\documentclass[reqno]{amsart}
\usepackage{hyperref}
\usepackage{amssymb}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2011(2011), No. 08, pp. 1--18.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2011 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2011/08\hfil Existence of global solutions]
{Existence of global solutions to the 2-D subcritical dissipative
quasi-geostrophic equation  and persistency of the initial
regularity}

\author[R. May, E. Zahrouni \hfil EJDE-2011/08\hfilneg]
{May Ramzi, Ezzeddine Zahrouni}  % in alphabetical order

\address{May Ramzi \newline
D\'epartement de Math\'ematiques,
Facult\'e des Sciences de Bizerte, Tunisie}
\email{ramzi.may@fsb.rnu.tn}

\address{Ezzeddine Zahrouni \newline
 D\'epartement de Math\'ematiques,
 Facult\'e des Sciences de Monastir, Tunisie}
\email{Ezzeddine.Zahrouni@fsm.rnu.tn}

\thanks{Submitted August 10, 2010. Published January 15, 2011.}
\subjclass[2000]{35Q35, 76D03}
\keywords{Quasi-geostrophic equation; Besov Spaces}

\begin{abstract}
 In this article, we prove that if the initial data $\theta_0$ and
 its Riesz transforms ($\mathcal{R}_1(\theta_0)$ and
 $\mathcal{R}_2(\theta_0)$) belong to the space
 $$
 (\overline{S(\mathbb{R}^2))} ^{B_{\infty }^{1-2\alpha ,\infty }},
 \quad 1/2<\alpha <1,
 $$
 then the 2-D Quasi-Geostrophic
 equation with dissipation $\alpha$ has a unique global in time
 solution $\theta$. Moreover, we show that if in addition
 $\theta_0 \in X$ for some functional space $X$ such as
 Lebesgue, Sobolev and  Besov's spaces then the solution $\theta$
 belongs to the space $C([0,+\infty [,X)$.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{definition}[theorem]{Definition}


\section{Introduction and statement of main results}


In this article, we are study the initial value-problem
for the two-dimensional quasi-geostrophic equation with
sub-critical dissipation
\begin{equation}
\begin{gathered}
\partial _{t}\theta +(-\Delta )^{\alpha }\theta
+ \nabla.(\theta u)=0\quad \text{on }\mathbb{R}_{\ast }^{+}\times
 \mathbb{R}^2 \\
\theta (0,x)=\theta _0(x),\quad x\in \mathbb{R}^2
\end{gathered} \label{QGalpha}
\end{equation}
where $\alpha \in ]\frac{1}{2},1[$ is a fixed parameter and $\nabla $
denotes the divergence operator with respect to the space variable
$x\in \mathbb{R}^2$. The scalar function $\theta $ represents
the potential temperature. The velocity $u=(u_1,u_2)$ is divergence free and
determined from $\theta $ through the Riesz transforms
\begin{equation*}
u=\mathcal{R}^{\bot }(\theta )\equiv (-\mathcal{R}_2(\theta ),
\mathcal{R}_1(\theta )).
\end{equation*}
The non local operator $(-\Delta )^{\alpha }$ is defined
through the Fourier transform,
\begin{equation*}
\mathcal{F}((-\Delta )^{\alpha }f)(\xi )=| \xi
| ^{2\alpha }\mathcal{F}(f)(\xi )
\end{equation*}
where $\mathcal{F}(f)$ is the Fourier transform of $f$ defined by
\begin{equation*}
\mathcal{F}(f)(\xi )=\hat{f}(\xi )=\int_{\mathbb{R}^2}f(x)e^{-i\langle
x,\xi \rangle }dx.
\end{equation*}
To study the existence of  solutions to \eqref{QGalpha}, we
follow the Fujita-Kato method. Thus we convert \eqref{QGalpha}
into the fixed point problem:
\begin{equation}
\label{mild}
\theta (t)=e^{-t(-\Delta )^{\alpha }}\theta _0 + {\mathcal{B}}_{\alpha}
[\theta ,\theta ] (t).
\end{equation}
Here $(e^{-t(-\Delta )^{\alpha }})_{t>0}$ is the
semi-group defined by
\begin{equation*}
\mathcal{F}(e^{-t(-\Delta )^{\alpha }}f)(\xi
)=e^{-t| \xi | ^{2\alpha }}\mathcal{F}(f)(\xi )
\end{equation*}
and $\mathcal{B}_{\alpha}$ is the bi-linear operator
\begin{equation}
\label{opB} {\mathcal{B}_{\alpha}}[\theta _1,\theta
_2] (t)=-{\mathcal{L}}_{\alpha }(\theta
_1\mathcal{R}^{\bot }(\theta _2))
\end{equation}
where, for $v=(v_1,v_2)$,
\begin{equation}
\label{linear} {\mathcal{L}}_{\alpha }(v)(t)=\int_0^{t}\nabla .
e^{-(t-s)(-\Delta )^{\alpha }}v ds.
\end{equation}
In the sequel,  by a mild solution on $]0,T[$ to \eqref{QGalpha}
with data $\theta _0$, we mean a function $\theta $ belonging to
the space $L_{\rm loc}^2([0,T[,F_2)$ and satisfying
in ${\mathcal D}'(]0,T[\times \mathbb{R}^2)$ the equation \eqref{mild}
where $F_2$ is the
completion of $S(\mathbb{R}^2)$ with respect to the norm
\begin{equation*}
\| f\| _{F_2}\equiv \sup_{x_0\in \mathbb{R}^2}(
\| 1_{B(x_0,1)}f\| _2+\| 1_{B(x_0,1)}\mathcal{
R}^{\bot }(f)\| _2).
\end{equation*}
One of the main properties of
\eqref{QGalpha} is the following scaling invariance property:
If $\theta $ is a solution of \eqref{QGalpha} with
data $\theta _0$ then, for any $\lambda >0$,
the function
$\theta _{\lambda }(t,x)\equiv \lambda ^{2\alpha -1}\theta
(\lambda ^{2\alpha }t,\lambda x)$
is a solution of \eqref{QGalpha} with data $\theta _{0,\lambda
}(x)\equiv \lambda ^{2\alpha -1}\theta _0(\lambda x)$.
This leads us to introduce the following notion of
\textit{super-critical space}: A Banach space $X$ will be
 called \textit{super-critical space} if $S(\mathbb{R}
^2)\hookrightarrow X\hookrightarrow S(\mathbb{R}^2)$
and there exists a constant $C_{X}\geq 0$ such that
for all $f\in X$,
\begin{equation*}
\sup_{0<\lambda \leq 1}\lambda ^{2\alpha -1}\|
f(\lambda .)\| _{X}\leq C_{X}\| f\| _{X}.
\end{equation*}
For instance, the Lebesgue space $L^{p}(\mathbb{R}^2)$
(respectively, the Sobolev space $H^{s}(\mathbb{R}^2)$)
is \textit{super-critical space} if
$p\geq p_{c}\equiv \frac{2}{2\alpha -1}$
(respectively, $s\geq s_{c}\equiv 2-2\alpha $).
Moreover, one can easily prove that the Besov space $B_{\infty
}^{1-2\alpha ,\infty }(\mathbb{R}^2)$ is the greatest
\textit{super-critical space}. The first purpose of this paper, is to
prove the global existence of smooth solutions of the equations
\eqref{QGalpha} for initial data in a \textit{super-critical space}
$\mathbf{\tilde{B}}^{\alpha }$ closed
to the space $B_{\infty }^{1-2\alpha ,\infty }(\mathbb{R}^2)$.
Our space $ \mathbf{\tilde{B}}^{\alpha }$ is the completion
of $S(\mathbb{R}^2)$ with respect to the norm
\begin{equation*}
\| f\| _{\mathbf{\tilde{B}}^{\alpha }}\equiv \|
f\| _{B_{\infty }^{1-2\alpha ,\infty }}+\| \mathcal{R}
^{\bot }(f)\| _{B_{\infty }^{1-2\alpha ,\infty }}.
\end{equation*}
Before setting precisely our global existence result, let us
recall some known results in this direction: in \cite{wu2},  Wu
proved that for any initial data $\theta _0$ in the space
$L^{p}(\mathbb{R}^2)$ with $p > p_{c} = \frac{2}{2\alpha-1}$
the equations \eqref{QGalpha}
has a unique global solution $\theta $ belonging to the space
$L^{\infty }([0,+\infty [ ,L^{p}(\mathbb{R}^2))$.
Similarly,  Constantin and  Wu \cite{constw} showed the global
existence and uniqueness for arbitrary initial data in the Sobolev
space $H^{s}(\mathbb{R}^2)$ where $s>s_{c} = 2 - 2\alpha$.
However, we notice that these results don't cover the limit
cases $p=p_{c}$ and $s=s_{c}$,  that are critical regularity
exponents.

We recall that global solutions are obtained  under smallness
size assumption on the initial data by several authors. For instance,
one can quote the results of  Wu \cite{wu1} for
$\theta_0 \in  {\dot B}_{p}^{s_p, \infty}(\mathbb{R}^2)$  (critical spaces)
with $s_p = \frac 2p - (2\alpha -1)$,  Niche and Schonbek, \cite{NS}
for $\theta _0 \in L^{p_{c}}(\mathbb{R}^2)$,   with
$p_c = \frac{2}{2\alpha-1}$,  Lemari\'e-Rieusset and Marchand
\cite{lemarie-marchand} for
$\theta_0 \in L^{\frac{2}{2\alpha - 1},\infty}(\mathbb{R}^2)$
and finally the work  May and  Zahrouni \cite{mayzah1}
where they considered initial data in the greatest critical
homogeneous Besov space
$ {\dot B}_{\infty}^{-(2\alpha-1), \infty}(\mathbb{R}^2)$.
 The later one contains all the preceding critical spaces. Indeed,
we have
\[
 {\dot L}^{p,s_p}(\mathbb{R}^2)
\subset {\dot B}_{p}^{s_p, \infty}(\mathbb{R}^2)
\subset {\dot B}_{\infty}^{-(2\alpha-1), \infty}(\mathbb{R}^2).
\]
Our space of initial data $\mathbf{\tilde{B}}^{\alpha }$
introduced above contains all known critical spaces,
in particular we have
\[
{\dot B}_{\infty}^{-(2\alpha-1), \infty}(\mathbb{R}^2)
\subset  \mathbf{\tilde{B}}^{\alpha }.
\]
Now we give our first result overcoming the above mentioned
smallness assumption. Our global existence result reads as follows.

\begin{theorem}\label{theo1}
Let $\nu =1-(1/2\alpha)$. For any initial data
$\theta _0\in \mathbf{\tilde{B}}^{\alpha }$,
equation \eqref{QGalpha} has a unique global solution
$\theta $ belonging to the space
$\cap _{T>0}\mathbf{E}_T^{\nu}$, where
$\mathbf{E}_T^{\nu}$ is the completion of
$C_{c}^{\infty }(]0,T]\times \mathbb{R}^2)$ with respect to the
norm
\begin{equation*}
\| v\| _{\mathbf{E}_T^{\nu }}\equiv
\sup_{0<t\leq
T}t^{\nu }(\| v(t)\| _{\infty }+\| \mathcal{R}
^{\bot }(v)(t)\| _{\infty }).
\end{equation*}
Moreover,
\begin{equation*}
\theta \in C([0,+\infty [ ,\mathbf{\tilde{B}}^{\alpha }).
\end{equation*}
\end{theorem}

The proof of the above theorem  is far from being a direct
consequence of an application of a Fixed Point Theorem.
We will establish a local existence result and  will be able
to get global existence that is essentially based on a new
adapted version of the well-known maximal principle
(Lemma \ref{lemme6}) that we stated and proved in second section.

