-\Delta_{-1})u_{N}$ has $B^{\alpha}_{p,\infty}$ regularity, and it is not
obvious that the function product is well defined in $B^{0}_{p,\infty}\times
B^{\alpha}_{p,\infty}$. This is the purpose of the following Lemma.

######  Lemma 3.6.

Consider $\beta>0$ and $q\in[1,+\infty]$ such that the inequality

(15) $\frac{2}{1+\frac{\beta}{d}}<q$

holds. Then, for all $(f,h)\in B^{0}_{q,\infty}\times B^{\beta}_{q,\infty}$,
the product $fh$ is well defined as an element of $B^{\beta-d/q}_{q,\infty}$
and we have the inequality

$\|fh\|_{B^{\beta-d/q}_{q,\infty}}\lesssim\|f\|_{B^{0}_{q,\infty}}\|h\|_{B^{\beta}_{q,\infty}}$

###### Proof (of the Lemma).

The proof is based on the Bony decomposition for the product $fh$, which reads

$fh=T_{f}(h)+\mathcal{T}_{h}(f)+\mathcal{R}(f,h).$

The two paraproducts create no issue. Proposition A.18 yields

$\begin{split}\|T_{f}(h)+\mathcal{T}_{h}(f)\|_{B^{\beta-d/q}_{q,\infty}}&\lesssim\|f\|_{B^{-d/q}_{\infty,\infty}}\|h\|_{B^{\beta}_{q,\infty}}+\|f\|_{B^{0}_{q,\infty}}\|h\|_{B^{\beta-d/q}_{\infty,\infty}}\\\
&\lesssim\|f\|_{B^{0}_{q,\infty}}\|h\|_{B^{\beta}_{q,\infty}},\end{split}$

thanks to the embedding properties of Besov spaces (see Proposition A.9 in the
appendix). However, the remainder $\mathcal{R}(f,h)$ is more difficult to deal
with, because it requires simultaneous conditions on the regularity and
intergrability exponents of $f$ and $h$ simply to be defined. There are two
distinct settings. The first one is when $q\geq 2$, in which case Proposition
A.18 and the condition $\beta>0$ insure that the remainder operator

$\mathcal{R}:B^{0}_{q,\infty}\times
B^{\beta}_{q,\infty}\operatorname*{\longrightarrow}B^{\beta}_{\frac{q}{2},\infty}\subset
B^{\beta-d/q}_{q,\infty}$

is bounded. In the case where we on the contrary have $q<2$, we increase the
integrability indices by means of embeddings of Besov spaces. Let $r\geq q$
whose precise value we will fix later on. We have $B^{\beta}_{q,\infty}\subset
B^{\sigma}_{r,\infty}$ for

$\sigma:=\beta-d\left(\frac{1}{q}-\frac{1}{r}\right).$

For the remainder operator $\mathcal{R}$ to be defined on
$B^{0}_{q,\infty}\times B^{\sigma}_{r,\infty}$, Proposition A.18 requires that
$\sigma>0$ and $1/q+1/r\leq 1$. We take the optimal value: set the value of
$r$ such that

(16) $\frac{1}{q}+\frac{1}{r}=1.$

Then the condition $\sigma>0$ can be written

$\frac{2}{q}<1+\frac{\beta}{d},$

which is an equivalent form of (15). ∎

######  Remark 3.7.

In fact, it is also possible to make sense of the product $fh$ when
$q=q_{0}:=2\left(1+\beta/d\right)^{-1}$, assuming that $(f,h)\in L^{q}\times
H^{\beta}_{q}$, where $H^{\beta}_{q}$ is the potential Sobolev space
associated to the norm $\|({\rm Id}-\Delta)^{s/2}h\|_{L^{q}}$. In that case,
the refined embeddings of Proposition A.14 give, as $q<2$, the inclusion
$L^{q}\times H^{\beta}_{q}\subset B^{0}_{q,2}\times B^{\beta}_{q,2}$, and
Proposition A.18 then insures the remainder operator is bounded on
$B^{0}_{q,2}\times
B^{\sigma}_{r,2}\operatorname*{\longrightarrow}B^{0}_{1,\infty}$, even though
$\sigma=0$.

However, this will not be useful in the sequel as the strict inequality in (7)
is necessary to provide compactness in appropriate spaces later on. In other
words, the non-optimal result in Lemma 3.6 is not responsible for our being
unable to construct weak solutions in the critical space $L^{p_{0}}$ with
$p_{0}=2(1+\alpha/d)^{-1}$.

Thanks to Lemma 3.6, we may bound the second part of the product
$\rho_{N}u_{N}$. We have, under condition (7) from the statement of the
Theorem, and with (13),

$\big{\|}\rho_{N}({\rm
Id}-\Delta_{-1})u_{N}\big{\|}_{B^{\alpha-d/p}_{p,\infty}}\lesssim\|\rho_{N}\|_{B^{0}_{p,\infty}}\|({\rm
Id}-\Delta_{-1})u_{N}\|_{B^{\alpha}_{p,\infty}}\lesssim\|\rho_{0}\|_{L^{p}}^{2}.$

By noticing that $\alpha-d/p<0$ since $p<d/\alpha$ by assumption, we may use
the embeddings $L^{p}\subset B^{0}_{p,\infty}\subset
B^{\alpha-d/p}_{p,\infty}$ to see that the whole product $\rho_{N}u_{N}$
remains uniformly bounded in the space
$L^{\infty}(B^{\alpha-d/p}_{p,\infty})$. This yields the bound

$\|\partial_{t}\rho_{N}\|_{B^{\alpha-1-d/p}_{p,\infty}}\leq\|\rho_{N}u_{N}\|_{B^{\alpha-d/p}_{p,\infty}}\lesssim\|\rho_{0}\|_{L^{p}}^{2}.$

Finally, this estimate allows us to bound the time derivatives of both low and
high frequency components $\partial_{t}\Delta_{-1}u_{N}$ and
$\partial_{t}({\rm Id}-\Delta_{-1})u_{N}$ of the velocity field: for all
$k>0$,

(17)
$\|\partial_{t}\Delta_{-1}u_{N}\|_{L^{\infty}(B^{k}_{\infty,1})}\lesssim\|\partial_{t}\rho_{N}\|_{L^{\infty}(B^{\alpha-1-d/p}_{p,\infty})}\lesssim\|\rho_{0}\|_{L^{p}}^{2}$

and

$\big{\|}\partial_{t}({\rm
Id}-\Delta_{-1})u_{N}\big{\|}_{L^{\infty}(B^{-s}_{p,\infty})}\lesssim\big{\|}\partial_{t}\rho_{N}\big{\|}_{L^{\infty}(B^{\alpha-1-d/p}_{p,\infty})}\lesssim\|\rho_{0}\|_{L^{p}}^{2},$

where we have set $-s=2\alpha-1-d/p<0$. As a consequence, both low and high
frequency components of the velocity field are bounded in spaces that provide
time compactness, namely, for all $T>0$ and $k>0$,

(18) $\|\Delta_{-1}u_{N}\|_{W^{1,\infty}_{T}(B^{k}_{\infty,1})}+\big{\|}({\rm
Id}-\Delta_{-1})u_{N}\big{\|}_{W^{1,\infty}_{T}(B^{-1-d}_{p,\infty})}\lesssim\|\rho_{0}\|_{L^{p}}^{2}+\|\rho_{0}\|_{L^{p}}.$

Our last estimate will be an interpolation inequality in order to get
simultaneous time and space compactness on $({\rm Id}-\Delta_{-1})u_{N}$ (this
is already done for $\Delta_{-1}u_{N}$). Consider $\theta\in]0,1[$ whose
precise value we will fix later on. We define the quantity
$\sigma:=-s\theta+(1-\theta)\alpha$ and, for simplicity of notation, note
$V_{N}:=({\rm Id}-\Delta_{-1})u_{N}$. Then, for all $t_{1},t_{2}\geq 0$, we
have by Proposition A.10,

$\begin{split}\big{\|}V_{N}(t_{2})-V_{N}(t_{1})\big{\|}_{B^{\sigma}_{p,\infty}}&\lesssim\big{\|}V_{N}(t_{2})-V_{N}(t_{1})\big{\|}_{B^{-s}_{p,\infty}}^{\theta}\big{\|}V_{N}(t_{2})-V_{N}(t_{1})\big{\|}_{B^{\alpha}_{p,\infty}}^{1-\theta}\\\
&\lesssim|t_{2}-t_{1}|^{\theta}\big{\|}V_{N}\big{\|}_{W^{1,\infty}_{T}(B^{-s}_{p,\infty})}^{\theta}\big{\|}V_{N}\big{\|}_{L^{\infty}_{T}(B^{\alpha}_{p,\infty})}^{1-\theta},\end{split}$

so that the functions $V_{N}=({\rm Id}-\Delta_{-1})u_{N}$ are uniformly
bounded in the space $C^{0,\theta}_{T}(B^{\sigma}_{p,\infty})$.

STEP 3: convergence to a solution. Let us summarize the uniform bounds we have
obtained so far. We have, for every $N\geq 1$, $k>0$ and $T>0$,

(19)
$\|\rho_{N}\|_{L^{\infty}(L^{p})}+\|\Delta_{-1}u_{N}\|_{W^{1,\infty}_{T}(B^{k}_{\infty,1})}+\big{\|}({\rm
Id}-\Delta_{-1})u_{N}\big{\|}_{C^{0,\theta}_{T}(B^{\sigma}_{p,\infty})}\lesssim\|\rho_{0}\|_{L^{p}}+\|\rho_{0}\|_{L^{p}}^{2}.$

Here, $\sigma\in]-s,\alpha[$ can be chosen positive and as close to $\alpha$
as desired while still keeping the Hölder exponent positive $\theta>0$. In
addition, if, as in the statement of Theorem 3.1, the initial datum is $L^{q}$
for some $q\in[1,+\infty]$, then we also have

(20)
$\|\rho_{N}\|_{L^{\infty}(L^{q})}=\|\rho_{0,N}\|_{L^{q}}\leq\|\rho_{0}\|_{L^{q}}.$

Recall that $p>1$ because $\alpha<d$, so that all spaces involved (except
possibly $L^{q}$) have separable preduals. We deduce the convergence of the
sequences $(\rho_{N})_{N}$, $(\Delta_{-1}u_{N})_{N}$ and $\big{(}({\rm
Id}-\Delta_{-1})u_{N}\big{)}_{N}$, up to an extraction, in the following
topologies (here $K\subset\mathbb{R}^{d}$ is an arbitrary compact set): as
$N\rightarrow+\infty$, we have

(21)
$\begin{split}&\rho_{N}\operatorname*{\,\,\rightharpoonup\,\,}^{*}\rho\qquad\text{in
}L^{\infty}(L^{p})\\\ &\Delta_{-1}u_{N}\rightarrow u_{LF}\qquad\text{in
}L^{\infty}_{T}(L^{\infty}_{\rm loc})\\\ &({\rm
Id}-\Delta_{-1})u_{N}\rightarrow u_{HF}\qquad\text{in
}L^{\infty}_{T}(B^{\beta}_{p,\infty}(K)).\end{split}$

In the lines above, $\beta<\sigma$ is an exponent whose value we will chose
later, and the functions $u_{LF}$ and $u_{HF}$ are the limits of the Low
Frequency and High Frequency parts. We accordingly set $u:=u_{LF}+u_{HF}$ so
that $u_{N}\operatorname*{\longrightarrow}u$ in, say,
$\mathcal{D}^{\prime}(]0,T[\times\mathbb{R}^{d})$. Convergence of the initial
data $\rho_{0,N}$ follows from the fact that $p<+\infty$ (by assumption), and
so (recall that $0\leq\Psi\leq 1$ is a cut-off function),

$\begin{split}\|\rho_{0,N}-\rho_{0}\|_{L^{p}}&\leq\big{\|}\Psi(x/N)({\rm
Id}-S_{N})\rho_{0}(x)\big{\|}_{L^{p}}+\big{\|}(1-\Psi(x/N))\rho_{0}(x)\big{\|}_{L^{p}}\\\
&\leq\big{\|}({\rm
Id}-S_{N})\rho_{0}(x)\big{\|}_{L^{p}}+o(1)\operatorname*{\longrightarrow}_{N\rightarrow+\infty}0.\end{split}$

As weak convergence of the densities insures that
$\partial_{t}\rho_{N}\operatorname*{\longrightarrow}\partial_{t}\rho$ in
$\mathcal{D}^{\prime}(\mathbb{R}_{+}^{*}\times\mathbb{R}^{d})$, the only real
problem is the convergence of the product $\rho_{N}u_{N}$. We will once again
use decomposition (14). On the one hand, the convergence results (21) above
make it clear that

(22) $\rho_{N}\Delta_{-1}u_{N}\operatorname*{\,\,\rightharpoonup\,\,}^{*}\rho
u_{LF}\qquad\text{in }L^{\infty}(L^{p}).$

The treatment of the other part of the product is more delicate. For this, we
fix a test function $\psi\in\mathcal{D}(]0,+\infty[\times\mathbb{R}^{d})$ and
study the brackets

(23) $\Big{\langle}\rho_{N}({\rm Id}-\Delta_{-1})u_{N}-\rho
u_{HF},\psi\Big{\rangle}=\Big{\langle}\rho_{N}\big{(}({\rm
Id}-\Delta_{-1})u_{N}-u_{HF}\big{)},\psi\Big{\rangle}+\big{\langle}\rho-\rho_{N},\psi
u_{HF}\big{\rangle}.$

Concerning the first bracket, we fix a function
$\varphi\in\mathcal{D}(]0,+\infty[\times\mathbb{R}^{d})$ such that
$\varphi\psi=\psi$. In order to use the strong convergence of the $({\rm
Id}-\Delta_{-1})u_{N}$ from (21), we wish to apply the product Lemma 3.6 to
$B^{0}_{p,\infty}\times B^{\beta}_{p,\infty}$. For this, we need $\beta$ to be
close enough to $\alpha$ that

(24) $\frac{2}{1+\frac{\alpha}{d}}<\frac{2}{1+\frac{\beta}{d}}<p,$

and this is always possible due to the strict inequality in (7). Under this
condition, and using the fact that $B^{\beta-d/p}_{p,\infty}$ has (pre)dual
$B^{d/p-\beta}_{p^{\prime},1}$, we obtain

