# On a 1D nonlocal transport equation with nonlocal velocity and subcritical
or supercritical diffusion

Omar Lazar

###### Abstract

We study a 1D transport equation with nonlocal velocity with subcritical or
supercritical dissipation. For all data in the weighted Sobolev space
$H^{k}(w_{\lambda,\kappa})\cap L^{\infty},$ where $k=\max(0,3/2-\alpha)$ and
$w_{\lambda,\kappa}$ is a given family of Muckenhoupt weights, we prove a
global existence result in the subcritical case $\alpha\in(1,2)$. We also
prove a local existence theorem for large data in
$H^{2}(w_{\lambda,\kappa})\cap L^{\infty}$ in the supercritical case
$\alpha\in(0,1)$. The proofs are based on the use of the weighted Littlewood-
Paley theory, interpolation along with some new commutator estimates.

##  1 Introduction

In this paper, we are interested in the following 1D transport equation with
nonlocal velocity which was introduced by Córdoba, Córdoba and Fontelos in
[14] :

$\ (\mathcal{T}_{\alpha})\ :\\\ \left\\{\begin{aligned}
&\partial_{t}\theta+\theta_{x}\mathcal{H}\theta+\nu\Lambda^{\alpha}\theta=0\hskip
56.9055pt\\\ &\theta(0,x)=\theta_{0}(x)\end{aligned}\right.$

where, $\mathcal{H}$ denotes the Hilbert transform defined by

$\mathcal{H}\theta\equiv\frac{1}{\pi}PV\int\frac{\theta(y)}{y-x}\ dy$

and

$\Lambda^{\alpha}\theta\equiv(-\Delta)^{\alpha/2}\theta=C_{\alpha}P.V.\int_{\mathbb{R}^{n}}{\frac{\theta(x)-\theta(x-y)}{|y|^{1+\alpha}}dy}$

where $C_{\alpha}>0$ is a positive constant and $0\leq\alpha<2$.

This equation is related to a simple scalar model introduced by Constantin,
Lax and Majda (see [10]) in order to get a better understanding of the 3D
Euler equation written in terms of the vorticity $w$ (that is the curl of the
velocity, and the velocity is determined by $v_{x}=\mathcal{H}w$), namely the
following 1D equation

$\partial_{t}\omega=\omega\mathcal{H}\omega.$

In [10], the authors proved that most of the solutions blow-up in finite time.
More recently Okamoto, Sakajo and Wunsch [29] introduced a generalization of
the Constantin, Lax, Majda (CLM) and the De Gregorio’s model, namely

$\partial_{t}w+avw_{x}-w\mathcal{H}w=0.$

The case $a=-1,a=0,a=1$ being respectively the Córdoba, Córdoba and Fontelos
model (blow-up of regular solutions can occure as shown in [14], [26], and
[31]), the CLM model (most solutions blow-up in finite time) and the De
Gregorio’s model (it is conjecured that solutions exist globally in time and
numerical evidence presented in [29] are in line with the conjecture). This
nonlocal transport equation can also be seen as a 1D model of the dissipative
surface quasi geostrophic equation introduced in [9] written in a divergence
form. It is therefore closely related to the incompressible 3D Euler equation
written in terms of vorticity (see [9]). Another motivation is the link
between this equation and the Birkoff-Rott equation which modelises the
evolution of vortex sheets with surface tension (see e.g.[14]). As usually,
one has to consider three cases depending on the value of $\alpha\in(0,2)$,
namely the cases $\alpha<1$, $\alpha=1$, $\alpha>1$ which are respectively
called supercritical, critical and subcritical cases. In the inviscid case
($\eta=0$), Córdoba Córdoba and Fontelos have shown in [14] that there exists
a class of smooth initial data for which the solutions blow-up in finite time.
This result has been extended to a slightly dissipative case by Li and Rodrigo
[26] who proved the existence of a class of smooth data such that the
solutions blow-up in finite time when the dissipation rate is in the range
$\alpha\in(0,1/2)$. These results have been proved in four differents way by
Silvestre and Vicol [31] for $\alpha\in[0,1/2)$ for equation
$(\mathcal{T}_{\alpha})$. It is still an open problem to know whether regular
solutions blow-up or exist globally in time in the case $1/2\leq\alpha<1$. The
eventual regularity of smooth solution in the spirit of previous works for the
supercritical SQG equation, has been obtained by Do in [16]. We also note that
the local well posedness in $H^{2}$ has been obtained by Bae and Granero-
Belinchón in [1] in the inviscd case. Both the subcritical and critical case
are now quite well understood, for instance, estimate are etablished in [17]
for data in $H^{3/2-\alpha}$ in the critical case using Littlewood-Paley
theory. Moreover, by adaptading for instance the method of Constantin and
Vicol introduced in [11] or the approach of Kiselev, Nazarov and Volberg [22]
one obtain the existence of global smooth solutions in the critical case.
Roughly speaking, most of the results known in the critical and subcritical
case for the surface quasi-geostrophic equation turn out to work for this 1D
model. However, it seems that constructing global $L^{2}$ solutions in the
supercritical or even in the critical case is still not known. Indeed, one can
easily derive a nice energy estimate (see [14]) but the lack of compactness
prevents one from passing to the weak limit in the nonlinearity and therefore
the existence remains an open problem (we refer to [11] or [24] for more
details).

In this article, we study equation $(\mathcal{T}_{\alpha})$ with subcritical
or supercritical diffusion. We extended the class of initial data of some
recent existence results (essentially some of those in [17], and [1]) to the
weighted setting. By considering data in weighted Lebesgue or Sobolev spaces
with a weight which has the property to be sufficently decaying at infinity
say like $|x|^{-\lambda}$, with $\lambda>0$, one is allowed to consider the
problem $(\mathcal{T}_{\alpha})$ with an initial data that behaves for
instance like $\theta_{0}(x)\sim|x|^{-1/2}$ at infinity, and therefore does
not necessarily belong to $L^{2}$. In order to avoid integrability problems
close to the origin one may consider a weight of the form
$w_{\lambda}(x)=(1+|x|)^{-\lambda}$. Then, since $(\mathcal{T}_{\alpha})$
involves the Hilbert transform, it could be interesting to choose
$w_{\lambda}$ so that the Hilbert transform is a continuous operator in the
weighted Lebesgue spaces $L^{p}(w_{\lambda})$, this is equivalent to choose
the weight in the Muckenkoupt class $\mathcal{A}_{p}$ (see 3.2 for the
definition). Such a weight has not to be integrable far from the origin by
definition (it has to obey a reverse Hölder inequality), hence one has to
choose $\lambda<1$. Therefore, a possible family of weights that one could
consider is for instance the one defined by
$w_{\lambda}(x)=(1+|x|)^{-\lambda}$ with $\lambda\in(0,1)$. Such kind of
weights have been already considered by Farwig and Sohr in [19] in the study
of the Stokes problem. More recently, in [24], Lemarié-Rieusset and the author
have studied the critical case for equation $\mathcal{T}_{\alpha}$ with data
in some weighted Sobolev spaces where the weight is given by
$w_{\lambda}(x)=(1+x^{2})^{-\lambda/2}$. As a matter of fact, if one would
need to control say $\kappa>1$ derivatives of the weight, then it would be
better to use a more general class of radially symmetric weights of the type
$w_{\lambda,\kappa}(x)=(1+|x|^{\kappa})^{-\lambda/\kappa}$ with
$\lambda\in(0,1)$ which are still Mukenhoupt weights. In this article, we
would need to control at least two derivatives of the weight thus we shall
assume the integer $\kappa\geq 2$ to be even (in order to avoid
differentiability issues at $-1$). We shall therefore consider the general
family of Muckenhoupt weights given by
$w_{\lambda,\kappa}(x)=(1+\left|x\right|^{\kappa})^{-\lambda/\kappa}$. Note
that the weights considered in [19] and [24] correspond respectively to the
cases $\kappa=1$ and $\kappa=2$.

When one try to do a priori estimates in such kind of weighted spaces some
difficulties appear. One of the main issue is to control some extra
commutators that involve the fractional differential operator
$(-\Delta)^{\alpha}$ and the weight $w_{\lambda,\kappa}$. Another obstacle is
that the usual Sobolev embedding from $H^{k}$ into $L^{\infty}$, $k>n/2$ is
not any more true in the weighted setting that is why, it must be important to
always specify that the data lie in $L^{\infty}$ even if the data is in a
sufficiently regular weighted Sobolev space. However, one may still use
weighted Sobolev’s embedding with intermediate exponents (different from 1 and
$\infty$), this is done for instance by Fabes, Kenig and Serapioni in [18] or
in Maz’ya’s book [27].

We shall prove a global existence result for data in the weighted Sobolev
spaces $H^{k}({w_{\lambda,\kappa}})$ with $k=\max(0,3/2-\alpha)$ for
subcritical values of $\alpha$. Beside, we prove a local existence theorem for
data in $H^{2}({w_{\lambda,\kappa}})$ in the supercritical case. In both
cases, the weight is defined by
$w_{\lambda,\kappa}(x)=(1+\left|x\right|^{\kappa})^{-\lambda/\kappa}$ where
$\lambda\in(0,1)$ in the subcritical case, and $\lambda\in(0,\alpha/2)$ in the
supercritical case, and $\kappa\geq 2$ is an even integer.

A key tool in the proof of those existence results is the use of some new
commutator estimates involving the family of weights $w_{\lambda,\kappa}$ and
the nonlocal operator $(-\Delta)^{\alpha/2}$ (in particular for supercritical
values of $\alpha$). These lemma can be used to treat other equations that
involve nonlocal fractional operators. Another tool is the Littlewood-Paley
theory in the weighted setting which allows to deal with data in Sobolev
space. It is worth saying that in the $H^{1}$ or $H^{1/2}$ case, one could
follow the approach of [24] in the subcritical case taking advantage of nice
cancellations and formulas involving the Hilbert transform. However, in this
paper we aim at stating a theorem that allows to deal with a bigger range of
Sobolev regularity. The use of the weighted Littlewood-Paley theory turned out
to be efficient to treat the low frequencies of the Hilbert transform which
are not continuous in $L^{\infty}$. Also, new commutator estimates are needed
to treat the supercritical or subcritical case.

The article is organized as follows. In the first section, we give the
statement of the main theorems. In the third section, we recall some tools and
important results that we shall use in the proof of our main theorems. In the
third part, we establish a priori estimates for positive initial data
$\theta_{0}\in H^{k}(w_{\lambda,\kappa}(x)dx)\cap L^{\infty}$ with
$k=\max(0,3/2-\alpha)$ where the case $k={0}$ and $k\neq 0$ are treated
separetly into two subsections. The last section is devoted to the proof of
the local existence for $H^{2}_{w}$ data in the supercritical case.

##  2 Main results

In the subcritical case, we prove the following global existence result for
arbitrary large data in weighted Sobolev spaces.

######  Theorem 2.1.

Assume that $1<\alpha<2$, then for all weights
$w_{\lambda,\kappa}(x)=(1+|x|^{\kappa})^{-\lambda/\kappa}$ with
$\lambda\in(0,1)$ and $\kappa\geq 2$ an even integer, and for all positive
initial data $\theta_{0}\in H^{k}(w_{\lambda,\kappa}(x)dx)\cap L^{\infty}$
with $k\in\max(0,3/2-\alpha)$, there exists at least one global solution
$\theta$ to the equation $\mathcal{T}_{\alpha},$ which verifies, for all
finite $T>0$

$\theta\in\mathcal{C}([0,T],H^{k}(w_{\lambda,\kappa}(x)dx))\cap
L^{2}([0,T],\dot{H}^{k+\alpha/2}(w_{\lambda,\kappa}(x)dx)).$

Moreover, for all $T<\infty$, we have

$\|\theta(T)\|^{2}_{H^{k}(w_{\lambda,\kappa})}\leq\|\theta_{0}\|^{2}_{H^{k}(w_{\lambda,\kappa})}e^{CT}$

The constant $C>0$ depends on $\|\theta_{0}\|_{L^{\infty}}$, $\lambda$,
$\kappa$ and $\nu$.