We can recover the results quoted above using our second
main result that is a persistency Theorem stating that,
the solution $\theta $ given by Theorem \ref{theo1} keeps
any further Besov or Lebesgue regularity of its initial data.
Precisely, our theorem states as follows.

\begin{theorem}\label{theo2}
Let $X$ be one of the following Banach spaces:
\begin{itemize}
\item $X=L^{p}(\mathbb{R}^2)$ with $1\leq p\leq \infty $;
\item $X=B_{p}^{s,q}(\mathbb{R}^2)$ with $s>-1$ and $1< p<\infty,
1\leq q\leq \infty$;
 \item $X=\dot{B}_{p}^{s,q}(\mathbb{R}^2)$
with $s>0$ and $1\leq p,q\leq \infty $.
\end{itemize}
Assume $\theta _0\in \mathbf{\tilde{B}}^{\alpha }\cap X$. Then
the mild solution $\theta $ of the equation \eqref{QGalpha} given
by Theorem \ref{theo1} belongs to the space $L_{\rm loc}^{\infty
}([0,+\infty [ ,X)$. Moreover,
if $\theta _0\in \mathbf{\tilde{B}}^{\alpha }\cap \overline{S(\mathbb{R}
^2)}^{X}$ then $\theta $ belongs to $C([0,+\infty [ ,\overline{S(
\mathbb{R}^2)}^{X})$.
\end{theorem}

As a consequence of the previous theorems, we have the following
theorem that generalizes the existence results of  Wu
\cite{wu2} and  Constantin and  Wu \cite{constw} recalled
above.

\begin{theorem}\label{theo3}
Let $X$ be the Lebesgue space $L^{p}(\mathbb{R}^2)$ with
$p\geq p_{c}= \frac{2}{2\alpha -1}$ or the Sobolev space
$H^{s}(\mathbb{R}^2)$ with $s\geq s_{c}=2-2\alpha $.
Assume $\theta _0\in X$. Then the equation \eqref{QGalpha}
with initial data $\theta _0$ has a unique global mild
solution $\theta $ belonging to the space $C([0,+\infty [ ,X)$.
\end{theorem}

We emphasize that the above stated results are new since the initial
data considered here are in the nonhomogeneous space
$ \mathbf{\tilde{B}}^{\alpha }$,  that is our knowledge the first
time employed in this context. Moreover, we are allowed to
obtain global solutions for this initial data without assuming
any smallness assumption on its size. Thus we have a better
results than those of  Wu \cite{wu1} and \cite{constw}.
As a by product of our method we are able to extend the result
of Wu to a large class of $L^p$ spaces, for which we have also
obtained the uniqueness issue. We focus on the fact that we have
established the propagation of any further regularity of initial
data belonging to  $\mathbf{\tilde{B}}^{\alpha }$.

Our next challenge is to extend the use of our method to the
critical Quasi-geostrophic equations.

The remainder of this paper is as follows: in section $2$ we
recall some definitions and we give some useful Lemmas that will
be used in this paper. In section $3$, we prove Theorem
\ref{theo1}. Section {4} is devoted to the proof of Theorem
\ref{theo2} and in section $4$, we will prove Theorem \ref{theo3}.


\section{Preliminaries}
\subsection{Notation}
In this subsection, we introduce some notation that will be used
frequently in this paper.
\begin{enumerate}
\item Let $X$ be a Banach space such that
$S(\mathbb{R}^2)\hookrightarrow X\hookrightarrow S'(\mathbb{R}^2)$.
We denote by $X_{\mathcal{R}}$ the space
\[
X_{\mathcal{R}}=\{f\in X:\mathcal{R}^{\perp }(f)\in X^2\}
\]
endowed with the norm
\[
\| f\| _{X_{\mathcal{R}}}=\| f\|_{X}+\| \mathcal{R}^{\perp }(f)\| _{X}.
\]
We recall that $\mathcal{R}^{\perp }(f)=(-\mathcal{R}_2f,\mathcal{R}
_1f)$ where $\mathcal{R}_1$ and $\mathcal{R}_2$ are
Riesz transforms.

\item Let $T>0$, $r\in [ 1,\infty ]$ and $X$ be a Banach
space. $L_T^{r}X$ denotes the space $L^{r}([0,T[,X)$. In particular,
$L_T^{r}L^{p}$ will denote the space
$L^{r}([0,T[,L^{p}(\mathbb{R}^2))$.

\item Let $X$ be a Banach space, $T>0$ and $\mu \in \mathbb{R}$.
we denote by $L_{\mu }^{\infty }([0,T],X)$ the space of
functions $f:]0,T]\to X$ such that
\[
\| f\| _{L_{\mu }^{\infty }([0,T],X)}\equiv
\sup_{0<t\leq
T}t^{\mu }\| f(t)\| _{X}<\infty \quad\text{and}\quad
\lim_{t\to 0}t^{\mu }\| f(t)\| _{X}=0.
\]
The sub-space $C_{\mu }^0([0,T],X)$ of $L_{\mu }^{\infty
}([0,T],X)$ is defined by
\[
C_{\mu }^0([0,T],X)\equiv L_{\mu }^{\infty }([0,T],X)\cap
C(]0,T],X).
\]

\item Let $A$ and $B$ be two reals functions. The notation
$A\lesssim B$ means that there exists a constant $C$, independent
of the effective parameters of $A$ and $B$, such that $A\leq CB$.
\end{enumerate}

\subsection{Besov spaces}\label{section2}

The standard definition of Besov spaces passes through the
Littlewood-Paley dyadic decomposition \cite{bergh}.
\cite{frazier}, and \cite{lemarie}. To this end, we take an
arbitrary function $\psi \in \mathcal{S}(\mathbb{R}^2) $ whose Fourier transform
$\hat{\psi}$ is such that
$\operatorname{supp}( {\hat{\psi}} ) \subset \{\xi,  \frac{1}{2}
\leq |\xi| \leq 2 \}$,
and for
$\xi \neq 0$, $\sum_{j \in \mathbb{Z}} \hat{\psi}(\frac{\xi}{2^j}) = 1$,
and define $\varphi \in \mathcal{S}(\mathbb{R}^2) $ by
$\hat{\varphi}(\xi) = 1 - \sum_{ j \geq 0 } \hat{\psi}(\frac{\xi}{2^j})$.
For  $j \in \mathbb{Z} $, we write  $\varphi_j(x) = 2^{2j} \varphi( 2^j x )$
 and $\psi_j(x) = 2^{2j} \psi(2^j x)$
and we denote  the convolution operators  $S_j$  and $\Delta_j$,
respectively, the convolution operators by
$\varphi_j$ and $\psi_j$.

\begin{definition} \rm
Let  $  1 \leq p, q \leq \infty$, $s \in \mathbb{R}$.
\\
1. A tempered distribution  $ f$  belongs to the (inhomogeneous)
Besov space  $ B_p^{s,q} $   if and only if
\[
\| f \|_{B_p^{s,q}} \equiv
\|S_0 f\|_p + (\sum_{j > 0 } 2^{jsq} \| \Delta_j f  \|_p^q )^{\frac 1q}
< \infty.
\]
 2. The homogeneous Besov space  $ \dot{B}_p^{s,q} $
is the space of $f\in \mathcal{S}'(\mathbb{R}^2)/_{\mathbb{R}[X]}$ such
that
\begin{align*}
\| f \|_{\dot{B}_p^{s,q}} \equiv (\sum_{j \in \mathbb{Z} } 2^{jsq}
\| \Delta_j f  \|_p^q )^{\frac 1q}<\infty,
\end{align*}
Where $\mathbb{R}[X] $  is the space of polynomials \cite{peetre}.
\end{definition}

An equivalent definition more adapted to the Quasi-geostrophic
equations involves the semigroup $(e^{-t(-\Delta )^{\alpha }})_{t>0}$.

\begin{proposition}\label{prop1}
If   $ s < 0$  and $ q = \infty$. Then
\begin{gather}
\label{car01}
 f \in \dot{B}_p^{s,\infty} \Longleftrightarrow \sup_{t > 0 }
 t^{\frac{-s}{2\alpha}} \| e^{-t (-\Delta)^{\alpha}} f\|_p < \infty,
\\
\label{car02}
 f \in \;B_p^{s,\infty} \Longleftrightarrow
 \forall T > 0,\quad \sup_{ 0 < t < T } t^{\frac{-s}{2\alpha}}
\| e^{-t (-\Delta)^{\alpha}} f\|_p  \leq C_T.
\end{gather}
\end{proposition}

The proof the above proposition can be easily done by following the
same lines as in the proof in \cite[Theorem 5.3]{lemarie} in the
case of the heat Kernel. One can see also the proof
in \cite[Proposition 2.1]{miao}.

\subsection{Intermediate results}

 We shall frequently use the following estimates on the
operator   $  e^{-t{(-\Delta)^{\alpha}}}$.

\begin{proposition}\label{prop2}
For  $ t > 0$,  we set  ${\mathcal K}_{t}$  the kernel of  $
e^{-t{(-\Delta)^{\alpha}}}$. Then for all $r\in [1,\infty]$ we have
\begin{gather}
\label{est1}
\|  {\mathcal K}_{t} \|_r   = C_{1r}t^{\sigma_r},\\
\label{est1a}
\|\nabla {\mathcal K}_{t}\|_r  =   C_{2r}  t^{\sigma_r-\frac{1}{2\alpha}},\\
\label{est1b} \|{\mathcal R}_j \nabla {\mathcal K}_{t} \|_r  =
C_{3r} t^{\sigma_r-\frac{1}{2\alpha}},
\end{gather}
where $\sigma_r=\frac{1}{\alpha}(\frac{1}{r}-1)$ and $C_{1r},
C_{2r}$ and $ C_{3r}$ are constants independent of $t$.
\end{proposition}

\begin{proof} 
This propostion was previously proved in \cite{wu2}.
Equalities \eqref{est1} and \eqref{est1a} can be found in
\cite{miao}. Estimate \eqref{est1b} can be obtained
by following the same argument as in \cite[Proposition 11.1]{lemarie}.
\end{proof}

Following the work of  Lemari\'e-Rieusset, we introduce the
notion of shift invariant functional space.