$\left|\Big{\langle}\rho_{N}\big{(}({\rm
Id}-\Delta_{-1})u_{N}-u_{HF}\big{)},\psi\Big{\rangle}\right|\leq\|\psi\|_{L^{1}_{T}(B^{d/p-\beta}_{p^{\prime},1})}\,\|\rho_{N}\|_{L^{\infty}(B^{0}_{p,\infty})}\\\
\times\Big{\|}\big{(}({\rm
Id}-\Delta_{-1})u_{N}-u_{HF}\big{)}\varphi\Big{\|}_{L^{\infty}_{T}(B^{\beta}_{p,\infty})}\;\operatorname*{\longrightarrow}_{N\rightarrow+\infty}0.$

For the second bracket in (23), since we know that $\rho_{N}$ converges to
$\rho$ in the weak-$(*)$ topology of $L^{\infty}_{T}(L^{p})$, we must make
sure that $u_{HF}\psi$ lies in the predual $L^{1}(L^{p^{\prime}})$. Here, it
is useful to consider two cases. Firstly, if $p\geq 2$, then we must have
$p^{\prime}\leq 2\leq p$, and since $\beta>0$, the function $u_{HF}$ must be
locally $B^{\beta}_{p,\infty}\subset L^{p}\subset L^{p^{\prime}}_{\rm loc}$.
Secondly, if $p\leq 2\leq p^{\prime}$, we must check that
$B^{\beta}_{p,\infty}\subset L^{p^{\prime}}$. As $p^{\prime}\geq p$,
embeddings of Besov spaces give the chain of inclusions

$B^{\beta}_{p,\infty}\subset
B^{\beta-d\left(\frac{1}{p}-\frac{1}{p^{\prime}}\right)}_{p^{\prime},\infty}\subset
L^{p^{\prime}},$

as long as the condition

$\beta-d\left(\frac{1}{p}-\frac{1}{p^{\prime}}\right)>0$

holds. But this is equivalent to (24), which we already enforced on $\beta$.
Therefore, in both cases, we have shown that $u_{HF}\psi\in
L^{1}(L^{p^{\prime}})$, so that we indeed have

$\big{\langle}\rho-\rho_{N},\psi
u_{HF}\big{\rangle}\operatorname*{\longrightarrow}0\qquad\text{as
}N\rightarrow+\infty.$

Together with (22), this shows convergence of the whole product
$\rho_{N}u_{N}$ to

$\rho_{N}u_{N}\operatorname*{\longrightarrow}\rho u_{LF}+\rho u_{HF}=\rho
u\qquad\text{in }\mathcal{D}^{\prime}(]0,+\infty[\times\mathbb{R}^{d}).$

Finally, we have to check that $(\rho,u)$ is indeed a solution of the
fractional Stokes problem. Because of the convergence properties (21), we
gather that the Fourier transforms of $u_{N}$ and $\rho_{N}$ converge locally,
e.g. in the sense of distributions:

$(\widehat{u_{N}},\widehat{\rho_{N}})\operatorname*{\longrightarrow}(\widehat{u},\widehat{\rho})\qquad\text{in
}\mathcal{D}^{\prime}(]0,+\infty[\times\mathbb{R}^{d}).$

Let us fix $\phi(t,\xi)\in\mathcal{D}(]0,+\infty[\times\mathbb{R}^{d})$ that
is supported away from $\xi=0$. Then $T_{N}\phi=\phi$ for $N$ large enough. By
noting $m(\xi)$ the symbol of the operator $\mathbb{P}g$, we see that

$\langle\widehat{u_{N}},\phi\rangle=\left\langle\widehat{\rho_{N}}(t,\xi),\frac{m(\xi)}{|\xi|^{\alpha}}\phi(t,\xi)\right\rangle\operatorname*{\longrightarrow}_{N\rightarrow+\infty}\left\langle\widehat{\rho}(t,\xi),\frac{m(\xi)}{|\xi|^{\alpha}}\phi(t,\xi)\right\rangle,$

where the brackets refer to the
$\mathcal{D}^{\prime}(]0,+\infty[\times\mathbb{R}^{d})\times\mathcal{D}(]0,+\infty[\times\mathbb{R}^{d})$
duality. Uniqueness of the limit implies that

$\widehat{u}(t,\xi)=\frac{m(\xi)}{|\xi|^{\alpha}}\widehat{\rho}(\xi)\qquad\text{in
}\mathcal{D}^{\prime}\big{(}]0,+\infty[\times(\mathbb{R}^{d}\setminus\\{0\\})\big{)},$

so that $u$ is indeed equal, up to a polynomial, to the unique
$\mathcal{S}^{\prime}_{h}$ solution of the Stokes equation with datum $\rho$,
as given by Proposition 2.1, and we only need to check that
$u(t)\in\mathcal{S}^{\prime}_{h}$ for almost all $t>0$. This follows from
Remark A.13 and the fact that the sequence $(u_{N})_{N}$ is uniformly bounded
in a homogeneous Besov space:

$\|u_{N}\|_{\dot{B}^{\alpha}_{p,\infty}}\lesssim\|\rho_{N}\|_{L^{p}},$

so that we must have $u\in L^{\infty}(\dot{B}^{\alpha}_{p,\infty})$, and hence
$u(t)\in\mathcal{S}^{\prime}_{h}$ for almost all $t>0$.

STEP 4: $L^{q}$ estimates. Finally, we only have to prove the uniform $L^{q}$
bounds (8) hold for the solution. If $q>1$, it is a simple consequence of the
$L^{q}$ boundedness (20) of the approximate solutions, their weak-$(*)$
convergence and the Banach-Steinhaus theorem. When $q=1$ however, the space
$L^{1}$ does not have a separable predual, so the argument fails. Instead, we
view $L^{1}$ as a subset of the space $\mathcal{M}$ of finite measures, whose
predual is the space $C_{0}$ of continuous functions that tend to zero at
infinity. Applying the Banach-Steinhaus theorem to $\mathcal{M}$ gives the
bound

$\|\rho\|_{\mathcal{M}}\leq\|\rho_{0}\|_{L^{1}}.$

But we know that $\rho$ also is an $L^{p}$ function with $1<p<+\infty$.
Consequently, $\rho$ is locally integrable and the $L^{1}$ and $\mathcal{M}$
norms therefore agree for $\rho$. In other words

$\|\rho\|_{L^{1}}=\|\rho\|_{\mathcal{M}}\leq\|\rho_{0}\|_{L^{1}}.$

This remark ends the proof. ∎

##  4 Local well-posedness

In this Section, we are concerned with proving local well-posedness of (1) by
means of a fixed-point argument. This result is mainly interesting when
$\alpha<1$, as we will have means of proving global well-posedness when
$\alpha\geq 1$. In particular, this means that we do not worry too much about
the case $d=2$, where the value $\alpha=2$ is of special interest, although
Theorem 4.1 could be adapted to fit that setting.

######  Theorem 4.1.

Consider a dimension $d\geq 2$ as well as indices $s>0$ and $p\in[1,+\infty]$
such that

$p<\frac{d}{\alpha},\qquad\text{ and }\qquad s\geq 1-\alpha.$

For all initial datum $\rho_{0}\in B^{s}_{\infty,1}\cap L^{p}$, there exists a
time $T>0$ such that the system (1) has a unique solution in that space
$\rho\in C^{0}([0,T[;B^{s}_{\infty,1}\cap L^{p})$. Moreover, the lifespan
$T^{*}$ of this solution satisfies the inequality

(25) $T^{*}\geq\frac{C}{\|\rho_{0}\|_{B^{s}_{\infty,1}\cap L^{p}}}.$

Finally, the lifespan $T^{*}$ is finite if and only if

$\int_{0}^{T^{*}}V(t)\,{\rm d}t=+\infty,$

where the quantity $V(t)$ is defined by $V(t)=\|\nabla u\|_{L^{\infty}}$ if
$s<1$ and $V(t)=\|\nabla u\|_{L^{\infty}}+\|\nabla\rho\|_{L^{\infty}}$
otherwise.

Let us make a few remarks concerning Theorem 4.1, and in particular about the
lifespan of the local solution.

######  Remark 4.2.

A direct consequence of the continuation criterion in the Theorem above is
that the problem (1) is globally well-posed when $\alpha>1$, under appropriate
conditions on the initial datum. The following Corollary explains why this is
the case.

Note that, for technical reasons, the 2D Transport-Stokes system $\alpha=d=2$
is excluded (in that case, we would need $p<1$). However, this case of special
interest will be covered by Theorem 6.5 below, which is more general than
Theorem 4.1 on the question of global solutions when $\alpha\geq 1$.

Similarly, the case $\alpha=1$, which is especially relevant as system (1)
then behaves nearly identically to the 2D Euler system, is not in the scope of
Corollary 4.3. Just as the 2D Euler equations, the setting $\alpha=1$
necessitates different methods which we will employ to prove the more advanced
results of Theorem 6.5 below.

######  Corollary 4.3.

With the notation of Theorem 4.1, we make the additional assumptions that
$s<1$ and $\alpha>1$. Then the unique solution provided by Theorem 4.1 is in
fact global.

###### Proof (of the Corollary).

Let $\rho\in C^{0}([0,T^{*}[;L^{p}\cap B^{s}_{\infty,1})$ be the unique
solution given by Theorem 4.1, where $T^{*}$ is, as above, the lifespan of the
solution. Then, the velocity field has Lipschitz regularity $u\in
L^{1}([0,T^{*}[;W^{1,\infty})$, as Proposition 2.2 gives

(26)
$\|u\|_{B^{1}_{\infty,1}}\lesssim\|\rho\|_{L^{p}}+\|\rho\|_{B^{1-\alpha}_{\infty,1}}.$

As a consequence, the conservation of Lebesgue norms holds: for every
$q\in[1,+\infty]$, we have $\|\rho(t)\|_{L^{q}}=\|\rho_{0}\|_{L^{q}}$. In
particular, if we choose $q>d/(\alpha-1)$, then the solution will be
$L^{\infty}(L^{q})$ provided $\rho_{0}\in L^{q}$. Let us show that this is
indeed the case: because $\rho_{0}\in B^{s}_{\infty,1}\subset L^{\infty}$ and

$p<\frac{d}{\alpha}<\frac{d}{1-\alpha}<q<+\infty,$

interpolation between $L^{p}$ and $L^{\infty}$ shows that we must have
$\|\rho_{0}\|_{L^{q}}\leq\|\rho_{0}\|_{L^{p}}^{\theta}\|\rho_{0}\|_{L^{\infty}}^{1-\theta}$
with $\theta=p(\alpha-1)/d$. Together with (26), the embeddings

$L^{q}\subset B^{0}_{q,\infty}\subset B^{-d/q}_{\infty,\infty}\subset
B^{1-\alpha}_{\infty,1},$

show that the Lipschitz norm of $u$ is always bounded
$\|u\|_{B^{1}_{\infty,1}}\lesssim\|\rho_{0}\|_{L^{q}}$, so that the integral

$\int_{0}^{T}\|\nabla u\|_{L^{\infty}}\,{\rm d}t\lesssim
T\|\rho_{0}\|_{L^{q}}$

remains finite for every $T>0$. By the continuation criterion of Theorem 4.1,
we deduce that the solution is global $T^{*}=+\infty$. ∎

######  Remark 4.4.

Due to the global existence of weak solutions in Theorem 3.1 above, explosion
of solutions should be thought of as a loss of regularity (and therefore
possible loss of uniqueness) at the time of blow-up. In that spirit, note that
the finiteness of the usual $\|\nabla u\|_{L^{1}_{T}(L^{\infty})}$ norm is not
enough to provide continuation of solutions with $s\geq 1$. If
$\|\nabla\rho\|_{L^{\infty}}$ blows-up but not $\|\nabla u\|_{L^{\infty}}$,
there may be a loss of regularity at the singularity, but while still
retaining uniqueness of solutions.

######  Remark 4.5.

Finally, as is usually the case with such equations, the continuation
criterion implies that the lifespan of the $C^{0}_{T}(B^{s}_{\infty,1})$
solution is essentially independent of $s$. Therefore, the lower bound (25)
can be reformulated as

$T^{*}\geq\frac{C}{\|\rho_{0}\|_{B^{1-\alpha}_{\infty,1}\cap L^{p}}}\text{ if
}s<1,\qquad\text{or}\qquad
T^{*}\geq\frac{C}{\|\rho_{0}\|_{B^{1}_{\infty,1}\cap L^{p}}}\text{ if }s\geq
1.$

######  Remark 4.6.

By taking into account the structure of the equations implied by gravity
stratification, we will be able to exhibit a continuation which involves the
derivative of the density in a single direction $\partial_{d}\rho$. We refer
to Theorem 5.1 below.

###  4.1 A Priori Estimates

######  Proposition 4.7.

Assume that $s\geq 1-\alpha$ satisfies $s>-1$ and that $\alpha<1$ or
$p<d/(\alpha-1)$. For any regular solution $\rho$ of (1) associated to the
initial datum $\rho_{0}$, there exists a time $T>0$ such that

$\|\rho\|_{L^{\infty}_{T}(B^{s}_{\infty,1}\cap
L^{p})}\lesssim\|\rho_{0}\|_{B^{s}_{\infty,1}\cap L^{p}}.$

In addition, inequality (25) holds for $T$.

###### Proof.

Finding a priori estimates for problem (1) is based on the theory of transport
equations in Besov spaces. The idea is to bound every dyadic block
$\Delta_{j}\rho$ of the solution in $L^{\infty}$ to find a $B^{s}_{\infty,1}$
estimate. Fix a $j\geq-1$ and apply the operator $\Delta_{j}$ to the transport
equation in (1). We have:

(27)
$\big{(}\partial_{t}+u\cdot\nabla\big{)}\Delta_{j}\rho=\big{[}u\cdot\nabla,\Delta_{j}\big{]}\rho.$

Now, Lemma A.19 lets us bound the commutator in the righthand side in terms of
$\rho$ and $\nabla u$. Let us treat separately the case wheres $s<1$ and
$s\geq 1$.

