\documentclass[reqno]{amsart} \usepackage{amssymb} \usepackage{hyperref} \AtBeginDocument{{\noindent\small {\em Electronic Journal of Differential Equations}, Vol. 2005(2005), No. 63, pp. 1--22.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu (login: ftp)} \thanks{\copyright 2005 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2005/63\hfil Quasi-geostrophic equations] {Quasi-geostrophic equations with initial data in Banach spaces of local measures} \author[S. Gala\hfil EJDE-2005/63\hfilneg] {Sadek Gala} \date{} \thanks{Submitted April 24, 2005. Published June 15, 2005.} \subjclass[2000]{35Q35, 35A07} \keywords{Quasi-geostrophic equation; local spaces; mild solutions; \hfill\break\indent self-similar solutions} \begin{abstract} This paper studies the well posedness of the initial value problem for the quasi-geostrophic type equations \begin{gather*} \partial_{t}\theta+u\nabla\theta+( -\Delta) ^{\gamma}\theta =0 \quad \text{on }\mathbb{R}^{d}\times] 0,+\infty[\\ \theta( x,0) =\theta_{0}(x), \quad x\in\mathbb{R}^{d} \end{gather*} where $0<\gamma\leq1$ is a fixed parameter and the velocity field $u=(u_{1},u_{2},\dots,u_{d})$ is divergence free; i.e., $\nabla u=0)$. The initial data $\theta_{0}$ is taken in Banach spaces of local measures (see text for the definition), such as Multipliers, Lorentz and Morrey-Campanato spaces. We will focus on the subcritical case $1/2<\gamma\leq1$ and we analyse the well-posedness of the system in three basic spaces: $L^{d/r,\infty}$, $\dot {X}_{r}$ and $\dot {M}^{p,d/r}$, when the solution is global for sufficiently small initial data. Furtheremore, we prove that the solution is actually smooth. Mild solutions are obtained in several spaces with the right homogeneity to allow the existence of self-similar solutions. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{definition}[theorem]{Definition} \newtheorem{remark}[theorem]{Remark} \section{Introduction} We analyse the well-posedness of initial value problems for the quasi-geostrophic equations in the subcritical case. Mild solutions are obtained in several spaces with the right homogeneity to allow the existence of self-similar solutions. Singularities, global existence and long time behavior for models in fluid mechanics have become an important topics in the mathematical community in the last decades. Understanding these features in incompressible Navier-Stockes (NS) and Euler equations in 3D in yet unsolved. Lower dimentional models have been deduced, not only for the simplification that might bring up in the mathematical approaches, but also because of the great resemblance that some of these models have with respect to the original incompressible NS equations and the mathematical insight their study may produce. The dissipative 2DQG equations are derived from general quasi-geostrophic equations (see \cite{P}) in the special case of constant potential vorticity and constant buoyancy frequency (see \cite{CMT}). Quasi-geostrophic equations are not only important as a simplification of 3DNS but \ also since they appear as a natural reduction for vertically stratified flows (see \cite{P}). Well posedness in several spaces and long time behavior of the solutions of the dissipative QG equations in different cases have been studied in (\cite{CL}, \cite{CMT}, \cite{CCW}, \cite{J}, \cite{W1}, \cite{W2}). The objective of this paper is to study the well posedness of the initial value problem for 2-dimensional of the quasi-geostrophic type equations \begin{gather} \partial_{t}\theta+u\nabla\theta+( -\Delta) ^{\gamma}\theta =0 \quad \text{on }\mathbb{R}^{d}\times] 0,+\infty[ , \label{eq1.1.1} \\ \theta( x,0) =\theta_{0}(x), \quad x\in\mathbb{R}^{d} \label{eq1.1.2} \end{gather} where $0<\gamma\leq1$ is a fixed parameter. The velocity field $u=(u_{1},u_{2},\dots ,u_{d})$ is divergence free; i.e., $\nabla u=0)$ and determined from the potential temperature $\theta$ by linear combinations of Riesz transforms; i.e.,. $u_{k}=\sum _{j=1}^{d}a_{jk}\mathcal{R}_{j}\theta, \quad 1\leq k\leq d$ where $\mathcal{R}_{j}=\partial_{j}( -\Delta) ^{-\frac{1}{2}}$. Let us remark that suitable choices of $a_{jk}$ assures that the velocity field is divergence free, that we will assume throughout this paper. The Riesz potential operator $( -\Delta) ^{\gamma}$ is defined as usual through the Fourier transform as $\widehat{( -\Delta) ^{\gamma}f( \xi) }=| \xi| ^{2\gamma}\widehat{f}( \xi)$ It is well-know that QG equation is very similar to the three dimensional Navier-Stokes equations (see \cite{CMT}). Besides its similarity to the three dimensional fluid equations, (see \cite{P}). The case $\gamma>\frac{1}{ 2}$ is called sub-critical, and the case $\gamma=\frac{1}{2}$ is critical, while the case $0\leq\gamma<\frac{1}{2}$ is super-critical, respectively. Our aim is to show existence and uniqueness and regularity of solutions for the quasi-geostrophic equations in the sub-critical case (\ref{eq1.1.1})-(\ref{eq1.1.2}) when the initial data is taken in Banach spaces of local measures, such as Multipliers, Lorentz and Morrey-Campanato spaces. \section{Shift-invariant spaces of local measures} We consider in this paper a special class of shift-invariant spaces of distributions was introduced by Lemari\'{e}-Rieusset in his work \cite{Lem}. The spaces which are invariant under pointwise multiplication with bounded continuous functions. \begin{definition} \begin{itemize} \item[(A)] A shift-invariant Banach spaces of test functions is a Banach space $E$ such that we have the continuous embeddings $\mathcal{D}(\mathbb{R}^{d}) \subset E\subset\mathcal{D}'( \mathbb{R}^{d})$ and so that \begin{itemize} \item[(a)] for all $x_{0}\in\mathbb{R}^{d}$ and for all $f\in E$, $f( x-x_{0}) \in E$ and $\| f\| _{E}=\| f( x-x_{0}) \| _{E}$ \item[(b)] for all $\lambda>0$, there exists $C_{\lambda}>0$ so that for all $f\in E$, $f( \lambda x) \in E$ and $\| f( \lambda x) \| _{E}\leq C_{\lambda }\| f\| _{E}$ \item[(c)] $\mathcal{D}( \mathbb{R}^{d})$ is dense in $E$. \end{itemize} \item[(B)] A shift-invariant Banach spaces of distributions is a Banach space $E$ which is the topological dual of a shift-invariant Banach of test functions space $E^{(0)}$ of smooth elements of $E$ is defined as the closure of $\mathcal{D}( \mathbb{R}^{d})$ in $E$. \item[(C)] A shift-invariant Banach space of local measures is a shift-invariant Banach space of distributions $E$ so that for all$\mathcal{\ }f\in E$ and all $g\in\mathcal{S}( \mathbb{R}^{d})$, we have $fg\in E$ and $\| fg\| _{E}\leq C_{E}\| f\| _{E}\| g\| _{L^{\infty}}$ where $C_{E}$ is a positive constant which depends neither on $f$ nor on $g$. \end{itemize} \end{definition} \begin{remark} \rm An easy consequence of hypothesis (a) is that a shift-invariant Banach space of test functions $E$ satisfies $\mathcal{S}( \mathbb{R}^{d}) \subset E$ and a consequence of hypothesis (b) is that $E\hookrightarrow \mathcal{S}'( \mathbb{R}^{d})$. Similarly, we have for a shift-invariant Banach space of distributions $E$ that $\mathcal{S}( \mathbb{R}^{d}) \subset E^{(0)}\subset E\hookrightarrow\mathcal{S} '( \mathbb{R}^{d})$. In particular, $E^{(0)}$ is a shift-invariant Banach space of test functions. \end{remark} A shift-invariant Banach spaces of distributions are adapted to convolution with integrable kernels. \begin{lemma}[Convolution in shift-invariant spaces of distributions] \label{lem2.1} If $E$ is a shift-invariant Banach space of test functions or of distributions and $\varphi\in\mathcal{S}( \mathbb{R}^{d})$, then for all $f\in E$, we have $f\ast\varphi\in E$ and $$\| f\ast\varphi\| _{E}\leq\| f\| _{E}\| \varphi\| _{L^{1}} \label{eq2.4}$$ Moreover, the convolution may be extended into a bounded bilinear operator from $E\times L^{1}$ to $E$ and we have for all $f\in E$ and for all $g\in L^{1}$, the inequality $\Vert f\ast g\Vert _{E}\leq\Vert f\Vert _{E}\Vert g\Vert _{L^{1}}$ \end{lemma} The proof of the above lemma can be found in \cite{Lem}. Now, we introduce the functional spaces relevant to our study of solutions of the Cauchy problem for system (\ref{eq1.1.1})-(\ref{eq1.1.2}), we list some facts about convolution and we discuss the notion of solution in these spaces. \subsection{Pointwise multipliers $\protect\dot {X}_{r}$} In this section, we give a description of the multiplier space $\dot {X}^{r}$ introduced recently by Lemari\'{e}-Rieusset in his work \cite{Lem}. The space $\dot {X}^{r}$of pointwise multipliers which map $L^{2}$ into $\dot {H}^{-r}$ is defined as follows: \begin{definition} % def. 2.2 For $0\leq r0. \end{gather*} The following imbeddings hold \begin{gather*} L^{\frac{d}{t}} \subset X^{r},\quad 0\leq r<\frac{d}{2},\; 0\leq t\leq r.\\ L^{\frac{d}{r}} \subset\dot {X}^{r},\quad 0\leq r<\frac{d}{2}\,. \end{gather*} We now turn to another way of introducing capacity. \subsubsection*{Capacitary measures and capacitary potentials} In this section, we give the notion of capacity which will develop in term of Riesz and Bessel potentials. \begin{definition} Let$d\geq2$and$x\in\mathbb{R}^{d}. The Riesz kernel is defined by k_{\alpha}(x)=C(d,\alpha)| x| ^{\alpha-d},\quad 0<\alphaR}} \frac{dy}{| y| ^{d-\alpha}| x-y| ^{d-\beta}} &\approx {\int_{| y| >R}} \frac{dy}{| y| ^{2d-\alpha-\beta}} \\ & \approx {\int_{R}^{\infty}} t^{d-1-2d+\alpha+\beta}dt= {\int_{R}^{\infty}} t^{\alpha+\beta-d-1}dt, \end{align*} which is convergent if and only if \alpha+\beta0 is defined as that function whose Fourier transform is \[ \widehat{G_{r}}(x)=( 2\pi) ^{-d/2}( 1+|x| ^{2}) ^{-r/2} It is known thatG_{r}$is a positive, integrable function which is analytic except at$x=0$. Similar to the Riesz kernel, we have $G_{r}\ast G_{s}=G_{r+s},\quad r,s\geq 0 \,.$ \begin{remark} \label{rmk2} \rm The Riesz potential leads to many important applications, but for the purpose of investigating Sobolev functions, the Bessel potential is more suitable. For an analysis of the Bessel kernel, we refer the reader to \cite{St1}. \end{remark} The Riesz capacity$\mathop{\rm cap}( e;\dot {H}^{r}) $of a compact set$e\subset\mathbb{R}^{d}$is defined by (see \cite{AH}) $\mathop{\rm cap}( e;\dot {H}^{r}) =\inf\{ \| u\| _{\dot {H}^{r}}^{2}\text{ : }u\in C_{0}^{\infty}( \mathbb{R} ^{d}) \text{, }u\geq1\text{ sur }e\}$ The Bessel capacity$\mathop{\rm cap}( e;H^{r}) $of a compact set$e\subset\mathbb{R}^{d}$is defined in a similar way, with the kernel$k_{r}$replaced by$G_{r}$. Since$G_{r}(x)\leq k_{r}(x)$($x\in\mathbb{R}^{d}$), it follows immediately from definitions that for$01$, or $$\frac{1}{| B| } {\int_{B}} w(x)dx\leq K \mathop{\rm ess\, inf}_{x\in B}w(x)) , \label{eq5.2}$$ if$p=1$. The infinimum over all such constants$C$is called the$A_{p}$constant of$w$. We denote by$A_{p}$,$1\leq p<\infty$, the set of all$A_{p}$weights. Below we list some simple, but useful properties of$A_{p}$weights. \begin{proposition} \label{pro1.3} \begin{enumerate} \item If$w\in A_{p}$,$1\leq p<\infty$, then since$w(x)^{-\frac{1}{p-1}}$is locally integrable, when$p>1$, and$\frac{1}{w}$is locally bounded, when$p=1$, we have$L^{p}(w(x)dx)\subset L_{\rm loc}^{1}( \mathbb{R}^{d})$. \item Note that if$w$is a weight, then, by writing$1=w^{\frac{1}{p}}w^{-\frac{1}{p}}$, H\"{o}lder's inequality implies that, for every ball$B$$1\leq\Big( \frac{1}{| B| } {\int_{B}}w(x)dx\Big) \Big( \frac{1}{| B| } {\int_{B}}w(x)^{\frac{-1}{p-1}}dx\Big) ^{p-1}$ when$p>1$and similarly for the expression that gives the$A_{1}$condition.It follows that if$w\in A_{p}$, then the constant of$w$is$\geq1$. \item If$w\in A_{p}$, where$10$and the constant is independent of$e$. \end{itemize} \end{lemma} The measure$\mu _{e}$associated with$e$is called the capacitary (equilibrium) measure of$e$. We will also need the asymptotic (see \cite{AH}) $$\label{eq1.5} \begin{gathered} G_{r}(x) \simeq | x| ^{r-d}, \quad \text{if } d\geq 3, \; | x| \to 0; \\ G_{r}(x) \simeq | x| ^{\frac{r-d}{2} }e^{-| x| },\quad \text{if } d\geq 2, \; | x| \to +\infty \,. \end{gathered}$$ Sometimes, it will be more convenient to use a modified kernel $\overset{\sim }{G_{r}}(x)=\max ( G_{r}(x),1)$ which does not have the exponential decay at$\infty $. Obviously, both$ G_{r}$and$\overset{\sim }{G_{r}}$are positive non-increasing radial kernels. Moreover,$\overset{\sim }{G_{r}}$has the doubling property $\overset{\sim }{G_{r}}(2s)\leq \overset{\sim }{G_{r}}(s)\leq c(d)\overset{ \sim }{G_{r}}(2s)$ The corresponding modified potential is $\overset{\sim }{P}(x)=\overset{\sim }{G_{r}}\ast \mu (x)$ The rest of the proof of theorem \ref{thMV} is based on the following proposition. \begin{proposition} \label{poids} Let$d\geq 2$and let$0<\delta <\frac{d}{d-2r}$. Then$\overset{\sim }{P}$lies in the Muckenhoupt class$A_{1}$on$\mathbb{R}^{d}$, i.e., $M\big( \overset{\sim }{P}^{\delta }(x)\big) \leq C(\delta ,d) \overset{\sim }{P}^{\delta }(x),\ dx\text{ p.p}$ where$M$denotes the Hardy-Littlewood maximal operator on$\mathbb{R}^{d}$, and the corresponding$A_{1}-$bound$C(\delta ,d)$depends only on$d$and$\delta $. \end{proposition} \begin{proof} Let$k:\mathbb{R}_{+}\to \mathbb{R}_{+}$be a non-increasing function which satisfies the doubling condition $k(2s)\leq ck(s),\quad s>0\,.