\begin{definition} \label{def2.2}\rm
A Banach space  $X$  is called \emph{shift invariant functional space}
if
\begin{itemize}
\item ${\mathcal S}(\mathbb{R}^2)\hookrightarrow X\hookrightarrow
{\mathcal S}^{'}(\mathbb{R}^2)$,
\item for all $\varphi \in  {\mathcal S}(\mathbb{R}^2)$ and
$f\in X$, $\| \varphi \ast f\| _{X}\leq C_X \| \varphi\| _1\| f\| _{X}$.
\end{itemize}
\end{definition}

\begin{remark} \label{rmk2.1} \rm
The Lebesgue spaces, the inhomogeneous Besov spaces $B_p^{s,q}$,
with $s\in \mathbb{R}, 1\leq p,q\leq \infty$, and the homogeneous
Besov spaces $\dot{B}_p^{s,q}$, with $s>0, 1\leq p,q\leq \infty$,
 are shift invariant functional spaces.
\end{remark}

The proof of Theorem \ref{theo1} requires the following lemmas.

\begin{lemma}\label{lemma0}
Let  $X$  be a shift invariant functional space. If
$ f \in X$  then
\begin{equation} \label{X0}
\sup_{t > 0 } \| e^{-t(-\Delta)^{\alpha}} f  \|_X
\leq C_X \|  f \|_X.
\end{equation}
Moreover, if   $ f \in \overline{\mathcal
S(\mathbb{R}^2)}^X$ , then
 $e^{-t(-\Delta)^{\alpha}}f \in C(] 0, \infty [,
\; \overline{\mathcal S(\mathbb{R}^2)}^X)$ and
$ e^{-t(-\Delta)^{\alpha}}  f \to  f $ in $X$ as $t\to 0^+$.
\end{lemma}

\begin{proof}
One obtain easily \eqref{X0} from \eqref{est1}.
Let us prove the last assertion. For $ t >0$,
we denote by $\mathcal{K}_{t}$ the
kernel of the operator $e^{-t(-\Delta)^{\alpha}}$. Then
$\mathcal{K}_{t}(.)=t^{-\frac{1}{\alpha
}}\mathcal{K}(t^{-\frac{1}{2\alpha }}.)$ where
$\mathcal{K}=\mathcal{K}_{t=1}$. Since $\mathcal{K}\in
L^{1}(\mathbb{R}^2)$ and $\int \mathcal{K}(x)dx=1$, there exists
a sequence $(\mathcal{K}_{(n)})_n\in (
C_{c}^{\infty }(\mathbb{R}^2))^{N}$ such that for all
$n$, $\int \mathcal{K}_{(n)}(x)dx=1$ and
 $(\mathcal{K}_{(n)})_n\to \mathcal{K}$ in
$L^{1}(\mathbb{R}^2)$. Let $(f_n)_n$
be a sequence in $C_{c}^{\infty }(\mathbb{R}^2)$ satisfying
$(f_n)_n\to f$ in $X$. Now we consider the functions
$(u_n)_n$ and $u$ defined on
$\mathbb{R}^{+\ast }\times \mathbb{R}^2$ by
\[
u(t,x)= \mathcal{K}_{t}\ast f \quad \text{and}\quad
u_n(t,x)=\mathcal{K}_{(n),t}\ast f_n
\]
where $\mathcal{K}_{(n),t}(.)
=t^{-\frac{1}{\alpha }}\mathcal{K}_{(n)}(t^{-
\frac{1}{2\alpha }}.)$ and $\ast $ denotes the convolution in
$\mathbb{R}^2$.

One can easily verify that for all $n$, the function
$\hat{u}_n(t,\xi )=
\mathcal{\hat{K}}_{(n)}(t^{\frac{1}{2\alpha }}\xi )\hat{f}_n(\xi )$
belongs to the space $C(\mathbb{R}^{+\ast },S(\mathbb{R}^2))$ and
satisfies $\hat{u}_n(t,.)\to \hat{f}_n$ in $S(\mathbb{R}^2)$
as $t$ goes to $0^{+}$. This implies that for all $n$, $u_n$ can be
extended to a function in $C(\mathbb{R}^{+},S(\mathbb{R}^2))$ with
$f_n$ as value at $t=0$. Consequently, to conclude the proof
of the Lemma,
we just need to show that the sequence $(u_n)_n$ converges
to $u$ in the space $L^{\infty }(\mathbb{R}^{+},X)$. To do
this, we notice that for any $t>0$ and any $n\in \mathbb{N}$ we have
\[
u_n(t)-u(t)=\mathcal{K}_{(n),t}\ast (f_n-f)+(\mathcal{K}_{(n),t}-
\mathcal{K}_{t})\ast f.
\]
Hence,
\begin{align*}
\| u_n(t)-u(t)\| _{X}
&\leq \| \mathcal{K}_{(n),t}\| _1\| f_n-f\| _{X}+\|
\mathcal{K}_{(n),t}-\mathcal{K}_{t}\| _1\| f\|_{X} \\
&\leq C \| f_n-f\| _{X}+\| \mathcal{K}_{(n)}-\mathcal{K
}\| _1\| f\| _{X},
\end{align*}
which leads to the desired result.
\end{proof}

The next lemma will be useful in the sequel.

\begin{lemma}\label{lemma1}
Let $X$ be a shift invariant functional space, $T>0$ and $\mu <1$.
Then, for all $f\in L_{\mu }^{\infty }([0,T],X)$, the function
$\mathcal{L}_{\alpha }(f)$
belongs to $L_{\mu '}^{\infty }([0,T],X_{\mathcal{R}})$
and satisfies
\[
\| \mathcal{L}_{\alpha }(f)\| _{L_{\mu'}^{\infty }([0,T],
X_{\mathcal{R}})}\leq C\|f\| _{L_{\mu }^{\infty }([0,T],X)}
\]
where $\mu '=\mu -1+\frac{1}{2\alpha }$ and $C$ is a
constant depending only on $\mu$, $\alpha $ and $X$. Moreover, if
$f$ belongs to
$L_{\mu}^{\infty }([0,T],\overline{S(\mathbb{R}^2)}^{X})$ then
$\mathcal{L}_{\alpha }(f)$ belongs to
$C_{\mu '}^0([0,T],(\overline{ S(\mathbb{R}^2)}^{X})_{\mathcal{R}})$.
\end{lemma}

\begin{proof}
 The first assertion is a an immediate consequence
of estimates (\ref{est1a})-(\ref{est1b}). The last assertion can
be easily proved by using the previous lemma and the Lebesgue's
dominated convergence theorem, we
left details to the reader.
\end{proof}

\begin{lemma}\label{lemma2}
Let  $T > 0$.  Then the following assertions hold:
\begin{enumerate}
\item The linear operator $e^{-t(-\Delta)^{\alpha}}$ is continuous
from  $\mathbf{\tilde{B}}^{\alpha }$  to  $\mathbb{E}_T^{\nu} $.
\item The bilinear operator $\mathcal{B}_{\alpha}$ is continuous
from  $\mathbb{E}_T^{\nu} \times \mathbb{E}_T^{\nu} \to \;\mathbb{E}_T^{\nu}$
and its norm is independent of $T$.
\end{enumerate}
\end{lemma}

\begin{proof}
 The first assertion follows from the
characterization of Besov spaces by the  kernel
 $e^{-t(-\Delta)^{\alpha}}$  and the definition of
 $\mathbf{\tilde{B}}^{\alpha }$
The second assertion, is a direct consequence of the previous
lemma and the fact that
$\mathbb{E}_T^{\nu}=C_{\nu }^0([0,T],(C_0(\mathbb{R}^2))_{\mathcal{R}})$
\end{proof}

The following lemma, which is a direct consequence of the
preceding one, will be useful in the proof of Theorem \ref{theo2}.

\begin{lemma}\label{lemma3}
Let  $ \theta_0 \in \tilde{B}^{\alpha}$.
  The sequence $\phi_n(\theta_0)$ defined by
\begin{gather*}
\phi_0(\theta_0)  = e^{-t(-\Delta)^{\alpha}}\theta_0,\;\\
\phi_{n+1}(\theta_0)  = e^{-t(-\Delta)^{\alpha}}\theta_0 + {\mathcal
B}_{\alpha}[\phi_n(\theta_0),\phi_n(\theta_0)],
\end{gather*}
belongs to  $ \cap_{T > 0} \mathbb{E}_T^{\nu}$.
Moreover, there exists a constant  $ \mu_0 > 0 $
(depending only on  $\alpha$ ) such that if for some $T > 0$
we have  $ \|\phi_0(f)\|_{\mathbb{E}_T^{\nu}} \leq \mu_0$  then  for all
$n \in \mathbb{N}^{*}$,
\begin{gather}
\label{suite1}
\|\phi_n(\theta_0)\|_{\mathbb{E}_T^{\nu}}   \leq   2 \|\phi_0(\theta_0)\|_{\mathbb{E}_T^{\nu}},\\
\label{suite2} \|\phi_{n+1}(\theta_0) -
\phi_n(\theta_0)\|_{\mathbb{E}_T^{\nu}}  \leq  \frac{1}{2^n}.
\end{gather}
In particular, the sequence $(\phi_n(\theta_0))_n $ converges in
the space $\mathbb{E}_T^{\nu}$ and its limit $\theta$ is a mild solution
to the equation \eqref{QGalpha} with initial data $\theta_0$.
\end{lemma}

 The following elementary lemma will play a crucial role
in this paper.

\begin{lemma}[Gronwall type Lemma]\label{lemma5}
 Let $T > 0$, $c_1, c_2 \geq 0$, $\kappa \in ] 0, 1 [$ and
 $ f\in  L^{\infty}(0,T)$  such that for all
$ t \in [ 0 , T ]$,
\[
f(t) \leq c_1 + c_2 \int_0^t \frac{f(s)}{(t-s)^{\kappa}} ds.
\]
Then for all $t \in [ 0 , T ]$,
\begin{equation} \label{ineg_lem5}
f(t) \leq 2 c_1 e^{\nu t},
\end{equation}
where  $ \nu = \nu_{\kappa,c_2} > 0$.
\end{lemma}

\begin{proof}
Let $\nu > 0$ to be precise in the sequel and consider the
function  $g$  defined on  $ [ 0 , T ]$   by
$$
g(t) = \sup_{0 < s < t } e^{-\nu s } f(s).
$$
Clearly, we have
$$
g(t) \leq   c_1 + c_2 \int_0^t \frac{e^{-\nu(t-s)}}{(t-s)^{\kappa}}
g(s) ds,
 \leq  c_1 + c_2 \gamma_{\kappa} \nu^{\kappa-1} g(t),
$$
where   $\gamma_{\kappa}  =  \int_0^{\infty}
\frac{e^{-t}}{t^{\kappa}}$. Thus, if we choose $\nu >0$ such that
$ c_2 \gamma_{\kappa} \nu^{\kappa-1}  = \frac 12$, we obtain
 the estimate \eqref{ineg_lem5}.
\end{proof}

\begin{lemma}[Maximal Principle]\label{lemme6}
Let $\theta $ be a mild solution of \eqref{mild}
belonging to the space $C([0,T],(C_0(\mathbb{R}^2))_{\mathcal{R}})$.
Then  for all  $t\in [ 0,  T ]$,  we have
\begin{gather}
\label{max1}
\|\theta(t)\|_{\infty} \leq \|\theta_0\|_{\infty}, \\
\label{max2}
 \|{\mathcal{R}^{\bot }}(\theta)(t)\|_{\infty}
\leq 2\|{\mathcal{R}^{\bot }}( \theta_0)\|_{\infty} e^{\eta t},
\end{gather}
 where  $ \eta  = \eta_{\alpha, \|\theta_0\|_{\infty}} > 0$.
\end{lemma}


\begin{proof}
 The inequality \eqref{max1} is proved in
\cite{resnick}, \cite{const2001} and \cite{wu2}, for sufficiently
smooth solution $\theta $. To prove it in our case, we will
proceed by linearization of the equations and regularization of
the initial data. We consider a sequence of
\emph{linear system}
\begin{equation}
\begin{gathered}
\partial _{t}v-(-\Delta)^{\alpha} v+\nabla .(u_nv)=0 \\
v(0,.)=\theta _n(.).
\end{gathered}  \label{QGLn}
\end{equation}
where $(\theta _n)_n$ is a given sequence in
$C_{c}^{\infty }(\mathbb{R}^2)$ converging to $\theta (0)$
in the space $L^{\infty }(\mathbb{R}^2)$ and
$u_n=\omega _n\ast \mathcal{R}^{\bot }(\theta )$
with $\omega _n(.)=n^2\omega (n.)$ where
$\omega \in C_{c}^{\infty }(\mathbb{R}^2)$ and $\int \omega dx=1$.