First case. In the first case $s<1$, applying Lemma A.19 to our setting gives
the following inequality:

$2^{js}\left\|\big{[}u\cdot\nabla,\Delta_{j}\big{]}\rho\right\|_{L^{p}}\lesssim
c_{j}(t)\|\nabla
u\|_{L^{\infty}}\|\rho\|_{B^{s}_{\infty,1}}=c_{j}(t)V(t)\|\rho\|_{B^{s}_{\infty,1}},$

where the sequences $\big{(}c_{j}(t)\big{)}_{j\geq-1}$ have unit norm in the
space $\ell^{1}(j\geq-1)$ and where $V(t)$ is defined as in the statement of
Theorem 4.1. In order to have an upper bound that depends only on the unknown
$\rho$, we express the derivatives of the velocity field as the image of the
density $\nabla u=\nabla(-\Delta)^{-\alpha/2}\mathbb{P}(\rho g)$ by a Fourier
multiplier of degree $1-\alpha$ and use Proposition 2.2 to obtain

(28) $V(t)=\|\nabla
u\|_{L^{\infty}}\lesssim\|\rho\|_{L^{p}}+\|\rho\|_{B^{1-\alpha}_{\infty,1}}\lesssim\|\rho_{0}\|_{L^{p}}+\|\rho\|_{B^{s}_{p,r}}.$

Here, we have used the fact that, for regular solutions, the $L^{p}$ norms are
simply transported by the (incompressible) flow of $u$. Overall, we may write
the following inequality:

(29)
$2^{js}\left\|\big{[}u\cdot\nabla,\Delta_{j}\big{]}\rho\right\|_{L^{p}}\lesssim
c_{j}(t)V(t)\|\rho\|_{B^{s}_{\infty,1}}\lesssim
c_{j}(t)\big{(}\|\rho_{0}\|_{L^{p}}+\|\rho\|_{B^{s}_{\infty,1}}\big{)}^{2}.$

Second case. In the second case, where $s\geq 1$, an additional term appears
in the commutator inequalities of Lemma A.19, so that we instead get

$\begin{split}2^{js}\left\|\big{[}u\cdot\nabla,\Delta_{j}\big{]}\rho\right\|_{L^{p}}&\lesssim
c_{j}(t)\Big{(}\|\nabla
u\|_{L^{\infty}}\|\rho\|_{B^{s}_{\infty,1}}+\|\nabla\rho\|_{L^{\infty}}\|\nabla
u\|_{B^{s-1}_{\infty,1}}\Big{)}\\\
&=c_{j}(t)V(t)\Big{(}\|\rho\|_{B^{s}_{\infty,1}}+\|\nabla
u\|_{B^{s-1}_{\infty,1}}\Big{)},\end{split}$

where, once again, the sequences $\big{(}c_{j}(t)\big{)}_{j\geq-1}$ have unit
norm in the space $\ell^{1}(j\geq-1)$ and the quantity $V(t)$ is the one
defined in Theorem 4.1. Now, because $s\geq 1$, the space
$B^{s}_{\infty,1}\subset B^{1}_{\infty,1}\subset W^{1,\infty}$ is embedded in
the space of Lipschitz functions and
$\|\nabla\rho\|_{L^{\infty}}\lesssim\|\rho\|_{B^{s}_{\infty,1}}$. We may
therefore write an inequality similar to (28), namely

$V(t)=\|\nabla
u\|_{L^{\infty}}+\|\nabla\rho\|_{L^{\infty}}\lesssim\|\rho\|_{L^{p}}+\|\rho\|_{B^{s}_{\infty,1}}.$

Likewise, we have

$\|\nabla u\|_{B^{s-1}_{\infty,1}}\lesssim\|\nabla
u\|_{B^{\alpha+s-1}_{\infty,1}}\lesssim\|\rho\|_{L^{p}}+\|\rho\|_{B^{s}_{\infty,1}},$

so that the whole commutator is bounded by an inequality that is nearly
identical to (29),

(30)
$2^{js}\left\|\big{[}u\cdot\nabla,\Delta_{j}\big{]}\rho\right\|_{L^{p}}\lesssim
c_{j}(t)V(t)\Big{(}\|\rho_{0}\|_{L^{p}}+\|\rho\|_{B^{s}_{\infty,1}}\Big{)}\lesssim
c_{j}(t)\big{(}\|\rho_{0}\|_{L^{p}}+\|\rho\|_{B^{s}_{\infty,1}}\big{)}^{2}.$

Note that both inequalities in (30) also hold in the first case, where $s<1$.

Inequality (30), which, recall, is valid regardless of whether $s<1$ or not,
allows us to perform an $L^{\infty}$ estimate on equation (27). We obtain, for
all $j\geq-1$ and $0\leq t\leq T$,

(31) $\begin{split}2^{js}\|\Delta_{j}\rho(t)\|_{L^{\infty}}&\leq
2^{js}\|\Delta_{j}\rho_{0}\|_{L^{\infty}}+C\int_{0}^{T}c_{j}(t)V(t)\Big{(}\|\rho\|_{L^{p}}+\|\rho\|_{B^{s}_{\infty,1}}\Big{)}\,{\rm
d}t\\\ &\leq
2^{js}\|\Delta_{j}\rho_{0}\|_{L^{\infty}}+C_{0}\int_{0}^{T}c_{j}(t)\Big{(}\|\rho\|_{L^{p}}+\|\rho\|_{B^{s}_{\infty,1}}\Big{)}^{2}\,{\rm
d}t.\end{split}$

In the above, $C_{0}$ is a fixed constant which we will use in further
estimates. Taking the sum over all $j\geq-1$ provides a integral inequality
involving the norms of the solution. In order to simpify notations further, we
set

$E(t)=\|\rho\|_{L^{p}}+\|\rho\|_{B^{s}_{\infty,1}}=\|\rho_{0}\|_{L^{p}}+\|\rho\|_{B^{s}_{\infty,1}},$

so that

(32) $E(T)\leq E(0)+C_{0}\int_{0}^{T}V(t)E(t)\,{\rm d}t\leq
E(0)+C_{0}\int_{0}^{T}E(t)^{2}\,{\rm d}t.$

Incidentally, note that if $V(t)$ is integrable on $[0,T]$, then $E(t)$ cannot
blow-up. In order to deduce upper bounds on the solution from this estimate,
we work on a time interval $[0,T_{0}]$ where the integral of the previous
inequality is not too large. More precisely, set

$T_{0}:=\sup\left\\{T>0,\quad\int_{0}^{T}E(t)^{2}\,{\rm d}t\leq
E(0)\right\\},$

so that for all $T\leq T_{0}$, we must have $E(t)\leq(1+C_{0})E(0)$. In
particular, so long as $T\leq T_{0}$, the integral in (32) cannot be larger
than

$\int_{0}^{T}E(t)^{2}\,{\rm d}t\leq(1+C_{0})^{2}E(0)^{2},$

which implies that $T_{0}$ is nonzero and bounded from below by

$T_{0}\geq\frac{(1+C_{0})^{-1}}{E(0)}.$

This fact ends the proof. ∎

###  4.2 Stability Estimates

The purpose of this paragraph is to prove stability estimates that will serve
to implement a fixed point argument, thus constructing unique solutions. We
will have to study a system of the form

$\begin{cases}\big{(}\partial_{t}+u\cdot\nabla\big{)}\delta\rho=-(\delta
u\cdot\nabla)\rho\\\ \delta u=(-\Delta)^{\alpha/2}\mathbb{P}(\delta\rho
g),\end{cases}$

where $(\rho,u)$ is a solution of (1) and $(\delta\rho,\delta u)$ are, loosely
stated, perturbations of this solution. Our work in this Section is summarized
in the following Proposition.

######  Proposition 4.8.

Assume that the pair $(s,p)$ fulfills the assumptions of Theorem 4.1, that is
$s>0$, $s\geq 1-\alpha$ and $p<d/p$. Consider $\rho\in B^{s}_{\infty,1}\cap
L^{p}$ and $\delta\rho\in B^{s-1}_{\infty,1}\cap B^{-1}_{p,\infty}$ associated
to $\delta u$ through $\delta u=(-\Delta)^{-\alpha/2}\mathbb{P}(\delta\rho
g)$. Then the product $\operatorname*{\rm{div}}(\rho\delta u)$ belongs to
$B^{s-1}_{\infty,1}\cap B^{-1}_{p,\infty}$ and the estimate

$\big{\|}\operatorname*{\rm{div}}(\rho\delta
u)\big{\|}_{B^{s-1}_{\infty,1}\cap
B^{-1}_{p,\infty}}\lesssim\|\rho\|_{B^{s}_{\infty,1}\cap
L^{p}}\|\delta\rho\|_{B^{s-1}_{\infty,1}\cap L^{p}}$

holds.

###### Proof.

We start by finding $B^{s-1}_{\infty,1}$ estimates for the product. Our
argument is based on the Bony decomposition: owing to the fact that $\delta u$
is divergence free, we may write

$\operatorname*{\rm{div}}(\rho\delta u)=\mathcal{T}_{\delta
u_{k}}(\partial_{k}\rho)+\mathcal{T}_{\partial_{k}\rho}(\delta
u_{k})+\partial_{k}\mathcal{R}(\delta u_{k},\rho),$

where, here and below, there is an implicit summation on the repeated index
$k=1,...,d$. Let us start by finding bounds for the velocity perturbation
$\delta u$ which involve only $\delta\rho$. By using Proposition 2.2 we find
that

(33) $\|\delta
u\|_{B^{s+\alpha-1}_{\infty,1}}\lesssim\|\delta\rho\|_{B^{-1}_{p,\infty}}+\|\delta\rho\|_{B^{s}_{\infty,1}}.$

We now look at the paraproducts. The first one is straightforward to bound
with Proposition A.18, as $\delta u\in B^{0}_{\infty,1}\subset L^{\infty}$ is
a bounded function (recall that $s+\alpha-1\geq 0$). We may therefore use (33)
to obtain

$\|\mathcal{T}_{\delta
u_{k}}(\partial_{k}\rho)\|_{B^{s-1}_{\infty,1}}\lesssim\|\nabla\rho\|_{B^{s-1}_{\infty,1}}\|\delta
u\|_{L^{\infty}}\lesssim\|\rho\|_{B^{s}_{\infty,1}}\Big{(}\|\delta\rho\|_{B^{-1}_{p,\infty}}+\|\delta\rho\|_{B^{s}_{\infty,1}}\Big{)}.$

For the second one, we must consider whether $s\geq 1$, in which case
$\partial_{k}\rho\in L^{\infty}$ or not. If indeed we have $s\geq 1$, then the
paraproduct assumes the regularity of $\delta u$ and

$\begin{split}\|\mathcal{T}_{\partial_{k}\rho}(\delta
u_{k})\|_{B^{s-1}_{\infty,1}}&\leq\|\mathcal{T}_{\partial_{k}\rho}(\delta
u_{k})\|_{B^{s+\alpha-1}_{\infty,1}}\\\
&\lesssim\|\rho\|_{B^{s}_{\infty,1}}\|\delta
u\|_{B^{s+\alpha-1}_{\infty,1}}\\\
&\lesssim\|\rho\|_{B^{s}_{\infty,1}}\Big{(}\|\delta\rho\|_{B^{-1}_{p,\infty}}+\|\delta\rho\|_{B^{s}_{\infty,1}}\Big{)}.\end{split}$

However, in the case where $s<1$, the derivatives $\partial_{k}\rho$ induce a
loss of regularity, according to Proposition A.18, and the solution cannot be
simply to factor out the divergence $\partial_{k}\mathcal{T}_{\rho}(\delta
u_{k})$, as we would not be making full use of the regularity of $\rho$. By
use of Proposition A.18, we have

$\|\mathcal{T}_{\partial_{k}\rho}(\delta
u_{k})\|_{B^{2s+\alpha-2}_{\infty,1}}\lesssim\|\nabla\rho\|_{B^{s-1}_{\infty,\infty}}\|\delta
u\|_{B^{s+\alpha-1}_{\infty,1}},$

but given that $s\geq 1-\alpha$, we must have $2s-2+\alpha\geq s-1$, and we
still obtain the desired bound:

$\|\mathcal{T}_{\partial_{k}\rho}(\delta
u_{k})\|_{B^{s-1}_{\infty,1}}\lesssim\|\rho\|_{B^{s}_{\infty,1}}\Big{(}\|\delta\rho\|_{B^{-1}_{p,\infty}}+\|\delta\rho\|_{B^{s}_{\infty,1}}\Big{)}.$

We now only are left with the remainder $\partial_{k}\mathcal{R}(\delta
u_{k},\rho)$. In order for it to be properly defined (as an element of
$B^{0}_{\infty,\infty}$), it requires that the sum of the regularities of
$\partial_{k}\rho$ and $\delta u_{k}$ be nonnegative $2s-1+\alpha\geq 0$, or
in other words $s\geq\frac{1}{2}(1-\alpha)$. However, our assumptions bearing
on $s$ show that in fact $s>\frac{1}{2}(1-\alpha)$, as we either have
$\alpha<1$ and

$s\geq 1-\alpha>\frac{1}{2}(1-\alpha),$

or we have $\alpha\geq 1$ and

$s>0\geq\frac{1}{2}(1-\alpha)\geq 1-\alpha.$

In both cases, $2s-1+\alpha>0$ and we may apply Proposition A.18 to get

$\begin{split}\big{\|}\partial_{k}\mathcal{R}(\delta
u_{k},\rho)\big{\|}_{B^{s-1}_{\infty,1}}&\lesssim\|\mathcal{R}(\delta
u_{k},\rho)\|_{B^{2s+\alpha-1}_{\infty,1}}\\\
&\lesssim\|\rho\|_{B^{s}_{\infty,1}}\|\delta u\|_{B^{s+\alpha}_{\infty,1}}\\\
&\lesssim\|\rho\|_{B^{s}_{\infty,1}}\Big{(}\|\delta\rho\|_{B^{-1}_{p,\infty}}+\|\delta\rho\|_{B^{s}_{\infty,1}}\Big{)}.\end{split}$

Overall, we have shown that the quantity $\operatorname*{\rm{div}}(\rho\delta
u)$ is bounded in the space $B^{s-1}_{\infty,1}$ by

$\|\rho\|_{B^{s}_{\infty,1}}\Big{(}\|\delta\rho\|_{B^{-1}_{p,\infty}}+\|\delta\rho\|_{B^{s}_{\infty,1}}\Big{)}.$

It now only remains to find the $B^{-1}_{p,\infty}$ bound. Although it is
possible to do this by means of the Bony decomposition, a much faster way
exists: $\operatorname*{\rm{div}}(\rho\delta u)$ simply is the derivative of a
$L^{p}$ function, so

$\begin{split}\big{\|}\operatorname*{\rm{div}}(\rho\delta
u)\big{\|}_{B^{-1}_{p,\infty}}&\lesssim\|\rho\delta u\|_{B^{0}_{p,\infty}}\\\
&\lesssim\|\rho\delta u\|_{L^{p}}\\\ &\lesssim\|\rho\|_{L^{p}}\|\delta
u\|_{L^{\infty}}\\\
&\lesssim\|\rho\|_{L^{p}}\Big{(}\|\delta\rho\|_{B^{-1}_{p,\infty}}+\|\delta\rho\|_{B^{s}_{\infty,1}}\Big{)},\end{split}$

and this finally ends the proof of the Proposition.

