\documentclass[reqno]{amsart}
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\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2007(2007), No. 150, pp. 1--9.\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 2007 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2007/150\hfil Global solutions for quasi-geostrophic equation]
{Global well-posedness  for the 2D quasi-geostrophic  equation
 in a \\ critical Besov space}

\author[A. Stefanov\hfil EJDE-2007/150\hfilneg]
{Atanas Stefanov} 

\address{Atanas Stefanov \newline
Department of Mathematics, University of Kansas, Lawrence,
KS 66045, USA}
\email{stefanov@math.ku.edu}

\thanks{Submitted November 10, 2006. Published November 9, 2007.}
\thanks{Supported grant 0701802 from the NSF-DMS}
\subjclass[2000]{35Q35, 36D03, 35K55, 76B65}
\keywords{2D quasi-geostrophic equations}

\begin{abstract}
 We show that the  2D quasi-geostrophic equation has global and
 unique strong solution when the (large) data belongs in the
 critical scale invariant space
 $\dot{B}^{2-2\alpha}_{2, \infty}\cap L^{2/(2\alpha-1)}$.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newcommand{\norm}[2]{\|#1\|_{#2}}

\section{Introduction}

In this paper we are concerned with the mathematical properties
of the Cauchy problem for the
quasi-geostrophic equation in two spatial dimensions
\begin{equation}
\label{eq:1}
\begin{gathered}
\theta_t +\kappa(-\Delta)^{\alpha} \theta + (J(\theta) \cdot \nabla)\theta=0 \quad (t,x)\in
{\mathbf R^+}\times \mathbb{R}^2 \\
\theta (0,x)=\theta^0(x),
\end{gathered}
\end{equation}
where $\theta:\mathbb{R}^2\to \mathbb{R}$ is a scalar, real-valued function,
and $J(\theta)=(-R_2\theta,  R_1\theta)$, $\alpha\in[0,1]$ and  $R_1, R_2$ are
the Riesz transforms defined via the Fourier transform by
$\widehat{R_j f}(\xi)=\xi_j|\xi|^{-1} \hat{f}(\xi)$, see also  Section
\ref{sec:2.3} for additional details.

The physical meaning and derivation
of \eqref{eq:1} has been discussed extensively in the
literature. We refer the interested reader to the classical book of Pedlosky,
\cite{P}.

Depending on the value of the parameter $\alpha$, one distinguishes
between the subcritical case $\alpha>1/2$, the critical case $\alpha=1/2$, and the
supercritical case $\alpha<1/2$. It is known that the critical case
$\alpha=1/2$ is especially relevant from a physical point of view, as it is a
 direct analogue of the 3 D Navier-Stokes equations. On the other hand,
considering the family of equations \eqref{eq:1} with $\alpha\in[0,1]$
allows us  to understand better the influence
of the diffusion on the evolution.

An important scale invariance
associated with problem \eqref{eq:1} is that
$\theta^\lambda(t,x)=\lambda^{2\alpha-1} \theta(\lambda^{2\alpha}t, \lambda x)$
is a solution if $\theta$ is. It follows  that the
space $\dot{H}^{2-2\alpha}(\mathbb{R}^2)$ is critical for the problem at hand.
A heuristic argument can be made to show that a
 well-posedness theory for initial data in $H^s$, $s<2-2\alpha$
should not hold. Thus, we concentrate our attention to the case
$s\geq 2-2\alpha$.


 The theory for existence of solutions and their uniqueness  vary greatly,
according to the criticality of the
index $\alpha$. For the critical and supercritical case, the
question has been studied in \cite{CL,CC,Ju1,Ju2,Wu,Wu1} among others.
 The results are that when the data is large and belongs to $H^s$,
$s>2-2\alpha$, then one has at least a local solution, which may blow up
after finite time. For small data in the critical space
(or some Besov variant), Chae-Lee, \cite{CL} and then
J. Wu, \cite{Wu, Wu1}  have been able to show existence of
global solutions.

We would like to mention that the majority of these results have been
subsequently refined to include Besov spaces of initial data with the same level
of regularity and scaling as the corresponding Sobolev spaces. Also, various
uniqueness and blow-up criteria have been developed,
see for example Section \ref{sec:2} below.  However, the fundamental  question for
existence of global, smooth solutions in the supercritical case remains open.
We note that very recently,  in the critical case $\alpha=1/2$,
Kiselev, Nazarov and Volberg, \cite{KNV} have shown the
existence of global and smooth solutions for any smooth (large) initial data.
The smoothness assumption in \cite{KNV} is essentially
at the level of $H^2(\mathbb{R}^2)$, while the  critical case, the critical
Sobolev space is $H^1(\mathbb{R}^2)$.

In the subcritical case, $\alpha>1/2$, which is of main concern for us,
the quasi-geostrophic
equation is better understood. Local and global well-posedness results,
as well as $L^p$ decay estimates for the solution has been shown.