In the supercritical case, we have a local existence result of solutions for
arbitrary $H^{2}_{w}$ data.

######  Theorem 2.2.

Assume that $0<\alpha<1$, then for all data $\theta_{0}\in
H^{2}_{w_{\lambda,\kappa}}$ where the weight is given by
$w_{\lambda}(x)=(1+|x|^{\kappa})^{-\lambda/\kappa}$ with
$\lambda\in(0,\alpha/2)$. Then, there exists a time $T^{*}(\theta_{0})>0$ such
that $(\mathcal{T}_{\alpha})$ admits at least one solution that verifies

$\theta\in\mathcal{C}([0,T],H^{2}(w_{\lambda,\kappa}(x)dx))\cap
L^{2}([0,T],\dot{H}^{2+\frac{\alpha}{2}}(w_{\lambda,\kappa}(x)dx))$

for all $T\leq T^{*}$.

######  Remark 2.3.

In the proof of Theorem 2.1, we shall treat the case $k=0$ separetly, for the
other values of $k$ the proof is based on the use of the weighted Littlewood-
Paley decomposition where a careful treatment of the low frequencies of the
velocity is needed.

######  Remark 2.4.

Beside extending some previous results from [17] and [1] this article contains
also some new commutator estimates that could be of interest in the study of
more general nonlocal and nonlinear equations involving the operator
$\Lambda^{\alpha}$ that are contained in lemma 4.1.

##  3 Weighted Littlewood-Paley theory and Muckenhoupt’s class

Let $w$ be a positive and locally integrable function. A measurable function
$\theta$ is said to belong to the weighted Lebesgue spaces $L^{p}(w)$ (noted
also $L^{p}_{w}$ or $L^{p}(wdx)$) with $1\leq p<\infty$ if and only if

$\|\theta\|_{L^{p}(w)}:=\left(\int|\theta(x)|^{p}\ w(x)\
dx\right)^{1/p}<\infty,$

and we have $L^{\infty}=L^{\infty}_{w}$. We shall say that $f$ belongs to the
weighted inhomogeneous Sobolev space $H^{s}_{w}$ (or $H_{s}(w)$) with
$\left|s\right|<1/2$ if $f\in L^{2}_{w}$ and $\Lambda^{s}\theta\in L^{2}_{w}$,
it is endowed with the semi-norm

$\|\theta\|_{H^{s}_{w}}=\|\theta\|_{L^{2}_{w}}+\|\Lambda^{s}\theta\|_{L^{2}_{w}}.$

Anagolously, we defined the homogeneous weighted Sobolev space
$\dot{H}^{s}_{w}$ (or $\dot{H}_{s}(w)$) as the space such that the following
semi-norm is finite

$\|\theta\|_{\dot{H}^{s}_{w}}=\|\Lambda^{s}\theta\|_{L^{2}_{w}}.$

We will use the usual notation for the space-time norms, namely, we will say
that $\theta$ belongs to the space $L^{2}([0,T],\dot{H}^{s}(w(x)dx))$ if

$\int_{0}^{T}\int|\Lambda^{s}\theta(x,t)|^{2}\ w(x)dx\ dt<\infty.$

with the classical modification for $L^{\infty}([0,T],\dot{H}^{s}(w(x)dx))$,
that is

$\sup_{0<t<T}\int|\Lambda^{s}\theta(x,t)|^{2}\ w(x)dx<\infty.$

We shall also use the so-called Hardy-Littlewood maximal function defined as
follows. The Hardy-Littlewood maximal function of a locally integrable
function $\theta$ on $\mathbb{R}^{n}$ is defined by

$\mathcal{M}f(x)=\sup_{Q}\frac{1}{|Q|}\int_{Q}|\theta(y)|\ dy,$

where the supremum is taken over all cubes $Q$ of $\mathbb{R}^{n}$ centered at
the point $x\in\mathbb{R}^{n}$ and $|Q|$ stands for the Lebesgue measure of
the cube $Q$. One of the remarkable properties of the maximal Hardy-Littlewood
function is that it is continuous operator from $L^{p}$ to $L^{p}$ for all
$1<p\leq\infty$. For $p=1$, it is a continuous operator from $L^{1}$ to the
weak Lebesgue space $L^{1,\infty}$ endowed with the norm
$\|\theta\|_{L^{1,\infty}}=\sup_{\lambda>0}\left\\{{\lambda}^{-1}|\\{x\in\mathbb{R}^{n},|\theta(x)|>\lambda\\}|\right\\}$.

The extension of this very useful continuity property to weighted Lebesgue
spaces goes back to Muckenhoupt [28]. He proved that a necessary and
sufficient condition on $w$ which ensured the continuity of the Hardy-
Littlewood maximal function on $L^{p}(w)$, $1<p\leq\infty$. More precisely,
the Muckenhoupt theorem [28] states that there exists a constant $C_{1}>0$
such that

$\quad\quad\quad\quad\quad\quad\quad\quad\int(\mathcal{M}\theta(x))^{p}w(x)\
dx\leq C_{1}(w)\int|\theta(x)|^{p}w(x)\ dx\quad\quad\quad\quad\quad\quad\quad$
(3.1)

if and only if, there exists a constant $C_{2}(w)>0$ such that, for all cubes
$Q$ in $\mathbb{R}^{n}$,

$\quad\quad\quad\quad\quad\quad\quad\quad\sup_{Q}\left(\frac{1}{|Q|}\int_{Q}w\
dx\right)\left(\frac{1}{|Q|}\int_{Q}w^{\frac{1}{1-p}}\ dx\right)^{p-1}\leq
C_{2}(w),\quad\quad\quad\quad\quad\quad$ (3.2)

where $|Q|$ is the Lebesgue measure of the arbitrary cubes $Q$ with edges
parallel to the coordinate axes. Those weights $w$ statisfying 3.2 are said to
belong to the $\mathcal{A}_{p}(\mathbb{R}^{n})$ class of Muckenhoupt ([28]).
The necessary condition is not difficult to obtain, it suffices to set
$\theta(x)=w(x)^{\frac{1}{1-p}}\mathds{1}_{Q}(x)$ in 3.1. As for the
sufficient condition we refer to [28] or [8].

Another remarkable property of the $\mathcal{A}_{p}$ weights is that the
Hilbert transform is a continuous operator on the space $L^{p}(w)$, with
$1<p<\infty$ if and only if the weight $w$ belongs the $\mathcal{A}_{p}$ class
of Muckenhoupt that is those which verify 3.2 (see [20]). More generally, this
property holds for all Calderón-Zygmund operators $T$ (see for instance [13],
[8], [32]), namely, there exists a constant $C(T,w)>0$ which depends on the
operator $T$ and on the weight $w$, such that

$\|T(f)\|_{L^{p}(w)}\leq C(T,w)\|w\|_{\mathcal{A}_{p}}\|f\|_{L^{p}(w)}.$

As mentioned in the introduction, the family of weights
$w_{\lambda,\kappa}(x)=(1+|x|^{\kappa})^{-\lambda/\kappa}$ with $0<\lambda<1$
and $\kappa\geq 2$ being an even integer, that we consider in the article
belongs to the $(\mathcal{A}_{p})_{p\in[2,\infty)}$ class of Muckenhoupt.
Therefore, one can use all the aforementioned useful continuity results.

We finish this section by recalling that, as in the unweighted setting (see
for instance [25], [4], [2]), the weighted Sobolev $H^{s}_{w}$ spaces can be
defined through Littlewood-Paley theory when
$w\in\mathcal{A}_{\infty}=\cap_{p>1}\mathcal{A}_{p}$ (see [23]). Even in the
case $w\in{\mathcal{A}_{p,loc}}$ (that is considering only small cubes in the
supremum in 3.2) one still have a satisfactory Littlewood-Paley theory as
shown in [30]. The construction is as follows, fix a function
$\phi_{0}\in\mathcal{D}(\mathbb{R}^{n})$ that is non-negative and radial and
such that ${\phi}_{0}(\xi)=1$, if $|\xi|\leq 1/2$ and ${\phi}_{0}(\xi)=0$ if
$|\xi|\geq 1$. Then, from this fixed function $\phi_{0}$ we define $\psi_{0}$
so that ${\psi}_{0}(\xi)=\phi_{0}(\xi/2)-\phi_{0}(\xi)$ (which is supported in
a corona). For $j\in\mathbb{Z}$, we define the distributions
$S_{j}f=\mathcal{F}^{-1}(\phi_{0}(2^{-j}\xi)\hat{f}(\xi))$ and
$\Delta_{j}f=\mathcal{F}^{-1}(\psi_{0}(2^{-j}\xi)\hat{f}(\xi))$ and we get the
so-called (inhomogeneous) Littlewood-Paley decompositon of
$f\in\mathcal{S}^{\prime}(\mathbb{R}^{n})$ that is for all $K\in\mathbb{Z}$ we
have the following inequality in $\mathcal{S}^{\prime}(\mathbb{R}^{n})$

$f=S_{K}f+\sum_{j\geq K}\Delta_{j}f.$ (3.3)

The homogeneous decomposition is obtained through a passage to the limit in
equality 3.3 as $K\rightarrow-\infty$ in the
$\mathcal{S}^{\prime}(\mathbb{R}^{n})$ topology and we obtain

$f=\sum_{j\in\mathbb{Z}}\Delta_{j}f.$ (3.4)

Equality 3.4 is called the homogeneous decomposition and has to be considered
modulo polynomials, indeed, $\Delta_{j}f=0\Leftrightarrow$ $f$ is a
polynomial. Then, we define the homogeneous weighted Sobolev spaces
$\dot{H}^{s}_{w}$ for $|s|<n/2$ as follows

$f\in\dot{H}^{s}_{w}\Longleftrightarrow f=\sum_{j\in\mathbb{Z}}\Delta_{j}f,\ \
{\text{in}}\ \mathcal{S}^{\prime}(\mathbb{R}^{n})\ \ {\text{and}}\ \
\sum_{j\in\mathbb{Z}}2^{2j}\|\Delta_{j}f\|^{2}_{L^{2}_{w}}<\infty.$

In order to deal with weighted $L^{p}$-estimate of derivatives that involves
the operators $\Delta_{j}$ or $S_{j}$ we shall use the so-called Bernstein’s
inequality. It says that, for all $f\in\mathcal{S}^{\prime}(\mathbb{R}^{n})$
and couple $(j,s)\in\mathbb{Z}\times\mathbb{R}$, and for all $1\leq p\leq
q\leq\infty$ and all weights $w\in A_{\infty}$, we have

$\|\Lambda^{s}\Delta_{j}f\|_{L^{p}_{w}}\lesssim
2^{js}\|\Delta_{j}f\|_{L^{p}_{w}}\ \ {\text{also}}\ \
\|\Delta_{j}f\|_{L^{q}_{w}}\lesssim
2^{{j}(\frac{n}{p}-\frac{n}{q})}\|\Delta_{j}f\|_{L^{p}_{w}}\ \text{and}\
\|\Lambda^{s}S_{j}f\|_{L^{p}_{w}}\lesssim 2^{js}\|S_{j}f\|_{L^{p}_{w}}$

For two distributions $f$ and $g$ that are in
$\mathcal{S}^{\prime}(\mathbb{R}^{n})$, we may write the paraproduct as
follows

$fg=\sum_{q\in\mathbb{Z}}S_{q+1}f\Delta_{q}g+\sum_{j\in\mathbb{Z}}\Delta_{j}fS_{j}g.$

To establish the existence of at least one solution to equation
$(\mathcal{T}_{\alpha})$ it is often useful to truncate the initial data using
a function $\psi_{R}$ that is a smooth, positive and compactly supported
function in $B_{2R}=[-2R,2R]$. To construct $\psi_{R}$, we consider a function
$\psi\in[0,1]$ such that $\psi(x)=1$ if $|x|\leq 1$, and $0$ if $|x|\geq 2$
and then, for $R>0$ we define

$\psi_{R}(x)\equiv\psi(x/R).$ (3.5)

This truncation function $\psi_{R}$ will be used throughout the paper when
making energy estimates. Note that the constants that will appear in the
estimations will always depend on harmless quantities, these constants will be
sometimes hidden into the symbol $\lesssim$. Before proving the results, we
need to introduce some important lemmas. This is the aim of the next section.