Let $n\in \mathbb{N}$. By converting the system  \eqref{QGLn}
into the integral equation
\begin{equation}
v(t)=e^{-t(-\Delta)^{\alpha}}\theta _n-\int_0^{t}\nabla
.e^{-(t-s)(-\Delta)^{\alpha}}(u_nv)ds \label{IQGLn}
\end{equation}
and by following a standard method, one can easily prove that
the system   \eqref{QGLn}
 has a unique global solution $v_n\in \cap _{k\in
\mathbb{N}}C^{\infty }([0,T],H^{k}(\mathbb{R}^2))$.
Hence we are allowed
to make the following computations: Let $p\in [ 2,\infty [ $.
For any $t\in [ 0,T]$ we have
\begin{align*}
\frac{1}{p}\frac{d}{dt}\| v_n(t)\| ^{p}
&=-\int ((-\Delta)^{\alpha} v)v| v|
^{p-2}dx-\int \nabla.(u_nv)v| v| ^{p-2}dx \\
&\equiv I_1(t)+I_2(t).
\end{align*}
Firstly, a simple integration by parts implies that
$I_2(t)=-I_2(t)$ and so
\[
I_2(t)=0.
\]
Secondly, by the positivity Lemma  (see \cite{resnick} and
\cite{cordoba2004}), we have
\[
I_1(t)\leq 0.
\]
Therefore,
\[
\sup_{t\in [ 0,T]}\| v_n(t)\| _{p}\leq \|
\theta _n\| _{p}.
\]
Letting $p\to +\infty $, yields
\[
\sup_{t\in [ 0,T]}\| v_n(t)\| _{\infty }\leq
\| \theta _n\| _{\infty }.
\]
Consequently, to obtain  \eqref{max1}, we just need to
show that the sequence $(v_n)_n$ converges to the
function $\theta $ in the space $L^{\infty }([0,T],L^{\infty
}(\mathbb{R}^2))$. To do this, we consider the sequence
$(w_n)_n=(v_n-\theta )_n$. Let $t\in [ 0,T]$ and
$n\in \mathbb{N}$. We have
\begin{align*}
w_n(t) &= e^{-t(-\Delta)^{\alpha}}(w_n(0))
-\int_0^{t}\nabla .e^{-(t-s)(-\Delta)^{\alpha}}((u_n-\mathcal{R}^{\bot }
(\theta ))v_n)ds \\
&\quad -\int_0^{t}\nabla .e^{-(t-s)(-\Delta)^{\alpha}}(\mathcal{R}^{\bot
}(\theta )w_n)ds.
\end{align*}
Thus, by using the Young inequality and Proposition \ref{prop2},
we easily get
\[
\| w_n(t)\| _{\infty }\leq \| \theta _n-\theta
(0)\| _{\infty }+C_{\alpha }T^{\nu }A_nB_n+C_{\alpha }M_{\theta
}\int_0^{t}\frac{\| w_n(s)\| _{\infty }}{(
t-s)^{1/2\alpha }}ds
\]
where $C_{\alpha }$ is a constant depending only on $\alpha $,
\begin{gather*}
A_n=\sup_{0\leq t\leq T}\| u_n(t)-\mathcal{R}^{\bot
}(\theta )(t)\| _{\infty },\\
B_n=\sup_{0\leq t\leq T}\| v_n(t)\| _{\infty },\\
M_{\theta }=\sup_{0\leq t\leq T}\| \mathcal{R}^{\bot
}(\theta )(t)\| _{\infty }.
\end{gather*}
Applying Lemma \ref{lemma5}, we obtain
\[
\sup_{0\leq t\leq T}\| w_n(t)\| _{\infty }\leq C[
\| \theta _n-\theta (0)\| _{\infty }+C_{\alpha }T^{\nu
}A_nB_n]
\]
where $C$ is a constant depending on $\alpha ,T$ and $\theta $
only.

Therefore, to obtain the desired conclusion, we just have to notice that the
sequence $(B_n)_n$ is bounded and that $A_n\to 0$ as $
n\to \infty $ thanks to the uniform continuity of the function $
\mathcal{R}^{\bot }(\theta )$\ on $[0,T]\times \mathbb{R}^2$,
which is a consequence of the fact $\mathcal{R}^{\bot }(\theta
)\in C([0,T], C_0(\mathbb{R}^2))$

Now, let us establish the inequality \eqref{max2}.
For any $t\in [ 0,T]$, we have
\[
\mathcal{R}^{\bot }(\theta )(t)=e^{-t(-\Delta)^{\alpha}}(
\mathcal{R}^{\bot }(\theta
)(0))-\int_0^{t}\mathcal{R}^{\bot }\nabla .e^{-(t-s)(-\Delta)^{\alpha}}(
\mathcal{R}^{\bot }(\theta )\theta )ds.
\]
Applying the Young inequality and \eqref{est1b}, we obtain
\[
\| \mathcal{R}^{\bot }(\theta )(t)\| _{\infty
}\leq \| \mathcal{R}^{\bot }(\theta )(0)\|
_{\infty }+C\|
\theta (0)\| _{\infty }\int_0^{t}\frac{\| \mathcal{R}^{\bot }
(\theta )(s)\| _{\infty }}{(t-s)^{1/2\alpha
}}ds
\]
where the constant $C$  depends only on $\alpha$.
Hence, Lemma \ref{lemma5} leads the desired inequality.
\end{proof}

\section{Proof of Theorem \ref{theo1}}

According to Lemma \ref{lemma2}, there exists $T>0$ such that
$\| e^{-t(-\Delta )^{\alpha }}\theta
_0\| _{\mathbf{E}_T^{\nu }}\leq \mu _0$ where $\mu
_0$ is the real defined by Lemma \ref{lemma3}. Therefore, the
same lemma ensures  that the equation \eqref{QGalpha}  with
initial data $\theta _0$ has a mild solution $\theta $ belonging
to the space $\mathbf{E}_T^{\nu }$. Following a standard
arguments (see for example \cite{lemarie} and \cite{canonne}), the
uniqueness of the solution $\theta $ can be easily deduced from
the continuity of the operator $\mathcal{B}_{\alpha }$ on the
space $\mathbf{E}_T^{\nu }$.
Hence, there exists a unique maximal solution,
$$
\theta \in \cap _{0<T<T^{\ast }}\mathbf{E}_T^{\nu }.
$$
where  $T^{*}$  is the maximal time existence. Let us show that
$\theta \in C([0, T^{*}), \mathbf{\tilde{B}}^\alpha )$.
Thanks to the embedding,
$$
(C_0(\mathbb{R}^2))_{\mathcal{R}} \subset \mathbf{\tilde{B}}^\alpha ,
$$
and Lemma \ref{lemma0}, we just need to prove the continuity of
$N(\theta)(t) = {\mathcal B}_{\alpha }[\theta,\theta](t)$
at $t = 0^+$ in the space $\mathbf{\tilde{B}}^\alpha$.  Furthermore,
we show that
$$
\lim_{t\to 0^+} N(\theta )(t) = 0,\quad \text{in}\quad
\mathbf{\tilde{B}^\alpha }.
$$
For that, we use Proposition \ref{prop2}, the Young inequality and
 estimates \eqref{est1a}-\eqref{est1b}, to obtain
\begin{align*}
\| N(\theta )(t)\|_{\mathbf{\tilde{B}}^\alpha }
 \lesssim
  \sup_{ 0 < t^{'} < 1}
 { t^{'}}^{\nu}
  \int_0^t ( t + t^{'} -\tau )^{-\frac{1}{2\alpha }} \tau^{-2\nu} d\tau \quad \|\theta \|_{\mathbb{E}_t^{\nu}}^2 \lesssim \|\theta \|_{\mathbb{E}_t^{\nu}}^2.
\end{align*}
Since $\|\theta \|_{\mathbb{E}_t^{\nu}}$ goes to $0$ as $t$ goes $0^+$
we obtain the desired result.

It remains to show that the solution $\theta $ is global, that is
 $ T^{\ast}=\infty $.  We argue by contradiction. If  $T^{\ast
}< \infty $ then, from  Lemma \ref{lemma3}, we must have
for all  $0 < t_0 < T^{*}$,
$$
 \| e^{t(-\Delta)^{\alpha}}\theta (t_0)\| _{\mathbf{E}_{T^{\ast }-t_0}^{\nu }}
\geq \mu_0,
$$
 which yields by the Young inequality
\begin{equation} \label{est6b}
\| \theta (t_0)\| _{\infty
}+\| {\mathcal R}^{\bot }(\theta)(t_0)\|
_{\infty }\geq \frac{c}{(T^{\ast }-t_0)^{\nu }},
\end{equation}
where $c>0$ is a universal constant. Which contradicts the Maximal
Principle (Lemma \ref{lemma5}).

\section{Proof of Theorem \ref{theo2}}\label{sectreg}

 Along this section, we consider  $\theta_0$ a given initial data
 belonging to the space $\mathbf{\tilde{B}}^\alpha$ and we denote
by $\theta$ the solution to  \eqref{QGalpha} given by Theorem
 \ref{theo1}. We will establish the persistency of the
regularity of the initial data. That is, if moreover
$\theta_0 \in X $ for a suitable Banach spaces $X$ then
the solution $\theta \in C([0,\infty), X)$.