∎

###  4.3 Proof of Theorem4.1: Iterative Scheme

Let us now finish proving Theorem 4.1. We will construct a solution of (1) by
means of an iterative scheme, and prove its uniqueness.

###### Proof (of Theorem 4.1).

STEP 1: iterative scheme. Let us define a sequence $(\rho_{n})_{n\geq 1}$ of
approximate solutions in the following way. For the initial data, set
$\rho_{0,n}=S_{n}\rho_{0}$ and define $\rho_{1}(t,x):=\rho_{0,1}(x)$ and, for
$n\geq 1$, $\rho_{n+1}$ is the unique global in time solution of

(34) $\begin{cases}\big{(}\partial_{t}+u_{n}\cdot\nabla\big{)}\rho_{n+1}=0\\\
u_{n}=(-\Delta)^{\alpha/2}\mathbb{P}(\rho_{n}g),\end{cases}\qquad\text{with
initial datum }\rho_{n+1}(0)=\rho_{0,n+1}.$

For notational convenience, we call $X:=B^{s}_{\infty,1}\cap L^{p}$ the space
in which the estimates for the solution are made, and
$Y:=B^{s-1}_{\infty,1}\cap B^{-1}_{p,\infty}$ the one in which we make the
stability estimates. Firstly, by performing mutatis mutandi the same
computations as those of Proposition 4.7, we obtain from (34) that, for all
$n\geq 1$ and $T>0$,

$\|\rho_{n+1}\|_{L^{\infty}_{T}(X)}\lesssim\|\rho_{0}\|_{X}+\int_{0}^{T}\|\rho_{n}\|_{X}\|\rho_{n+1}\|_{X}\,{\rm
d}t.$

Similarly, by the same arguments as those of Proposition 4.7, by defining the
times $T_{n}>0$ by

$T_{n+1}:=\sup\left\\{T>0,\qquad\int_{0}^{T}\|\rho_{n}\|_{X}\|\rho_{n+1}\|_{X}\,{\rm
d}t\leq\|\rho_{0}\|_{X}\right\\},$

we see that $\|\rho_{n+1}\|_{L^{\infty}_{T_{n+1}}(X)}\lesssim\|\rho_{0}\|_{X}$
and an inductive argument allows us to show that the times $T_{n}$ are
uniformly bounded from below by

$T_{n}\geq\frac{C}{\|\rho_{0}\|_{X}}:=T_{0}.$

STEP 2: convergence. We must now show that the sequence $(\rho_{n})$ is
convergent in some space (in $Y$ in fact). To do so, we define the difference
functions

$\delta\rho_{n}:=\rho_{n+1}-\rho_{n}\qquad\text{ and }\qquad\delta
u_{n}:=u_{n+1}-u_{n},$

and proceed identically with the initial data $\delta\rho_{0,n}$. We agree
that $\delta\rho_{0}(t,x)=0$ and $\delta\rho_{0,0}(x)=0$. The difference
functions solve the following set of equations:

$\begin{cases}\big{(}\partial_{t}+u_{n}\big{)}\delta\rho_{n}=-\operatorname*{\rm{div}}(\rho_{n}\delta
u_{n-1})\\\ \delta
u_{n-1}:=(-\Delta)^{-\alpha/2}\mathbb{P}(\delta\rho_{n-1}g).\end{cases}$

In order to show that $(\rho_{n})_{n}$ is Cauchy, we will show that the series

$\rho_{n}-\rho_{1}:=\delta\rho_{n-1}+\cdots+\delta\rho_{1}$

is normally convergent in the Banach space $Y$. Now, because of our
assumptions on $s>0$, the norms $Y=B^{s-1}_{\infty,1}\cap B^{-1}_{p,\infty}$
are propagated by the transport equation: since $u$ is divergence free,
Theorem A.21 insures that we have, for all $T\leq T_{0}$,

(35)
$\begin{split}\|\delta\rho_{n}\|_{L^{\infty}_{T}(Y)}&\lesssim\Big{(}\|\delta\rho_{0,n}\|_{Y}+\|\operatorname*{\rm{div}}(\rho_{n}\delta
u_{n-1})\|_{L^{1}_{T}(Y)}\Big{)}\exp\left\\{C\int_{0}^{T}\|\nabla
u_{n}\|_{L^{\infty}\cap B^{s-2}_{\infty,1}}\,{\rm d}t\right\\}\\\
&\lesssim\left(\|\delta\rho_{0,n}\|_{Y}+\|\rho_{n}\|_{L^{\infty}_{T}(X)}\int_{0}^{T}\|\delta\rho_{n-1}\|_{Y}\,{\rm
d}t\right)\exp\left\\{C\int_{0}^{T}\|\nabla u_{n}\|_{L^{\infty}\cap
B^{s-2}_{\infty,1}}\,{\rm d}t\right\\},\end{split}$

where this last inequality comes from Proposition 4.8. As we go on, we will no
longer be interested in precise inequalities (we only need to show that
$(\delta\rho_{n})_{n}$ is summable in $Y$), so we will treat all quantities
bounded by $\|\rho_{n}\|_{L^{\infty}_{T_{0}}(X)}\lesssim\|\rho_{0}\|_{X}$ as
irrelevant constants. The previous inequalities then simplify to

$\|\delta\rho_{n}\|_{L^{\infty}_{T}(Y)}\lesssim\|\delta\rho_{0,n}\|_{Y}+\int_{0}^{T}\|\delta\rho_{n-1}\|_{Y}\,{\rm
d}t.$

In order to use these inequalities, we must study the norms
$\|\delta\rho_{0,n}\|_{Y}$ for the initial data more closely. We have

$\|\delta\rho_{0,n}\|_{B^{s-1}_{\infty,1}}=\big{\|}(S_{n+1}-S_{n})\rho_{0}\big{\|}_{B^{s-1}_{\infty,1}}=2^{n(s-1)}\|\Delta_{n}\rho_{0}\|_{L^{\infty}}\leq
2^{-n}\|\rho_{0}\|_{B^{s}_{\infty,1}}.$

We may make a similar computation with
$\|\delta\rho_{0,n}\|_{B^{-1}_{p,\infty}}$. Overall, we get that
$\|\delta\rho_{0,n}\|_{Y}=O(2^{-n})$. By plugging this into our estimate and
setting $r_{n}(T):=\|\delta\rho_{n}\|_{L^{\infty}_{T}(Y)}$, we have

$\forall T\leq T_{0},\qquad
r_{n}(T)\leq\frac{C_{0}}{2^{n}}+C_{0}\int_{0}^{T}r_{n-1}(t)\,{\rm d}t,$

where $C_{0}>0$ is a fixed constant whose precise value depends on
$\|\rho_{0}\|_{X}$. By repeated use of this last inequality and remembering
that $r_{1}(T)=r_{1}$ does not depend on time, we find that, for all $n\geq
1$,

$\begin{split}r_{n}(T)&\leq\sum_{k=0}^{n-2}\frac{C^{k+1}T^{k}}{2^{n-k}k!}\,+\,\frac{C^{n-1}T^{n-1}}{(n-1)!}r_{1}\\\
&\leq\frac{C}{2^{n}}\sum_{k=0}^{n-2}\frac{(2CT)^{k}}{k!}\,+\,\frac{(CT)^{n-1}}{(n-1)!}r_{1}\\\
&\leq\frac{C}{2^{n}}e^{2CT}+\frac{(CT)^{n-1}}{(n-1)!}r_{1}.\end{split}$

Now, this proves that the series $\sum\delta\rho_{n}$ is normally convergent
in $Y$, and therefore the sequence $(\rho_{n})$ is certainly convergent in
$L^{\infty}_{T_{0}}(Y)$. Let $\rho$ be the associated limit. We have, as
$n\rightarrow+\infty$,

(36) $\rho_{n}\operatorname*{\longrightarrow}\rho\qquad\text{in
}L^{\infty}_{T_{0}}(Y).$

We define the associated velocity field by
$u=(-\Delta)^{-\alpha/2}\mathbb{P}(\rho g)\in
L^{\infty}_{T_{0}}(B^{s-1+\alpha}_{\infty,1})$. Note that, although we have
shown the convergence of the $\rho_{n}$ in $L^{\infty}_{T_{0}}(Y)$, the
sequence is uniformly bounded in $L^{\infty}_{T_{0}}(X)$. By Proposition A.12,
we see that the solution also lies in this space.

STEP 3: the limit is a solution. We have to check that the function $\rho\in
L^{\infty}_{T_{0}}(Y)$ we have constructed is indeed a solution of problem
(1). First of all, it is clear from the convergence of the $\rho_{n}$ shown
above that, as $n\rightarrow+\infty$,

$\begin{split}&\rho_{0,n}\operatorname*{\longrightarrow}\rho_{0}\qquad\text{in
}\mathcal{D}^{\prime}(\mathbb{R}^{d}),\\\
&\partial_{t}\rho_{n}\operatorname*{\longrightarrow}\partial_{t}\rho\qquad\text{in
}\mathcal{D}^{\prime}(]0,T_{0}[\times\mathbb{R}^{d}),\end{split}$

so that we just have to study the convergence of the product $\rho_{n}u_{n}$.
We use the following product lemma, whose proof flows directly from
Proposition A.18.

######  Lemma 4.9.

Consider $s>0$ such that $s\geq 1-\alpha$. Then the function product
$(f,h)\longmapsto fh$ is bounded in the following topologies

(37) $\begin{split}&B^{s}_{\infty,1}\times
B^{s+\alpha-1}_{\infty,1}\operatorname*{\longrightarrow}B^{-\alpha}_{\infty,1}\\\
&B^{s-1}_{\infty,1}\times
B^{s+\alpha}_{\infty,1}\operatorname*{\longrightarrow}B^{-\alpha}_{\infty,1}.\end{split}$

###### Proof (of the Lemma).

The proof is based on the Bony decomposition of a product $fh$,

$fh=\mathcal{T}_{f}(h)+\mathcal{T}_{h}(f)+\mathcal{R}(f,h).$

Firstly, we see that the remainder $\mathcal{R}$ is bounded in both topologies
(37), according to Proposition A.18, since the sum of the regularity indices
satisfies

$s+(s+\alpha-1)=(s-1)+(s+\alpha)=2s+\alpha-1\geq s>0\geq-\alpha.$

Concerning the paraproduct $\mathcal{T}_{f}(h)$, we must distinguish between
the cases where the regularity index of $f$ is, or not, nonnegative. Overall,
Proposition A.18 provides the following continuity properties for the
paraproduct:

$(f,h)\in B^{s}_{\infty,1}\times
B^{s+\alpha-1}_{\infty,1}\longmapsto\mathcal{T}_{f}(h)\in\begin{cases}B^{2s+\alpha-1}_{\infty,1}\subset
B^{-\alpha}_{\infty,1}\quad\text{if }s<0,\\\ B^{s-1+\alpha}_{\infty,1}\subset
B^{-\alpha}_{\infty,1}\quad\text{if }s\geq 0,\end{cases}$

and

$(f,h)\in B^{s-1}_{\infty,1}\times
B^{s+\alpha}_{\infty,1}\longmapsto\mathcal{T}_{f}(h)\in\begin{cases}B^{2s+\alpha-1}_{\infty,1}\subset
B^{-\alpha}_{\infty,1}\quad\text{if }s<1,\\\ B^{s+\alpha}_{\infty,1}\subset
B^{-\alpha}_{\infty,1}\quad\text{if }s\geq 1.\end{cases}$

For the paraproduct $\mathcal{T}_{h}(f)$, things are much quicker, since $h$
is either an element of $B^{s+\alpha-1}_{\infty,1}$ or
$B^{s+\alpha}_{\infty,1}$, and is in any case bounded $h\in L^{\infty}$.
Therefore, the paraproduct $\mathcal{T}_{h}(f)$ has its value in
$B^{s}_{\infty,1}\subset B^{-\alpha}_{\infty,1}$ or $B^{s-1}_{\infty,1}\subset
B^{-\alpha}_{\infty,1}$.

In all cases, the remainder and both paraproducts have range in
$B^{-\alpha}_{\infty,1}$, and this ends proving the Lemma. ∎

With product Lemma 4.9, we may finish proving that $\rho$ is a solution of
(1). As we have explained, all we have to do is study the convergence of the
product $\rho_{n}u_{n}$. We have,

$\begin{split}\|\rho_{n}u_{n}-\rho
u\|_{B^{-\alpha}_{\infty,1}}&\leq\|\rho_{n}(u_{n}-u)\|_{B^{-\alpha}_{\infty,1}}+\|(\rho_{n}-\rho)u\|_{B^{-\alpha}_{\infty,1}}\\\
&\lesssim\|\rho_{n}\|_{B^{s}_{\infty,1}}\|u_{n}-u\|_{B^{s+\alpha-1}}+\|\rho_{n}-\rho\|_{B^{s-1}_{\infty,1}}\|u\|_{B^{s+1}_{\infty,1}}\\\
&\lesssim\|\rho_{n}\|_{X}\|\rho_{n}-\rho\|_{Y}+\|\rho_{n}-\rho\|_{Y}\|\rho\|_{X}.\end{split}$

From the remarks at the end of STEP 2, we know that the solution $\rho$
belongs to $L^{\infty}_{T_{0}}(X)$ and from the a priori estimates of STEP 1,
we know that the approximate solutions are uniformly bounded in the same
space. The convergence (36) of the $\rho_{n}$ and the previous inequalities
show the convergence of the product
$\rho_{n}u_{n}\operatorname*{\longrightarrow}\rho u$ in the space
$L^{\infty}_{T_{0}}(B^{-\alpha}_{\infty,1})$ and hence, as
$n\rightarrow+\infty$,

$\operatorname*{\rm{div}}(\rho_{n}u_{n})\operatorname*{\longrightarrow}\operatorname*{\rm{div}}(\rho
u)\qquad\text{in }\mathcal{D}^{\prime}([0,T_{0}[\times\mathbb{R}^{d}).$

This proves that $\rho$ is indeed a solution of the transport equation in (1).
Reasoning exactly as in the proof of Theorem 3.1, see especially the end of
STEP 3, we see that $u$ is a solution of the fractional Stokes problem.