##  4 Commutator estimates in the subcritical or supercritical cases.

In this section, we prove a commutator estimate involving the family of
$(\mathcal{A}_{p})_{p\in(1,\infty)}$ Muckenhoupt weights
$w_{\lambda,\kappa}(x)=(1+|x|^{\kappa})^{-\lambda/\kappa}$, $0<\lambda<1$,
$\kappa\geq 2$ being an even integer. More precisely, in our a priori
estimates we shall need to control some extra commutators of the type
$T_{w_{\lambda,\kappa}}(f)\equiv[\Lambda^{\alpha/2},w_{\lambda,\kappa}]f$. We
shall need to use that such a commutator is continuous operator from
$L^{2}(w_{\lambda,\kappa})$ to $L^{2}(w^{-1}_{\lambda,\kappa})$ in both the
subcritical and supercritical cases. In the supercritical case, we need
$\lambda$ to be smaller than $\alpha/2$ whereas in the subcritical case we
allow $\lambda\in(0,1)$. We shall also need another lemma that gives an
$L^{\infty}$ bound for the singular operator
$\Lambda^{\alpha}w_{\lambda,\kappa}$ with $\alpha\in(0,2)$. We shall prove
both lemmas in $\mathbb{R}^{n}$. The constant $C(n,\alpha,\lambda,\kappa)>0$
that will appear throughout this section can be different from a line to
another but we shall keep the same notation for the sake of simplicity.

######  Lemma 4.1.

Let us consider the family of weights given by
$w_{\lambda,\kappa}(x)=(1+\left|x\right|^{\kappa})^{-\lambda/\kappa}$, then
have the following two estimates :

$\bullet$ If $\alpha\in(0,1)$, then for all $\lambda$ such that
$0<\lambda<\alpha/2$ and for all even integer $\kappa\geq 2$, the operator

$\displaystyle T_{w_{\lambda,\kappa}}:$ $\displaystyle
L^{2}(w_{\lambda,\kappa})$ $\displaystyle\longrightarrow
L^{2}(w^{-1}_{\lambda,\kappa})$ $\displaystyle f$
$\displaystyle\longmapsto[\Lambda^{\alpha/2},w_{\lambda,\kappa}]f$

is continuous. That is, for all $f\in L^{2}(w_{\lambda,\kappa})$, there exists
a constant $C>0$ which depends on $\alpha,\lambda,\kappa$, and $n$ such that,

$\int\left|[\Lambda^{\alpha/2},w_{\lambda,\kappa}]f\right|^{2}\frac{dx}{w_{\lambda,\kappa}}\leq
C\int|f|^{2}w_{\lambda,\kappa}\ dx.$

$\bullet$ If $1<\alpha<2$, then for all $\lambda\in(0,1)$ and for all even
integer $\kappa\geq 2$, the operator $T_{w_{\lambda,\kappa}}$ is continuous
from $L^{2}(w_{\lambda,\kappa})$ to $L^{2}(w^{-1}_{\lambda,\kappa}).$

######  Remark 4.2.

Note that in the supercritical case, the weight $w_{\lambda,\kappa}$ appearing
in the commutator $T_{w_{\lambda,\kappa}}$ should be denoted
$w_{\lambda(\alpha),\kappa}$ since for each fixed $\alpha$ we have a different
weight, but for the sake of simplicity we shall not write the dependence on
$\alpha$.

Proof of lemma 4.1. We first remark that

$\displaystyle\left|\left[\Lambda^{\alpha/2},w_{\lambda,\kappa}\right]f\right|\leq
C({\alpha},n)P.V.\int\frac{|w_{\lambda,\kappa}(x)-w_{\lambda,\kappa}(y)|}{|x-y|^{n+{\alpha/2}}}|f(y)|\
dy.$

Then, we shall use the following lemma

######  Lemma 4.3.

For all $(x,y)\in\mathbb{R}^{n}\times\mathbb{R}^{n}$, for all
$\lambda\in(0,1)$, and for all even integer $\kappa\geq 2$, the following
inequality holds

$|w_{\lambda,\kappa}({x})-w_{\lambda,\kappa}({y})|\leq
C(\lambda,\kappa)\min\left(|{x}-{y}|,|{x}-{y}|^{\lambda/2}\right)\sqrt{w_{\lambda,\kappa}({x})w_{\lambda,\kappa}({y})}.$

Proof of lemma 4.3. We split the integral into two pieces. If $x$ and $y$ are
such that : $|x-y|>1$. Then, if $|x-y|>\frac{|x|}{2}$ and
$|x-y|>\frac{|y|}{2}$ then, since
$w_{\lambda,\kappa}({x})-w_{\lambda,\kappa}({y}))\leq
w_{\lambda,\kappa}({x}+w_{\lambda,\kappa}({y})$, it suffices to estimate
$w_{\lambda,\kappa}(x)+w_{\lambda,\kappa}(y)$ and in this case

$\displaystyle w_{\lambda,\kappa}(x)+w_{\lambda}(y)$ $\displaystyle\leq$
$\displaystyle\sqrt{{w_{\lambda,\kappa}(x)}{w_{\lambda,\kappa}(y)}}\left(\frac{1}{\sqrt{w_{\lambda,\kappa}(x)}}+\frac{1}{\sqrt{w_{\lambda,\kappa}(y)}}\right)$
$\displaystyle\leq$ $\displaystyle
C(\lambda,\kappa)|x-y|^{\lambda/2}\sqrt{{w_{\lambda,\kappa}(x)}{w_{\lambda,\kappa}(y)}}$

Where we have used that
$(1+2^{\kappa}|x-y|^{\kappa})^{\lambda/\kappa}>w^{-1}_{\lambda,\kappa}(x)$
then
$w_{\lambda,\kappa}(x)^{-1/2}\leq(1+2^{\kappa}|x-y|^{\kappa})^{\lambda/2\kappa}\leq
C(\lambda,\kappa)|x-y|^{{\lambda}/{2\kappa}}$. By symmetry, we also have
$w_{\lambda,\kappa}(y)^{-1/2}\leq(1+2^{\kappa}|x-y|^{\kappa})^{\lambda/2\kappa}\leq
C(\lambda,\kappa)|x-y|^{\lambda/2}$.

Otherwise, $|x-y|\leq\frac{|x|}{2}$ or $|x-y|\leq\frac{|y|}{2}$. In this case,
we have that $w_{\lambda,\kappa}(x)$ and $w_{\lambda,\kappa}(y)$ are
comparable, in the sense that, one may find a constant $C(\lambda,\kappa)>0$
so that $w_{\lambda,\kappa}(x)\approx C(\lambda,\kappa)w_{\lambda,\kappa}(y)$.
Therefore,

$w_{\lambda,\kappa}(x)+w_{\lambda,\kappa}(y)\leq
C(\lambda,\kappa)w_{\lambda,\kappa}(x)\leq
C(\lambda,\kappa)\sqrt{w_{\lambda,\kappa}(x)w_{\lambda,\kappa}(y)}\leq
C(\lambda,\kappa)|x-y|^{\lambda/2}\sqrt{w_{\lambda,\kappa}(x)w_{\lambda,\kappa}(y)}$

Finally, if $|x-y|<1$, then

$\displaystyle|w_{\lambda,\kappa}(x)-w_{\lambda,\kappa}(y)|\leq
C|x-y|\sup_{z\in[x,y]}|\nabla w_{\lambda,\kappa}(z)|$ $\displaystyle\leq$
$\displaystyle C(\lambda,\kappa)|x-y|\sup_{z\in[x,y]}|w_{\lambda,\kappa}(z)|$
$\displaystyle\leq$ $\displaystyle
C(\lambda,\kappa)|x-y|\sqrt{w_{\lambda,\kappa}(x)w_{\lambda,\kappa}(y)}.$

∎

Therefore, using lemma 4.3, we infer that

$\displaystyle\left|[\Lambda^{\alpha/2},w_{\lambda,\kappa}]f\right|\leq
C(\lambda,\kappa,n)\sqrt{w_{\lambda,\kappa}(x)}\int\min\left(\frac{1}{|x-y|^{n-1+\frac{\alpha}{2}}},\frac{1}{|x-y|^{{n-\lambda+\frac{\alpha}{2}}}}\right)\sqrt{w_{\lambda,\kappa}(y)}|f(y)|\
dy\hskip 5.69046pt$ (4.2)

Let us set

$\chi(x)\equiv\min\left(\frac{1}{|x|^{n-1+\frac{\alpha}{2}}},\frac{1}{|x|^{{n-\lambda+\frac{\alpha}{2}}}}\right),$

In the supercritical case, we have $0<\alpha<1$, therefore, if $\lambda$ is
such that $0<\lambda<\alpha/2$, then we observe that

$\chi(x)=\frac{1}{|x|^{{n-\lambda+\frac{\alpha}{2}}}}\mathds{1}_{|x|>1}\in
L^{1}(\mathbb{R}^{n}),$

and,

$\chi(x)=\frac{1}{|x|^{n-1+\frac{\alpha}{2}}}\mathds{1}_{|x|\leq 1}\in
L^{1}(\mathbb{R}^{n}).$

Hence, by convolution
$x\mapsto\left(\chi*\sqrt{w_{\lambda,\kappa}}f\right)(x)\in
L^{2}(\mathbb{R}^{n})$ and we have for $C=C(\alpha,\lambda,\kappa,n)>0$

$\displaystyle\left\|\int\min\left(\frac{1}{|x-y|^{n-1+\frac{\alpha}{2}}},\frac{1}{|x-y|^{{n-\lambda+\frac{\alpha}{2}}}}\right)\sqrt{w_{\lambda,\kappa}(y)}\
f(y)\ dy\right\|_{L^{2}}$ $\displaystyle\leq$ $\displaystyle
C\|B\|_{L^{1}}\left\|\sqrt{w_{\lambda,\kappa}(y)}f(y)\right\|_{L^{2}}$ (4.3)
$\displaystyle\leq$ $\displaystyle C\|f\|_{L^{2}(w_{\lambda,\kappa})}$

Then, using inequality 4.2 we conclude that the commutator $f\mapsto
T_{w_{\lambda,\kappa}}(f)$ is a continuous operator from
$L^{2}(w_{\lambda,\kappa})\rightarrow L^{2}(w^{-1}_{\lambda,\kappa})$ for
supercritical values of $\alpha$. As for the subcritical case, namely
$1<\alpha<2$, we have that, for all $\lambda\in(0,1)$, the function
$x\mapsto\chi(x)\in L^{1}(\mathbb{R}^{n})$. Therefore inequality 4.3 still
hold, and we may conclude as before.

∎

###  4.1 Control of the $L^{\infty}$ norm of the nonlocal operator
$\Lambda^{\alpha}w_{\lambda}$, with $\alpha\in(0,2)$.

We shall state and prove a general lemma that deals with a bound for
$\Lambda^{\alpha}w_{\lambda}$ valid for all values of $\alpha\in(0,2)$. As a
matter of fact, we shall only use the estimate for $\alpha=1$. More precisely,
we shall only use it in the study of the $L^{2}$ norm, and more precisely when
we will integrate by parts and send the $\Lambda$ onto the weight
$w_{\lambda,\kappa}$ just after the use of the Córdoba and Córdoba inequality
(see equation 5.1).