$ It is easy to see that the radial weight$k(| x| )\in A_{1}$if and only if $$\int_{0}^{R}k^{\delta }(t)t^{d-1}dt\leq c R^{d}k(R),\quad R>0 \label{eq5.6}$$ Moreover, the$A_{1}$-bound of$k$is bounded by a constant which depends only on$C$in the preceding estimate and the doubling constant$c$(see \cite{St2}). It follows that $\overset{\sim }{G}_{r}(s)\simeq | s| ^{r-d}\quad \text{if } d\geq 3 \text{ for } 00 with the constant C is independent of e. \end{itemize} Now, it follows from a proposition \ref{poids} that \varphi ^{\delta }\in A_{1}. Hence, by (\ref{eq 5.2}), \[ \int_{\mathbb{R}^{d}}| g(x)| ^{2}\varphi ^{\delta }dx\leq K\int_{\mathbb{R}^{d}}| h(x)| ^{2}\varphi ^{\delta }dx$ Applying this inequality with$(i)$and$(ii)$, we get $\int_{e}| g(x)| ^{2}dx\leq \int_{\mathbb{R }^{d}}| g(x)| ^{2}\varphi ^{\delta }dx\leq C\int_{ \mathbb{R}^{d}}| h(x)| ^{2}\varphi ^{\delta }dx=C\int_{0}^{B}\int_{\varphi \geq t}| h(x)| ^{2}dxt^{\delta -1}dt\,.$ By$(\ref{eq 5.1})$and$($iii$)$, $\int_{\varphi \geq t}| h(x)| ^{2}dx\leq Ccap\left\{ \varphi \geq t\right\} \leq \frac{C}{t}\mathop{\rm cap}( e)\,.$ Hence, $\int_{e}| g(x)| ^{2}dx\leq C\int_{0}^{B}t^{-1}\mathop{\rm cap}( e) t^{\delta -1}dt=C\mathop{\rm cap}( e) \int_{0}^{B}t^{\delta -2}dt\,.$ Clearly, for all$0\leq r1$so that$0<\delta <\frac{d}{d-2r}$. Then $\int_{0}^{B}t^{\delta -2}dt=\frac{B^{\delta -1}}{\delta -1}<\infty$ which concludes$\int_{e}| g(x)| ^{2}dx\leq C\mathop{\rm cap}( e)$. \end{proof} We will need the boundedness of the Riesz transforms$\mathcal{R} _{j}f=f\ast\frac{x_{j}}{| x| ^{d+1}}$($j=1,2,\dots$) in the spaces of functions defined by the capacitary condition (\ref{eq 5.1}). \begin{corollary}\label{co.1} Let$0\leq r0}{\sup}R^{d/q-d/p}( \int _{B(x,R)}| f(y)| ^{p}dy) ^{1/p}\label{eq1.15} It is easy to check the following properties \begin{gather*} \| f(\lambda x)\| _{M_{p,q}} =\frac{1}{\lambda^{\frac{d}{q}} }\| f\| _{M_{p,q}}, \quad 0<\lambda\leq1.\\ \| f(\lambda x)\| _{\dot{M}_{p,q}} =\frac {1}{\lambda^{\frac{d}{q}}}\| f\| _{\dot{M}_{p,q}}, \; \lambda>0 \end{gather*} We shall assume the following classical results \cite{K}. \begin{itemize} \item [(a)] For $1\leq p\leq p'$, $p\leq q\leq+\infty$ and for all functions $f$ in $\dot{M}_{p,q}\cap L^{\infty}$, $\| f\| _{\dot {M}_{p',q\frac{p'}{p}}} \leq\| f\| _{L^{\infty}}^{1-\frac{p}{p'}}\|f\| _{\dot{M}_{p,q}}^{\frac{p}{p'}}.$ \item[(b)] For $p,q,p',q'$ so that $\frac{1}{p}+\frac {1}{p'}\leq1$, $\frac{1}{q}+\frac{1}{q'}\leq1$, $f\in \dot {M}_{p,q}$, $g\in\dot {M}_{p',q'}$. Then $fg\in\dot{M}_{p",q"}$ with $\frac{1}{p}+\frac {1}{p'}=\frac{1}{p"}$, $\frac{1}{q}+\frac{1}{q'}=\frac{1}{q"}$. \item[(c)] For $1\leq p\leq d$ and all $\lambda>0$, we have $\lambda f(\lambda x)\| _{\dot{M}_{p,d}}=\| f\| _{\dot {M}_{p,d}}$. \item[(d)] If $p's\right\} | \] where$s>0$. Given a real function$\lambda_{f}(s)$, we define its rearrangement$f^{\ast}(t)$as $f^{\ast}(t)=\inf\left\{ s>0:\lambda_{f}(s)\leq t\right\} \text{, }t>0$ It is easy to check that$f^{\ast}$and$\lambda_{f}(s)$are non-negative and non-increasing functions. Moreover, if$\lambda_{f}$is strictly decreasing and continuous, then$f^{\ast}$is the inverse function of$\lambda_{f}$and both$f^{\ast}$and$fhave the same distribution function. From this fact, we deduce that \begin{aligned} \int_{\mathbb{R}^{d}}| f(x)| ^{p}dx & =\int_{\mathbb{R}^{d}}\int_{0}^{| f(x)| }pt^{p-1}dtdx=\int_{0}^{\infty}\int_{\mathbb{R} ^{d}}\kappa_{\lambda_{f}(t)}pt^{p-1}dtdx \\ & =\int_{0}^{\infty}pt^{p-1}\lambda_{f}(t)dt=\int_{0}^{\infty }pt^{p-1}\lambda_{f^{\ast}}(t)dt \\ & =\int_{0}^{\infty}( t^{\frac{1}{p}}f^{\ast}(t)) ^{p}\frac{dt}{t} \end{aligned}\label{eq3.51} We may now introduce the Lorentz spaces. %\begin{definition}[Lorentz spaces] The Lorentz spacesL^{p,q}( \mathbb{R}^{d}) $is defined as the set of all functions$f$such that$\| f\|_{L^{p,q}}^{\ast}<+\infty$, with $\| f\| _{L^{p,q}}^{\ast}= \begin{cases} \Big( \frac{q}{p}\int_{0}^{\infty}\big[ t^{\frac{1}{p}}f^{\ast }(t) \big] ^{q}\frac{dt}{t}\Big) ^{\frac{1}{q}}&\text{if }00}\big[ t^{\frac{1}{p}}f^{\ast}(t)\big] & \text{if }00$ which can be computed$\left[ \text{H}\right] $as $f^{\ast\ast}(t)=\sup_{m(E)\geq t} \big[ \frac{1}{m(E)} \int_{E}| f(x)| dx\text{ }\big]\,.$ Hence, we define the norm $\| f\| _{L^{p,q}}= \begin{cases} \Big( \frac{q}{p}\int_{0}^{\infty}\big[ t^{\frac{1}{p}}f^{\ast\ast }(t)\big] ^{q}\frac{dt}{t}) ^{1/q}, & \text{if }10}\big[ t^{\frac{1}{p}}f^{\ast\ast}(t)\big], &\text{if } 10} t|\{ x\in\mathbb{R}^{d}:| f(x)| >t\} | ^{1/p}$ Lorentz spaces have the same scaling relation as$L^{p}$spaces, i.e., for all$\lambda>0$, we have $\| f(\lambda x)\| _{L^{p,q}}=\lambda^{-d/p}\| f\| _{L^{p,q}}$ where$1\leq p<+\infty$,$1\leq q\leq+\infty$. We first need an interpolation result in Lorentz spaces \cite{Lem}. \begin{lemma}[Interpolation of linear operator] \label{proA} Let$p_{0}$,$p_{1}\in[ 1,+\infty]$with$p_{0}\neq p_{1}$. Let$\theta\in] 0,1[ $and let$\frac{1}{p}=\frac{1-\theta}{p_{0}}+\frac{\theta}{p_{1}}$. If$T$is a linear operator defined from$L^{p_{0}}+L^{p_{1}}$to Banach space$E$such that$T\in\mathcal{L}(L^{p_{0}},E) $(with the operator norm$M_{0}$) and$T\in\mathcal{L}( L^{p_{1}},E) $( with the operator norm$M_{1}$), then$T$is bounded from$L^{p,\infty}$to$E$and the operator norm$M$is controlled by $M\leq C( p_{0},p_{1},\text{ }\theta) M_{0}^{1-\theta}M_{1}^{\theta}.$ \end{lemma} \begin{proof} We just write$f=g+h$with $\big( \frac{M_{0}}{M_{1}}\big) ^{\theta}\| g\| _{L^{p_{0}}}+\big( \frac{M_{0}}{M_{1}}\big) ^{\theta-1}\| h\| _{L^{p_{1}}}\leq C\| f\| _{L^{p,\infty}}$ Then, we have$Tf=Tg+Thwhere \begin{align*} \| Tf\| _{E} & \leq\| Tg\| _{E}+\| Th\| _{E} \\ & \leq C\| f\| _{L^{p,\infty}}( M_{0}( \frac{M_{0} }{M_{1}}) ^{-\theta}+M_{1}\big( \frac{M_{0}}{M_{1}}\big) ^{1-\theta}) \\ & =2CM_{0}^{1-\theta}M_{1}^{\theta}\| f\| _{L^{p,\infty}} \end{align*} which proves the lemma. \end{proof} Now, we deal with the continuity of operators in Lorentz spaces and in particular continuity of the Riesz transform inL^{p,\infty}$,$11$. \end{corollary} \begin{proof} We know that the Riesz transform is continuous in$L^{p}$,$p>1$. Now, take$10$, we define$\varphi_{\delta}(x)=\frac {1}{\delta^{d}}\varphi( \frac{x}{ \delta}) $. If$10, we have \begin{align*} ( \varphi_{\delta}\ast f-f) (x) & =\int _{\mathbb{R}^{d}}\varphi_{\delta}(y)[ f(x-y)-f(x)] dy \\ & =\int _{\mathbb{R}^{d}}\varphi(t)[ f(x-t\delta)-f(x)] dt\,. \end{align*} Next, taking the norm\| .