\subsection{Propagation of the  $L^p$  regularity}

In this subsection we will prove the propagation of the initial
$  L^p  $ regularity. Precisely, we prove the following proposition.

\begin{proposition}\label{propLp}
Let   $ X  =  L^p $     with   $ p \in [ 1, \infty]$.
If  $\theta_0 \in X$  then  $ \theta$ belongs to
$\bigcap_{T >0}L^{\infty}([ 0, T], X) $. Moreover, if   $ \theta_0 \in
\overline{S(\mathbb{R}^2)}^X $  then    $ \theta  \in  C([ 0,
\infty ), \overline{S(\mathbb{R}^2)}^X )$
\end{proposition}

\begin{proof}
 Assume   $ \theta_0 \in  X $  and let  $  T >0$.
 We consider the Banach spaces
  $  \mathbf{Z}_1 = \mathbb{E}_T^{\nu} $ and
$   \mathbf{Z}_2 = L^{\infty}([ 0, T], X) $  endowed respectively
with the  norms
$$
\| v \|_{\mathbf{Z}_1} = \sup_{0 < t < T}
e^{-\lambda t} t^{\nu} \| v(t)\|_{\infty} \quad\text{and}\quad
\| v \|_{\mathbf{Z}_2} = \sup_{0 < t < T} e^{-\lambda t}\| v(t)\|_{p},
$$
where $\lambda > 0$ to be fixed later. We consider the linear
integral equation,
\begin{equation} \label{linpsi}
v = \Psi _{\theta }(v)
\equiv e^{t(-\Delta)^{\alpha}}\theta _0+ \mathcal{B}_{\alpha }[\theta ,v].
\end{equation}
Let $k\in \{1;2\}$. According to Lemma \ref{lemma1}, the
affine functional  $\Psi_{\theta} :  \mathbf{Z}_{k} \to
\mathbf{Z}_{k}$  is continuous. Let us estimate the norm of its
linear part
\[
K_{\theta }(v)=\mathcal{B}_{\alpha }[\theta ,v].
\]
Let $\varepsilon >0$ to be chosen later. A direct computation
using \eqref{est1a} gives
\begin{align*}
\| K_{\theta }\| _{\mathcal{L}(\mathbf{Z}_1)}
&= \sup_{\| v\| _{\mathbf{Z}_1}\leq 1}\| K_{\theta
}(v)\| _{\mathbf{Z}_1} \\
&\leq C_1\sup_{0<t<T}t^{\nu }\int_0^{t}(t-\tau )^{-\frac{1}{2\alpha }
}\tau ^{-2\nu }e^{-\lambda (t-\tau )}\| \theta \| _{\mathbf{
E}_{\tau }^{\nu }}d\tau  \\
&\leq C_2\Big(\| \theta \|
_{\mathbf{E}_{\varepsilon
}^{\nu }}\sup_{0<t<\varepsilon }t^{\nu }\int_0^{t}(t-\tau )^{-\frac{1}{
2\alpha }}\tau ^{-2\nu }d\tau +T^{\nu }\varepsilon ^{-2\nu
}\|
\theta \| _{\mathbf{E}_T^{\nu }}\lambda ^{-\nu }\Gamma (\nu )\Big)\\
&\leq C_{3}\Big(\| \theta \|
_{\mathbf{E}_{\varepsilon
}^{\nu }}+T^{\nu }\varepsilon ^{-2\nu }\lambda ^{-\nu }\| \theta \| _{
\mathbf{E}_T^{\nu }}\Big).
\end{align*}
where the constants $C_1,C_2,C_{3}$ depend only on
$\alpha $.
Similarly, we prove the estimate
 \[
\| K_{\theta }\|
_{\mathcal{L}(\mathbf{Z}_2)}\leq C(\| \theta
\| _{\mathbf{E}_{\varepsilon }^{\nu }}+T^{\nu
}\varepsilon ^{-2\nu }\lambda ^{-\nu }\| \theta \| _{
\mathbf{E}_T^{\nu }}),
\]
where $C$ is a constant depending only on $\alpha $. Since
 $\|\theta\|_{\mathbb{E}_{\epsilon}^{\nu}} \to 0$  as
 $\epsilon \to 0^+$,   one can choose, successively,
 $\epsilon $  small enough
 and  $\lambda $  large enough so that $\Psi_{\theta}$
becomes a contraction on  $\mathbf{Z}_1$  and
$ \mathbf{Z}_2$  and therefore on  $\mathbf{Z}_1\cap \mathbf{Z}_2$.
Let  $v_1$ and $v_{1,2}$ be the unique fixed point of
$\Psi_{\theta}$ respectively in  $\mathbf{Z}_1$  and
$\mathbf{Z}_1\cap \mathbf{Z}_2$.  Now, since
$\mathbf{Z}_1 \cap \mathbf{Z}_2 \subset \mathbf{Z}_1$
then  $v_1 = v_{1,2}$.
 Moreover, by construction $\theta$ is a fixed point of
$\Psi_{\theta}$ in  $\mathbf{Z}_1$  thus  $\theta = v_1 = v_{1,2}$
and hence  $ \theta \in  L^{\infty}([ 0, T], X)$.

The proof of the last statement of the proposition is identically
similar, we have only to replace  $\mathbf{Z}_2$   by
$ C([ 0, T] ,\overline{S(\mathbb{R}^2})^X )$.
\end{proof}

\subsection{Propagation of  ${\dot{B}}^{s,q}_p$  regularity for  $ s > 0$}

In this section, we prove an abstract result, which implies in
particular the persistence  of the  ${\dot{B}}^{s,q}_p$  regularity for
$s>0$. Our result states as follows.

\begin{proposition}\label{propositionabstraite}
Let $X$ be a shift invariant functional space such that for
a constant $C$ and all $f,g\in X\cap L^{\infty}(\mathbb{R}^2)$,
\begin{equation}
\| fg\| _{X}\leq C(\| f\|
_{\infty }\| g\| _{X}+\| g\|
_{\infty }\| f\| _{X}).
\end{equation}
If the initial data $\theta _0$ is in $X_{\mathcal{R}}$ then the
solution $\theta $ belongs to $\cap_{T > 0} L^{\infty} ([0,T],
X_{\mathcal R}) $. Moreover, if $\theta _0$ belongs to
$(\overline{S(\mathbb{R}^2)}^X)_{\mathcal{R}}$ then $\theta $
belongs to $ C(\mathbb{R}^{+},
(\overline{S(\mathbb{R}^2)}^X)_{\mathcal{R}})$).
\end{proposition}

The proof of this proposition relies essentially on the two following
lemmas. The first one is an elementary compactness lemma.

\begin{lemma}\label{compact}
Let $\lambda >0$ and $K$ a compact subset of
$\mathbf{\tilde{B}}^{\alpha }$.
Then there exists $\delta =\delta (K,\lambda )>0$ such that
for all $f\in K$,
\[
\| e^{-t(-\Delta)^{\alpha} }f\| _{\mathbb{E}_{\delta }^{\nu }}\leq \lambda.
\]
\end{lemma}

\begin{proof}
For $n\in \mathbb{N}^{\ast }$, we set
\[
V_n=\big\{ f\in \mathbf{\tilde{B}}^{\alpha },\;
\|e^{-t(-\Delta)^{\alpha} } f\| _{\mathbb{E}_{1/n}^{\nu }}<\lambda \big\} .
\]
We claim that for all $n \in \mathbb{N}^*$, $ V_n$ is an open
subset of $\mathbf{\tilde{B}}^{\alpha }$ and
$\cup _nV_n =\mathbf{\tilde{B}}^{\alpha }$. This follows
easily from the continuity of the linear operator $e^{-t(-\Delta)^{\alpha}} $
 from $\mathbf{\tilde{B}}^{\alpha }$ into $\mathbb{E}_T^{\nu }$ for
all $T>0$ and the propriety: For all $f \in \mathbf{\tilde{B}}^{\alpha }$,
\[
\lim_{T\to 0}\| e^{-t(-\Delta)^{\alpha}} f \|_{\mathbb{E}_T^{\nu }}=0.
\]
Thus, since $K$ is a compact subset of
$\mathbf{\tilde{B}}^{\alpha }$, there exists a finite subset
$I \subset \mathbb{N}^{\ast }$
such that $K \subset \cup _{I}V_n = V_{n^{\ast }}$ where
$n^{\ast }= \max( n \in I)$. Hence, we conclude that the choice
$\delta =1/n^{\ast }$ is suitable.
\end{proof}

 The second lemma establishes a local in time
propagation of the $X$ regularity.

\begin{lemma}\label{localpersistence}
 Let $X$ be as in Prop. \ref{propositionabstraite}.
If $\theta _0$ belongs to $X_{\mathcal{R}}$ (resp.
$(\overline{S(\mathbb{R}^2)}^X)_{\mathcal{R}} $) then there exists
$\delta =\delta(X,\alpha )>0$ such that the solution
$\theta \in L^{\infty }([0,\delta ] ,X_{\mathcal{R}})$
(resp. $C([0,\delta ],(\overline{S(\mathbb{R}^2)}^X)_{\mathcal{R}})$.
Moreover, the time  $ \delta $ is bounded below by,
\[
\sup \big\{ T>0, \; \| e^{-t(-\Delta)^{\alpha} }\theta _0\|
_{\mathbb{E}_T^{\nu }}\leq \mu \big\},
\]
where $\mu $ is a non negative constant depending on $X$ and $\alpha $
only.
\end{lemma}

\begin{proof}
 Let us consider the case of $\theta _0\in X_{\mathcal{R}}$.
The proof in the other case is similar.
Let $\mu \in ] 0,\mu _0[ $ to be chosen later and let $T>0$
such that $\| e^{-t(-\Delta)^{\alpha}}\theta _0\| _{\mathbf{E}_T^{\nu }}\leq \mu.$
According to Lemma \ref{lemma3}, the
sequence $(\phi_n(\theta _0))_n$ converges in
$\mathbb{E}_T^{\nu }$ to the solution $\theta $ and satisfies the
following estimates
\begin{gather}
\label{est20a}
\sup_n\| \phi _n(\theta _0)\| _{\mathbf{E}_T^{\nu}}
\leq \mu \\
\label{est20b}
\forall n \in \mathbb{N},\quad \| \phi _{n+1}(\theta _0)-\phi _n(\theta
_0)\| _{\mathbf{E}_T^{\nu }} \leq 2^{-n}.
\end{gather}
Then, to conclude we just need to show that $(\phi
_n(\theta _0))_n$ is a Cauchy sequence in the Banach
space $\mathbf{Z}_{\mathcal R}=L^{\infty }([0,T]
,X_{\mathcal{R}})$ endowed with its natural norm,
\[
\| v\| _{\mathbf{Z}_{\mathcal
R}}=\sup_{0<t<T}(\| v(t)\| _{X}+\|
{\mathcal{R}^{\perp }}(v)(t)\| _{X}).
\]
Firstly, using Lemma \ref{lemma1} and the fact that
$(\phi_n(\theta _0))_n\in \mathbf{E}_T^{\nu }$, we infer
inductively that the sequence $(\phi _n(\theta_0))_n$ belongs
to the space $\mathbf{Z }_{\mathcal{R}}$. Secondly, once again
the Lemma \ref{lemma1}, implies that the
sequence $(\omega _{n+1})_n\equiv
(\phi_{n+1}(\theta _0)-\phi_n(\theta _0))_n$ satisfies
the  inequality
\begin{align*}
\| \omega _{n+1}\| _{\mathbf{Z}_{\mathcal R}}
&\leq C(\| \phi _n(\theta _0)\| _{\mathbf{Z}_{\mathcal R}}+\|
\phi _{n-1}(\theta _0)\| _{\mathbf{Z}_{\mathcal R}})\|
\omega _n\| _{\mathbf{E}_{\delta }^{\nu }} \\
&\quad+C(\| \phi _n(\theta _0)\| _{\mathbf{E}
_T^{\nu }}+\| \phi _{n-1}(\theta _0)\| _{\mathbf{E}
_{\delta }^{\nu }})\| \omega _n\| _{\mathbf{Z}
_{\mathcal R}},
\end{align*}
where $C=C(X,\alpha )>0$.
This inequality combined with the estimates
\eqref{est20a}-\eqref{est20b} yields
\[
\| \omega _{n+1}\| _{\mathbf{Z}_{\mathcal R}}\leq C(\frac{1}{2}
)^{n}(\| \phi _n(\theta _0)\| _{\mathbf{Z}
_{\mathcal R}}+\| \phi _{n-1}(\theta _0)\| _{\mathbf{Z}
_{\mathcal R}})+4C\mu \| \omega _n\| _{\mathbf{Z}
_{\mathcal R}}
\]
Finally, if we choose $\mu > 0 $ such that $4C\mu <1$ one can
conclude the proof by using the following lemma which is inspired
from \cite{furioli}.
\end{proof}