######  Lemma 4.4.

For all $\alpha\in(0,2)$, for all family of weights
$w_{\lambda,\kappa}(x)=(1+|x|^{\kappa})^{-\lambda/\kappa}$, where
$0<\lambda<1$ and $\kappa\geq 2$ an even integer, there exists a constant
$C=C(\alpha,\lambda,\kappa,n)>0$ such that the following estimate holds

$|\Lambda^{\alpha}w_{\lambda,\kappa}|\leq Cw_{\lambda,\kappa}(x)$

Proof of lemma 4.4. It suffices to write

$\displaystyle\Lambda^{\alpha}w_{\lambda,\kappa}$
$\displaystyle=\frac{1}{2}C(n,\alpha)\int_{|x-y|\leq
1}\frac{2w_{\lambda,\kappa}(x)-w_{\lambda,\kappa}(2x-y)-w_{\lambda,\kappa}(y)}{|x-y|^{1+\alpha}}\
dy$ $\displaystyle\ +\
C(n,\alpha)\int_{|x-y|>1}\frac{w_{\lambda,\kappa}(x)-w_{\lambda,\kappa}(y)}{|x-y|^{1+\alpha}}\
dy$ $\displaystyle\equiv I_{1}+I_{2}$

For $I_{1}$, we see that, using a second order Taylor-expansion

$|2w_{\lambda,\kappa}(x)-w_{\lambda,\kappa}(2x-y)-w_{\lambda,\kappa}(y)|\leq|x-y|^{2}\sup_{z\in[x,y]}|\nabla^{2}w_{\lambda,\kappa}(z)|,$

together with the following inequality,

$|\nabla^{2}w_{\lambda,\kappa}(x)|\leq Cw_{\lambda,\kappa}(x),$

allow us to conclude that,

$|I_{1}(x)|\leq Cw_{\lambda,\kappa}(x).$

As for the second integral $I_{2}$, we split the set $\Omega(x)=\\{y|\
|x-y|>1\\}$ into two regions. Namely, we intersect the set $\Omega(x)$ with

$\Omega_{1}(x)=\\{|x-y|>\frac{|x|}{2}\\}\cap\\{|x-y|>\frac{|y|}{2}\\}\ \
\text{or}\ \
\Omega_{2}(x)=\\{|x-y|\leq\frac{|x|}{2}\\}\cup\\{|x-y|\leq\frac{|y|}{2}\\},$

and we write

$\displaystyle I_{2}$ $\displaystyle=$ $\displaystyle C(n,\alpha)\
w_{\lambda,\kappa}(x)\int_{\Omega\cap\Omega_{1}}\frac{1-\frac{w_{\lambda,\kappa}(y)}{w_{\lambda,\kappa}(x)}}{|x-y|^{1+\alpha}}\
dy+C(n,\alpha,\kappa)\
w_{\lambda,\kappa}(x)\int_{\Omega\cap\Omega_{2}}\frac{1-\frac{w_{\lambda,\kappa}(y)}{w_{\lambda,\kappa}(x)}}{|x-y|^{1+\alpha}}\
dy$ $\displaystyle\equiv$ $\displaystyle I_{2,a}+I_{2,b}.$

For the first integral, we observe that
$(1+|x|^{\kappa})^{\lambda/\kappa}\leq(1+2^{\kappa}|x-y|^{\kappa})^{\lambda/\kappa}\leq
C(\lambda,\kappa)|x-y|^{\lambda}$ and therefore,
$w^{-1}_{\lambda,\kappa}(x)\leq C(\lambda,\kappa)|x-y|^{\lambda}$. This allows
us to get

$\displaystyle|I_{2,a}|$ $\displaystyle\leq$ $\displaystyle
Cw_{\lambda,\kappa}(x)\left(\int_{\Omega\cap\Omega_{1}}\frac{1}{|x-y|^{1+\alpha}}+\int_{\Omega\cap\Omega_{1}}\frac{|w^{-1}_{\lambda,\kappa}(x)w_{\lambda,\kappa}(y)|}{|x-y|^{1+\alpha}}\
dy\right)$ $\displaystyle\leq$ $\displaystyle
Cw_{\lambda,\kappa}(x)\left(C(\alpha)+\int_{\Omega\cap\Omega_{1}}\frac{1}{|x-y|^{1+\alpha-\gamma}}\
dy\right)$ $\displaystyle\leq$ $\displaystyle Cw_{\lambda,\kappa}(x)$

where we used that $1\leq\alpha$ (critical or subcritical values) or
$2\lambda<\alpha<1$ (supercritical value) imply the convergence of the last
integral 4.1 because $0<\lambda<1$. For the last integral $I_{2,b}$, we use
that $w_{\lambda,\kappa}(x)$ and $w_{\lambda,\kappa}(y)$ are comparable in the
region $\Omega\cap\Omega_{1}$, so that

$|I_{2,b}|\leq w_{\lambda,\kappa}(x)\
\int_{\Omega\cap\Omega_{2}}\frac{1}{{|x-y|^{1+\alpha}}}\ dy\leq
Cw_{\lambda,\kappa}(x)$

∎

##  5 Global existence for data $\theta_{0}\in H^{k}(w_{\lambda,\kappa})\cap
L^{\infty}$

The aim of this section is to prove Theorem 2.1. We shall first treat the case
of an $L^{2}_{w}\cap L^{\infty}$ data and then the case of a data in
$H^{k}_{w}\cap L^{\infty}$ with $k\neq 0$. Indeed, this latter case required a
more precise analysis of the low frequencies of the velocity. In the energy
estimates, we shall omit to write the dependence of $w$ on $\lambda$ and
$\kappa$ for the sake of readibility.

###  5.1 The case $\theta_{0}\in L^{2}(w_{\lambda,\kappa})\cap L^{\infty}.$

In this subsection, we study the following initial value problem, for all
$1<\alpha<2$:

$\ (\mathcal{T}_{\alpha})\ :\\\ \left\\{\begin{aligned}
&\partial_{t}\theta+\theta_{x}\mathcal{H}\theta+\nu\Lambda^{\alpha}\theta=0\hskip
56.9055pt\\\ &0<\theta(0,x)=\theta_{0}(x)\in L^{2}(w)\cap
L^{\infty}\end{aligned}\right.$

Before going into the computations to get $\it{a\ priori}$ estimates, one need
to check the existence of at least a solution to equation
$(\mathcal{T}_{\alpha})$. To do so, we truncate the initial data by
multiplying by the function $x\mapsto\phi(x/R)$ that has been introduced at
the end of section 3 and we focus on equation $\mathcal{T}_{{\alpha},R}$
defined as follows

$\ (\mathcal{T}_{\alpha,R})\ :\\\ \left\\{\begin{aligned}
&\partial_{t}\theta_{R}+\theta_{x}\mathcal{H}\theta_{R}+\nu\Lambda^{\alpha}\theta_{R}=0\hskip
56.9055pt\\\ &0<\theta_{0,R}(x)=\theta_{0}(x)\phi(x/R)\in L^{2}\cap
L^{\infty}\end{aligned}\right.$

Let $\theta_{R}$ be a solution of $\mathcal{T}_{R}$, since the initial data is
in $L^{2}$ and is positive, then it is classical to prove that there exist at
least one solution $\theta_{R}\in L^{\infty}([0,T],L^{2})\cap
L^{2}([0,T],\dot{H}^{\alpha/2})$ (see [14]). Then multiplying equation
$\mathcal{T}_{R}$ by such a solution $\theta_{R}$ and integrating in space
give

$\partial_{t}\int\frac{\theta^{2}_{R}}{2}w\
dx=-\int\theta_{R}\partial_{x}\theta_{R}\mathcal{H}\theta_{R}w\ dx-\int
w\theta_{R}\Lambda^{\alpha}\theta_{R}\ dx,$

which can be rewritten as,

$\partial_{t}\int\frac{\theta^{2}_{R}}{2}w\
dx=-\int\theta_{R}\partial_{x}\theta_{R}\mathcal{H}\theta_{R}w\
dx-\int\Lambda^{\alpha/2}\theta_{R}[\Lambda^{\alpha/2},w]\theta_{R}\
dx-\int|\Lambda^{\alpha/2}\theta_{R}|^{2}w\ dx.$

An integration by parts, together with the fact that
$\partial_{x}\mathcal{H}=-\Lambda$, give

$\partial_{t}\int\frac{\theta^{2}_{R}}{2}w\
dx=-\int\theta^{2}_{R}\Lambda\theta_{R}w\ dx\ +\int\theta_{R}\mathcal{H}\theta
w_{x}\ dx\ -\int\Lambda^{\alpha/2}\theta_{R}[w,\Lambda^{\alpha/2}]\theta_{R}\
dx\ -\int|\Lambda^{\alpha/2}\theta_{R}|^{2}w\ dx.$

By using the Córdoba and Córdoba inequality [12] : $\Lambda\theta^{3}_{R}\leq
3\theta^{2}_{R}\Lambda\theta_{R}$ (the convexity of
$\theta\mapsto\theta^{3}_{R}$ is ensured by the positivity assumption on
$\theta_{R}$), and then integrating by parts yields

$\displaystyle\partial_{t}\int\frac{\theta^{2}_{R}}{2}w\
dx\leq-\frac{1}{3}\int\theta^{3}_{R}\Lambda w\
dx+\int\theta_{R}\mathcal{H}\theta_{R}w_{x}\ dx$ $\displaystyle-$
$\displaystyle\int\Lambda^{\alpha/2}\theta_{R}[w,\Lambda^{\alpha/2}]\theta_{R}\
dx$ $\displaystyle-$ $\displaystyle\int|\Lambda^{\alpha/2}\theta_{R}|^{2}w\
dx.$

Using lemma 4.4 and the fact that pointwisely we have the inequality
$\partial_{x}w<w$ (indeed, it suffices to see that
$\partial_{x}w_{\lambda,\kappa}(x)=-\lambda
x^{k-1}(1+|x|^{\kappa})^{-1}w_{\lambda,\kappa}(x)$ and since $0<\lambda<1$, we
obviously have ${-{\lambda}}x^{k-1}<1+|x|^{\kappa}$), we obtain

$\displaystyle\partial_{t}\int\frac{\theta^{2}_{R}}{2}w\ dx$
$\displaystyle\leq$
$\displaystyle\frac{1}{3}\int|\theta_{R}||\theta_{R}\sqrt{w}|^{2}\ dx\
+\int|\sqrt{w}\theta_{R}||\mathcal{H}\theta_{R}\ \sqrt{w}|\ dx\
+\int|\sqrt{w}\Lambda^{\alpha/2}\theta_{R}||[\Lambda^{\alpha/2},w]\theta_{R}|\frac{dx}{\sqrt{w}}$
$\displaystyle\ -\int|\Lambda^{\alpha/2}\theta_{R}|^{2}w\ dx.$

Recalling that $\|\theta_{R}\|_{L^{2}(w)}=\|\theta_{R}\sqrt{w}\|_{2}$ and
$\|\theta_{R}\|_{\dot{H}^{\alpha/2}(w)}=\|\sqrt{w}\Lambda^{\alpha/2}\theta_{R}\|_{2}$,
and using lemma 4.1 together with the $L^{\infty}$ maximum principle, and the
continuity of the Hilbert transform on $L^{2}(w)$ (because
$w\in\mathcal{A}_{2}$) we finally get

$\partial_{t}\int\frac{\theta^{2}_{R}}{2}w\ dx\leq
C\|\theta_{0,R}\|_{L^{\infty}}\|\theta_{R}\|^{2}_{L(w)}+\|\Lambda^{\alpha/2}\theta_{R}\|_{L^{2}(w)}\|\theta_{R}\|_{L^{2}(w)}-\|\Lambda^{\alpha/2}\theta_{R}\|^{2}_{L^{2}(w)}.$

Using Young’s inequality we infer that, for all $\eta>0$

$\partial_{t}\int\frac{\theta^{2}_{R}}{2}w\
dx\leq\left(\frac{2}{\eta}+C\|\theta_{0,R}\|_{L^{\infty}}\right)\|\theta\|^{2}_{L^{2}(w)}+\left(\frac{\eta}{2}-1\right)\|\Lambda^{\alpha/2}\theta_{R}\|^{2}_{L^{2}(w)}.$
(5.2)

By choosing $\eta$ sufficiently small (for instance $\eta<2$), we find

$\partial_{t}\int\frac{\theta^{2}_{R}}{2}w+C\|\Lambda^{\alpha/2}\theta_{R}\|^{2}_{L^{2}(w)}dx\leq
C\|\theta_{0}\|_{L^{\infty}}\|\theta_{R}\|^{2}_{L^{2}(w)}.$

Integrating in time $s\in[0,T]$ yields

$\|\theta_{R}(x,T)\|^{2}_{L^{2}(w)}+C\int_{0}^{T}\|\Lambda^{\alpha/2}\theta_{R}\|^{2}_{L^{2}(w)}\
ds\leq\|\theta_{0,R}\|^{2}_{L^{2}(w)}+C\|\theta_{0,R}\|_{L^{\infty}}\int_{0}^{T}\|\theta_{R}(x,s)\|^{2}_{L^{2}(w)}\
ds.$

Hence, by Gronwall’s inequality we get

$\sup_{0<t<T}\int\theta^{2}_{R}w\ dx<\infty\ \ \text{and}\ \
\int_{0}^{T}\int|\Lambda^{\alpha/2}\theta_{R}|^{2}w\ dx<\infty.$

In particular, we have, for all $T<\infty$

$\|\theta_{R}(T)\|^{2}_{L^{2}(w)}\leq\|\theta_{0,R}\|^{2}_{L^{2}(w)}e^{CT},$

where $C>0$ depends only on $\|\theta_{0,R}\|_{L^{\infty}}$, and $\beta$.
Then, the passage to the weak limit as $R\rightarrow\infty$ allows us to prove
the first statement of Theorem 2.1. Indeed, since $1<\alpha<2$, one has enough
compactness to pass to the weak limit in the nonlinear term. This can be done
for instance via Rellich’s compactness theorem [25] which provides the strong
convergence in $(L^{2}L^{2})_{loc}$ of $\theta_{R}$ and allows us to pass to
the weak limit in the nonlinear term. It is obvious that $\theta_{0,R}$
converges strongly in $L^{2}$ as well as in $L^{\infty}$ toward $\theta_{0}$.
This concludes the proof.