\| _{L^{p,q}}, we obtain \begin{align*} \| \varphi_{\delta}\ast f-f\| _{L^{p,q}} & \leq\int _{\mathbb{R}^{d}}\| \varphi(t)\left[ f(x-t\delta)-f(x)\right] \| _{L^{p,q}}dt \\ & \leq\| f(x-t\delta)-f(x)\| _{L^{p,q}}\,. \end{align*} Note that\| f(x-t\delta)-f(x)\| _{L^{p,q}}\leq2\| f\| _{L^{p,q}}$. Since$1\leq q<\infty$, then $\lim_{\delta\to 0} \| f(x-t\delta)-f(x)\| _{L^{p,q}}=0$ and we can use the Lebesgue dominated convergence theorem to conclude the proof. \end{proof} \section{The solution operator for the linear equation} Consider the solution operator for the linear equation $\partial_{t}\theta+( -\Delta) ^{\gamma}\theta=0,\quad \text{on } \mathbb{R}^{d}\times] 0,+\infty[$ where$\gamma\in[ 0,1] $. For a given initial data$\theta_{0}$, the solution of this equation is given by $\theta(t)=K_{\gamma}(t)\theta_{0}=e^{-t( -\Delta)^{\gamma}}\theta_{0},$ where$K_{\gamma}(t)=e^{-t( -\Delta) ^{\gamma}}$is a convolution operator with its kernel$k_{\gamma}$being defined through the Fourier transform $\widehat{k}_{\gamma}( x,t) =e^{-t| \xi| ^{2\gamma}}$ In particular,$k_{\gamma}$is the heat kernel for$\gamma=1$and the Poisson kernel for$\gamma=1/2$. The kernel$k_{\gamma}$possesses similar properties as the heat kernel does. For example, for$\gamma\in[ 0,1]$and$t>0$,$k_{\gamma}(x,t)is a nonnegative and non-increasing radial function and satisfies the following homogeneity properties \label{eq1.2.1} \begin{aligned} k_{\gamma}( x,t) & =t^{-\frac{d}{2\gamma}}k_{\gamma}( t^{- \frac{1}{2\gamma}}x,1) \\ ( \nabla_{x}k_{\gamma}) ( x,t) & =t^{-\frac {( d+1) }{2\gamma}}( \nabla_{x}k_{\gamma}) ( t^{-\frac{1}{ 2\gamma}}x,1) \end{aligned} This remarks lead to the following result. \begin{lemma} For allt>0$and$\gamma\in[ 0,1] $,$K_{\gamma}(t)=e^{-t( -\Delta) ^{\gamma}}$is a convolution operator with its kernel$k_{\gamma}\in L^{1}$. \end{lemma} \begin{proof} We have$\widehat{k}_{\gamma}( x,t) =e^{-t|\xi| ^{2\gamma}}\in L^{1}$; thus$k_{\gamma}$is a continuous bounded function. Hence $\| k_{\gamma}( .,t) \| _{L^{1}}=\widehat {k}_{\gamma}( 0,\xi) =1$ \end{proof} Furthermore, the operators$K_{\gamma}$and$\nabla K_{\gamma}$are bounded on$L^{\infty}$. To prove this fact, we need the following lemma. \begin{lemma} \label{prop2} For any$t>0$, the operators$K_{\gamma}$and$\nabla K_{\gamma }$are bounded operators from$L^{\infty}\ $to$L^{\infty}$. Furthermore, we have for any$u\in L^{\infty}$, \begin{gather} \| K_{\gamma}(t)u\| _{L^{\infty}}\leq\| u\| _{L^{\infty}}, \label{eq1.2.2}\\ \| \nabla K_{\gamma}(t)u\| _{L^{\infty}}\leq Ct^{-\frac {1}{ 2\gamma}}\| u\| _{L^{\infty}}, \label{eq1.2.3} \end{gather} where$C$is a constant depending on$\gamma$. \end{lemma} \begin{proof} To prove (\ref{eq1.2.2}), we have $\| K_{\gamma}(t)u\| _{L^{\infty}}\leq\| k_{\gamma }( .,t) \| _{L^{1}}\| u\| _{L^{\infty}}\,.$ Estimate (\ref{eq1.2.3}) can be proved similarly by using the identity $\partial_{x}k_{\gamma}(x,t)=t^{-\frac{1}{2\gamma}}\widetilde{k}_{\gamma}(x,t)$ where$\widetilde{k}_{\gamma}$is another radial function enjoying the same properties as$k_{\gamma}$does. \end{proof} Now, we introduce a basic tool for this papier: The Besov spaces$\overset{.}{B}_{\infty}^{-r,\infty}$for$r>0$. \subsection{Littlewood-Paley decomposition of tempered distribution} \textbf{Definition of Dyadics blocks} Let$\varphi\in\mathcal{D}(\mathbb{R}^{d})$be a non negative function so that$| \xi| \leq\frac{1}{2}\Rightarrow \varphi(\xi)=1 $and$| \xi| \geq1\Rightarrow\varphi (\xi)=0$. Let$ \psi(\xi)=\varphi(\frac{\xi}{2})-\varphi(\xi)$. Let$\Delta _{j}$and$S_{j}$be defined as the Fourier multipliers $\mathcal{F}( S_{j}f) =\varphi(\frac{\xi}{2^{j}})\mathcal{F}( f) \quad \text{and}\quad \mathcal{F}( \Delta_{j}f) =\psi(\frac{\xi}{ 2^{j}})\mathcal{F}( f)$ The distribution$\Delta_{j}f$is called the$j$-th dyadic block of the Littlewood-Paley decomposition of$f$. \smallskip For all$k\in\mathbb{Z}$and for all$f\in\mathcal{S}'(\mathbb{R}^{d})$, we have $f=S_{k}f+\sum_{j\geq k}\Delta_{j}f$ in$\mathcal{S}$'$(\mathbb{R}^{d})$. This equality is called the Littlewood-Paley decomposition of the distribution$f$. \ If, moreover,$ \underset{k\to -\infty}{\lim}S_{k}f=0$in$\mathcal{S}'$, then the equality $f=\sum_{j\in\mathbb{Z}} \Delta_{j}f.$ is called the homogeneous Littlewood-Paley decomposition of the distribution$f$. \noindent\textbf{Distributions vanishing at infinity.}\; We define the space of tempered distributions vanishing at infinity as the space $\mathcal{S}_{0}^{'}(\mathbb{R}^{d})=\{ f\in \mathcal{S}'(\mathbb{R}^{d}) \text{ \ so that \ }f=\sum_{j\in\mathbb{Z}}\Delta_{j}f\text{ \ in }\mathcal{S}'\}$ We may now define homogeneous Besov spaces in the following way: %\begin{definition} For$s\in] -\infty,+\infty[ $,$p$,$q\in[ 1,+\infty ]$, the homogeneous Besov spaces$\dot {B_{p}}^{s,q}$is defined as $\dot {B_{p}}^{s,q}=\{ f\in\mathcal{S}'/\mathbb{C} [ X] :2^{js}\| \Delta_{j}f\| _{L^{p}}\in l^{q}(\mathbb{Z})\} .$ equipped with the norm $\| f\| _{\dot {B_{p}}^{s,q}}=\Big( \sum_{j\in\mathbb{Z}} ( 2^{js}\| \Delta_{j}f\| _{L^{p}}) ^{q}\Big) ^{1/q}$ and$\mathbb{C}[ X] $denotes the set of all multinomials. \smallskip Similarly, the inhomogeneous Besov spaces$B_{p}^{s,q}$are defined by $B_{p}^{s,q}( \mathbb{R}^{d}) =\{ f\in\mathcal{S}^{\prime }:\| f\| _{B_{p}^{s,q}}<\infty\},$ where for$q\neq\infty$, $\| f\| _{B_{p}^{s,q}}=\Big( \sum_{j\in\mathbb{Z}} ( 2^{js}\| \Delta_{j}f\| _{L^{p}}) ^{q}\Big)^{1/q};$ and for$q=\infty$, $\| f\| _{B_{p}^{s,\infty}}=\sup{j\geq0}2^{js}\| \Delta_{j}f\| _{L^{p}}.$ If$s<0$and if$f_{j}$satisfies$\mathop{\rm supp}f_{j}\subset\{ \xi : \frac{2^{j}}{2}\leq| \xi| \leq2.2^{j}\} $and$( 2^{js}\| f_{j}\| _{L^{p}}) _{j\in\mathbb{Z}}\in l^{\infty}(\mathbb{Z})$, then$\sum_{j\in\mathbb{Z}} f_{j}$converges in$\mathcal{S}'$. Indeed, we have$\sum_{j\leq 0}\| f_{j}\| _{L^{p}}<+\infty$, but for all$g\in\mathcal{S}$and all$k\in\mathbb{N}$:$\sum_{j\geq0} 2^{jk}\|\Delta_{j}g\| _{L^{p}}<+\infty$. We write for some$M\in\mathbb{N}$, $\| g\| _{L^{p'}}\leq\sum_{|\alpha| \leq M} \sum_{| \beta| \leq M}\big\| \xi^{\alpha}. \frac{\partial^{\beta}}{\partial\xi^{\beta}} \overset{\wedge}{g}\big\| _{L^{\infty}}\,.