\begin{lemma}\label{pointfixe}
 Let $(x_n)_n$ be a sequence in a normed vector space
$(Z,\| .\| )$. If there exist a constant
$\lambda \in[ 0,1[$ and $(\sigma _n)_n\in l^{1}(\mathbb{N})$
such that for all $n\in \mathbb{N}^{\ast }$,
\begin{equation}
\label{sequence}
\| x_{n+1}-x_n\| \leq \sigma _n(\|
x_n\| +\| x_{n-1}\| )+\lambda \|x_n-x_{n-1}\|,
\end{equation}
then the series $\sum_n\| x_{n+1}-x_n\| $ converges. In
particular, $(x_n)_n$ is a Cauchy sequence in $Z$.
\end{lemma}

\begin{proof}
Let us define the sequence $M_n=\sup_{k\leq n}\|
x_{k}\| $. It follows inductively from \eqref{sequence},
\begin{equation} \label{est21}
\| x_{n+1}-x_n\| \leq 2\sum_{k=0}^{n-1}\sigma
_{n-k}M_{n-k}\lambda ^{k}
\leq \varpi _nM_n,
\end{equation}
where $\varpi_n=2\sum_{k=0}^{n-1}\sigma _{n-k}\lambda ^{k}$.
Noticing that since $(\varpi _n)_n$ is a convolution of two
sequences in $l^{1}(\mathbb{N})$ then $(\varpi _n)_n$
belongs to $l^{1}(\mathbb{N})$. Therefore,  we just need to show
that the sequence $(M_n)_n$ is bounded. This is somehow
obvious. In fact, using the triangular inequality
$\|x_{n+1}\| \leq \| x_n\| +\|x_{n+1}-x_n\| $, \eqref{est21} yields
\[
M_{n+1}\leq  ( 1+ \varpi_n ) M_n.
\]
Which in turn implies
\[
M_n\leq \Pi _{k=0}^{n-1}(1+\varpi _{k})\leq e^{\sum_{k\geq
0}\varpi _n}.
\]
The proof is complete.
\end{proof}

Now let us show how the two previous lemmas allow to prove
Proposition \ref{propositionabstraite}.

\begin{proof}
As usual we consider only the case of
$\theta _0\in X_{\mathcal{R}}$. Let $T > 0$.
By Theorem \ref{theo1}, the solution $\theta $ is continuous from
$\mathbb{R}^{+}$ into $\mathbf{\tilde{B}}^{\alpha}$, then
$K \equiv \theta ([0,T] )$ is a compact subset of
$\mathbf{\tilde{B}}^{\alpha }$.
Therefore, by Lemma \ref{compact}, there exists $\delta >0$ such that
for all $\tau \in [0,T]$,
\begin{equation} \label{est22}
\| e^{-t(-\Delta)^{\alpha}}\theta (\tau )\| _{\mathbb{E}_{\delta}^{\nu }}\leq \mu_0,
\end{equation}
where $\mu_0 $ is the real given by Lemma \ref{localpersistence}.
Now, we consider a partition
$0=t_0<\dots <t_{N+1}=T$ of the interval $[0,T] $ such that
$\sup_{i}t_{i+1}-t_{i}\leq \frac{\delta }{2}$. We will show
inductively that
\begin{align}
\label{est22a} \theta \in L^{\infty }([
t_{i},t_{i+1}] ,X_{\mathcal{R}}),
\end{align}
which implies in turn the desired result
$\theta \in L^{\infty}([0,T] ,X_{\mathcal{R}})$.  First, by
Lemma \ref{localpersistence}, the claim  \eqref{est22a}
is true for $i=0$. Assume that, it is also true for
$i\leq N$. Then there exists $\tau _0$ in
$]t_{i},t_{i+1}[ $ such that $\tilde{\theta}_0 \equiv \theta
(\tau _0) \in X\cap \mathbf{\tilde{B}}^{\alpha }$. We notice
that $\tilde{\theta} \equiv  \theta(. + \tau_0) $ is the unique
solution given by Theorem \ref{theo1} of the Quasi-geostrophic
equation with initial data $\tilde{\theta}_0$. Then according
to Lemma \ref{localpersistence} and \eqref{est22}, we obtain
$ \theta \in L^{\infty }([\tau_0,\tau_0 + \delta ],X_{\mathcal{R}})$.
Hence, we are ready to conclude since
$[t_{i+1},t_{i+2}] \subset [\tau_0, \tau_0 +\delta ]$.
\end{proof}

\subsection{Propagation of $B_p^{s,q}$ regularity for $ s < 0$}

\begin{proposition}\label{propsnegative}
Let $X $ be $ B_p^{s,q}$ or $ \dot{B}_p^{s,q}$  with $ -1<s < 0 $
and $ 1 \leq p$, $q \leq \infty$. If $\theta_0 $ belongs to
$X_{\mathcal{R}}$ then the solution $\theta$ belongs to
$\cap_{T>0} L^{\infty}([0,T], X_{\mathcal{R}})$ and satisfies
$$
t^{-\frac{s}{2\alpha}}\theta \in \cap_{T>0} L^{\infty}([0,T],
(L^p)_{\mathcal{R}}).
$$
\end{proposition}

As in the case $ s > 0, $ by using the compactness
Lemma \ref{compact} we just need to prove the following
 local persistency result.

\begin{lemma} \label{lemme}
If $\theta_0  \in X_{\mathcal{R}}$ then there exists $\delta > 0 $
such that $\theta\in L^{\infty}([0,\delta],
X_{\mathcal{R}})$ and satisfies
$$
t^{-\frac{s}{2\alpha}}\theta \in  L^{\infty}([0,\delta],
(L^p)_{\mathcal{R}}).
$$
Moreover, the time $\delta $ is bounded below by
\[
\sup \big\{ T>0/\| e^{-t(-\Delta)^{\alpha} }\theta _0\|
_{\mathbb{E}_T^{\nu }}\leq \mu_0 \big\},
\]
where  $\mu_0 $  is given by Lemma \ref{lemma3}.
\end{lemma}

\begin{proof}
We consider only the case of $X =  B_p^{s,q}$. The
proof in the other case is similar. Let  $ T > 0$ such that
$$
\| e^{-t(-\Delta)^{\alpha} }\theta _0\| _{\mathbb{E}_T^{\nu}}\leq \mu_0.
$$
According to  Lemma \ref{lemma3} the sequence
$(\phi_n(\theta_0))_n$  satisfies
\begin{align}
\label{est35}
\|\phi_{n+1}(\theta_0)- \phi_n(\theta_0) \|_{\mathbb{E}_T^{\nu}}
 \leq \frac{1}{2^n},
\end{align}
and converges to the solution $\theta$ in  $\mathbb{E}_T^{\nu}$.
 Our first task is to prove that $(\phi_n(\theta_0))_n$ is
a Cauchy sequence in the space
$$
X_{\sigma,p}^T = \{  v  : ( 0 , T ] \to L^p \;
\|v\|_{X_{\sigma,p}^T } \equiv \sup_{0<t<T} t^{\frac{\sigma}{2\alpha}}
( \|v(t)\|_p + \|\mathcal{R}^{\perp } (v)(t) \|_p ) < \infty  \},
$$
where $\sigma = -s$.