∎

It is worth recalling that this is an open problem in the critical or super-
critical case and when the data is just in $L^{2}$ or $L^{2}_{w}$. Although
one can derive nice a priori estimates, it is still not clear whether one can
to pass to the limit because of the lack of compactness (see for instance
[31], [24]).

###  5.2 The case $k=\max(0,3/2-\alpha)$.

In this section, we study the following initial value problem

$\ (\mathcal{T}_{\alpha})\ :\\\ \left\\{\begin{aligned}
&\partial_{t}\theta+\theta_{x}\mathcal{H}\theta+\nu\Lambda^{\alpha}\theta=0\hskip
56.9055pt\\\ &0<\theta(0,x)=\theta_{0}(x)\in H^{k}(w)\cap
L^{\infty}.\end{aligned}\right.$

We shall focus on the following approximate family of equations (where $\phi$
have been introduced in section 3).

$\ (\mathcal{T}_{\alpha,R})\ :\\\ \left\\{\begin{aligned}
&\partial_{t}\theta_{R}+\partial_{x}\theta_{R}\mathcal{H}\theta_{R}+\nu\Lambda^{\alpha}\theta_{R}=0\hskip
56.9055pt\\\ &0<\theta_{0,R}(x)=\theta_{0}(x)\phi(x/R)\in H^{k}\cap
L^{\infty}.\end{aligned}\right.$

From theorem 3.1 of [14], we know that there exists at least one solution
$\theta_{R}\in L^{\infty}L^{2}\cap L^{2}H^{k+\alpha/2}$. For the sake of
readibility, we shall omit to write the index $R>0$ in the energy estimates.
In order to prove our result, we shall consider the evolution of the following
weighted Sobolev norm

$\|\theta\|^{2}_{H^{k}(w)}=\int|\Lambda^{k}\theta|^{2}\ w(x)dx\ +\
\int|\theta|^{2}\
w(x)dx=\|\sqrt{w}\Lambda^{k}\theta\|^{2}_{L^{2}}+\|\sqrt{w}\theta\|^{2}_{L^{2}}.$

The main goal is to show that, for all finite $T$, we have

$\|\theta(T)\|^{2}_{H^{k}(w)}+C\int_{0}^{T}\int|\Lambda^{\frac{\alpha}{2}+k}\theta|^{2}w\
dx\
ds\lesssim\|\theta_{0}\|^{2}_{H^{k}(w)}+\int_{0}^{T}\|\theta(x,s)\|^{2}_{H^{k}(w)}\
ds.$

In the previous section, we have already seen that, for all $\eta>0$

$\partial_{t}\int\frac{\theta^{2}}{2}w\
dx\leq\left(\frac{2}{\eta}+C\|\theta_{0}\|_{L^{\infty}}\right)\|\theta\|^{2}_{L^{2}(w)}+\left(\frac{\eta}{2}-1\right)\|\Lambda^{\alpha/2}\theta\|^{2}_{L^{2}(w)}$
(5.3)

Therefore, it suffices to focus on the evolution of the homogeneous Sobolev
norm. One can write it in terms of controlled commutators (via lemma 4.1) as
follows

$\displaystyle\frac{1}{2}\partial_{t}\int|\Lambda^{k}\theta|^{2}\ wdx$
$\displaystyle=$
$\displaystyle-\int\Lambda^{k}\theta\Lambda^{k}(\theta_{x}\mathcal{H}\theta)w\
dx-\nu\int\Lambda^{\frac{\alpha}{2}}(w\Lambda^{1/2}\theta)\
\Lambda^{\frac{\alpha}{2}+k}\theta\ dx$ $\displaystyle=$
$\displaystyle-\int\Lambda^{\sigma}(w\Lambda^{k}\theta)\Lambda^{k-\sigma}(\theta_{x}\mathcal{H}\theta)\
dx-\nu\int\sqrt{w}\Lambda^{\frac{\alpha}{2}+k}\theta\
[\Lambda^{\frac{\alpha}{2}},w]\Lambda^{k}\theta\ \frac{dx}{\sqrt{w}}$
$\displaystyle\hskip
113.81102pt-\nu\int|\Lambda^{\frac{\alpha}{2}+k}\theta|^{2}\ wdx$
$\displaystyle=$
$\displaystyle-\int\sqrt{w}\Lambda^{k-\sigma}(\theta_{x}\mathcal{H}\theta)\frac{1}{\sqrt{w}}[\Lambda^{\sigma},w]\Lambda^{k}\theta\
dx-\int\sqrt{w}\Lambda^{k+\sigma}\theta\sqrt{w}\Lambda^{k-\sigma}(\theta_{x}\mathcal{H}\theta)\
dx$ $\displaystyle-\nu\int\sqrt{w}\Lambda^{\frac{\alpha}{2}+k}\theta\
[\Lambda^{\frac{\alpha}{2}},w]\Lambda^{k}\theta\
\frac{dx}{\sqrt{w}}-\nu\int|\Lambda^{\frac{\alpha}{2}+k}\theta|^{2}\ wdx$
$\displaystyle=$ $\displaystyle(I)+(II)+(III)+(IV)$

To estimate $(I)$ and $(II)$ we use the dyadic Littlewood-Paley decomposition
in the weighted setting. We have, using the paraproduct formula

$\|\theta_{x}\mathcal{H}\theta\|_{\dot{H}^{k-\sigma}(w)}\leq\sum_{q\in\mathbb{Z}}2^{q(k-\sigma)}\|S_{q+1}\theta_{x}\Delta_{q}\mathcal{H}\theta\|_{L^{2}(w)}+\sum_{j\in\mathbb{Z}}2^{j(k-\sigma)}\|\Delta_{j}\theta_{x}S_{j}\mathcal{H}\theta\|_{L^{2}(w)}.$

To estimate the first sum, we use Bernstein’s inequality and the continuity of
the Hilbert transform on $L^{2}(w)$ because $w\in\mathcal{A}_{2}$, we get

$\displaystyle\sum_{q\in\mathbb{Z}}2^{q(k-\sigma)}\|S_{q+1}\theta_{x}\Delta_{q}\mathcal{H}\theta\|_{L^{2}(w)}$
$\displaystyle\leq$
$\displaystyle\sum_{q\in\mathbb{Z}}2^{q(k-\sigma)}\|S_{q+1}\theta_{x}\|_{L^{\infty}}\|\Delta_{q}\mathcal{H}\theta\|_{L^{2}(w)}$
$\displaystyle\leq$
$\displaystyle\|\theta\|_{L^{\infty}}\sum_{q\in\mathbb{Z}}2^{(k+1-\sigma)q}\|\Delta_{q}\mathcal{\theta}\|_{L^{2}(w)}$
$\displaystyle\leq$
$\displaystyle\|\theta_{0}\|_{L^{\infty}}\|\theta\|_{\dot{H}^{k+1-\sigma}(w)}.$

As for the second sum, we need to interpolate since we do not control the
$L^{\infty}$ norm of the low frequencies of $\mathcal{H}\theta$. More
precisely, we use the fact that $\theta\in L^{2}(w)\cap L^{\infty}$ then by
interpolation $\mathcal{H}\theta\in L^{s}(w)$ for all $2<s<\infty$ and we have

$\|\mathcal{H}\theta\|_{L^{s}(w)}\leq
C\|\theta\|^{1-2/s}_{L^{\infty}}\|\theta\|^{2/s}_{L^{2}(w)}.$

Then, if $r$ is the real such that

$\frac{1}{s}+\frac{1}{r}=\frac{1}{2},$

we obtain, using Hölder and then Bernstein’s inequality, that

$\displaystyle\sum_{j\in\mathbb{Z}}2^{j(k-\sigma)}\|\Delta_{j}\theta_{x}S_{j}\mathcal{H}\theta\|_{L^{2}(w)}$
$\displaystyle\leq$
$\displaystyle\sum_{j\in\mathbb{Z}}2^{j(k-\sigma)}\|\Delta_{j}\theta_{x}\|_{L^{r}(w)}\|S_{j}\mathcal{H}\theta\|_{L^{s}(w)}$
$\displaystyle\leq$ $\displaystyle
C\sum_{j\in\mathbb{Z}}2^{j(k-\sigma)}2^{j(\frac{1}{2}-\frac{1}{r})}2^{j}\|\Delta_{j}\theta\|_{L^{2}(w)}\|\mathcal{H}\theta\|_{L^{s}(w)}$
$\displaystyle\leq$ $\displaystyle
C\|\theta\|^{1-2/s}_{L^{\infty}}\|\theta\|^{2/s}_{L^{2}(w)}\sum_{j\in\mathbb{Z}}2^{j({k-\sigma+\frac{1}{s}+1})}\|\Delta_{j}\theta\|_{L^{2}(w)}$
$\displaystyle\leq$ $\displaystyle
C\|\theta_{0}\|^{1-2/s}_{L^{\infty}}\|\theta\|^{2/s}_{L^{2}(w)}\|\theta\|_{\dot{H}^{k+\frac{1}{s}+1-\sigma}(w)}.$

Therefore,

$\displaystyle\|\theta_{x}\mathcal{H}\theta\|_{\dot{H}^{k-\sigma}(w)}\leq\|\theta_{0}\|_{L^{\infty}}\|\theta\|_{\dot{H}^{k+1-\sigma}(w)}+\|\theta_{0}\|^{1-2/s}_{L^{\infty}}\|\theta\|^{2/s}_{L^{2}(w)}\|\theta\|_{\dot{H}^{k+\frac{1}{s}+1-\sigma}(w)}.$

Then, lemma 4.1 gives

$(I)\lesssim\|\theta_{0}\|_{L^{\infty}}\|\theta\|_{\dot{H}^{k}(w)}\|\theta\|_{\dot{H}^{k+1-\sigma}(w)}+\|\theta_{0}\|^{1-2/s}_{L^{\infty}}\|\theta\|_{\dot{H}^{k}(w)}\|\theta\|^{2/s}_{L^{2}(w)}\|\theta\|_{\dot{H}^{k+\frac{1}{s}+1-\sigma}(w)}.$

Choosing $s$ and $\sigma$ such that $\frac{1}{s}+1-\sigma<\frac{\alpha}{2}$
(note that it suffices to choose $s$ sufficiently big and
$\sigma>\frac{1}{2}$, actually we will choose
$\sigma=\frac{1}{2}+\frac{1}{s}$), and using the following interpolation
inequality valid for all $\gamma\in(0,1)$