$ We use the semi-group$e^{-t( -\Delta) ^{\gamma}}$operating on shift-invariant Banach spaces of distributions, it will be very useful to characterize the action of the kernel$k_{\gamma}(t)$on Besov spaces associated with such spaces \cite[Theorem 5.3, p.44-45]{Lem}. \begin{proposition} Let$s<0$and$T>0$. There exists a constant$C_{T}>0$such that $$\dot {B_{p}}^{s,q}=\big\{ f\in\mathcal{S}':e^{-t(-\Delta) ^{\gamma}}f\in L^{p}, \Big({\int_{0}^{T}} ( t^{-\frac{s}{2}}\| e^{-t( -\Delta) ^{\gamma}}f\| _{L^{p}})^{q} \frac{dt}{t}\Big) ^{1/q}\leq C_{T}\big\} \label{eq1}$$ for$p$,$q\in [ 1,+\infty ]$. Moreover, the norm $\| e^{-t( -\Delta) ^{\gamma}}f\| _{L^{p}}+\| t^{-\frac{s}{2}}( e^{-t( -\Delta) ^{\gamma}}f) \| _{L^{q}(] 0,\infty[ ,\frac{dt}{t},L^{p}) }$ and the norm$\| f\| _{\dot {B}_{p}}^{s,q}$are equivalent. \end{proposition} Let us return to the condition$E\hookrightarrow\dot {B}_{\infty }^{-r,\infty}$. One of the purposes of this section is, if possible, to show under a characterization (\ref{eq1}) that $f\in\dot {B}_{\infty}^{-r,\infty}\Leftrightarrow t^{\frac{r}{2\gamma} }e^{-t( -\Delta) ^{\gamma}}f\in L^{\infty}( ] 0,T[ ,L^{\infty}) ,\quad \text{for all }T>0$ That is, we want to know whether $\| f\| _{\dot {B}_{\infty}^{-r,\infty}}\sim\| e^{-t( -\Delta) ^{\gamma}}f\| _{L^{\infty}}+\| t^{-\frac{s}{2}}( e^{-t( -\Delta) ^{\gamma}}f) \| _{L^{\infty}( ] 0,\infty[ ,L^{\infty}) }$ for every function$f\in\dot {B}_{\infty}^{-r,\infty}$and all$t>0$. The proof uses exactly the same ideas as in \cite[of theorem 5.4 in]{Lem} and we omit the details here. \begin{lemma} \label{lem3.7} Let$E$a Banach space satisfying$\mathcal{S}\hookrightarrow E\hookrightarrow \mathcal{S}'$, for all$x_{0}\in\mathbb{R}^{d}$;$\|f\|_{E}=\|f( x-x_{0})\| _{E}$and so that $\sup_{0<\lambda\leq1} \lambda^{r}\| f( \lambda x) \| _{E}\leq C\| f\| _{E}\quad \text{with }r\in\mathbb{R}\,.$ Then $E\hookrightarrow B_{\infty}^{-r,\infty}\,.$ Similarly, if $\underset{0<\lambda}{\sup}\lambda^{r}\| f( \lambda x) \| _{E}\leq C\| f\| _{E}\quad \text{with }r\in\mathbb{R},$ then $E\hookrightarrow\dot {B}_{\infty}^{-r,\infty}\,.$ \end{lemma} \begin{proof} We shall prove only the first statement; the proof of the second is similar. We may assume that $\sup_{0<\lambda\leq1} \lambda^{r}\| f( \lambda x) \| _{E}\leq C_{}\| f\| _{E}\quad \text{with } r\in\mathbb{R}$ Using the Littlewood-Paley decomposition, we have by noting$ S_{0}f=\varphi\ast f$$\| S_{0}f\| _{L^{\infty}} =\sup_{x\in\mathbb{R}^{d}} |{\int_{\mathbb{R}^{d}}}\varphi(y)f( x-y) dy| \leq\sup_{x\in\mathbb{R}^{d}} \| \varphi\| _{L^{1}}\| f\| _{E}$ Moreover, for all$j\geq0, we have \begin{align*} 2^{-jr}\| \Delta_{j}f\| _{L^{\infty}} & =2^{-jr}\|\Delta_{0}f( \frac{.}{2^{j}}) ( 2^{j}x) \| _{L^{\infty}} \\ & =2^{-jr}\| \Delta_{0}f( \frac{.}{2^{j}}) \|_{L^{\infty}}\\ &\leq C2^{-jr}\| f( \frac{.}{2^{j}}) \| _{E} \leq C\| f\| _{E} \end{align*} \end{proof} We now establish estimates for the operatorK_{\gamma}(t)$in Banach spaces of local measures. \begin{proposition} \label{prop3.8} For any$t>0$, the operators$K_{\gamma }(t)$and$\nabla K_{\gamma }(t)$are bounded operators in Banach spaces of local measures and depend on$t$continuously, where$\nabla $denotes the space derivative. Furthermore, we have for$u\in E$, \begin{gather*} \| K_{\gamma }(t)u\| _{E}\leq \| u\| _{E},\\ \| \nabla K_{\gamma }(t)u\| _{E}\leq Ct^{-\frac{1}{2\gamma } }\| u\| _{E} \end{gather*} \end{proposition} The statement of the above proposition is easy to check. \section{Well Posedness in Banach spaces of local measures} The result of Wu's theorem may be generalized in a direct way to the setting of spaces of local measures, such as Lorentz spaces, Multipliers spaces or Morrey-Campanato spaces. Let us introduce suitable functional spaces to analyse the Cauchy problem for system (\ref{eq1.1.1})-(\ref{eq1.1.2}) based on the spaces of measures local. \noindent\textbf{Definition} Let$\mathcal{E}_{E}$be the space of all function$f(x,t)$, with$t>0$and$ x\in\mathbb{R}^{2}$, such that $f(x,t)\in L^{\infty}( ( 0,\infty) ,E)$ and therefore the norm in$\mathcal{E}_{E}$is defined by $\| f\| _{\mathcal{E}_{E}}=\sup_{t>0}\|f(.,t)\| _{E}$ Let us also define by$\mathcal{E}_{\infty}$the space of all functions$ f(x,t)$, with$t>0$and$x\in\mathbb{R}^{2}$, such that \begin{gather*} f(x,t)\in L^{\infty}( ( 0,\infty) ,E),\\ t^{\frac{r}{2\gamma}}f(x,t)\in L^{\infty}( ( 0,\infty),L^{\infty}), \end{gather*} where$r=2\gamma-1<1$. The norm in$\mathcal{E}_{\infty}$is $\| f\| _{\mathcal{E}_{\infty}}=\sup_{t>0} \| f(.,t)\| _{E}+\sup_{t>0} t^{\frac{r}{2\gamma} }\| f(.,t)\| _{L^{\infty}}\,.$ Set $\mathcal{B}_{\infty}=\{ \theta/\text{ }t^{\frac{r}{2\gamma}}\theta\in L^{\infty}( ( 0,\infty) ,L^{\infty})\}.$ Let us make precise the notion of mild solution. \begin{definition} \label{def3.4} \rm A global mild solution of the system (\ref{eq1.1.1})-(\ref{eq1.1.2}) in$\mathcal{E}_{E}$and$\mathcal{E}_{\infty}$is a function$\theta(t)$in the corresponding space satisfying $$\theta(t)=K_{\gamma}(t)\theta_{0}(t)-B( \theta,\theta) (t)=K_{\gamma}(t)\theta_{0}(t)-\int _{0}^{t}K_{\gamma}(t-s)\nabla( \theta u) (s)ds \label{eq1.3.4}$$ and$\theta(t)\to \theta_{0}$as$t\to 0^{+}$, where the limit is taken in the weak-$^{\ast}$topology of$E$. \end{definition} We prove the following well-posedness results for mild solutions. \begin{theorem}[ well-posedness] \label{theorem3.3} Let$\theta_{0}\in E$and$\gamma<1$for$d=2$. There exists$\delta>0$such that if$\| \theta_{0}\| _{E}<\delta$, then the initial value problem for \eqref{eq1.1.1}-\eqref{eq1.1.2} has a global mild solution$\theta(x,t)\in\mathcal{E}_{E}$. Moreover, if$ \| \theta\| _{\mathcal{E}_{E}}<2\delta$, then the solution is unique in$\mathcal{E}_{E}$. \end{theorem} Well posedness theorem will be a consequence of the following lemma for generic Banach spaces \cite{Lem}: \begin{lemma} \label{lem10} Let$F$be a Banach space with norm$\| .\| _{F}$and$B:F\times F\to F$be a continuous bilinear operator, i.e., there exists$K>0$such that for all$x,y\in F$, we have $\| B(x,y)\| _{F}\leq K\| x\|_{F}\| y\| _{F}$ Given$0<\delta<\frac{1}{4K}$and$y\in F$,$y\neq0$, such that$\| y\| _{F}<\delta$, there exists a solution$x\in F$for the equation$x=y+B(x,x)$such that$\| x\| _{F}<2\delta$. The solution is unique in the ball$\overline{B}( 0,2\delta) $. Moreover, the solution depends continuously on$y$in the following sense : if$\| \widetilde{y}\| _{F}<\delta$,$\widetilde{x}=\widetilde{y}+B( \widetilde{x},\widetilde{x}) $, and$\| \widetilde {x}\| _{F}<2\delta$, then $\| x-\widetilde{x}\| _{F}\leq\frac{1}{1-4K\delta}\| y-\widetilde{y}\| _{F}$ \end{lemma} As a consequence of the previous lemma, one needs to verify the continuity of the bilinear terms in the integral form of the QG equation to obtain the well-posedness theorem for the solutions of the integral equation. The weak-$ ^{\ast}$continuity at$0^{+}$will finally end the proof of well-posedness of mild solutions. \subsection*{Proof of theorem \protect\ref{theorem3.3}} To proceed, we write the QG equation into the integral form $$\theta(t)=K_{\gamma}(t)\theta_{0}-\int _{0}^{t}K_{\gamma}(t-s)( u\nabla\theta) (s)ds \label{eq3.2}$$ We observe that$( u\nabla\theta) =\nabla.