\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 $$\begin{gathered} \partial _{t}\theta +(-\Delta )^{\alpha }\theta + \nabla.(\theta u)=0\quad \text{on }\mathbb{R}_{\ast }^{+}\times \mathbb{R}^2 \\ \theta (0,x)=\theta _0(x),\quad x\in \mathbb{R}^2 \end{gathered} \label{QGalpha}$$ 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: $$\label{mild} \theta (t)=e^{-t(-\Delta )^{\alpha }}\theta _0 + {\mathcal{B}}_{\alpha} [\theta ,\theta ] (t).$$ 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 $$\label{opB} {\mathcal{B}_{\alpha}}[\theta _1,\theta _2] (t)=-{\mathcal{L}}_{\alpha }(\theta _1\mathcal{R}^{\bot }(\theta _2))$$ where, for $v=(v_1,v_2)$, $$\label{linear} {\mathcal{L}}_{\alpha }(v)(t)=\int_0^{t}\nabla . e^{-(t-s)(-\Delta )^{\alpha }}v ds.$$ 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_{0-1$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 $\| f\| _{L_{\mu }^{\infty }([0,T],X)}\equiv \sup_{0 0 } 2^{jsq} \| \Delta_j f \|_p^q )^{\frac 1q} < \infty.$ 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 $$\label{X0} \sup_{t > 0 } \| e^{-t(-\Delta)^{\alpha}} f \|_X \leq C_X \| f \|_X.$$ 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} LetX$be a shift invariant functional space,$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$t \in [ 0 , T ]$, $$\label{ineg_lem5} f(t) \leq 2 c_1 e^{\nu t},$$ 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} $$\begin{gathered} \partial _{t}v-(-\Delta)^{\alpha} v+\nabla .(u_nv)=0 \\ v(0,.)=\theta _n(.). \end{gathered} \label{QGLn}$$ 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 $$v(t)=e^{-t(-\Delta)^{\alpha}}\theta _n-\int_0^{t}\nabla .e^{-(t-s)(-\Delta)^{\alpha}}(u_nv)ds \label{IQGLn}$$ 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 thatI_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 $\| w_n(t)\| _{\infty }\leq \| \theta _n-\theta (0)\| _{\infty }+C_{\alpha }T^{\nu }A_nB_n+C_{\alpha }M_{\theta }\int_0^{t}\frac{\| w_n(s)\| _{\infty }}{( t-s)^{1/2\alpha }}ds$ whereC_{\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, $$\theta \in \cap _{00 is a universal constant. Which contradicts the Maximal Principle (Lemma \ref{lemma5}). \section{Proof of Theorem \ref{theo2}}\label{sectreg} Along this section, we consider \theta_0 a given initial data belonging to the space \mathbf{\tilde{B}}^\alpha and we denote by \theta the solution to \eqref{QGalpha} given by Theorem \ref{theo1}. We will establish the persistency of the regularity of the initial data. That is, if moreover \theta_0 \in X for a suitable Banach spaces X then the solution \theta \in C([0,\infty), X). \subsection{Propagation of the L^p regularity} In this subsection we will prove the propagation of the initial L^p regularity. Precisely, we prove the following proposition. \begin{proposition}\label{propLp} Let X = L^p with p \in [ 1, \infty]. If \theta_0 \in X then \theta belongs to \bigcap_{T >0}L^{\infty}([ 0, T], X) . Moreover, if \theta_0 \in \overline{S(\mathbb{R}^2)}^X then \theta \in C([ 0, \infty ), \overline{S(\mathbb{R}^2)}^X ) \end{proposition} \begin{proof} Assume \theta_0 \in X and let T >0. We consider the Banach spaces \mathbf{Z}_1 = \mathbb{E}_T^{\nu} and \mathbf{Z}_2 = L^{\infty}([ 0, T], X) endowed respectively with the norms$$ \| 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, $$\label{linpsi} v = \Psi _{\theta }(v) \equiv e^{t(-\Delta)^{\alpha}}\theta _0+ \mathcal{B}_{\alpha }[\theta ,v].