$\|\theta\|_{\dot{H}^{k+\frac{1}{s}+1-\sigma}(w)}\leq\|\theta\|^{1-\gamma}_{\dot{H}^{k+\frac{1}{s}+1-\sigma-\gamma}(w)}\|\theta\|^{\gamma}_{\dot{H}^{k+\frac{1}{s}+2-\sigma-\gamma}(w)},$

we obtain

$\displaystyle(I)$ $\displaystyle\leq$
$\displaystyle\|\theta_{0}\|_{L^{\infty}}\|\theta\|_{\dot{H}^{k}(w)}\|\theta\|_{\dot{H}^{k+\frac{\alpha}{2}-\frac{1}{s}}(w)}+\|\theta_{0}\|^{1-2/s}_{L^{\infty}}\|\theta\|_{\dot{H}^{k}(w)}\|\theta\|^{2/s}_{L^{2}(w)}\|\theta\|_{\dot{H}^{k+\frac{1}{s}+1-\sigma}(w)}$
$\displaystyle\lesssim$
$\displaystyle\|\theta\|_{H^{k}(w)}\|\theta\|_{H^{k+\frac{\alpha}{2}-\frac{1}{s}}(w)}+\|\theta\|^{2/s}_{L^{2}(w)}\|\theta\|_{H^{k}(w)}\|\theta\|^{1-\gamma}_{H^{k+\frac{1}{s}+1-\sigma-\gamma}(w)}\|\theta\|^{\gamma}_{H^{k+\frac{1}{s}+2-\sigma-\gamma}(w)}.$

Then, we use that $H^{k}({w})\hookrightarrow
H^{k+\frac{1}{s}+1-\sigma-\gamma}({w})$ (note that it suffices to choose
$\gamma=1-{\varepsilon}$ with $\epsilon$ small enough, so that
$\epsilon+\frac{1}{s}<\sigma$) moreover we have
$H^{k+\frac{1}{s}+2-\sigma-\gamma}(w)\hookrightarrow
H^{k+\frac{\alpha}{2}}(w)$ (since $\sigma=\frac{1}{2}+\frac{1}{s}$ and we
choose $s$ big enough and $\epsilon$ small enough so that
$\frac{1}{2}+\epsilon<\frac{\alpha}{2}$ therefore
$\frac{1}{s}+\epsilon+1-\sigma<\frac{\alpha}{2}$). We get

${}(I)\lesssim\|\theta\|_{H^{k}(w)}\|\theta\|_{H^{k+\frac{\alpha}{2}-\frac{1}{s}}(w)}+\|\theta\|^{2/s}_{L^{2}(w)}\|\theta\|^{2-\gamma}_{H^{k}(w)}\|\theta\|^{\gamma}_{H^{k+\frac{\alpha}{2}}(w)}.$
(5.4)

Then, we shall repeatedly use Young’s inequality, first with the exponent
$p_{1}=\frac{1}{1-\gamma}>1$, we obtain

$\displaystyle(I)$ $\displaystyle\lesssim$
$\displaystyle\|\theta\|_{H^{k}(w)}\|\theta\|_{H^{k+\frac{\alpha}{2}-\frac{1}{s}}({w})}+(1-\gamma)\|\theta\|^{\frac{1}{s(1-\gamma)}}_{L^{2}({w})}\|\theta\|_{H^{k}({w})}+\gamma\|\theta\|_{H^{k+\frac{\alpha}{2}}(w)}\|\theta\|^{\frac{1}{s\gamma}}_{L^{2}({w})}\|\theta\|^{\frac{1}{\gamma}}_{H^{k}(w)},$

and then in the 3 terms of the above inequality (respectively, with the
exponents $p_{2}=2$ for the first two ones, and $p_{3}=1+\gamma>1$ for the
last term), we find that for all $\mu_{1}>0$

$\displaystyle(I)$ $\displaystyle\lesssim$
$\displaystyle\frac{1}{2\mu_{1}}\|\theta\|^{2}_{H^{k}(w)}+\frac{\mu_{1}}{2}\|\theta\|^{2}_{H^{k+\frac{\alpha}{2}}(w)}+\frac{(1-\gamma)}{2}\|\theta\|^{\frac{2}{s(1-\gamma)}}_{L^{2}(w)}+\frac{(1-\gamma)}{2}\|\theta\|^{2}_{H^{k}(w)}$
$\displaystyle\ +\
\frac{\mu^{1+\gamma}_{2}}{(1+\gamma)}\|\theta\|^{1+\gamma}_{H^{k+\frac{\alpha}{2}}(w)}+\frac{\gamma}{(1+\gamma)\mu^{1+\frac{1}{\gamma}}_{2}}\|\theta\|^{\frac{1}{s(1+\gamma)}}_{L^{2}(w)}\|\theta\|^{\frac{1}{1+\gamma}}_{H^{k}(w)}.$

Then, using once again Young’s inequality in the last term of the previous
inequality (with the exponent $p_{4}=2+2\gamma>1$) we finally get

$\displaystyle(I)$ $\displaystyle\lesssim$
$\displaystyle\frac{1}{2\mu_{1}}\|\theta\|^{2}_{\dot{H}^{k}(w)}+\frac{\mu_{1}}{2}\|\theta\|^{2}_{H^{k+\frac{\alpha}{2}}(w)}+\frac{(1-\gamma)}{2}\|\theta\|^{\frac{2}{s(1-\gamma)}}_{L^{2}(w)}+\frac{(1-\gamma)}{2}\|\theta\|^{2}_{H^{k}(w)}$
$\displaystyle\ +\
\frac{\mu^{1+\gamma}_{2}}{(1+\gamma)}\|\theta\|^{1+\gamma}_{H^{k+\frac{\alpha}{2}}(w)}+\frac{\gamma(1+2\gamma)}{2\mu_{2}^{1+\frac{1}{\gamma}}(1+\gamma)^{2}}\|\theta\|^{\frac{2}{s(1+2\gamma)}}_{L^{2}(w)}+\frac{\gamma}{2\mu_{2}^{1+\frac{1}{\gamma}}(1+\gamma)^{2}}\mu_{2}^{1+\frac{1}{\gamma}}\|\theta\|^{2}_{H^{k}(w)}.$

Since $s$ is chosen big enough, we have $\frac{2}{s(1+2\gamma)}<2$ and
$\frac{2}{s(1-\gamma)}<2$ (note that the first inequality holds for instance
if $s>1$, whereas the second is verified if $s>\frac{2}{\epsilon}$). Actually,
in the estimation of $(II)$ below we will need to assume that
$\sigma=\frac{1}{s}+\frac{1}{2}\leq\frac{\alpha}{2}$; thus all those
conditions imply that $s>\max(\frac{2}{\epsilon},\frac{2}{\alpha-1})$.
Futhermore, we obviously have that $1+\gamma<2$, hence we obtain,

$\displaystyle(I)$ $\displaystyle\lesssim$
$\displaystyle\frac{1}{2\mu_{1}}\|\theta\|^{2}_{H^{k}(w)}+\frac{\mu_{1}}{2}\|\theta\|^{2}_{H^{k+\frac{\alpha}{2}}(w)}+\frac{(1-\gamma)}{2}\|\theta\|^{2}_{L^{2}(w)}+\frac{(1-\gamma)}{2}\|\theta\|^{2}_{H^{k}(w)}$
$\displaystyle\ +\
\frac{\mu^{1+\gamma}_{2}}{(1+\gamma)}\|\theta\|^{2}_{H^{k+\frac{\alpha}{2}}(w)}+\frac{\gamma(1+2\gamma)}{2\mu_{2}^{1+\frac{1}{\gamma}}(1+\gamma)^{2}}\|\theta\|^{2}_{L^{2}(w)}+\frac{\gamma}{2\mu_{2}^{1+\frac{1}{\gamma}}(1+\gamma)^{2}}\|\theta\|^{2}_{H^{k}(w)}.$

Hence, we have obtained

$\displaystyle(I)$ $\displaystyle\lesssim$
$\displaystyle\left(\frac{\mu_{1}}{2}+\frac{\mu^{1+\gamma}_{2}}{1+\gamma}\right)\|\theta\|^{2}_{H^{k+\frac{\alpha}{2}}({w)}}+\left(\frac{1}{2\mu_{1}}+\frac{1-\gamma}{2}+\frac{\gamma}{2\mu_{2}^{1+\frac{1}{\gamma}}(1+\gamma)^{2}}\right)\|\theta\|^{2}_{H^{k}(w)}$
$\displaystyle+$
$\displaystyle\left(\frac{1-\gamma}{2}+\frac{\gamma(1+2\gamma)}{2\mu_{2}^{1+\frac{1}{\gamma}}(1+\gamma)^{2}}\right)\|\theta\|^{2}_{L^{2}(w)}.$

For $(II)$, we have

$(II)\leq\|\theta_{0}\|_{L^{\infty}}\|\theta\|_{\dot{H}^{k+\sigma}(w)}\|\theta\|_{\dot{H}^{k+1-\sigma}(w)}+\|\theta_{0}\|^{1-2/s}_{L^{\infty}}\|\theta\|_{\dot{H}^{k+\sigma}(w)}\|\theta\|^{2/s}_{L^{2}(w)}\|\theta\|_{\dot{H}^{k+\frac{1}{s}+1-\sigma}(w)}.$

Therefore, using the fact that $s$ is chosen so that
$\sigma=\frac{1}{s}+\frac{1}{2}\leq\frac{\alpha}{2}$ and that
$1-\gamma=\epsilon$, we get

$\displaystyle(II)$
$\displaystyle\lesssim\|\theta\|_{\dot{H}^{k+\sigma}(w)}\|\theta\|_{\dot{H}^{k+1-\sigma}(w)}+\|\theta\|_{\dot{H}^{k+\sigma}(w)}\|\theta\|^{2/s}_{L^{2}(w)}\|\theta\|^{1-\gamma}_{\dot{H}^{k+\frac{1}{s}+1-\sigma-\gamma}(w)}\|\theta\|^{\gamma}_{\dot{H}^{k+\frac{1}{s}+2-\sigma-\gamma}(w)}$
$\displaystyle\lesssim\|\theta\|_{H^{k+\frac{\alpha}{2}}(w)}\|\theta\|_{H^{k+\frac{1}{2}-\frac{1}{s}}(w)}+\|\theta\|_{H^{k+\frac{\alpha}{2}}(w)}\|\theta\|^{2/s}_{L^{2}(w)}\|\theta\|^{1-\gamma}_{H^{k+\frac{1}{s}+1-\sigma-\gamma}(w)}\|\theta\|^{\gamma}_{H^{k+\frac{1}{s}+2-\sigma-\gamma}(w)}.$

Then, using the interpolation inequality

$\|\theta\|_{H^{k+\frac{1}{2}-\frac{1}{s}}(w)}\leq\|\theta\|^{1-\frac{1}{s}}_{H^{k+\frac{1}{2}}(w)}\|\theta\|^{\frac{1}{s}}_{H^{k-\frac{1}{2}}(w)},$

we obtain,

$\displaystyle(II)\lesssim\|\theta\|^{2-\frac{1}{s}}_{H^{k+\frac{\alpha}{2}}}\|\theta\|^{1/s}_{H^{k}(w)}+\|\theta\|_{H^{k+\frac{\alpha}{2}}(w)}\|\theta\|^{2/s}_{L^{2}(w)}\|\theta\|^{1-\gamma}_{H^{k+\frac{1}{s}+1-\sigma-\gamma}(w)}\|\theta\|^{\gamma}_{H^{k+\frac{1}{s}+2-\sigma-\gamma}(w)}.$

Then, we use Young’s inequality in the first term of the above inequality
(with the exponent $p_{4}=2s>1$) and we estimate the last term as we did for
$(I)$ (in particular, we use the same embeddings), we get, for all
$\mu_{3}>0$,

$\displaystyle(II)$
$\displaystyle\lesssim\frac{2s-1}{2s}{\mu_{3}^{\frac{2s}{2s-1}}}\|\theta\|^{2}_{H^{k+\frac{\alpha}{2}}(w)}+\frac{1}{2s\mu_{3}^{2s}}\|\theta\|^{2}_{H^{k}(w)}+\|\theta\|^{2/s}_{L^{2}(w)}\|\theta\|^{1-\gamma}_{H^{k}(w)}\|\theta\|^{1+\gamma}_{H^{k+\frac{\alpha}{2}}(w)}$