( u\theta )$because$\nabla.u=0$. The nonlinear term can then alternatively written as $$B(u,\theta)(t)=\int _{0}^{t}\nabla K_{\gamma}(t-s)( u\theta) (s)ds \label{eq1.3.2}$$ We start by a series of lemmas in order to prove the continuity of the bilinear term$B( \theta,u) $defined by (\ref{eq1.3.2}). We will solve (\ref{eq1.3.2}) in$\mathcal{E}_{\infty}$and the following estimates for the operator$B$acting on this type of spaces will be used. The three properties of$K_{\gamma}(t)$that we shall use are the following obvious remarks. \begin{lemma} Given$h\in E$. We have \begin{gather*} \| K_{\gamma}(t-s)h\| _{E}\leq\| h\| _{E}, \\ \| K_{\gamma}(t-s)h\| _{L^{\infty}}\leq C( t-s) ^{-\frac{r}{2\gamma}}\| h\| _{\dot {B}_{\infty }^{-r,\infty}},\\ \| K_{\gamma}(t-s)\nabla.\| _{L^{1}}\leq C( t-s)^{-\frac{1}{2\gamma}} \end{gather*} \end{lemma} The proof of the above lemma is obvious. \begin{proposition} \label{p1} Let$\frac{1}{2}<\gamma\leq1$and$t>0$. Assume that$u$and$\theta$are in$\mathcal{E}_{\infty}$. Then the operator$B$is bounded in$\mathcal{E}_{\infty}with $\| B(u,\theta)\| _{\mathcal{E}_{\infty}}\leq C_{\gamma}\| u\| _{\mathcal{E}_{\infty}}\| \theta\|_{\mathcal{E}_{\infty}}$ \end{proposition} \begin{proof} Applying (\ref{eq1.2.3}) from proposition \ref{prop2}, we estimate \begin{align*} \| B(u,\theta)\| _{L^{\infty}} & \leq {\int_{0}^{t}} \| K_{\gamma}(t-s)\nabla( \theta u) (s)\| _{L^{\infty}}ds\\ & \leq C {\int_{0}^{t}} ( t-s) ^{-\frac{1}{2\gamma}}\| ( \theta u) (s)\| _{L^{\infty}}ds\\ & \leq C {\int_{0}^{t}} ( t-s) ^{-\frac{1}{2\gamma}}s^{-\frac{r}{\gamma}}( s^{\frac{r}{2\gamma}}\| \theta(s)\| _{L^{\infty}}) ( s^{\frac{r}{2\gamma}}\| u(s)\| _{L^{\infty}}) ds\\ & \leq C\| \theta\| _{\mathcal{B}_{\infty}}\| u\| _{\mathcal{B}_{\infty}} {\int_{0}^{t}} ( t-s) ^{-\frac{1}{2\gamma}}s^{-\frac{r}{\gamma}}ds \end{align*} where in the last line we used the continuity of the Riesz transform. Therefore, $\| B(u,\theta)\| _{L^{\infty}}\leq CI(t)\| \theta\| _{\mathcal{B}_{\infty}}\| u\| _{\mathcal{B}_{\infty}}$ where the integralI(t)in the right-hand side can be computed as \begin{align*} I(t) & = {\int_{0}^{t}} ( t-s) ^{-\frac{1}{2\gamma}}s^{-\frac{r}{\gamma}}ds=t^{-\frac {1}{2\gamma}+1-\frac{r}{\gamma}} {\int_{0}^{1}} ( 1-z) ^{-\frac{1}{2\gamma}}z^{-\frac{r}{\gamma}}dz\\ & =t^{-\frac{r}{2\gamma}}\Gamma( \frac{r}{2\gamma}) \Gamma( \frac{1-r}{2\gamma}) \quad \text{with }00 is the existence of \ constant $C\geq0$ so that for all $f\in E$, we have $\sup_{\lambda>0} \lambda\| f( \lambda x) \| _{E}\leq C\| f\| _{E}\,.$ \end{remark} \begin{proposition} \label{prop4.7} If $\theta$ and $u$ belongs to $\mathcal{E}_{\infty}$, then there exists $C=C( d,\gamma)$ such that $\| B( \theta,u) \| _{E}\leq C\underset{t>0}{\sup} \| \theta(t)\| _{E}( t^{\frac{r}{2\gamma}}\underset{t>0 }{\sup}\| u(t)\| _{L^{\infty}}) .$ \end{proposition} \begin{proof} The boundedness of $B$ from $\mathcal{E}_{E}\times\mathcal{B}_{\infty}$ to $\mathcal{E}_{E}$ and from $\mathcal{B}_{\infty}\times\mathcal{E}_{E}$ to $\mathcal{E}_{E}$ is obvious, since $\| K_{\gamma}(t-s)\nabla.\| _{\mathcal{L}( E\times E,E) }\leq C( t-s) ^{-\frac{1}{2\gamma}}$ and the invariance of $E$ under convolution with $L^{1}$ kernels. Thus, we get \begin{align*} \| B( \theta,u) \| _{E} & \leq C\int _{0}^{t}\| K_{\gamma}(t-s)\nabla( \theta u) (s)\| _{E}ds \\ & \leq C\int _{0}^{t}\| \nabla k_{\gamma}(t-s)\| _{L^{1}}\| ( \theta u) (s)\| _{E}ds \\ & \leq C\int _{0}^{t}( t-s) ^{-\frac{1}{2\gamma} }\| u(s)\| _{E}\| \theta(s)\| _{L^{\infty}}ds \\ & \leq C\int _{0}^{t}( t-s) ^{-\frac{1}{2\gamma}}s^{- \frac{r}{2\gamma}}\| u(s)\| _{E}( s^{\frac{r}{2\gamma} }\| \theta (s)\| _{L^{\infty}}) ds \\ & \leq C\| u\| _{\mathcal{E}_{E}}\| \theta \| _{\mathcal{B}_{\infty}}, \end{align*} where the continuity of the Riesz transform in $E$ has been used, which gives the result, since $\int _{0}^{t}( t-s) ^{-\frac{1}{2\gamma}}s^{-\frac{r}{ 2\gamma}}ds=\Gamma ( \frac{r}{2\gamma}) \Gamma( \frac{1}{ 2\gamma})$ where $\Gamma$ being Euler's Gamma function. \end{proof} A direct application of Lemma \ref{lem10} in $E$ completes the proof of the well-posedness of the integral equation. To finish the proof of the well-posedness of mild solutions in $E$ by applying Lemma \ref{lem10}, it remains to check that the linear part of the integral equation can be bounded in terms of the initial data and that solutions of the integral equation are indeed mild solutions by definition \ref{def3.4}. Following the arguments in \cite[lemma 6]{Bar}, it can be proved in a completely analogous way that $B(u,\theta)(t)\to 0$ when $t\to 0^{+}$ weakly-star in the topology of $E$. Now, it remains to verify that the linear part takes the initial data and is bounded in the corresponding space. \begin{proposition}\label{lem3.13} If $\theta_{0}\in E$, then $K_{\gamma}(t)\theta_{0}\in \mathcal{E}_{E}$, \begin{gather*} \| K_{\gamma}(t)\theta_{0}\| _{\mathcal{E}_{E}}\leq C\| \theta_{0}\| _{E},\\ K_{\gamma}(t)\theta_{0}\to \theta_{0}\text{ when }t\to 0^{+}, \end{gather*} where the limit is taken in the weak-star topology of $E$. Moreover, if $\theta_{0}\in E\cap L^{\infty}$, then $\| K_{\gamma}(t)\theta_{0}\| _{\widetilde{E}}\leq C( \| \theta_{0}\| _{E}+\| \theta_{0}\| _{L^{\infty}})$ where $\widetilde{E}=L^{\infty}( ( 0,\infty) ,E\cap L^{\infty})$. \end{proposition} \begin{proof} Using (\ref{eq2.4}), we deduce \begin{gather*} \| K_{\gamma}(t)\theta_{0}\| _{E}\leq C\| k_{\gamma }(t)\| _{L^{1}}\| \theta_{0}\| _{E}=C\| \theta_{0}\| _{E},\\ \| K_{\gamma}(t)\theta_{0}\| _{L^{\infty}}\leq C\| k_{\gamma}(t)\| _{L^{1}}\| \theta_{0}\| _{L^{\infty }}=C\| \theta_{0}\| _{L^{\infty}}\,. \end{gather*} To prove the weak-star continuity, note that for all $\varphi \in \mathcal{S} ( \mathbb{R}^{d})$, we have \begin{align*} | \langle K_{\gamma }(t)\theta _{0}-\theta _{0},\varphi \rangle | & =\big| \int ( e^{-t| \xi | ^{2\gamma }}-1) \widehat{\theta }_{0}\widehat{\varphi } d\xi \big| \\ & \leq t \sup_{\xi \in \mathbb{R}^{d}} \frac{( 1-e^{-t| \xi | ^{2\gamma }}) }{t| \xi | ^{2\gamma }} \| \theta _{0}\| _{E}\|| \xi | ^{2\gamma }\widehat{\varphi }\| _{L^{1}( \mathbb{R}^{d}) } \end{align*} which converges to $0$ as $t\to 0^{+}$, because $\sup_ {\xi \in \mathbb{R}^{d}} \frac{( 1-e^{-t| \xi | ^{2\gamma }}) }{t| \xi | ^{2\gamma }} \leq \sup_{z>0} e^{-z}=1,.