STEP 4: uniqueness of the solution. The uniqueness of the solution we have
constructed pretty much flows from the stability estimates we used to prove
convergence of the iterative scheme. Consider a given time $T>0$ and two
solutions $\rho_{1},\rho_{2}\in L^{\infty}_{T}(X)$ related to the initial data
$\rho_{0,1},\rho_{0,2}\in X$. Then, by defining the difference function
$\delta\rho:=\rho_{2}-\rho_{1}$, the difference of initial data
$\delta\rho_{0}:=\rho_{0,2}-\rho_{0,1}$, and the associated difference of
velocity fields $\delta u$, we find that $\delta\rho$ is a (weak) solution of
the problem

$\begin{cases}\big{(}\partial_{t}+u_{2}\cdot\nabla\big{)}\delta\rho=-\operatorname*{\rm{div}}(\rho_{1}\delta
u)\\\ \delta u=(-\Delta)^{-\alpha/2}\mathbb{P}(\delta\rho g).\end{cases}$

Using Theorem A.21 and the estimates therein provides an inequality which is
nearly identical to (35), namely

$\begin{split}\|\delta\rho\|_{L^{\infty}_{T}(Y)}&\lesssim\Big{(}\|\delta\rho_{0}\|_{Y}+\|\operatorname*{\rm{div}}(\rho_{1}\delta
u)\|_{L^{1}_{T}(Y)}\Big{)}\exp\left\\{C\int_{0}^{T}\|\nabla
u_{2}\|_{L^{\infty}\cap B^{s-2}_{\infty,1}}\,{\rm d}t\right\\}\\\
&\lesssim\left(\|\delta\rho_{0}\|_{Y}+\|\rho_{1}\|_{L^{\infty}_{T}(X)}\int_{0}^{T}\|\delta\rho\|_{Y}\,{\rm
d}t\right)\exp\left\\{C\int_{0}^{T}\|\nabla u_{2}\|_{L^{\infty}\cap
B^{s-2}_{\infty,1}}\,{\rm d}t\right\\}.\end{split}$

If $\delta\rho_{0}=0$, then the identity of both solutions $\delta\rho=0$
follows from Grönwall’s lemma.

STEP 5: proof of the continuation criterion. By standard arguments, it is
enough to show that the solution $\rho$ remains bounded in $X$ as long as
$V(t)$ remains integrable. Recall from inequality (32) that the norm
$E(t)=\|\rho(t)\|_{X}$ of the solution fulfills the integral inequality

$\forall T>0,\qquad E(T)\leq E(0)+C_{0}\int_{0}^{T}V(t)E(t)\,{\rm d}t.$

A direct application of Grönwall’s lemma gives the desired boundedness.

STEP 6: time continuity of the solution. We set $T:=T_{0}$ for the sake of
simplicity. So far, we have constructed a solution in the space
$L^{\infty}_{T}(X)$, but the claim that it is continuous in time with respect
to the $X=B^{s}_{\infty,1}\cap L^{p}$ topology remains unproved. The arguments
are fairly standard, but will be refered to later on, so we give a concise
proof. We start by studying continuity in $L^{p}$, which rests on the
following Lemma (the Friedrichs commutator lemma).

######  Lemma 4.10 (see also Lemma II.1 in [16]).

Consider $T>0$, a divergence free vector field $u\in L^{1}_{T}(W^{1,\infty})$
and a function $\rho\in L^{\infty}_{T}(L^{p})$ for some $p\in[1,+\infty[$.
Then, if $(K_{\epsilon})_{\epsilon>0}$ is a mollification sequence with
$K_{\epsilon}(y)=\epsilon^{-d}K(\epsilon^{-1}y)$, we have

$\big{[}u\cdot\nabla,K_{\epsilon}*\big{]}\rho\operatorname*{\longrightarrow}_{\epsilon\rightarrow
0^{+}}0\qquad\text{in }L^{1}_{T}(L^{p}).$

In the above, $K_{\epsilon}*$ denotes the convolution operator $f\mapsto
K_{\epsilon}*f$.

###### Proof.

We start by showing that the commutator remains bounded in $L^{p}$ as
$\epsilon\rightarrow 0^{+}$. By integration by parts, we see that

$\big{[}u\cdot\nabla,K_{\epsilon}*\big{]}\rho(x)=\int\rho(y)\Big{(}u(x)-u(y)\Big{)}\cdot\nabla
K_{\epsilon}(x-y){\rm d}y.$

By using the mean value theorem, we know the difference of velocities is
bounded by $|x-y|\|\nabla u\|_{L^{\infty}}$. Therefore, the Hausdorff-Young
convolution inequality gives the upper bound

$\begin{split}\left\|\big{[}u\cdot\nabla,K_{\epsilon}*\big{]}\rho\right\|_{L^{p}}&\lesssim\|\nabla
u\|_{L^{\infty}}\|\rho\|_{L^{p}}\int|y||\nabla K_{\epsilon}(y)|{\rm d}y\\\
&\lesssim\|\nabla u\|_{L^{\infty}}\|\rho\|_{L^{p}}\int|y||\nabla K(y)|{\rm
d}y.\end{split}$

This shows that the commutator remains bounded in $L^{1}_{T}(L^{p})$ as
$\rightarrow 0^{+}$. On the other hand, we may note that it must converge to
zero in $L^{1}_{T}(L^{p})$ if the density is regular enough, say, $\rho\in
L^{\infty}_{T}(W^{1,p})$. Therefore, we fix a sequence $(\rho_{n})_{n}$ of
$L^{\infty}_{T}(W^{1,p})$ functions such that
$\rho_{n}\operatorname*{\longrightarrow}\rho$ in the space
$L^{\infty}_{T}(L^{p})$ (this is possible because we have assumed
$p<+\infty$). Therefore, our commutator is bounded by

$\left\|\big{[}u\cdot\nabla,K_{\epsilon}*\big{]}\rho\right\|_{L^{p}}\lesssim\|\nabla
u\|_{L^{\infty}}\|\rho_{n}-\rho\|_{L^{p}}+\left\|\big{[}u\cdot\nabla,K_{\epsilon}*\big{]}\rho_{n}\right\|_{L^{p}},$

and so

$\operatorname*{\overline{\lim}}_{\epsilon\rightarrow
0^{+}}\left\|\big{[}u\cdot\nabla,K_{\epsilon}*\big{]}\rho\right\|_{L^{p}}\lesssim\|\nabla
u\|_{L^{\infty}}\|\rho_{n}-\rho\|_{L^{p}}\operatorname*{\longrightarrow}_{n\rightarrow+\infty}0.$

This concludes the proof of the Lemma. ∎

Let us return to the equation. By taking a mollification sequence
$(K_{\epsilon})_{\epsilon>0}$ as in Lemma 4.10 above, we see that the function
$\rho_{\epsilon}:=K_{\epsilon}*\rho$ is a (weak) solution of the transport
equation

$\begin{cases}\partial_{t}\rho_{\epsilon}+\operatorname*{\rm{div}}(u\rho_{\epsilon})=\big{[}u\cdot\nabla,K_{\epsilon}*\big{]}\rho\\\
\rho_{\epsilon}(0)=K_{\epsilon}*\rho_{0}.\end{cases}$

Because the function $\rho_{\epsilon}$ lies in $L^{\infty}_{T}(W^{1,p})$ for
each fixed value of $\epsilon$, we deduce from the previous equation that
$\rho_{\epsilon}\in C^{0}_{T}(L^{p})$. In addition, thanks to Lemma 4.10, the
sequence $(\rho_{\epsilon})$ is Cauchy in that space, and so $\rho\in
C^{0}_{T}(L^{p})$.

We now have to show that the solution is continuous in the $B^{s}_{\infty,1}$
topology. Here, we follow the ideas in [4], see in particular the proof of
Theorem 3.19 and pp. 138–139 therein.

From Lemma 4.9, we deduce that the product $\rho u$ lies in the space
$L^{\infty}_{T}(B^{-1-\alpha}_{\infty})$. Therefore $\partial_{t}\rho\in
L^{\infty}_{T}(B^{-1-\alpha}_{\infty,1})$, and in particular all the
Littlewood-Paley blocks of $\rho$ are continuous with respect to time:
$\Delta_{j}\rho\in C^{0}_{T}(L^{\infty})$, and we have $S_{n}\rho\in
C^{0}_{T}(B^{s}_{\infty,1})$. To conclude, we must show that the sequence
$(S_{n}\rho)_{n}$ is in fact Cauchy in the space
$C^{0}_{T}(B^{s}_{\infty,1})$. We have

$\|\rho-S_{n}\rho\|_{B^{s}_{\infty,1}}\lesssim\sum_{j\geq
n-1}2^{js}\|\Delta_{j}\rho\|_{L^{\infty}}.$

By using inequality (31) from the proof of Proposition 4.7, we may bound the
sum above: there is a family $\big{(}c_{j}(t)\big{)}_{j\geq-1}$ of sequences
lying in the unit ball of $\ell^{1}(j\geq-1)$ such that

$\|\rho-S_{n}\rho\|_{L^{\infty}_{T}(B^{s}_{\infty,1})}\lesssim\sum_{j\geq
n-1}2^{js}\|\Delta_{j}\rho_{0}\|_{L^{\infty}}+\int_{0}^{T}\|\rho\|_{X}^{2}\sum_{j\geq
n-1}c_{j}(t)\,{\rm d}t.$

We infer from the dominated convergence theorem that the righthand side of the
above inequality converges to zero as $n\rightarrow+\infty$, and so the
sequence $(S_{n}\rho)_{n}$ is indeed Cauchy in the space
$C^{0}_{T}(B^{s}_{\infty,1})$. This shows that $\rho\in C^{0}_{T}(X)$, and
concludes the proof of Theorem 4.1.

∎

##  5 A Directional Continuation Criterion

######  Theorem 5.1.

Assume that $\alpha<1$. Under the assumptions and notations of Theorem 4.1,
the unique solution $\rho\in C^{0}([0,T[;B^{1-\alpha}_{\infty,1}\cap L^{p})$
may be continued beyond time $T$ provided that

(38)
$\int_{0}^{T}\big{\|}\partial_{d}\rho\big{\|}_{B^{-\alpha}_{\infty,1}}\,{\rm
d}t<+\infty$

###### Proof.

As we have explained in the introduction above, the proof of this continuation
criterion is based on the precise form of the velocity field due to gravity
stratification. By writing the Leray projection operator explicitly, the
fractional Stokes equation reads

$u=(-\Delta)^{-\alpha/2}\Big{(}\rho
g+\nabla(-\Delta)^{-1}\partial_{d}\rho\Big{)},$

so that the product $u\cdot\nabla$ involves two different terms which factor
linear expressions of $\partial_{d}\rho$. More precisely, we have

(39)
$u\cdot\nabla\rho=(-\Delta)^{-\alpha/2}\rho.\,\partial_{d}\rho+\nabla\rho\cdot\nabla(-\Delta)^{-1-\alpha/2}\partial_{d}\rho.$

In order to prove the continuation criterion, we will need to revisit the
usual theory of transport equations in Besov spaces and highlight, at each
step, how the special structure (39) will affect the estimates. Our goal, as
in STEP 5 of the proof of Theorem 4.1, will be to show that the local solution
$\rho$ of Theorem 4.1 remains bounded in $X=B^{1-\alpha}_{\infty,1}\cap L^{p}$
on every time interval $[0,T]$ on which condition (38) holds. The continuation
of the solution will then follow by standard arguments.

First of all, because the velocity field $u$ associated to the solution is
Lipschitz $u\in L^{1}_{T}(B^{1}_{\infty,1})\subset L^{1}_{T}(W^{1,\infty})$,
the $L^{p}$ norm of the solution is transported by the flow:

$\forall t,\qquad\|\rho(t)\|_{L^{p}}=\|\rho_{0}\|_{L^{p}}.$

We therefore may focus on showing that the solution remains bounded in
$B^{1-\alpha}_{\infty,1}$ under condition (38). We therefore study the
Littlewood-Paley blocks $\Delta_{j}\rho$ for $j\geq-1$, which solve the
equation

(40)
$\big{(}\partial_{t}+u\cdot\nabla\big{)}\Delta_{j}\rho=\big{[}u\cdot\nabla,\Delta_{j}\big{]}\rho.$

Usually, when dealing with transport equations, we use the standard estimates
of Lemma A.19 to handle the commutator. However, in order to fully take
advantage of the structure of the product $u\cdot\nabla\rho$, we must go
through its proof in detail and isolate the contribution of the $d$-th
derivative $\partial_{d}\rho$. Let us write the Bony decomposition for the
products involved in the commutator. We have:

$\big{[}u\cdot\nabla,\Delta_{j}\big{]}\rho=\big{[}\mathcal{T}_{u_{k}},\Delta_{j}\big{]}\partial_{k}\rho+\mathcal{T}_{\Delta_{j}\partial_{k}\rho}(u_{k})+\mathcal{R}(u_{k},\Delta_{j}\partial_{k}\rho)-\Delta_{j}\mathcal{T}_{\partial_{k}\rho}(u_{k})-\Delta_{j}\mathcal{R}(\partial_{k}\rho,u_{k}).$

We will deal with each of those terms in the order in which they appeared in
the previous line.