Since we have $1+\gamma=2-\epsilon$, the previous inequality becomes

$\displaystyle(II)\lesssim\frac{(2s-1)}{2s}{\mu_{3}^{\frac{2s}{2s-1}}}\|\theta\|^{2}_{H^{k+\frac{\alpha}{2}}(w)}+\frac{1}{2s\mu^{2s}_{3}}\|\theta\|^{2}_{H^{k}(w)}+\|\theta\|^{2/s}_{L^{2}(w)}\|\theta\|^{\epsilon}_{H^{k}(w)}\|\theta\|^{2-\epsilon}_{H^{k+\frac{\alpha}{2}}(w)}.$

The last term of the previous inequality can be estimated as we did for in
inequality 5.4. Indeed, it suffices to switch the norms from $\dot{H}^{k}$ to
$\dot{H}^{k+\frac{\alpha}{2}}$ and to replace $\gamma$ by $\epsilon$ (this is
allowed since the only condition we used throughout these steps was
$\gamma>0$) we analogously infer that, for all $\mu_{4}>0$

$\displaystyle(II)$
$\displaystyle\lesssim\left(\frac{(2s-1)}{2s}{\mu_{3}^{\frac{2s}{2s-1}}}+\frac{\mu^{1+\epsilon}_{4}}{1+\epsilon}\right)\|\theta\|^{2}_{H^{k+\frac{\alpha}{2}}(w)}+\left(\frac{1}{2s\mu^{2s}_{3}}+\frac{1-\epsilon}{2}+\frac{\epsilon}{2\mu_{4}^{1+1/\epsilon}(1+\epsilon)^{2}}\right)\|\theta\|^{2}_{\dot{H}^{k}(w)}$
$\displaystyle\ +\
\left(\frac{1-\epsilon}{2}+\frac{\epsilon(1+2\epsilon)}{2\mu_{4}^{1+1/\epsilon}(1+\epsilon)^{2}}\right)\|\theta\|^{2}_{L^{2}(w)}.$

For $(III)$, we have that, for all $\mu_{5}>0$

$(III)\lesssim\|\theta\|_{\dot{H}^{k+\frac{\alpha}{2}}(w)}\|\theta\|_{\dot{H}^{k}(w)}\lesssim\frac{\mu_{5}}{2}\|\theta\|^{2}_{\dot{H}^{k+\frac{\alpha}{2}}(w)}+\frac{1}{2\mu_{5}}\|\theta\|^{2}_{\dot{H}^{k}(w)}.$

Then, using the bound 5.3, and, for all $i\in[0,5]$, we choose $\mu_{i}$ and
$\eta$ small enough (recall that $0<1-\gamma=\epsilon<1$) so that

$\frac{1}{2}\partial_{t}\|\theta\|^{2}_{H^{k}(w)}\leq
C\|\theta\|^{2}_{H^{k}(w)}.$

Integrating in time $s\in[0,T]$ and using Grönwall’s inequality, we conclude
that for all $T<\infty$

$\|\theta(T)\|^{2}_{H^{k}(w)}\leq\|\theta_{0}\|^{2}_{H^{k}(w)}e^{CT},$

where $C$ depends only on $\beta$ and $\|\theta_{0}\|_{L^{\infty}}$. Moreover,
we also obtain

$\frac{1}{2}\partial_{t}\|\theta\|^{2}_{H^{k}(w)}+\int|\Lambda^{\frac{\alpha}{2}+k}\theta|^{2}\
wdx\leq C(\beta,\|\theta_{0}\|_{L^{\infty}})\|\theta\|^{2}_{H^{k}(w)}.$

Integrating in time $s\in[0,T]$, we obtain

$\|\theta(T)\|^{2}_{H^{k}(w)}+C\int_{0}^{T}\int|\Lambda^{\frac{\alpha}{2}+k}\theta|^{2}w\
dx\
ds\lesssim\|\theta_{0}\|^{2}_{H^{k}(w)}+\int_{0}^{T}\|\theta(x,s)\|^{2}_{H^{k}(w)}\
ds.$

Hence, for all $T<\infty$

$\int_{0}^{T}\int|\Lambda^{\frac{\alpha}{2}+k}\theta|^{2}w\ dx\ ds<\infty.$

We conclude that for all $T<\infty$, we have $\theta\in
L^{\infty}([0,T],H^{k}(w(x)dx))\cap
L^{2}([0,T],\dot{H}^{k+\alpha/2}(w(x)dx))$.

To conclude the proof of Theorem 2.1, we pass to the weak limit as
$R\rightarrow\infty$. To do so, we consider a sequence of solutions
$(\theta_{0,m})_{m\in\mathbb{N}^{*}}$. The strong convergences of the
truncated initial data in $H^{2}_{w}$ and in $L^{\infty}$ are straighforward.
The a priori estimates of the previous section allows us to get that the
sequence $\theta_{m}$ is bounded in the space
$L^{\infty}([0,T],H^{\alpha/2}({w\hskip 0.28133ptdx}))\cap
L^{2}([0,T],H^{k+\frac{\alpha}{2}}({w\hskip 0.28133ptdx}))$ for all
$0<T<\infty$. Then, if $\phi(x,t)\in\mathcal{D}((0,\infty]\times\mathbb{R})$,
we get that $\phi\theta_{m}$ is bounded in
$L^{2}([0,T],H^{k+\frac{\alpha}{2}}({w\hskip 0.28133ptdx}))$. In order to
apply Rellich’s theorem we need a bound on $\partial_{t}({\phi}\theta_{m})$ or
equivatlently on the quantity
$\theta_{m}\partial_{t}\phi+\phi\partial_{t}\theta_{m}$. It suffices to focus
on

$\phi\partial_{t}\theta_{m}=-\phi\partial_{x}(\theta_{m}\mathcal{H}\theta_{m})+\theta_{m}\Lambda\theta_{m}-\Lambda^{\alpha}\theta_{m}.$

Using the a priori $L^{2}([0,T],H^{k+\frac{\alpha}{2}}({w\hskip
0.28133ptdx}))$ bound and the $L^{\infty}([0,T],L^{\infty}({w\hskip
0.28133ptdx}))$ bound on $\theta_{m}$ along with the continuity of the Hilbert
transform on Sobolev spaces one easily infer that $\phi\partial_{t}\theta_{m}$
is bounded in $L^{2}([0,T],H^{k-\frac{\alpha}{2}}({w\hskip 0.28133ptdx}))$.
Therefore, the Rellich compactness theorem (see [25]) allows us to get the
existence of a subsquence $\theta_{m_{n}}$ that converges strongly in
$L^{2}_{loc}((0,\infty)\times\mathbb{R})$ toward a function $\theta$. Using
that $\theta_{{m_{n}}}$ is a bounded sequence in
$L^{\infty}([0,T],H^{k}(w(x)dx))$ and on
$L^{2}([0,T],\dot{H}^{k+\alpha/2}(w(x)dx))$ whose dual spaces are separable
Banach spaces, we obtain the *-weak convergence when $m_{n}\to+\infty$ of the
subsequence $\theta_{m_{n}}$ toward $\theta$ in the spaces
$L^{\infty}([0,T],H^{k}({w\hskip 0.28133ptdx}))$ and
$L^{2}([0,T],H^{k+\frac{\alpha}{2}}({w\hskip 0.28133ptdx}))$. Then, using the
strong $L^{2}_{loc}((0,\infty)\times\mathbb{R})$ one can pass to the weak
limit in the nonlinear term, and by standard procedure we also have the
convergence of the linear terms in
$\mathcal{D}^{\prime}((0,\infty]\times\mathbb{R})$ and we get that the limit
is a solution to $\mathcal{T}_{\alpha}$ in the sense of distribution. ∎

##  6 Local existence in $H^{2}(w_{\lambda,\kappa})$ in the case $0<\alpha<1$

In this section, we prove Theorem 2.2. For $0<\alpha<1$, we consider the
following Cauchy problem,

$\ (\mathcal{T}_{\alpha})\ :\\\ \left\\{\begin{aligned}
&\partial_{t}\theta_{R}+\partial_{x}\theta_{R}\mathcal{H}\theta_{R}+\nu\Lambda^{\alpha}\theta_{R}=0\hskip
56.9055pt\\\ &0<\theta_{0,R}(x)=\theta_{0}(x)\in H^{2}(w_{\lambda,\kappa})\cap
L^{\infty}\end{aligned}\right.$

Since the usual Sobolev embedding $H^{s}({w})(\mathbb{R}^{n})\hookrightarrow
L^{\infty}(\mathbb{R}^{n})$ with $s>n/2$ is not anymore true for general
$\mathcal{A}_{\infty}$ weights, one cannot argue as in [1] for instance.
However, thanks to the pointwise inequality

$\left(\int\frac{|\theta(x)-\theta(y)|^{p}}{|x-y|^{n+sp}}\ dy\right)^{1/p}\leq
C\left[\mathcal{M}\left(|\theta-\theta(x)|^{p}\right)(x)\right]^{\frac{1-s}{p}}\times\left[\mathcal{M}(|\nabla\theta|^{q})(x)\right]^{\frac{s}{q}}$
(6.1)

valid for all $s\in(0,1)$, $p\in[1,\infty)$ and $q\geq\frac{pn}{n+p}$, one
recover the weighted Gagliardo-Niremberg and the weighted Sobolev inequalities
(with $p\neq\infty$). In particular, we can recover the weighted Sobolev
embedding type $\dot{H}^{2+s/2}(w)\hookrightarrow\dot{W}^{2,\frac{2}{1-s}}(w)$
for all $s\in(0,1)$. For a proof of 6.1, we refer to the book of Maz’ya [27],
p 641.

We shall study the following truncated equation and get a priori estimates
uniformly in $R>0$

$\ (\mathcal{T}_{\alpha,R})\ :\\\ \left\\{\begin{aligned}
&\partial_{t}\theta_{R}+\partial_{x}\theta_{R}\mathcal{H}\theta_{R}+\nu\Lambda^{\alpha}\theta_{R}=0\hskip
56.9055pt\\\ &0<\theta_{0,R}(x)=\theta_{0}(x)\phi_{R}(x)\in H^{2}\cap
L^{\infty}\end{aligned}\right.$

Where the function $\phi_{R}(x)$ have been introduced in the end of section 3.
Then, via ([1], Theorem 3.2) we know that there exists a solution $\theta_{R}$
to equation $(\mathcal{T}_{\alpha,R})$ that is sufficiently regular for the
product $\theta_{xx}\partial_{t}\theta_{xx}$ to make sense. We shall prove the
following energy estimate, for all $R>0$

$\partial_{t}\|\theta_{R}\|_{H^{2}({w})}^{2}+C\|\theta_{R}\|^{2}_{\dot{H}^{2+\frac{\alpha}{2}}}\lesssim\|\theta_{R}\|^{2}_{H^{2}({w})}+\|\theta_{R}\|^{4}_{H^{2}({w})}+\|\theta_{R}\|^{\frac{16}{3}}_{H^{2}(w)}$
(6.2)

We shall omit to write the dependence on $R$ for the sake of readibility. The
control of the $L^{2}(w)$ norm is easy to obtain, indeed we have

$\displaystyle\partial_{t}\int\frac{\theta^{2}}{2}w\
dx+\int|\Lambda^{\alpha/2}\theta|^{2}w\ dx$ $\displaystyle=$
$\displaystyle-\int\theta^{2}\Lambda\theta w\ dx\
+\int\theta^{2}\mathcal{H}\theta w_{x}\ dx$ $\displaystyle-$
$\displaystyle\int\Lambda^{\alpha/2}\theta[\Lambda^{\alpha/2},w]\theta\ dx$

Then, using the first commutator estimate of lemma 4.1 available for
supercritical values of $\alpha$ along with the weighted Sobolev embedding
$\dot{H}^{1/6}({w})\hookrightarrow L^{3}({w})$ yield the following estimate

$\displaystyle\partial_{t}\int\frac{\theta^{2}}{2}w\
dx+\int|\Lambda^{\alpha/2}\theta|^{2}w\lesssim\|\theta\|^{2}_{\dot{H}^{1/6}({w})}\|\theta\|_{\dot{H}^{7/6}({w})}+\|\theta\|^{3}_{\dot{H}^{1/6}({w})}+\|\theta\|^{2}_{H^{\alpha/2}(w)}\lesssim\|\theta\|^{3}_{H^{2}(w)}$

Note that, in the evolution of the weighted $L^{2}$ norm, we do not need any
positivity assumption on the data since the term on the right hand side of 6
can be estimated directly via a weighted Sobolev embedding. Indeed, we are
dealing with $H^{2}_{w}$ estimates (actually an $H^{1}_{w}$ estimate allows
one to get rid of the positivity assumption).