$ Since $\widehat{\varphi }\in \mathcal{S}( \mathbb{R}^{d})$ and $R>0$, $\| | \xi | ^{2\gamma }\widehat{\varphi } \| _{L^{1}( B( 0,R) ) }<\infty , \quad \text{when }\gamma >0.$ \end{proof} A direct application of lemma \ref{lem10} in $\mathcal{E}_{E}$ completes the proof of the well-posedness of mild solution in spaces of local measures. We generalize Wu's \cite{W1} theorem to shift-invariant Banach spaces of local measures by using the regularizing of the kernel $K_{\gamma }(t)$. \begin{theorem} \label{th.lem} Let $E$ be a shift-invariant Banach space of local measures. \begin{itemize} \item[(a)] Let $\mathcal{E}_{E}=L^{\infty}( ( 0,\infty),E)$ and $\mathcal{B}_{\infty}=\{ \theta/t^{\frac{r}{2\gamma}}\theta\in L^{\infty}( ( 0,\infty) ,L^{\infty})\}$. The bilinear operator $B$ defined by $B( \theta,u) (t)=\int _{0}^{t}K_{\gamma}(t-s)\nabla( \theta u) (s)ds$ is bounded from $\mathcal{E}_{E}\times\mathcal{B}_{\infty}$ to $\mathcal{E}_{E}$ and from $\mathcal{B}_{\infty}\times\mathcal{E}_{E}$ to $\mathcal{E}_{E}$. Moreover, $B( \theta,u) \in C( ( 0,\infty),E)$ and converges$^{\ast}$ -weakly to $0$ as $t$ goes to $0$. If $\lim_{t\to 0} t^{\frac{r}{2\gamma}}\|\theta\| _{L^{\infty}}=0$, if $\theta\in\mathcal{B}_{\infty}$ and $u\in\mathcal{E}_{E}$, then the convergence is strong. \item[(b)] If $E$ is continuously embedded in the Besov space $\dot {B}_{\infty}^{-r,\infty}$ for some $r\leq1$, then the bilinear operator $B$ is bounded as well from $( \mathcal{E}_{E}\cap\mathcal{B}_{\infty})\times \mathcal{B}_{\infty}$ to $\mathcal{B}_{\infty}$\ and from $\mathcal{B}_{\infty}\times( \mathcal{E}_{E}\cap\mathcal{B}_{\infty })$ to $\mathcal{B}_{\infty}$. Hence, there exists a constant $\delta_{E}>0$ so that for all $\theta_{0}\in E$ (with $\nabla.\theta_{0}=0$) with $\| \theta_{0}\| _{E}<\delta_{E}$, then the initial value problem for \eqref{eq1.1.1}-\eqref{eq1.1.2} with initial data $\theta_{0}$ has a global mild solution $\theta(x,t)\in\mathcal{E}_{E}\cap \mathcal{B}_{\infty}$. \end{itemize} \end{theorem} For the proof of the above theorem, we use the same ideas as in \cite[Theorem 17.2]{Lem}. Then we can get the desired result, but we omit the details here. \begin{theorem}[Regularization] \label{theorem3.4} Under the assumptions of previous theorem, there exists $0<\epsilon<\delta$ such that if $\| \theta _{0}\|_{E}<\epsilon$, then the solution of theorem \ref{theorem3.3}, $\theta(x,t)\in\mathcal{E}_{\infty}$. Moreover, if $\| \theta\| _{\mathcal{E}_{\infty}}<2\epsilon$, then the solution is unique in this class. \end{theorem} \begin{proof} We just need to show the continuity of the bilinear form in the regularizing norm $\underset{t>0}{\sup}t^{\frac{r}{2\gamma}}\| .\|_{\infty}$. For this, we have by proposition \ref{p1}, $\| B(u,\theta)\| _{L^{\infty}}\leq Ct^{-\frac{r}{2\gamma} }\| \theta\| _{\mathcal{B}_{\infty}}\| u\| _{\mathcal{B}_{\infty}}\,;$ therefore, $\sup_{t>0} t^{\frac{r}{2\gamma}}\| B(u,\theta)\| _{L^{\infty}}\leq C\| \theta\| _{\mathcal{B}_{\infty}}\| u\| _{\mathcal{B}_{\infty}}\,.$ This completest the proof of continuity of the bilinear form. \end{proof} A direct application of lemma \ref{lem10} in $\mathcal{E}_{\infty}$ completes the proof of the well-posedness of the integral equation. Initial data are taken in the same sense as in theorem \ref{theorem3.3} and therefore, we have completed the proof of regularization in the solutions of theorem \ref{theorem3.3}. We now show that the solution obtained in the previous theorem is actually smooth. We adapt the arguments of Kato in \cite{K} for proving that the constructed solutions are $C^{\infty}$-smooth instantly. \begin{proposition} Let $\theta(x,t)\in\mathcal{E}_{\infty}$ be the unique global mild-solution in theorem \ref{theorem3.4}. Then $$\partial_{t}^{k}\partial_{x}^{m}\theta(x,t)\in C( ( 0,\infty ) ,\widetilde{E}) \label{eq3.8}$$ As a consequence, the solution $\theta(x,t)$ is infinitely smooth in space and time. \end{proposition} \begin{proof} We will just give the main steps of the proof since this is a small variation of arguments found in \cite{K}. The proof is done by induction. We will focus in the case $k=0$ since the same induction argument one can derive the estimates for the temporal derivatives (see \cite{K} for details). Let $\theta$ be the solution of theorem \ref{theorem3.4}. Denote $\beta =\theta(\tau)$ for any $\tau>0$. Let us remark that we have local existence, $T-\tau$ small enough, for the mild formulation $$\theta(t)=K_{\gamma}(t-\tau)\beta-\int _{\tau}^{t}\nabla K_{\gamma}(t-s)( \theta R[ \theta] ) (s)ds \label{eq3.9}$$ in the space $L^{\infty}\big( ( \tau,T) ,\widetilde{E}\big)$. Denote $B_{\tau}(\theta,\theta)=\int _{\tau}^{t}\nabla K_{\gamma}(t-s)( \theta R[ \theta] ) (s)ds\,.$ Let us perform an induction argument on $m$. The assertion (\ref{eq3.8}) for $k=m=0$ is true by theorem \ref{theorem3.4}. Let us assume that (\ref{eq3.8}) is true for all $m\leq N-1$, $k=0$. Our aim is to show that the mild formulation (\ref{eq3.9}) has a solution in the space $\mathcal{X}_{N}$ defined by \begin{gather*} \partial_{x}^{j}\theta(x,t) \in L^{\infty}( [ \tau,T[ , \widetilde{E}) , \quad j=0,1,2,\dots N-1, \\ ( t-\tau) ^{\frac{1}{2\gamma}}\partial_{x}^{N}\theta(x,t) \in L^{\infty}([ \tau,T[ ,\widetilde{E}) \end{gather*} endowed with the norm $\| \theta\| _{\mathcal{X}_{N}} =\sup_{t\in(\tau,T) } \big[( t-\tau) ^{\frac{1}{2\gamma}}( \| \partial_{x}^{N}\theta\| _{L^{\infty}}+\| \partial_{x}^{N}\theta \| _{E}) +\sum _{j=0}^{d}( \| \partial_{x}^{j}\theta\| _{L^{\infty}}+\| \partial_{x}^{j}\theta\| _{E}) \big]$ Note that $K_{\gamma}(t-\tau)\beta\in\mathcal{X}_{N}$ since by induction hypothesis, we have $\partial_{x}^{j}\theta(\tau)\in\widetilde{E}$ for $j=0,1,2,\dots N-1$ and by convolution \begin{align*} \| \partial_{x}^{N}K_{\gamma}(t-\tau)\beta\| _{L^{\infty}} & =\| ( \partial_{x}k_{\gamma}(t-\tau)) \ast\partial _{x}^{N-1}\beta\| _{L^{\infty}} \\ & \leq C\| ( \partial_{x}k_{\gamma}(t-\tau)) \| _{L^{1}}\| \partial_{x}^{N-1}\beta\| _{L^{\infty}} \\ & \leq C( t-\tau) ^{\frac{1}{2\gamma}}\| \partial _{x}^{N-1}\beta\| _{L^{\infty}} \end{align*} and \begin{align*} \| \partial_{x}^{N}K_{\gamma}(t-\tau)\beta\| _{E} & =\| ( \partial_{x}k_{\gamma}(t-\tau)) \ast\partial _{x}^{N-1}\beta\| _{E} \\ & \leq C\| ( \partial_{x}k_{\gamma}(t-\tau)) \| _{L^{1}}\| \partial_{x}^{N-1}\beta\| _{E} \\ & \leq C( t-\tau) ^{\frac{1}{2\gamma}}\| \partial _{x}^{N-1}\beta\| _{L^{\infty}} \end{align*} Now, given $t\in( \tau,T)$ and $\theta$, $u\in\mathcal{X}_{N}$, we have \begin{gather*} \| B_{\tau}(\theta,u)(t)\| _{L^{\infty}}\leq C( T-\tau) ^{\frac{r}{2\gamma}}\sup_{t\in[ \tau,T[ } \| \theta(t)\| _{L^{\infty}}\sup_{t\in[ \tau,T[ }\| u(t)\| _{L^{\infty}}, \\ \| B_{\tau}(\theta,u)(t)\| _{E}\leq C( T-\tau) ^{ \frac{r}{2\gamma}}\sup_{t\in[ \tau,T[ }\|\theta(t)\| _{L^{\infty}} \sup_{t\in[ \tau,T[ }\| u(t)\| _{E} \end{gather*} Therefore, given $\theta$, $u\in\mathcal{X}_{N}$, we conclude that \begin{gather*} \| \partial_{x}^{j}B_{\tau}(\theta,u)(t)\| _{L^{\infty}}\leq C( T-\tau) ^{\frac{r}{2\gamma}}\sup_{t\in [ \tau,T[ } \| \theta(t)\| _{L^{\infty}} \sup_{t\in [ \tau,T[ }\| \partial_{x}^{j}u(t)\| _{L^{\infty}}, \\ \| \partial_{x}^{j}B_{\tau}(\theta,u)(t)\| _{E}\leq C( T-\tau) ^{\frac{r}{2\gamma}}\sup_{t\in[ \tau,T[ } \| \theta(t)\| _{L^{\infty}}\sup_{t\in[ \tau,T[ } \| \partial_{x}^{j}u(t)\| _{E} \end{gather*} for all $j=0,1,2,\dots N-1$. Now, we observe that $\partial_{x}^{N}B_{\tau }(\theta,u)$ is a linear combination of terms of the form $\partial_{j,i,x}^{N}B_{\tau}(\theta,u)(t)=C\int _{\tau}^{t}\partial_{x}\nabla K_{\gamma}(t-s)( \partial^{j}\theta R[ \partial^{i}\theta] ) (s)ds$ with $j+i=N-1$. Thus, we can get the estimate \begin{gather*} \| \partial_{x}^{N}B_{\tau}(\theta,u)(t)\| _{L^{\infty}}\leq C( T-\tau) ^{\frac{r}{2\gamma}-\frac{1}{2\gamma} }\sup_{t\in[ \tau,T[ }\| \partial_{x}^{j}\theta (t)\| _{L^{\infty}}\sup_{t\in[ \tau,T[ } \| \partial_{x}^{j}u(t)\| _{L^{\infty}}, \\ \| \partial_{x}^{N}B_{\tau}(\theta,u)(t)\| _{E}\leq C( T-\tau) ^{\frac{r}{2\gamma}-\frac{1}{2\gamma}} \sup_{t\in[\tau,T[ } \| \partial_{x}^{j}\theta(t)\| _{L^{\infty}}\sup_{t\in[ \tau,T[ }\| \partial_{x}^{j}u(t)\| _{E}\,. \end{gather*} Finally we conclude that $\| B_{\tau}(\theta,u)(t)\| _{\mathcal{X}_{N}}\leq C( T-\tau) ^{\frac{r}{2\gamma}}\| \theta\| _{\mathcal{X} _{N}}\| u\| _{\mathcal{X}_{N}}\,.$ By choosing $T-\tau$ small enough, we are allowed to use lemma \ref{lem10} to obtain a solution $\widetilde{\theta}(x,t)$ in $\mathcal{X}_{N}$ of the mild formulation (\ref{eq3.9}). Owing to the uniqueness of the mild formulation (\ref{eq3.9}) in $L^{\infty}([ \tau,T[ ,\widetilde {E})$, we have finally shown that $\widetilde{\theta} (x,t)=\theta (x,t)$ in $( \tau,T)$. Now, it is easy to perform all these arguments in a recursive finite collection of equal-length time intervals $A_{i}$ such that $( 0,\infty) =\cup_{i}A_{i}$ $\$\ where $A_{i}=( \tau_{i},T_{i})$ concluding that $\partial_{x}^{j}\theta\in C( ( 0,\infty) ,\widetilde {E} ) \quad \text{for all }j=0,1,2,\dots N.$ As said above, the induction on temporal derivatives is done analogously and thus $\theta(x,t)$ has derivatives of all orders in $\widetilde{E}$. \end{proof} \section{Self-Similar solutions in $E$.} Assuming that $\theta(x,t)$ is a smooth solution of the QG equation (\ref{eq1.1.1})-(\ref{eq1.1.2}), it is straightforward to check that $\theta_{\lambda}(x,t)=\lambda^{2\gamma-1}\theta(\lambda^{2\gamma}x,\lambda t)$ is also a solution of (\ref{eq1.1.1})-(\ref{eq1.1.2}). In fact, we can look for particular solutions of the system (\ref{eq1.1.1})-(\ref {eq1.1.2}) satisfying $$\theta_{\lambda}(x,t)=\theta(x,t) \label{eq4.1}$$ for any $t>0$, $x\in\mathbb{R}^{2}$ and $\lambda>0$. These particular solutions are called self-similar solutions of the system and it is clear that taking $t\to 0^{+}$ formally in (\ref{eq4.1}), $\theta(x,0)$ should be a homogeneous function of degree $( 1-2\gamma)$. This remark gives the hint that a suitable space to find self-similar solutions should be one containing homogeneous functions with that exponent. Moreover, in case such a self-similar solution exists, its norm is invariant by scaling transformation, i.e., $\theta(x,t)\to \theta_{\lambda}(x,t)=\lambda^{2\gamma-1}\theta (\lambda^{2\gamma}x,\lambda t)$ Now we show the existence and uniqueness of global in time non trivial self-similar solutions in Banach spaces of local measures with the right homogeneity. \begin{theorem} \label{thm5.1} Let $\theta_{0}\in E$. Assume that $\theta_{0}$ is a homogeneous function of degree $( 1-2\gamma)$. Then, if $\| \theta _{0}\| _{E}<\delta$, the solution $\theta(x,t)$ provided by theorem \ref{theorem3.3} is self-similar. The solution is unique if $\| \theta\| _{\mathcal{E}_{E}}<2\delta$. Moreover, if the initial data is small, $\| \theta_{0}\|_{E}<\epsilon$, the previous unique self-similar solution becomes regularized. \end{theorem} \begin{proof} All well-posedness theorem has been proved by ultimately using Lemma \ref{lem10}. One can check (see \cite{Lem}) that the solution is obtained by a successive approximation method. In fact, defining the recursive sequence \begin{gather*} \theta_{1}(x,t) =K_{\gamma}(t)\theta_{0}(x) \\ \theta_{k+1}(x,t) =\theta_{1}(x,t)-B(\theta_{k},\theta_{k}) \end{gather*} when $k=1,2,\dots$. It is easy to verify that $\theta_{1}(x,t)$ satisfies the self-similarity property $\theta_{1}(x,t)=\lambda^{2\gamma-1}\theta_{1}(\lambda^{2\gamma}x,\lambda t)\,.$ A simple induction argument proves that $\theta_{k}$ has this property for all $k$. Therefore, as the mild solution $\theta(x,t)$ is obtained as the limit of the sequence $\{ \theta_{k}\}$, we have that $\theta(x,t)$ must verify $\theta(x,t)=\lambda^{2\gamma-1}\theta(\lambda^{2\gamma}x,\lambda t)$ for all $\lambda>0$, all $t>0$ and $x\in\mathbb{R}^{2}$. The assertion of uniqueness results from well-posedness theorem. \end{proof} \subsection*{Acknowledgement} The author wants to thank the anonymous referee for his/her helpful comments, also to Professor P. G.\ Lemari\'{e}-Rieusset for his constant encouragement and suggestions. \begin{thebibliography}{00} \bibitem{AH} Adams, D. R., and Hedberg, L. 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France} \email{Sadek.Gala@maths.univ-evry.fr} \section*{Addendum: April 4, 2006.} After publication, the managing editors were informed that key results in this manuscript were `borrowed" from the article below, without the proper citation. The reader should be aware of this claim and decide on her/his own about the originality of this article. \begin{itemize} \item[[1]] Jose A. Carrillo, Lucas C. F. Ferreira; \emph{Self-similar solutions and large time asymptotics for the dissipative quasi-geostrophic equations}, HYKE Preprints, 2005-10, http://www.hyke.org/preprint/index.php \end{itemize} \end{document}