First term: $\big{[}\mathcal{T}_{u_{k}},\Delta_{j}\big{]}\partial_{k}\rho$. We
start by writing the explicit meaning of the paraproduct in the commutator:

$\big{[}\mathcal{T}_{u_{k}},\Delta_{j}\big{]}\partial_{k}\rho=\sum_{|j-m|\leq
4}\Big{\\{}\big{[}S_{m-1}(-\Delta)^{-\alpha/2}\rho,\Delta_{j}\big{]}\Delta_{m}\partial_{d}\rho+\big{[}S_{m-1}\nabla(-\Delta)^{-1-\alpha/2}\partial_{d}\rho,\Delta_{j}\big{]}\Delta_{m}\nabla\rho\Big{\\}}.$

The sum in the equation above is supported on the indices $m$ which are not
too far from $j$, namely $|j-m|\leq 4$, as can be seen from writing explicitly
the commutators in the sum. For example,

$\big{[}S_{m-1}(-\Delta)^{-\alpha/2}\rho,\Delta_{j}\big{]}\Delta_{m}\partial_{d}\rho=S_{m-1}(-\Delta)^{-\alpha/2}\rho.\Delta_{j}\Delta_{m}\partial_{d}\rho-\Delta_{j}\big{(}S_{m-1}(-\Delta)^{-\alpha/2}\rho.\Delta_{m}\partial_{d}\rho\big{)}.$

In the above, the operator $\Delta_{j}\Delta_{m}$ is zero if $|j-m|>4$, and
the product $S_{m-1}(-\Delta)^{-\alpha/2}\rho.\Delta_{m}\partial_{d}\rho$ is
spectrally supported in an annulus of size roughly $2^{m}$, so that it is
canceled by $\Delta_{j}$ when $|j-m|>4$. Now, to estimate both commutators
above, we apply Lemma A.20. On the one hand, we get

$\left\|\big{[}S_{m-1}(-\Delta)^{-\alpha/2}\rho,\Delta_{j}\big{]}\Delta_{m}\partial_{d}\rho\right\|_{L^{\infty}}\lesssim
2^{-j}\|\nabla
S_{m-1}(-\Delta)^{-\alpha/2}\rho\|_{L^{\infty}}\|\Delta_{j}\partial_{d}\rho\|_{L^{\infty}}.$

By assumption, we have $\alpha<1$, so that the operator
$\nabla(-\Delta)^{-\alpha/2}$ is a Fourier multiplier whose symbol is a
homogeneous function of degree $1-\alpha>0$. An application of Proposition 2.2
therefore shows that the operator

$\nabla(-\Delta)^{-\alpha/2}\mathbb{P}g:B^{1-\alpha}_{\infty,1}\operatorname*{\longrightarrow}B^{0}_{\infty,1}\subset
L^{\infty}$

is bounded, so that the previous inequality becomes, for $|j-m|\leq 4$,

(41)
$\begin{split}\left\|\big{[}S_{m-1}(-\Delta)^{-\alpha/2}\rho,\Delta_{j}\big{]}\Delta_{m}\partial_{d}\rho\right\|_{L^{\infty}}&\lesssim
2^{-j}\|\rho\|_{B^{1-\alpha}_{\infty,1}}\|\Delta_{j}\partial_{d}\rho\|_{L^{\infty}}\\\
&\lesssim
c_{j}(t)2^{-j(1-\alpha)}\|\rho\|_{B^{1-\alpha}_{\infty,1}}\|\partial_{d}\rho\|_{B^{-\alpha}_{\infty,1}},\end{split}$

where, as usual, the sequences $\big{(}c_{j}(t)\big{)}_{j\geq-1}$ have unit
norm in $\ell^{1}(j\geq-1)$. Similarly, we have, for $|j-m|\leq 4$,

(42)
$\begin{split}\left\|\big{[}S_{m-1}\nabla(-\Delta)^{-1-\alpha/2}\partial_{d}\rho,\Delta_{j}\big{]}\Delta_{m}\nabla\rho\right\|_{L^{\infty}}&\lesssim
2^{-j}\big{\|}S_{m-1}\nabla^{2}(-\Delta)^{-1-\alpha/2}\partial_{d}\rho\big{\|}_{L^{\infty}}\|\Delta_{m}\nabla\rho\|_{L^{\infty}}\\\
&\lesssim
c_{j}(t)2^{-j(1-\alpha)}\|\rho\|_{B^{1-\alpha}_{\infty,1}}\big{\|}\nabla^{2}(-\Delta)^{-1-\alpha/2}\partial_{d}\rho\big{\|}_{L^{\infty}}.\end{split}$

In the previous upper bound, the operator
$\nabla^{2}(-\Delta)^{-1-\alpha/2}\partial_{d}$ is of order $1-\alpha>0$.
However, we wish to have an estimate involving $\partial_{d}\rho$, so we must
set aside the last derivative $\partial_{d}$ and work with the order $-\alpha$
operator $\nabla^{2}(-\Delta)^{-1-\alpha/2}$, which requires integrability to
be defined, by Proposition 2.1. We have

$\begin{split}\big{\|}\nabla^{2}(-\Delta)^{-1-\alpha/2}\partial_{d}\rho\big{\|}_{L^{\infty}}&\lesssim\big{\|}\Delta_{-1}\nabla^{2}(-\Delta)^{-1-\alpha/2}\partial_{d}\rho\big{\|}_{L^{\infty}}+\big{\|}({\rm
Id}-\Delta_{-1})\nabla^{2}(-\Delta)^{-1-\alpha/2}\partial_{d}\rho\big{\|}_{L^{\infty}}\\\
&\lesssim\|\Delta_{-1}\partial_{d}\rho\|_{L^{p}}+\|\partial_{d}\rho\|_{B^{-\alpha}_{\infty,1}}\\\
&\lesssim\|\rho\|_{L^{p}}+\|\partial_{d}\rho\|_{B^{-\alpha}_{\infty,1}}.\end{split}$

By putting together (41) and (42), we obtain a bound for the whole commutator,

(43)
$\left\|\big{[}\mathcal{T}_{u_{k}},\Delta_{j}\big{]}\partial_{k}\rho\right\|_{L^{\infty}}\lesssim
c_{j}(t)2^{-j(1-\alpha)}\|\rho\|_{B^{1-\alpha}_{\infty,1}}\Big{(}\|\rho_{0}\|_{L^{p}}+\|\partial_{d}\rho\|_{B^{-\alpha}_{\infty,1}}\Big{)}.$

Second Term: $\mathcal{T}_{\Delta_{j}\partial_{k}\rho}(u_{k})$. Let us write
the paraproduct explicitly in order to clarify the meaning of this term:

$\mathcal{T}_{\Delta_{j}\partial_{k}\rho}(u_{k})=\sum_{m\geq-1}\Big{\\{}S_{m-1}\Delta_{j}\partial_{d}\rho.\Delta_{m}(-\Delta)^{-\alpha/2}\rho+S_{m-1}\Delta_{j}\nabla\rho\cdot\Delta_{m}\nabla(-\Delta)^{-1-\alpha/2}\partial_{d}\rho\Big{\\}}.$

First of all, note that the operator $S_{m-1}\Delta_{j}$ is nonzero only when
$m\geq j$. In addition, because $S_{-1}=0$, none of the terms with the
$\Delta_{m}$ blocks involve low frequencies, and so we will be able to use
Lemma A.4 on them. By taking advantage of these two remarks, we see that

$\begin{split}\|\mathcal{T}_{\Delta_{j}\partial_{k}\rho}(u_{k})\|_{L^{\infty}}&\leq\sum_{m\geq
j}\Big{\\{}\big{\|}S_{m-1}\Delta_{j}\partial_{d}\rho\big{\|}_{L^{\infty}}\big{\|}\Delta_{m}(-\Delta)^{-\alpha/2}\rho\big{\|}_{L^{\infty}}\\\
&\qquad\qquad\qquad\qquad\qquad+\big{\|}S_{m-1}\Delta_{j}\nabla\rho\big{\|}_{L^{\infty}}\big{\|}\Delta_{m}\nabla(-\Delta)^{-1-\alpha/2}\partial_{d}\rho\big{\|}_{L^{\infty}}\Big{\\}}\\\
&\lesssim\sum_{m\geq
j}\Big{\\{}2^{-m\alpha}\big{\|}S_{m-1}\Delta_{j}\partial_{d}\rho\big{\|}_{L^{\infty}}\big{\|}\Delta_{m}\rho\big{\|}_{L^{\infty}}\\\
&\qquad\qquad\qquad\qquad\qquad+2^{-m(1+\alpha)}\big{\|}S_{m-1}\Delta_{j}\nabla\rho\big{\|}_{L^{\infty}}\big{\|}\Delta_{m}\partial_{d}\rho\big{\|}_{L^{\infty}}\Big{\\}}.\end{split}$

Let us bound first the terms featuring the $\Delta_{j}$ blocks. Since the
operator $S_{m-1}:L^{\infty}\operatorname*{\longrightarrow}L^{\infty}$ is
bounded, we may ignore it and factor the resulting upper bounds out of the
sum:

$\begin{split}\|\mathcal{T}_{\Delta_{j}\partial_{k}\rho}(u_{k})\|_{L^{\infty}}&\lesssim
2^{j\alpha}c_{j}(t)\|\partial_{d}\rho\|_{B^{-\alpha}_{\infty,1}}\sum_{m\geq
j}2^{-m\alpha}\|\Delta_{m}\rho\|_{L^{\infty}}\\\
&\qquad\qquad\qquad+2^{j\alpha}c_{j}(t)\|\rho\|_{B^{1-\alpha}_{\infty,1}}\sum_{m\geq
j}2^{-m(1+\alpha)}\|\Delta_{m}\partial_{d}\rho\|_{L^{\infty}}.\end{split}$

For the terms with the $\Delta_{m}$ blocks, straightforward Bernstein
inequalities provide

(44)
$\begin{split}\|\mathcal{T}_{\Delta_{j}\partial_{k}\rho}(u_{k})\|_{L^{\infty}}&\lesssim
2^{j\alpha}c_{j}(t)\|\partial_{d}\rho\|_{B^{-\alpha}_{\infty,1}}\sum_{m\geq
j}2^{-m}c_{m}(t)\|\rho\|_{B^{1-\alpha}_{\infty,1}}\\\
&\qquad\qquad\qquad+2^{j\alpha}c_{j}(t)\|\rho\|_{B^{1-\alpha}_{\infty,1}}\sum_{m\geq
j}2^{-m}c_{m}(t)\|\partial_{d}\rho\|_{B^{-\alpha}_{\infty,1}}.\\\ &\lesssim
2^{-j(1-\alpha)}c_{j}(t)\|\partial_{d}\rho\|_{B^{-\alpha}_{\infty,1}}\|\rho\|_{B^{1-\alpha}_{\infty,1}}.\end{split}$

Third term: $\mathcal{R}(u_{k},\Delta_{j}\partial_{k}\rho)$. As before, let us
give an explicit form for this term:

$\mathcal{R}(u_{k},\Delta_{j}\partial_{k}\rho)=\sum_{|m-n|\leq
1}\Delta_{m}u_{k}.\Delta_{n}\Delta_{j}\partial_{k}\rho.$

In the above, because $m$ and $n$ are close and $\Delta_{n}\Delta_{j}$ is
nonzero only if $|n-j|\leq 1$, we see that the sum must be supported on those
indices such that $j-2\leq m,n\leq j+2$. Let $I_{j}$ be that range of indices.
We have:

$\big{\|}\mathcal{R}(u_{k},\Delta_{j}\partial_{k}\rho)\big{\|}_{L^{\infty}}\leq\sum_{m,n\in
I_{j}}\Big{\\{}\big{\|}\Delta_{m}(-\Delta)^{-\alpha/2}\rho\big{\|}_{L^{\infty}}\big{\|}\Delta_{n}\Delta_{j}\partial_{d}\rho\big{\|}_{L^{\infty}}\\\
+\big{\|}\Delta_{m}(-\Delta)^{-1-\alpha/2}\nabla\partial_{d}\rho\big{\|}_{L^{\infty}}\big{\|}\Delta_{n}\Delta_{j}\nabla\rho\big{\|}_{L^{\infty}}\Big{\\}}.$

On the one hand, we have to use $L^{p}$ regularity of the solution to deal
with the order negative operators $(-\Delta)^{-\alpha/2}$ and
$\nabla(-\Delta)^{-1-\alpha/2}\partial_{d}$. Thanks to Proposition 2.2, we
have

$\begin{split}&\big{\|}\Delta_{m}(-\Delta)^{-\alpha/2}\rho\big{\|}_{L^{\infty}}\lesssim
2^{-m}c_{m}(t)\Big{(}\|\rho\|_{L^{p}}+\|\rho\|_{B^{1-\alpha}_{\infty,1}}\Big{)}\\\
&\big{\|}\Delta_{m}(-\Delta)^{-1-\alpha/2}\nabla\partial_{d}\rho\big{\|}_{L^{\infty}}\lesssim
2^{-m}c_{m}(t)\Big{(}\|\rho\|_{L^{p}}+\|\partial_{d}\rho\|_{B^{-\alpha}_{\infty,1}}\Big{)}\end{split}$

which we plug into the estimate for the remainder and obtain

(45)
$\begin{split}\big{\|}\mathcal{R}(u_{k},\Delta_{j}\partial_{k}\rho)\big{\|}_{L^{\infty}}&\lesssim\sum_{m,n\in
I_{j}}\Big{\\{}2^{-m}c_{m}(t)\Big{(}\|\rho\|_{L^{p}}+\|\rho\|_{B^{1-\alpha}_{\infty,1}}\Big{)}2^{j\alpha}c_{j}(t)\|\partial_{d}\rho\|_{B^{-\alpha}_{\infty,1}}\\\
&\qquad\qquad\qquad+2^{-m}c_{m}(t)\Big{(}\|\rho\|_{L^{p}}+\|\partial_{d}\rho\|_{B^{-\alpha}_{\infty,1}}\Big{)}2^{j\alpha}c_{j}(t)\|\rho\|_{B^{1-\alpha}_{\infty,1}}\Big{\\}}\\\
&\lesssim 2^{-j(1-\alpha)}c_{j}(t)\|\rho\|_{L^{p}\cap
B^{1-\alpha}_{\infty,1}}\Big{(}\|\rho\|_{L^{p}}+\|\partial_{d}\rho\|_{B^{-\alpha}_{\infty,1}}\Big{)}.\end{split}$