The evolution of the homogeneous part is dealt as follows, we have

$\displaystyle\frac{1}{2}\partial_{t}\|\theta\|^{2}_{H^{2}_{\lambda}}+\int|\Lambda^{2+\frac{\alpha}{2}}\theta|^{2}\
w\ dx$ $\displaystyle=$ $\displaystyle-\frac{1}{2}\int
w_{x}(\theta_{xx})^{2}\mathcal{H}\theta-\frac{3}{2}\int
w(\theta_{xx})^{2}\Lambda\theta+\int w\theta_{xx}\theta_{x}\Lambda\theta_{x}$
$\displaystyle-\int\Lambda^{2+\frac{\alpha}{2}}\theta[\Lambda^{\alpha/2},w]\theta_{xx}$
$\displaystyle=$ $\displaystyle I_{1}+I_{2}+I_{3}+I_{4}$

In order to estimate $I_{1}$, we use Hölder’s inequality with
$\frac{1}{2}=\frac{1}{p}+\frac{1}{q}$ with $p=\frac{2}{1-\alpha}$ and
$q=\frac{2}{\alpha}$ and then Young’s inequality ($p^{-1}_{1}+p^{-1}_{2}=1$,
$p^{-1}_{3}+p^{-1}_{4}=1$). We shall also use the weighted Sobolev embeddings
$H^{\alpha/2}(w)\hookrightarrow L^{\frac{2}{1-\alpha}}(w)$ and
$H^{\frac{1}{2}-\frac{\alpha}{2}}(w)\hookrightarrow L^{\frac{2}{\alpha}}(w)$
that is valid for $\alpha\in(0,1)$ and the fact that
$w_{\lambda}\in\mathcal{A}_{\frac{2}{\alpha}}$. Therefore, we obtain

$\displaystyle I_{1}$ $\displaystyle\lesssim$
$\displaystyle\frac{1}{p_{1}\epsilon_{1}^{p_{1}}}\|\theta\|^{p_{1}}_{\dot{H}^{2}({w})}+\frac{\epsilon_{1}^{p_{2}}}{p_{2}}\|\theta_{xx}\|^{p_{2}}_{L^{\frac{2}{1-\alpha}}({w})}\|\mathcal{H}\theta\|^{p_{2}}_{L^{\frac{2}{\alpha}}({w})}$
$\displaystyle\lesssim$
$\displaystyle\frac{1}{p_{1}\epsilon_{1}^{p_{1}}}\|\theta\|^{p_{1}}_{\dot{H}^{2}({w})}+\frac{\epsilon_{1}^{p_{2}}}{p_{2}}\|\theta_{xx}\|^{p_{2}}_{H^{\alpha/2}(w)}\|\theta\|^{p_{2}}_{H^{\frac{1}{2}-\frac{\alpha}{2}}(w)}$
$\displaystyle\lesssim$
$\displaystyle\frac{1}{p_{1}\epsilon_{1}^{p_{1}}}\|\theta\|^{p_{1}}_{\dot{H}^{2}({w})}+\frac{\epsilon_{1}^{p_{2}}}{p_{3}p_{2}}\|\theta\|^{p_{3}p_{2}}_{\dot{H}^{2+\frac{\alpha}{2}}({w})}+\frac{\epsilon_{1}^{p_{2}}}{p_{4}p_{2}}\|\theta\|^{p_{2}p_{4}}_{H^{2}(w)}$

Then, we choose $p_{2}=1+\eta_{2}$ and $p_{3}=1+\eta_{3}$ where $\eta_{2}>0$,
$\eta_{3}>0$ are such that $\eta_{2}+\eta_{3}+\eta_{2}\eta_{3}\leq 1$ this
gives $p_{2}p_{3}\leq 2$. For instance, one may take $\eta_{2}=\eta_{3}=1/3$
hence $p_{2}=p_{3}=4/3$ and then $p_{1}=p_{4}=4$. Therefore, for all
$\epsilon_{1}$ we obtain

$I_{1}\lesssim\frac{1}{4\epsilon_{1}^{4}}\|\theta\|^{4}_{H^{2}({w})}+\frac{3\epsilon_{1}^{4/3}}{16}\|\theta\|^{\frac{16}{3}}_{H^{2}(w)}+\frac{9\epsilon_{1}^{4/3}}{16}\|\theta\|^{2}_{\dot{H}^{2+\frac{\alpha}{2}}({w})}.\\\
$

To estimate $I_{2}$ and $I_{3}$, we follow the same steps as before. We shall
keep the same constant to estimate these two terms.

We first observe that $I_{2}$ is slightly more singular than $I_{1}$, however,
due to $\Lambda=-\partial_{x}\mathcal{H}$ and the continuity of $\mathcal{H}$
on $L^{2/\alpha}(w)$ because $w\in\mathcal{A}_{\frac{2}{\alpha}}$, following
what we did for $I_{1}$, one gets

$\displaystyle I_{2}$ $\displaystyle\lesssim$
$\displaystyle\frac{1}{p_{1}\epsilon_{1}^{p_{1}}}\|\theta\|^{p_{1}}_{H^{2}({w})}+\frac{\epsilon_{1}^{p_{2}}}{p_{2}}\|\theta_{xx}\|^{p_{2}}_{L^{\frac{2}{1-\alpha}}({w})}\|\mathcal{H}\theta_{x}\|^{p_{2}}_{L^{\frac{2}{\alpha}}({w})}$
$\displaystyle\lesssim$
$\displaystyle\frac{1}{p_{1}\epsilon_{1}^{p_{1}}}\|\theta\|^{p_{1}}_{H^{2}({w})}+\frac{\epsilon_{1}^{p_{2}}}{p_{2}}\|\theta_{xx}\|^{p_{2}}_{H^{\alpha/2}(w)}\|\theta_{x}\|^{p_{2}}_{H^{\frac{1}{2}-\frac{\alpha}{2}}(w)}$
$\displaystyle\lesssim$
$\displaystyle\frac{1}{p_{1}\epsilon_{1}^{p_{1}}}\|\theta\|^{p_{1}}_{H^{2}({w})}+\frac{\epsilon_{1}^{p_{2}}}{p_{3}p_{2}}\|\theta\|^{p_{3}p_{2}}_{\dot{H}^{2+\frac{\alpha}{2}}({w})}+\frac{\epsilon_{1}^{p_{2}}}{p_{4}p_{2}}\|\theta\|^{p_{2}p_{4}}_{H^{\frac{3}{2}-\frac{\alpha}{2}}(w)}$

hence, since $H^{2}(w)\hookrightarrow H^{\frac{3}{2}-\frac{\alpha}{2}}(w)$ for
$\alpha\in(0,1)$, we finally get that, for all $\epsilon_{1}>0$

$I_{2}\lesssim\frac{1}{4\epsilon_{1}^{4}}\|\theta\|^{4}_{H^{2}({w})}+\frac{3\epsilon_{1}^{4/3}}{16}\|\theta\|^{\frac{16}{3}}_{H^{2}(w)}+\frac{9\epsilon_{1}^{4/3}}{16}\|\theta\|^{2}_{\dot{H}^{2+\frac{\alpha}{2}}({w})}.\\\
$

To estimate $I_{3}$, it suffices to observe that this term is as regular as
$I_{2}$. Roughly speaking, these two terms just differ from an Hilbert
transform since we can write

$I_{3}=-\int w\theta_{xx}\mathcal{H}\theta_{xx}\theta_{x}\ dx,$

therefore, using the continuity of $\mathcal{H}$ on $L^{\frac{2}{1-\alpha}}$
because $w\in\mathcal{A}_{\frac{2}{1-\alpha}}$, we obtain the same estimate as
$I_{2}$, that is, for all $\epsilon_{1}>0$

$I_{3}\lesssim\frac{1}{4\epsilon_{1}^{4}}\|\theta\|^{4}_{H^{2}({w})}+\frac{3\epsilon_{1}^{4/3}}{16}\|\theta\|^{\frac{16}{3}}_{H^{2}(w)}+\frac{9\epsilon_{1}^{4/3}}{16}\|\theta\|^{2}_{\dot{H}^{2+\frac{\alpha}{2}}({w})}.\\\
$

The term $I_{4}$ can be estimated thanks to the first commutator estimate of
lemma 4.1, then Young’s inequality yields

$|I_{4}|\leq\left|\int\sqrt{w}\Lambda^{2+\frac{\alpha}{2}}\theta\
[\Lambda^{\alpha/2},w]\frac{\theta_{xx}}{\sqrt{w}}\
dx\right|\lesssim\frac{\epsilon_{2}}{2}\|\theta\|^{2}_{\dot{H}^{2+\frac{\alpha}{2}}_{w}}+\frac{2}{\epsilon_{2}}\|\theta\|^{2}_{\dot{H}^{2}_{w}}.$

Finally, we have obtained

$\partial_{t}\|\theta\|_{H^{2}({w})}^{2}+\|\theta\|^{2}_{\dot{H}^{2+\frac{\alpha}{2}}}\lesssim\left(\frac{\epsilon_{2}}{2}+\frac{27\epsilon_{1}^{4/3}}{16}\right)\|\theta\|^{2}_{\dot{H}^{2+\frac{\alpha}{2}}_{w}}+\frac{2}{\epsilon_{2}}\|\theta\|^{2}_{H^{2}({w})}+\frac{3}{4\epsilon_{1}^{4}}\|\theta\|^{4}_{H^{2}({w})}+\frac{9\epsilon_{1}^{4/3}}{16}\|\theta\|^{\frac{16}{3}}_{H^{2}(w)}$

Then, choosing $\epsilon_{1}$ and $\epsilon_{2}$ sufficiently small enough
gives the desired a priori estimate 6.2.

The strong convergence of the initial data as $R\rightarrow\infty$ in $H^{2}$
and $L^{\infty}$ is straighforward. We just focus on the passage to the weak
limit in the nonlinearity. For all
$\eta(x,t)\in\mathcal{D}((0,\infty]\times\mathbb{R})$, we have that
$\eta\theta_{k}$ is bounded in
$L^{2}([0,T],\dot{H}^{2+\frac{\alpha}{2}}({w\hskip 0.28133ptdx}))$ and then
there exists a subsequence $\theta_{k_{n}}$ that converges weakly to some
$\theta$. Moreover, writting that
$\eta\partial_{t}\theta_{k}=-\eta\partial_{x}(\theta_{k}\mathcal{H}\theta_{k})+\eta\theta_{k}\Lambda\theta_{k}-\eta\Lambda^{\alpha}\theta_{k}$
we get, by the a priori bound in the space
$L^{2}([0,T^{*}],\dot{H}^{2+\frac{\alpha}{2}}({w\hskip 0.28133ptdx}))\cap
L^{\infty}([0,T^{*}],L^{\infty}({w\hskip 0.28133ptdx}))$ that
$\eta\partial_{t}\theta_{k}$ is bounded in
$L^{2}([0,T^{*}],\dot{H}^{1+\frac{\alpha}{2}}({w\hskip 0.28133ptdx}))$.
Therefore, one obtains the strong convergence of a subsequence
$\theta_{k_{n^{\prime}}}$ to a function $\theta$ in
$(L^{2}((0,T^{*}]\times\mathbb{R}))_{loc}$ which allows to pass to the limit
in the nonlinearity. Then, it is classical to prove that the limit is a
solution to the equation in the weak sense.
∎

Acknowledgment: The author thanks P-G. Lemarié-Rieusset for useful
discussions. He also thanks Diego Córdoba for useful discussions regarding
this model. He was supported by the ERC grant Stg-203138-CDSIF and the
National Grant MTM2014-59488-P from the Spanish government.

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