Thanks to the Besov  characterization \eqref{car02} and
 Lemma \ref{lemma1}, we can show inductively that
$(\phi_n(\theta_0))$ belongs to  $X_{\sigma,p}^T$ and satisfies
\begin{equation} \label{est35a}
\begin{aligned}
&\|\phi_{n+1}(\theta_0) - \phi_n(\theta_0) \|_{X_{\sigma,p}^T}\\
&\leq C \|\phi_n(\theta_0)  - \phi_{n-1}(\theta_0) \|_{\mathbb{E}_T^{\nu}}
\max( \|\phi_n(\theta_0)  \|_{X_{\sigma,p}^T} , \| \phi_{n-1}(\theta_0) \|_{X_{\sigma,p}^T}).
\end{aligned}
\end{equation}
Thus, By  \eqref{est35} and   Lemma \ref{pointfixe} we deduce that
$(\phi_n(\theta_0) )_n$ is a Cauchy sequence in $X_{\sigma,p}^T$.
Therefore its limit $\theta \in X_{\sigma,p}^T$. Now by a simple
computation using the characterization \eqref{car02} we deduce
that
$\theta \in L^{\infty}( [ 0, T_0], (B_p^{s,\infty})_{\mathcal{R}})$.
Moreover, for $\epsilon > 0$ such that
\begin{equation} \label{eps}
 -1 < s \pm \epsilon < 0,
\end{equation}
one can show that the nonlinear part
$N(\theta)(t) = {\mathcal{B}}_{\alpha}
[\theta ,\theta ] (t)$ satisfies
\begin{align}
\label{matin} \| N(\theta)(t) \|_{B_p^{s\pm\epsilon,\infty}}  + \|
{\mathcal R}^{\perp } N(\theta)(t) \|_{B_p^{s\pm\epsilon,\infty}}
\leq  C_{s,\epsilon} t^{-\pm\frac{\epsilon}{2\alpha}}
\|\theta\|_{\mathbb{E}_{t}^{\nu}} \|\theta\|_{X_{\sigma,p}^T}.
\end{align}
Indeed, we have $\tau \in ] 0, 1 [$,
\begin{equation}
\begin{aligned} \label{m3} %\label{m4}
&\tau^{-\frac{s\pm\epsilon}{2\alpha}} \| e^{-\tau(-\Delta)^{\alpha}}
 N(\theta)(t)\|_p \\
& \leq  C \int_0^t (t + \tau - r
 )^{-\frac{1}{2\alpha}}  \tau^{-\frac{s\pm\epsilon}{2\alpha}}
 r^{-\nu}  r^{-\frac{\sigma}{2\alpha}} dr \;
 \|\theta\|_{\mathbb{E}_{t}^{\nu}}
 \|\theta\|_{X_{\sigma,p}^T}, \\
&\leq  C \int_0^t (\frac{\tau}{t+\tau -r})^{-\frac{s\pm\epsilon}{2\alpha}}
 ( t + \tau -r)^{\frac{-1 -(s\pm\epsilon)}{2\alpha}} r^{-\nu}  r^{-\frac{\sigma}{2\alpha}} dr \; \|\theta\|_{\mathbb{E}_{t}^{\nu}}
 \|\theta\|_{X_{\sigma,p}^T}, \\
&\leq  C \int_0 ^t   ( t  -r)^{\frac{-1 - ( s
 \pm\epsilon)}{2\alpha}} r^{-\nu}  r^{-\frac{\sigma}{2\alpha}} dr
 \|\theta\|_{\mathbb{E}_{t}^{\nu}}
 \|\theta\|_{X_{\sigma,p}^T}, \\
&\leq  C  t ^{-\frac{\pm\epsilon}{2\alpha}}  \;
\|\theta\|_{\mathbb{E}_{t}^{\nu}}\|\theta\|_{X_{\sigma,p}^T},
\end{aligned}
\end{equation}
Where  we have used the facts that, $ 0 \leq
\frac{\tau}{t+\tau -r} \leq 1$,
$t + \tau - r \geq t - r$ and \eqref{eps}.
Similarly, we have the same estimate \eqref{m3} for the
${\mathcal R}^{\perp } N(\theta)(t)$. Hence, by Proposition
\ref{prop1} we obtain \eqref{matin}. Thus, by using the interpolation
inequality
\[
\| f\| _{B_{p}^{s,1}}\leq \Big(\|
f\| _{B_{p}^{s-\varepsilon ,\infty }}\Big)^{1/2}
\Big(\| f\| _{B_{p}^{s+\varepsilon ,\infty }}\Big)^{1/2}
\]
we obtain that for all $t\in ]0,T]$,
\begin{equation}
\| N(\theta )(t)\|
_{B_{p}^{s,1}}+\| \mathcal{R}^{\perp }N(\theta
)(t)\| _{B_{p}^{s,1}}\leq C
\|\theta\|_{\mathbb{E}_{t}^{\nu}}\|\theta\|_{X_{\sigma,p}^T}.  \label{mmm}
\end{equation}
Hence $N(\theta) \in L^{\infty}([ 0, T],
(B_p^{s,1})_{\mathcal R})$ which implies $\theta \in L ^{\infty}([
0, T], (B_p^{s,q})_{\mathcal R})$.
\end{proof}

\begin{remark}\label{remarque} \rm
By replacing the space $X_{\sigma ,p}^{T}$ by
$\tilde{X}_{\sigma ,p}^{T}\equiv C_{\frac{\sigma }{2\alpha }}^0([0,T],(
L^{p})_{\mathcal{R}})$ in the proof of Lemma
\ref{lemme}, one can show
that if $\theta _0$ is in $(\overline{S(\mathbb{R}^2)}
^{B_{p}^{s,q}})_{\mathcal{R}}$ with $-1<s<0$ and $1\leq
p,q\leq \infty $, then the solution $\theta $ belongs to the space
$\cap_{T>0}\tilde{X}_{\sigma ,p}^{T}$.
\end{remark}

\subsection{The case of null regularity $s = 0$}
In this subsection we aim to prove the following result.

\begin{proposition}
Let $X $ be $ B_p^{0,q}$ or $ \dot{B}_p^{0,q}$  with
$ 1 \leq p, q \leq \infty$. If $\theta_0\in  X$ then the solution
$$
\theta \in \cap_{T>0} L^{\infty}([0,T], X).
$$
\end{proposition}

Thanks to the following imbeddings
\begin{gather*}
\dot{B}_p^{0,1} \subset \dot{B}_p^{0,q}
\subset \dot{B}_p^{0,\infty}, \\
\dot{B}_p^{0,1} \subset B_p^{0,q}  \subset \dot{B}_p^{0,\infty},
\end{gather*}
the proof of the above proposition is an immediate consequence
of the following lemma.

\begin{lemma} \label{lem4.5}
If   $ \theta_0 \in \dot{B}_p^{0,\infty}$  then
$ N(\theta)= {\mathcal{B}}_{\alpha}[\theta ,\theta ] (t) \in
\cap_{T>0}L^{\infty}([0,T], \dot{B}_p^{0,1} )$.
\end{lemma}

\begin{proof}
 By using  Young's inequality we deduce that
\begin{align*}
\dot{B}_{p}^{0,\infty} \cap {\dot{B}}^{-(2\alpha-1),\infty }_{\infty} \subset  \dot{B}_{2p}^{\frac 12 -\alpha,\infty}.
\end{align*}
Observe that $s^{*} =  \frac 12 -\alpha < 0 $ and hence according
to the proof of Proposition \ref{propsnegative}
 and to the continuity of the Riesz transforms on homogeneous Besov
spaces, we have $\theta \in \cap_{T > 0} X_{\sigma^{*},2p}^T$  where
$ \sigma^{*} = \alpha - \frac 12$.
Let $ T > 0 $  and    $ 0 < \varepsilon < 2\alpha-1$. The basic
estimate
\begin{align*}
\| \sqrt{-\Delta}^{\pm\varepsilon}\; \nabla e^{-t{(-\Delta)^{\alpha}}} f \|_p
&  \leq  C_0 t^{-\frac{\pm\varepsilon + 1}{2\alpha}} \| f \|_p.
\end{align*}
yields immediately
\begin{align*}
\| (\sqrt{-\Delta})^{\pm \varepsilon} N(\theta)(t) \|_p  & \leq  C
  t^{-\frac{\pm\varepsilon}{2\alpha}}
\| \theta \|_{X_{\sigma^*,2p}^T}^2.
\end{align*}
Now, we use the interpolation result (see \cite[Theorem 6.3]{bergh})
\begin{align*}
[ (\sqrt{-\Delta})^{\varepsilon} L^p,
(\sqrt{-\Delta})^{-\varepsilon} L^p ]_{\frac{1}{2},1} =
\dot{B}_{p}^{0,1},
\end{align*}
to deduce
\begin{align}
\label{nonl2}  \| N(\theta)(t) \|_{
\dot{B}_{p}^{0,1}} & \leq  C  \| \theta \|_{X_{\sigma^*,2p}^T}^2,
\quad \forall  0 < t < T,
\end{align}
this implies
\begin{equation}
\label{nonlN}
N(\theta) \in L^{\infty}([0,T], \dot{B}_{p}^{0,1} ).
\end{equation}
\end{proof}

As in the context of the Navier-Stokes equations
\cite{cannone2000}, we observe thanks to \eqref{nonlN} and
\eqref{nonl2} that in the case  $ -1 < s \leq 0$, the
fluctuation term $N(\theta)$ is more regular than the tendency
$e^{-t(-\Delta)^{\alpha}}\theta_0$.  Moreover, we have the following result.

\begin{proposition} \label{prop4.5}
Let $X=B_{p}^{s,\infty }$ with $s]-1,0[$ and
$1\leq p\leq \infty $. If $\theta _0\in X_{\mathcal{R}}$
then $N(\theta )$ belongs to the space
$C([0,\infty [;(B_{p}^{0,1})_{\mathcal{R}})$.
\end{proposition}

\begin{proof}
 We consider the two cases:

\noindent\textbf{Case $s\in ]-1,0[$:}
According to Proposition
\ref{propsnegative}, $t^{-\frac{s}{2\alpha}}\theta \in
\cap_{T>0} L^{\infty}([0,T], (L^{p})_{\mathcal{R}})$. Then a
simple computation using that
$\theta \in \cap_{T>0} \mathbf{E}_T^{\nu}$ gives
$N(\theta )\in C(]0,\infty[;(L^{p})_{\mathcal{R}})$
which yields $N(\theta )\in C(]0,\infty[ ;(B_{p}^{0,1})_{\mathcal{R}})$
since $s<0$.  On the other hand, the estimate \eqref{mmm}
implies that $N(\theta)(t)\to 0$ in $(B_{p}^{s,1})_{\mathcal{R}}$
as $t$ goes to $0^{+}$. Thus, we obtain the desired result.

\noindent\textbf{Case $s=0$:}
By interpolation, $\theta _0\in (\overline{S(\mathbb{R}^2)}
^{B_{p_{\ast }}^{s_{\ast },\infty }})_{\mathcal{R}}$ where
$s_{\ast }= \frac{1}{2}-\alpha $ and $p_{\ast }=2p$.
Hence, according to Remark \ref{remarque}, the solution
$\theta $ belongs to $\cap_{T>0}\tilde{X}_{\sigma^* ,p^*}^{T}$  where
$\sigma^*=-s_*$. Let $\varepsilon \in
[ 0,2\alpha -1[$. A simple computation gives
\[
\sqrt{-\Delta }\,^{\mp \varepsilon }N(\theta )\in
\cap_{T>0}C^0_{\mp \varepsilon /(2\alpha)} ([0,T],(L^{p})_{\mathcal{R}})
\]
Hence, by interpolation we obtain
\[
N(\theta )\in \cap_{T>0}C([0,T],(B_{p}^{0,1})_{\mathcal{R}})\,.
\]
 \end{proof}

\begin{remark} \label{rmk4.2} \rm
Let $X=B_{p}^{s,q}$ with $-1<s\leq 0$ and $1\leq p,q\leq \infty $.
 If $\theta _0\in (\overline{S(\mathbb{R}^2)}^{X})_{\mathcal{R}}$ then
Lemma \ref{lemma0} and the preceding proposition imply that the
solution $\theta $ is in $C([0,\infty [ ;X_{R})$.
\end{remark}

\section{Proof of Theorem \ref{theo3}}\label{sectlp}

The existence part is a direct consequence of Theorem \ref{theo1},
Theorem \ref{theo2} and the following embedding (consequence of
Bernstein's inequality and the boundedness of the Riesz transforms
on Lebesgue's and Sobolev's spaces)
\begin{gather*}
L^{p}(\mathbb{R}^2) \subset \mathbf{\tilde{B}}_{\alpha}\quad
\forall  p\geq p_{c}, \\
H^{s}(\mathbb{R}^2) = B_2^{s,2}(\mathbb{R}^2)\subset \mathbf{\tilde{B}
}_{\alpha }\quad \forall  s\geq s_{c}.
\end{gather*}
Let us establish the uniqueness part. First we notice that since
for $s\geq s_{c}$,
\[
H^{s}(\mathbb{R}^2)\hookrightarrow H^{s_{c}}(\mathbb{R}^2)
\hookrightarrow L^{p_{c}}(\mathbb{R}^2).
\]
We just need to prove the uniqueness in the spaces
$(C([0,T],L^{p}(\mathbb{R}^2)))_{p\geq p_{c}}$.
This will be deduced from the following continuity result
of the bilinear operator $\mathcal{B}_{\alpha }$.