Fourth term: $\Delta_{j}\mathcal{T}_{\partial_{k}\rho}(u_{k})$. For this term,
we may directly use the results of Proposition A.18. Let us write the
paraproduct as

$\mathcal{T}_{\partial_{k}\rho}(u_{k})=\mathcal{T}_{\partial_{d}\rho}(-\Delta)^{-\alpha/2}\rho+\mathcal{T}_{\partial_{k}\rho}\partial_{k}(-\Delta)^{-1-\alpha/2}\partial_{d}\rho.$

In order to apply Proposition A.18, we must note that there are two cases,
depending on whether the derivatives $\partial_{k}\rho\in
B^{-\alpha}_{\infty,1}$ have negative regularity (when $\alpha>0$) or not
(when $\alpha=0$). In the first case, $\alpha>0$, the second and third
inequality of Proposition A.18 can be used, whereas in the second case we use
the first inequality. In both cases, the embedding
$B^{-\alpha}_{\infty,1}\subset B^{-\alpha}_{\infty,\infty}$ allows us to
write:

$\|\mathcal{T}_{\partial_{d}\rho}(-\Delta)^{-\alpha/2}\rho\|_{B^{1-\alpha}_{\infty,1}}\lesssim\|\partial_{d}\rho\|_{B^{-\alpha}_{\infty,1}}\|(-\Delta)^{-\alpha/2}\rho\|_{B^{1}_{\infty,1}}\lesssim\|\partial_{d}\rho\|_{B^{-\alpha}_{\infty,1}}\Big{(}\|\rho\|_{L^{p}}+\|\rho\|_{B^{1-\alpha}_{\infty,1}}\Big{)}$

and

$\begin{split}\|\mathcal{T}_{\partial_{k}\rho}(-\Delta)^{-1-\alpha/2}\partial_{k}\partial_{d}\rho\|_{B^{-\alpha}_{\infty,1}}&\lesssim\|\nabla\rho\|_{B^{-\alpha}_{\infty,1}}\|(-\Delta)^{-1-\alpha/2}\partial_{k}\partial_{d}\rho\|_{B^{1}_{\infty,1}}\\\
&\lesssim\|\rho\|_{B^{1-\alpha}_{\infty,1}}\Big{(}\|\rho\|_{L^{p}}+\|\partial_{d}\rho\|_{B^{-\alpha}_{\infty,1}}\Big{)}.\end{split}$

In terms of the size of the Littlewood-Paley blocks, this means that we have
an upper bound identical to the ones above:

(46)
$\big{\|}\Delta_{j}\mathcal{T}_{\partial_{k}\rho}(u_{k})\big{\|}_{L^{\infty}}\lesssim
2^{-j(1-\alpha)}c_{j}(t)\|\rho\|_{L^{p}\cap
B^{1-\alpha}_{\infty,1}}\Big{(}\|\rho\|_{L^{p}}+\|\partial_{d}\rho\|_{B^{-\alpha}_{\infty,1}}\Big{)}.$

Fifth term: $\Delta_{j}\mathcal{R}(\partial_{k}\rho,u_{k})$. Just as for the
previous term, we may simply use Proposition A.18 on the remainder

$\mathcal{R}(\partial_{k}\rho,u_{k})=\mathcal{R}\big{(}\partial_{d}\rho,(-\Delta)^{-\alpha/2}\rho\big{)}+\mathcal{R}\big{(}\partial_{k}\rho,\partial_{k}(-\Delta)^{-1-\alpha/2}\partial_{d}\rho\big{)}.$

Because we have assumed that $\alpha<1$, the remainder is always defined in
the $B^{-\alpha}_{\infty,1}\times
B^{1}_{\infty,1}\operatorname*{\longrightarrow}B^{1-\alpha}_{\infty,1}$
topology. Therefore, Proposition A.18 gives

$\begin{split}&\big{\|}\mathcal{R}\big{(}\partial_{d}\rho,(-\Delta)^{-\alpha/2}\rho\big{)}\big{\|}_{L^{\infty}}\lesssim\|\partial_{d}\rho\|_{B^{-\alpha}_{\infty,1}}\Big{(}\|\rho\|_{L^{p}}+\|\rho\|_{B^{1-\alpha}_{\infty,1}}\Big{)}\\\
&\big{\|}\mathcal{R}\big{(}\partial_{k}\rho,\partial_{k}(-\Delta)^{-1-\alpha/2}\partial_{d}\rho\big{)}\big{\|}_{B^{1-\alpha}_{\infty,1}}\lesssim\|\rho\|_{B^{1-\alpha}_{\infty,1}}\Big{(}\|\rho\|_{L^{p}}+\|\partial_{d}\rho\|_{B^{-\alpha}_{\infty,1}}\Big{)}.\end{split}$

In terms of the Littlewood-Paley blocks, this gives, just as before,

(47)
$\big{\|}\Delta_{j}\mathcal{R}(\partial_{k}\rho,u_{k})\big{\|}_{L^{\infty}}\lesssim
2^{-j(1-\alpha)}c_{j}(t)\|\rho\|_{L^{p}\cap
B^{1-\alpha}_{\infty,1}}\Big{(}\|\rho\|_{L^{p}}+\|\partial_{d}\rho\|_{B^{-\alpha}_{\infty,1}}\Big{)}.$

Conclusion: With inequalities (43), (44), (5), (46) and (47) put together, we
may finally use $L^{\infty}$ estimates on the transport equation (40) so as to
obtain, for $j\geq-1$ and $T>0$,

$\begin{split}2^{j(1-\alpha)}\|\Delta_{j}\rho\|_{L^{\infty}_{T}(L^{\infty})}&\lesssim
2^{j(1-\alpha)}\|\Delta_{j}\rho_{0}\|_{L^{\infty}}+\int_{0}^{T}\big{\|}\big{[}u\cdot\nabla,\Delta_{j}\big{]}\rho\big{\|}_{L^{\infty}}\,{\rm
d}t\\\ &\lesssim
2^{j(1-\alpha)}\|\Delta_{j}\rho_{0}\|_{L^{\infty}}+\int_{0}^{T}c_{j}(t)\|\rho\|_{L^{p}\cap
B^{1-\alpha}_{\infty,1}}\Big{(}\|\rho\|_{L^{p}}+\|\partial_{d}\rho\|_{B^{-\alpha}_{\infty,1}}\Big{)}\,{\rm
d}t,\end{split}$

and after summation on $j\geq-1$,

$\|\rho\|_{L^{\infty}_{T}(L^{p}\cap
B^{1-\alpha}_{\infty,1})}\lesssim\|\rho_{0}\|_{L^{p}\cap
B^{1-\alpha}_{\infty,1}}+\int_{0}^{T}\|\rho\|_{L^{p}\cap
B^{1-\alpha}_{\infty,1}}\Big{(}\|\rho\|_{L^{p}}+\|\partial_{d}\rho\|_{B^{-\alpha}_{\infty,1}}\Big{)}\,{\rm
d}t.$

By remembering that because the velocity field is Lipschitz, the $L^{p}$ norms
are conserved $\|\rho(t)\|_{L^{p}}=\|\rho_{0}\|_{L^{p}}$, we may use
Grönwall’s lemma on the previous integral inequality to show that $\rho$
remains indeed in the space $L^{\infty}_{T}(L^{p}\cap
B^{1-\alpha}_{\infty,1})$ as long as condition (38) holds. ∎

##  6 Global Existence and Uniqueness: $1\leq\alpha\leq d$

In this Section, we focus on finding global well-posedness of problem 1 when
$\alpha\geq 1$. We had already obtained such a result when $\alpha>1$ under a
mild (Hölder) regularity assumption in Corollary 4.3. However, this leaves
open the limit cases $\alpha=1$ and $\alpha=d$, in addition to letting us
wonder if it is possible to get rid of the additional regularity requirement.

To answer these questions, we will use two different setting: that of Lebesgue
spaces when $1<\alpha<d$ and that of Besov spaces when $\alpha\in\\{1,d\\}$.

The difference between these two settings is lies in the spaces in which it is
possible to prove stability estimates, and hence obtain uniqueness of a
solution. If $\rho_{1}$ and $\rho_{2}$ are two solutions, the difference
$\delta\rho=\rho_{2}-\rho_{1}$ solves

(48)
$\big{(}\partial_{t}+u_{1}\cdot\nabla\big{)}\delta\rho=-\operatorname*{\rm{div}}(\rho_{2}\delta
u).$

Now, for the sake of the argument, assume that the solutions are simply
bounded $\rho_{1},\rho_{2}\in C^{0}(L^{\infty})$. Then, we are faced with two
different problems. Firstly, the velocity field $u_{1}$ is not necessarily
Lipschitz, but $\log$-Lipschitz, and this means we have to deal with possible
loss of regularity in the estimates for the transport equations. Secondly, the
righthand side of (48) will have, at best, the regularity of $\nabla\rho$,
that is a $s=-1$ order of regularity. This is no problem when $\alpha>1$, as
the gain of regularity of order $\alpha$ allows for
$B^{-1-\epsilon}_{q,\infty}\subset B^{-\alpha}_{q,\infty}$ estimates in the
transport equation, thus giving order $s=-1$ bounds on $\delta\rho$, but it is
a major obstacle when $\alpha=1$.

Our way around this will be to treat the case $\alpha=1$ differently by
constructing solutions in the endpoint Besov-Lipschitz space
$B^{0}_{\infty,1}$, which is slightly smaller than $L^{\infty}$. In doing so,
we will use the method of Hmidi and Keraani [24] based on linear estimates for
the transport equation. Incidentally, we will also take advantage of estimates
of this type to construct solutions that also lie in the homogeneous Besov
space $\dot{B}^{0}_{\frac{d}{p},1}$, thus covering the special case
$d=\alpha$.

###  6.1 Well-Posedness in Critical Lebesgue Spaces

Firstly, we consider the case where $1<\alpha<d$, where we will be able to
prove global well-posedness in critical Lebesgue spaces $L^{d/(\alpha-1)}$.

######  Theorem 6.1.

Consider $d\geq 2$ and $\alpha\in]1,d[$. Let $p\in[1,d/\alpha[$ and set
$q=d/(\alpha-1)$. Then, for any initial datum $\rho_{0}\in L^{p}\cap L^{q}$,
problem (1) has a unique solution $\rho\in L^{\infty}(L^{p}\cap L^{q})$.

######  Remark 6.2.

In particular, when $d=3$ and $\alpha=2$, we recover the result of Mecherbet
and Sueur [30], who prove global well-posedness for initial data with $L^{3}$
regularity (and sufficient integrability).

######  Remark 6.3.

Note that the assumptions of Theorem 6.1 above are stronger than those of
Theorem 3.1 giving the existence of global weak solutions in Lebesgue spaces,
since the inequality

(49) $q=\frac{d}{\alpha-1}>\frac{2}{1+\frac{\alpha}{d}}$

holds as long as $\alpha<d+2$.

######  Remark 6.4.

It is not known whether the solution from Theorem 6.1 is continuous with
respect to time: as it will appear in the proof, the velocity field is only
$\log$-Lipschitz, so the proof of Lebesgue time continuity by means of the
Friedrichs commutator Lemma (Lemma 4.10), as in STEP 6 of the proof of Theorem
4.1, fails. However, if the initial datum is subcritical $\rho_{0}\in
L^{p}\cap L^{r}$ with $r>q=d/(\alpha-1)$, then the velocity field is Lipschitz
and it becomes possible to prove that $\rho\in C^{0}_{b}(L^{p}\cap L^{r})$.

###### Proof (of Theorem 6.1).

First of all, because $1<d/\alpha<q$ and $\rho_{0}\in L^{p}\cap L^{q}$,
interpolation of Lebesgue spaces will always allow us to assume that $p>1$.

Thanks to Remark 6.3, we see that the assumptions of Theorem 6.1 are enough to
provide the existence of a global weak solution: because of inequality (49),
it is always possible to find a $r>p$ such that

$\frac{2}{1+\frac{\alpha}{d}}<r<\frac{d}{\alpha}<\frac{d}{\alpha-1}=q,$

since the inequality $2\left(1+\frac{\alpha}{d}\right)^{-1}<\frac{d}{\alpha}$
holds al long as $\alpha<d$. Noting that $p<r<q$, interpolation of Lebesgue
spaces shows that the initial datum $\rho_{0}\in L^{r}\subset L^{p}\cap L^{q}$
satisfies the assumptions of Theorem 3.1, thus providing a global weak
solution $\rho\in L^{\infty}(L^{p}\cap L^{q})$ with

$\forall t\geq 0,\qquad\|\rho(t)\|_{L^{p}\cap
L^{q}}\leq\|\rho_{0}\|_{L^{p}\cap L^{q}}.$

We only have to show uniqueness of this solution. We therefore consider two
solutions $\rho_{1},\rho_{2}\in L^{\infty}(L^{p}\cap L^{q})$ related to the
same initial datum $\rho_{0,1}=\rho_{0,2}=\rho_{0}$ and form the difference
function $\delta\rho=\rho_{2}-\rho_{1}$. Let $\delta\rho_{0}=0$ be defined
accordingly. Then, $\delta\rho$ is a weak solution of the following system

(50)
$\begin{cases}\big{(}\partial_{t}+u_{2}\cdot\nabla\big{)}\delta\rho=-\operatorname*{\rm{div}}(\rho_{1}\delta
u)\\\ \delta u=(-\Delta)^{-\alpha/2}\mathbb{P}(\delta\rho g).\end{cases}$

The main difficulty we face at this point is that, because $\rho_{2}\in
L^{q}$, the associated velocity field $u_{2}$ may not be Lipschitz, so we must
be careful when using estimates related to the transport equation. As a matter
of fact, we may only find weaker regularity: thanks to the embeddings of
Proposition A.14,

$L^{q}\subset B^{0}_{q,2}\text{ if }q\leq 2\qquad\text{and}\qquad L^{q}\subset
B^{0}_{q,q}\text{ if }q\geq 2,$

we find that, by setting $r=2$ when $q\leq 2$ and $r=q$ when $q\geq 2$,

$\|u_{2}\|_{B^{1}_{\infty,r}}\lesssim\|u_{2}\|_{B^{\alpha-d/q}_{\infty,r}}\lesssim\|\rho_{2}\|_{L^{p}}+\|\rho_{2}\|_{B^{-d/q}_{\infty,r}}\lesssim\|\rho_{2}\|_{L^{p}}+\|\rho_{2}\|_{B^{0}_{q,r}}\lesssim\|\rho\|_{L^{p}\cap
L^{q}},$

so that $u_{2}$ belongs to a Besov space $B^{1}_{\infty,r}$ whose third index
if finite $r<+\infty$. Consequently, the velocity field is $\log$-Lipschitz
$u_{2}\in L^{\infty}(LL_{\beta})$ for $\beta=1-\frac{1}{r}>0$ (see the
discussion in the Appendix on these spaces). More precisely, we may separate
low and high frequencies to see that

$u_{2}=\Delta_{-1}u_{2}+({\rm Id}-\Delta_{-1})u_{2}\in
L^{\infty}_{T}(B^{k}_{\infty,1})+L^{\infty}_{T}(B^{\alpha}_{q,r})$

for all $k\geq 0$. Because $\alpha=1+\frac{d}{q}$, we have the embedding
$B^{\alpha}_{q,r}\subset LL^{q}_{\beta}$, so the velocity field $u_{2}$ does
indeed satisfy the assumptions of Theorem A.25, and we may perform estimates
on the transport equation, but at the price of a loss of regularity.