$$ 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 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), $$\| fg\| _{X}\leq C(\| f\| _{\infty }\| g\| _{X}+\| g\| _{\infty }\| f\| _{X}).$$ 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_{00. 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 }, $$\label{sequence} \| x_{n+1}-x_n\| \leq \sigma _n(\| x_n\| +\| x_{n-1}\| )+\lambda \|x_n-x_{n-1}\|,$$ 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}, $$\label{est21} \| x_{n+1}-x_n\| \leq 2\sum_{k=0}^{n-1}\sigma _{n-k}M_{n-k}\lambda ^{k} \leq \varpi _nM_n,$$ 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], $$\label{est22} \| e^{-t(-\Delta)^{\alpha}}\theta (\tau )\| _{\mathbb{E}_{\delta}^{\nu }}\leq \mu_0,$$ where \mu_0 is the real given by Lemma \ref{localpersistence}. Now, we consider a partition 0=t_0<\dots 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 0 such that $$\label{eps} -1 < s \pm \epsilon < 0,$$ 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{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} 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 $t\in ]0,T]$, $$\| N(\theta )(t)\| _{B_{p}^{s,1}}+\| \mathcal{R}^{\perp }N(\theta )(t)\| _{B_{p}^{s,1}}\leq C \|\theta\|_{\mathbb{E}_{t}^{\nu}}\|\theta\|_{X_{\sigma,p}^T}. \label{mmm}$$ 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 $-10}\tilde{X}_{\sigma ,p}^{T}$. \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 $$\label{nonlN} N(\theta) \in L^{\infty}([0,T], \dot{B}_{p}^{0,1} ).$$ \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 $-10$. There exists a constant $C$ independent of $T$ such that: \begin{itemize} \item For any $u,v$ in $L_T^{\infty }L^{p}$, $$\| \mathcal{B}_{\alpha }[u,v]\| _{L_T^{\infty }L^{p}}\leq C T^{\sigma }\| u\| _{L_T^{\infty }L^{p}}\| v\| _{L_T^{\infty }L^{p}}\text{,} \label{ESS}$$ where $\sigma =\frac{1}{\alpha }(\frac{1}{p_{c}}-\frac{1}{p})$; \item for any $u,v$ in $L_T^{\infty }L^{p_{c}}$, $$\| \mathcal{B}_{\alpha }[u,v]\| _{L_T^{q}L^{p_{c}}}+\| \mathcal{B}_{\alpha }[v,u]\| _{L_T^{q}L^{p_{c}}}\leq C\| u\| _{L_T^{\infty }L^{p_{c}}}\| v\| _{L_T^{q}L^{p_{c}}.} \,; \label{ESS2}$$ \item for any $u\in L_T^{\infty }L_{\mathcal{R}}^{\infty }$ and $v\in L_T^{q}L^{p_{c}}$, $$\| \mathcal{B}_{\alpha }[u,v]\| _{L_T^{q}L^{p_{c}}}+\| \mathcal{B}_{\alpha }[v,u]\| _{L_T^{q}L^{p_{c}}}\leq C\; T^{1-\frac{1}{2\alpha }}\| u\| _{L_T^{\infty }L_{\mathcal{R}}^{\infty }}\| v\| _{L_T^{q}L^{p_{c}}.}\,. \label{ESS3}$$ \end{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 $10$ 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 $t_{\ast }p_{c}$.} 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 $$\tilde{\theta}=\mathcal{B}_{\alpha }[\tilde{\theta}_1,\tilde{\theta}]- \mathcal{B}_{\alpha }[\tilde{\theta},\tilde{\theta}_2]. \label{equa}$$ 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} \bibitem{bergh} J.~Bergh and J.~L\"ofstrom. \newblock {\em Interpolation Spaces. An introduction}. \newblock Springer Verlag, Berlin, 1976. \bibitem{canonne}M.~Cannone. \newblock {\em Ondelettes, Paraproduits et Navier-Stokes}. \newblock Diderot Editeur, Arts et Sciences, Paris, (1995). \bibitem{cannone2000} M.~Cannone and F. 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