\begin{lemma}
Let $p\in ]p_{c},\infty [$, $q\in ]1,\infty [ $ and
$T>0$. There exists a constant $C$ independent of $T$ such that:
\begin{itemize}
\item For any $u,v$ in $L_T^{\infty }L^{p}$,
\begin{equation}
\| \mathcal{B}_{\alpha }[u,v]\| _{L_T^{\infty
}L^{p}}\leq C T^{\sigma }\| u\| _{L_T^{\infty
}L^{p}}\| v\| _{L_T^{\infty }L^{p}}\text{,}
\label{ESS}
\end{equation}
where $\sigma =\frac{1}{\alpha }(\frac{1}{p_{c}}-\frac{1}{p})$;

\item for any $u,v$ in $L_T^{\infty }L^{p_{c}}$,
\begin{equation}
\| \mathcal{B}_{\alpha }[u,v]\|
_{L_T^{q}L^{p_{c}}}+\| \mathcal{B}_{\alpha
}[v,u]\| _{L_T^{q}L^{p_{c}}}\leq C\|
u\| _{L_T^{\infty }L^{p_{c}}}\| v\|
_{L_T^{q}L^{p_{c}}.} \,; \label{ESS2}
\end{equation}

\item for any $u\in L_T^{\infty }L_{\mathcal{R}}^{\infty }$ and
$v\in
L_T^{q}L^{p_{c}}$,
\begin{equation}
\| \mathcal{B}_{\alpha }[u,v]\|
_{L_T^{q}L^{p_{c}}}+\| \mathcal{B}_{\alpha
}[v,u]\| _{L_T^{q}L^{p_{c}}}\leq C\;
T^{1-\frac{1}{2\alpha }}\| u\| _{L_T^{\infty
}L_{\mathcal{R}}^{\infty }}\| v\|
_{L_T^{q}L^{p_{c}}.}\,.  \label{ESS3}
\end{equation}
\end{itemize}
\end{lemma}

\begin{proof}
Estimate \eqref{ESS} follows easily from the continuity of the
Riesz transforms on the Lebesgue spaces $L^{r}(\mathbb{R}^2)$
with $1<r<\infty $, the Young and the H\"older inequality and the
estimate \eqref{est1a} on the $L^{r}(\mathbb{R}^2)$ norm of the
kernel of the operator $\nabla e^{-(t-s)(-\Delta )^{\alpha }} $.
Estimate \eqref{ESS2} is a consequence of the
continuity of the Riesz transforms on the space
$L^{p_{c}}(\mathbb{R}^2)$, the H\"older
inequality, the Sobolev embedding
\[
\| \frac{\nabla }{(-\Delta )^{\alpha
}}f\| _{p_{c}}\lesssim \| f\|
_{\frac{p_{c}}{2}}
\]
and the  maximal regularity property of the operator
$(-\Delta )^{\alpha }$,
\[
\| \int_0^{t}(-\Delta )^{\alpha
}e^{-(t-s)(-\Delta )^{\alpha }}vds\|
_{L_T^{q}L^{p_{c}}}\lesssim \| v\|
_{L_T^{q}L^{p_{c}}}
\]
which can be proved by following
\cite[Theorem 7.3]{lemarie}. Let us now
prove estimate \eqref{ESS3}. For any $t\in [ 0,T]$ we have
\begin{align*}
\| \mathcal{B}_{\alpha }[u,v](t)\| _{L^{p_{c}}}
&\lesssim \int_0^{t}\frac{1}{(t-s)^{1/2\alpha }}\|
\mathcal{R}^{\perp
}(u)(s)\| _{\infty } \| v(s)\| _{p_{c}}ds \\
&\lesssim \| \mathcal{R}^{\perp }(u)\|
_{L_T^{\infty }L^{\infty }}(1_{[0,T]}s^{-\frac{1}{2\alpha
}})\ast (1_{[0,T]}  \| v(s)\| _{p_{c}})(t)
\end{align*}
where the star $\ast $ denotes the convolution in $\mathbb{R}$.
Hence Young's inequality yields
\[
\| \mathcal{B}_{\alpha }[u,v]\|
_{L_T^{q}L^{p_{c}}}\lesssim \| \mathcal{R}^{\perp
}(u)\| _{L_T^{\infty }L^{\infty }}T^{1-\frac{1}{2\alpha
}}\| v\| _{L_T^{q}L^{p_{c}}}.
\]
Similarly, we obtain
\begin{align*}
\| \mathcal{B}_{\alpha }[v,u]\|
_{L_T^{q}L^{p_{c}}}
&\lesssim  T^{1-\frac{1}{2\alpha
}}\| u\| _{L_T^{\infty }L^{\infty }}\|
\mathcal{R}^{\perp }(v)\|
_{L_T^{q}L^{p_{c}}} \\
&\lesssim  T^{1-\frac{1}{2\alpha }}\| u\|
_{L_T^{\infty }L^{\infty }} \|
v\| _{L_T^{q}L^{p_{c}}}.
\end{align*}
Estimate (\ref{ESS3}) is then proved.
\end{proof}

Now we are ready to finish the proof of the uniqueness.
Let $p\geq p_{c}$ and $T>0$ be two reals number and let
$\theta _1$ and $\theta _2$ be two mild solutions
of the equation \eqref{QGalpha} with the same data $\theta _0$
such that $\theta _1,\theta _2\in C([0,T],L^{p}(\mathbb{R}^2))$.
We aim to show that $\theta _1=\theta _2$ on $[0,T]$. For this, we
will argue by contradiction. Then we suppose that $t_{\ast }<T$
where
\[
t_{\ast }\equiv \sup \{t\in [ 0,T]:\forall s\in [
0,t],\;\theta _1(s)=\theta _2(s)\}.
\]
To conclude, we need to prove that there exists
$\delta \in ]0,T-t_{\ast }]$
such that $\tilde{\theta}_1=\tilde{\theta}_2$ on $[0,\delta ]$, where
$\tilde{\theta}_1$ and $\tilde{\theta}_2$ are the functions defined on
$[0,T-t_{\ast }]$ by
\[
\tilde{\theta}_1(t)=\theta _1(t+t_{\ast}),\quad
\tilde{\theta}_2(t)=\theta _2(t+t_{\ast }).
\]
We deal separately with the sub-critical case and the critical
case.

\textbf{Case $p>p_{c}$.} Thanks to the continuity of
$\theta _1$ and $\theta _2$ on $[0,T]$, we have
$\theta_1(\tau _{\ast })=\theta _2(t_{\ast })$.
Hence, the functions $\tilde{\theta}_1$ and
$\tilde{\theta}_2$ are two mild solutions on
$[0,\delta _0\equiv T-t_{\ast }]$ of the equation \eqref{QGalpha}
with the same data $\theta _1(\tau _{\ast})$. Therefore,
the function $\tilde{\theta}\equiv \tilde{\theta}_1-\tilde{\theta}_2$
satisfies the equation
\begin{equation}
\tilde{\theta}=\mathcal{B}_{\alpha }[\tilde{\theta}_1,\tilde{\theta}]-
\mathcal{B}_{\alpha }[\tilde{\theta},\tilde{\theta}_2].
\label{equa}
\end{equation}
Thus, according to \eqref{ESS} we have for any
$\delta \in ]0,\delta _0]$,
\begin{align*}
\| \tilde{\theta}\| _{L_{\delta }^{\infty }L^{p}}
&\leq C\delta ^{\sigma }(\|
\tilde{\theta}_1\| _{L_{\delta }^{\infty
}L^{p}}+\| \tilde{\theta}_2\| _{L_{\delta
}^{\infty }L^{p}})\| \tilde{\theta}\|
_{L_{\delta }^{\infty }L^{p}} \\
&\leq C\delta ^{\sigma }(\| \theta _1\|
_{L_T^{\infty }L^{p}}+\| \theta _2\|
_{L_T^{\infty }L^{p}})\|
\tilde{\theta}\| _{L_{\delta }^{\infty }L^{p}},
\end{align*}
where $C>0$ is independent on $\delta $. Consequently,
for $\delta $ small enough, $\tilde{\theta}=0$ on
$[0,\delta ]$ which ends the proof in the sub-critical case.

\textbf{Case $p=p_{c}$.}
 Choose a fix real $q>1$ and let $\varepsilon >0$ to be chosen later.
By density of smooth functions in the space
$C([0,T],L^{p_c}(\mathbb{R}^2))$, one can decompose
$\tilde{\theta}_1$ and $\tilde{\theta}_2$ into
$\tilde{\theta}_1=u_1+v_1$ and $\tilde{\theta}_2=u_2+v_2$ with
\begin{gather}
\| u_1\| _{L_{\delta _0}^{\infty
}L^{p_{c}}}+\| u_2\| _{L_{\delta _0}^{\infty
}L^{p_{c}}} \leq \varepsilon ,
\label{es1} \\
\| v_1\| _{L_{\delta _0}^{\infty }L_{\mathcal{R}
}^{\infty }}+\| v_2\| _{L_{\delta _0}^{\infty }L_{
\mathcal{R}}^{\infty }} \equiv \mathcal{M}<\infty .  \label{es2}
\end{gather}
As in the previous case, the function
$\tilde{\theta}\equiv \tilde{\theta}_1-\tilde{\theta}_2$
satisfies
\begin{align*}
\tilde{\theta}
&=\mathcal{B}_{\alpha }[\tilde{\theta}_1,\tilde{\theta}]+
\mathcal{B}_{\alpha }[\tilde{\theta},\tilde{\theta}_2] \\
&= \mathcal{B}_{\alpha }[u_1,\tilde{\theta}]+\mathcal{B}_{\alpha }[\tilde{
\theta},u_2]+\mathcal{B}_{\alpha }[v_1,\tilde{\theta}]+\mathcal{B}
_{\alpha }[\tilde{\theta},v_2].
\end{align*}
Now by applying \eqref{ESS2}-\eqref{ESS3} and using
\eqref{es1}-\eqref{es2}
we obtain, for any $\delta \in ]0,\delta _0]$, the  estimate
\[
\| \tilde{\theta}\| _{L_{\delta }^{q}L^{p}}\leq
C(\varepsilon +\delta ^{1-\frac{1}{2\alpha
}}\mathcal{M})\| \tilde{\theta}\|
_{L_{\delta }^{q}L^{p}},
\]
where $C>0$ is a constant depending only on $\alpha$, $p$ and $q$.

Thus, by choosing $\varepsilon $ small enough, we conclude that
there exists $\delta \in ]0,\delta _0]$ such that
$\|\tilde{\theta}\|_{L_{\delta }^{q}L^{p}}=0$, which implies
that $\tilde{\theta}_1=\tilde{\theta}_2$ on $[0,\delta ]$.
The proof is then achieved.

\begin{remark} \label{rmk5.1} \rm
The idea of the proof of the uniqueness in the
critical case is inspired from Monniaux \cite{monniaux}.
\end{remark}

\subsection*{Acknowledgments}
We are  grateful to the anonymous referee for a number of helpful
comments that improved this article. 

\begin{thebibliography}{99}

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