Let us establish the framework in which we will use the estimaetes of Theorem
A.25. Our goal will be to find bounds on $\delta\rho$ by using the first
equation in (50). Since the derivative $\nabla\rho_{1}$ is involved in the
righthand side, we will have to work in a space contained in
$B^{-1}_{p,\infty}\cap B^{-1}_{q,\infty}$. For practical reasons that will
become clear later, we will conduct the estimates at a lower level of
regularity $\sigma_{1}<-1$. More precisely, we define a regularity index
$\sigma_{0}$

$\sigma_{0}:=-1-d\min\left\\{\frac{1}{q},\frac{1}{q^{\prime}},\frac{1}{p^{\prime}}\right\\},$

where, as usual, $p^{\prime}$ and $q^{\prime}$ are the conjuguate exponents of
$p$ and $q$. Then, since we have assumed $p>1$ and $\alpha=1+\frac{d}{q}>1$,
we see that $\sigma_{0}<-1$. We may therefore consider a $\sigma_{1}<-1$ and a
$\epsilon>0$, whose precise values we will fix later on, such that

$\sigma_{0}<\sigma_{1}-\epsilon<\sigma_{1}<-1.$

In order to use Theorem A.25 with initial datum $\delta\rho\in
B^{\sigma_{1}}_{p,\infty}\cap B^{\sigma_{1}}_{q,\infty}$, we need to check
that the righthand side term $-\operatorname*{\rm{div}}(\rho_{1}\delta u)$
satisfies

(51)
$\forall\eta\in[0,\epsilon],\qquad\big{\|}\operatorname*{\rm{div}}(\rho_{1}\delta
u)\big{\|}_{B^{-\sigma_{1}-\eta}_{b,\infty}}\lesssim\Big{(}\|\delta\rho\|_{B^{-\sigma_{1}-\eta}_{p,\infty}}+\|\delta\rho\|_{B^{-\sigma_{1}-\eta}_{q,\infty}}\Big{)}\|\rho_{1}\|_{L^{p}\cap
L^{q}},$

for $b$ taking the values $b=p$ and $b=q$.

We focus on proving inequality (51). By invoking the Bony decomposition for
the product $\operatorname*{\rm{div}}(\rho_{1}\delta u)$, we obtain

$\operatorname*{\rm{div}}(\rho_{1}\delta
u)=\mathcal{T}_{\partial_{k}\rho_{1}}(\delta u_{k})+\mathcal{T}_{\delta
u_{k}}(\partial_{k}\rho_{1})+\partial_{k}\mathcal{R}(\delta u_{k},\rho_{1})$

where, as usual, there is an implicit sum over the repeated index $k=1,...,d$.
The paraproducts create no special problem: Proposition A.18 immediately gives

$\begin{split}\|\mathcal{T}_{\partial_{k}\rho_{1}}(\delta
u_{k})\|_{B^{\sigma_{1}-\eta}_{b,\infty}}+\|\mathcal{T}_{\delta
u_{k}}(\partial_{k}\rho_{1})\|_{B^{\sigma_{1}-\eta}_{b,\infty}}&\leq\frac{C}{|1+\sigma_{1}-\eta|}\|\delta
u\|_{B^{1+\sigma_{1}-\eta}_{\infty,\infty}}\|\rho_{1}\|_{L^{p}\cap L^{q}}\\\
&\leq\frac{C}{|1+\sigma_{1}-\eta|}\Big{(}\|\delta\rho\|_{B^{\sigma_{1}-\eta}_{p,\infty}}+\|\delta\rho\|_{B^{\sigma_{1}-\eta}_{q,\infty}}\Big{)}\|\rho_{1}\|_{L^{p}\cap
L^{q}}.\end{split}$

Since we have chosen $\sigma_{1}<1$, the constant appearing in this inequality
is bounded by $|1+\sigma_{1}-\eta|^{-1}\leq|1+\sigma_{1}|^{-1}$ for all
$\eta\in[0,\epsilon]$, and we can safely ignore it from now on. On the other
hand, the remainder term $\mathcal{R}(\delta u_{k},\rho_{1})$ is a bit more
delicate to handle: it is only properly defined if the sum of the regularities
of $\delta u$ and $\rho_{1}$ is positive. For this, we take advantage of the
fact that $\delta u$ is the image of $\delta\rho$ by the operator
$(-\Delta)^{-\alpha/2}\mathbb{P}g$, hence giving it positive regularity as
$\alpha>0$, namely

$\|\Delta_{-1}\delta
u\|_{B^{k}_{\infty,1}}\lesssim\|\delta\rho\|_{B^{\sigma_{1}-\eta}_{p,\infty}}\qquad\text{and}\qquad\big{\|}({\rm
Id}-\Delta_{-1})\delta
u\big{\|}_{B^{\alpha+\sigma_{1}-\eta}_{q,\infty}}\lesssim\|\delta\rho\|_{B^{\sigma_{1}-\eta}_{q,\infty}},$

for any $k\geq 0$. For the low frequency terms, there is no problem.
Proposition A.18 gives

$\|\mathcal{R}(\Delta_{-1}\delta
u_{k},\rho_{2})\|_{B^{k}_{b,\infty}}\lesssim\|\Delta_{-1}\delta
u\|_{B^{k}_{\infty,1}}\|\rho_{1}\|_{L^{b}}\lesssim\|\delta\rho\|_{B^{\sigma_{1}-\eta}_{p,\infty}}\|\rho_{1}\|_{L^{b}}.$

For the high frequency part, we resort to a method nearly identical that used
when constructing global weak solutions in the proof of Theorem 3.1 above: we
want to consider the remainder as a
$B^{\alpha+\sigma_{1}-\eta}_{q,\infty}\times B^{0}_{b,\infty}$ operator. One
of two things may happen: either
$\frac{1}{\gamma}:=\frac{1}{q}+\frac{1}{b}\leq 1$ and Proposition A.18 yields

$\big{\|}\mathcal{R}\big{(}({\rm Id}-\Delta_{-1})\delta
u,\rho_{1}\big{)},\big{\|}_{B^{\alpha+\sigma_{1}-\eta}_{\gamma,\infty}}\lesssim\big{\|}({\rm
Id}-\Delta_{-1})\delta
u\big{\|}_{B^{\alpha+\sigma_{1}-\eta}_{q,\infty}}\|\rho_{1}\|_{B^{0}_{b,\infty}}$

since $\alpha+\sigma_{1}-\eta\geq\alpha+\sigma-\epsilon>0$ if we chose
$\sigma_{1}$ close enough to $-1$ and if we take $\epsilon$ small enough.
Then, the embedding $B^{\alpha+\sigma_{1}-\eta}_{\gamma,\infty}\subset
B^{1+\sigma_{1}-\eta}_{b,\infty}$ provides the desired inequality:

(52) $\big{\|}\partial_{k}\mathcal{R}\big{(}({\rm Id}-\Delta_{-1})\delta
u,\rho_{1}\big{)},\big{\|}_{B^{\sigma_{1}-\eta}_{b,\infty}}\lesssim\big{\|}({\rm
Id}-\Delta_{-1})\delta
u\big{\|}_{B^{\alpha+\sigma_{1}-\eta}_{q,\infty}}\|\rho_{1}\|_{B^{0}_{b,\infty}}.$

When we have $\frac{1}{q}+\frac{1}{b}>1$ instead, we must first take advantage
of the positive regularity of $({\rm Id}-\Delta_{-1})\delta u$ and the
embeddings of Proposition A.9 to reach a higher integrability index.
Precisely, we get $B^{\alpha+\sigma_{1}-\eta}_{q,\infty}\subset
B^{s}_{\gamma,\infty}$ with $\frac{1}{\gamma}+\frac{1}{b}=1$ and

(53)
$\begin{split}s&=\alpha+\sigma_{1}-\eta-d\left(\frac{1}{q}-\frac{1}{\gamma}\right)\\\
&=1+\sigma_{1}-\eta+d\left(1-\frac{1}{b}\right).\end{split}$

Because we have assumed that $q>p>1$, we can always take this quantity to be
positive $s\geq s_{0}>0$ by choosing $\sigma_{1}$ and $\epsilon$ such that
$|\sigma_{1}+1|+\epsilon$ is small enough. In that case, we may apply
Proposition A.18 to get

$\big{\|}\mathcal{R}\big{(}({\rm Id}-\Delta_{-1})\delta
u,\rho_{1}\big{)},\big{\|}_{B^{s}_{1,\infty}}\lesssim C(s_{0})\big{\|}({\rm
Id}-\Delta_{-1})\delta
u\big{\|}_{B^{\alpha+\sigma_{1}-\eta}_{q,\infty}}\|\rho_{1}\|_{B^{0}_{b,\infty}}.$

The embedding $B^{s}_{1,\infty}\subset B^{1+\sigma_{1}-\eta}_{b,\infty}$ shows
that (52) is also true in that case. Therefore, inequality (51) does indeed
hold for all $\eta\in[0,\epsilon]$ provided that we choose
$|\sigma_{1}+1|+\epsilon$ small enough.

We are at the end of the proof. A direct application of Theorem A.25 gives,
for any fixed $T>0$, after imposing again $\epsilon$ to be small enough,

$\sup_{t\in[0,T]}\Big{(}\|\delta\rho(t)\|_{B^{\sigma_{1}-\epsilon}_{p,\infty}}+\|\delta\rho(t)\|_{B^{\sigma_{1}-\epsilon}_{q,\infty}}\Big{)}\lesssim\|\delta\rho_{0}\|_{B^{\sigma_{1}}_{p,\infty}\cap
B^{\sigma_{1}}_{q,\infty}}\exp\left\\{\frac{C}{\epsilon^{r-1}}\left(\int_{0}^{T}\|\rho_{2}\|_{L^{p}\cap
L^{q}}\,{\rm d}t\right)^{r}\right\\}.$

This ends the proof of uniqueness. ∎

###  6.2 Well-Posedness in Besov Spaces: the Endpoint Cases

The proof of Theorem 6.1 fails in two cases of special interest: those where
$\alpha=d$ and where $\alpha=1$. By going back to the proof, we see that there
are two main reasons why these “endpoint” cases evade us.

On the one hand, the fact that we could take $p$ to satisfy $1<p<d/\alpha$ was
crucial to proving inequality (51) with $b=p$. If we had $p=1$ (and even then,
we could not take $d=\alpha$) then the regularity exponents (53) involved in
the products cannot be positive, as $\sigma_{1}<-1$.

On the other hand, and this is a similar problem, if $\alpha=1$, then we must
take $q=+\infty$. In that case, the estimates for the transport equation only
hold in function spaces of regularity at best $\sigma_{0}=-1$, which is the
best regularity exponent we can hope for stability estimates because of the
derivatives $\nabla\rho_{1}$ involved in the righthand side of the first
equation in (50). Using estimates which involve a loss of regularity is
therefore no longer possible, as it would put us at a $-1-\epsilon<\sigma_{0}$
regularity level.

The solution to both problems will be to work at a higher regularity level,
where we will be able to use global Besov-Lipschitz estimates on the velocity
field. Our main tool is a bound for solutions of the transport equation with
regularity $s=0$ that is linear with respect to the velocity field (see A.23).

######  Theorem 6.5.

Consider $d\geq 2$ and $\alpha\in[1,d]$. Let $p=d/\alpha$ and
$q=d/(\alpha-1)$. Then, for any initial datum
$\rho_{0}\in\dot{B}^{0}_{p,1}\cap B^{0}_{q,1}$, problem (1) has a unique
solution $\rho\in C^{0}(\dot{B}^{0}_{p,1}\cap B^{0}_{q,1})$.

###### Proof.

Strictly speaking, we have not yet proved existence of such a solution: so far
solutions have been constructed only for initial data that has some
integrability $\rho_{0}\in L^{r}$ with $r<d/\alpha$ (in addition to being
regular enough). However, once we show that global a priori and stability
estimates hold in $\dot{B}^{0}_{p,1}\cap B^{0}_{q,1}$, then it is a
straightforward thing to construct a solution by means of a compactness
argument (replicating the different steps in the proof of Theorem 3.1). For
the sake of conciseness, we will focus on the estimates and leave aside the
full proof of existence.

SPEP 1: a priori estimates. Assume that $\rho$ is a regular solution of (1)
with associated velocity field $u$. Start by noting that, thanks to
Proposition 2.2, the following Besov-Lipschitz bound holds:

$\|u\|_{B^{1}_{\infty,1}}\lesssim\|\rho\|_{\dot{B}^{0}_{p,1}}+\|\rho\|_{B^{0}_{q,1}}.$

Set $X=\dot{B}^{0}_{p,1}\cap B^{0}_{q,1}$. We then use Proposition A.23, which
provide, as both spaces $\dot{B}^{0}_{p,1}$ and $B^{0}_{q,1}$ have regularity
exponent $s=0$,

$\begin{split}\forall\,T>0,\qquad\|\rho\|_{L^{\infty}_{T}(X)}&\lesssim\|\rho_{0}\|_{X}\left(1+\int_{0}^{T}\|\nabla
u\|_{L^{\infty}}\,{\rm d}t\right)\\\