To summarize the latest results, Constantin and Wu, \cite{CW1}
have  shown global well-posedness for the inhomogeneous
version\footnote{That is, the authors also consider the equation
with right-hand side not necessarily equal to zero.} of
\eqref{eq:1} whenever the data is in $H^{s}: s>2-2\alpha$. For small
data, there are plethora of results, which we will not review
here, since we are primarily interested in the large data regime.
On the other hand, time-decay estimates for
$\norm{\theta(t)}{L^p}$ have been shown in \cite{CW1} and
\cite{Ju1}, see Section \ref{sec:2} below  for further details.
Finally, we mention a local well-posedness result for large data
in $H^{2-2\alpha}\cap L^2$, due to Ning Ju, \cite{Ju4}. Note that the
space $H^{2-2\alpha}$ is not scale invariant (due to the $L^2$ part
of it) and thus, such solutions cannot be rescaled to global ones.


In this work, we show that the quasi-geostrophic equation is
globally  well-posed in the critical space $\dot{B}^{2-2\alpha}_{2,
\infty}\cap L^{2/(2\alpha-1)}$, that is whenever the data $\theta^0$
belongs to the space, there is a global and unique\footnote{For
the uniqueness one has to assume in addition $\theta^0\in
L^2(\mathbb{R}^2)$}  solution in the same space.

\begin{theorem} \label{theo:1}
Let $\alpha\in (1/2,1)$.
Then for any
initial data $\theta^0\in \dot{B}^{2-2\alpha}_{2, \infty}(\mathbb{R}^2)\cap L^{2/(2\alpha-1)}(\mathbb{R}^2)$,
the quasi-geostrophic equation \eqref{eq:1} has a global solution
$$
\theta\in L^\infty([0,\infty); \dot{B}^{2-2\alpha}_{2, \infty}(\mathbb{R}^2)\cap L^{2/(2\alpha-1)}(\mathbb{R}^2) )
$$
Moreover, the solution satisfies  the {\it a priori} estimate
\begin{equation}
\label{eq:3}
\norm{\theta(t)}{\dot{B}^{2-2\alpha}_{2, \infty}
\cap L^{2/(2\alpha-1)}}\leq C_{\kappa, \alpha}(\norm{\theta^0}{\dot{B}^{2-2\alpha}_{2, \infty}\cap L^{2/(2\alpha-1)}}+
 \norm{\theta^0}{L^{2/(2\alpha-1)}}^{M(\alpha)}),
\end{equation}
for all $t>0$ and $M(\alpha)=\max(2, 1/(2\alpha-1))$.  In particular, the
norms remain bounded for $0<t<\infty$.

In addition, if $\theta_0\in L^2(\mathbb{R}^2)$, then $\theta\in L^2((0,\infty),
H^{\alpha}(\mathbb{R}^2))$, in fact
\begin{equation}
\label{eq:890}
\norm{\theta}{L^2,
H^{\alpha}(\mathbb{R}^2))}\leq \norm{\theta^0}{L^2(\mathbb{R}^2)}.
\end{equation}
For a fixed $T>0$, the  solution is  unique
class of weak solutions on $[0,T]$  satisfying
$\theta\in L^\infty([0,T], L^2(\mathbb{R}^2))\cap L^\infty([0,T],
L^{2/(2\alpha-1)})\cap L^2((0,T), H^{\alpha}(\mathbb{R}^2))$.
\end{theorem}

Several remarks are in order.
\begin{enumerate}
\item Note that
 global solutions exist and are unique in the space $\dot{B}^{2-2\alpha}_{2, \infty}(\mathbb{R}^2)
\cap L^{2/(2\alpha-1)}(\mathbb{R}^2)$, when the data is in the same scale invariant
space. Note that such space properly contains $\dot{H}^{2-2\alpha}(\mathbb{R}^2)$.  In other words,
taking data in $\dot{H}^{2-2\alpha}(\mathbb{R}^2)$ guarantees the existence of
global solution, but by \eqref{eq:3} we only know that the slightly smaller norm
$\|\theta(t)\|_{\dot{B}^{2-2\alpha}_{2, \infty}\cap L^{2/(2\alpha-1)}}$ stays bounded.
\item It is an interesting question, whether  Theorem \ref{theo:1} and
more precisely \eqref{eq:3} hold in  in the case of the Sobolev space $\dot{H}^{2-2\alpha}$ or even
for some Besov space in the form $B^{2-2\alpha}_{2, r}$ for some $r<\infty$.  We note that the main difficulty  is proving estimate \eqref{eq:3} for smooth solutions. Once \eqref{eq:3} is established,  one easily deduce the global existence and uniqueness by standard arguments.
\item The results in Theorem \ref{theo:1} apply may be readily extended to
${\mathbf T}^2$. We omit the details, as they  amount to
 a minor modification of the  proof presented below.
\end{enumerate}


\section{Preliminaries} \label{sec:2}

\subsection{The 2D quasigeostrophic equation -
existence and maximum principles}
We start this section by recalling the Resnick's theorem, \cite{R} for
existence of weak solutions. That is whenever $\theta^0\in L^2(\mathbb{R}^2)$ and
for any $T>0$, there exists a function
$\theta\in L^\infty([0,T], L^2(\mathbb{R}^2))\cap L^2[[0,T], H^\alpha(\mathbb{R}^2))$,
so that for any test function $\varphi$,
\begin{align*}
&\int_{\mathbb{R}^2} \theta(T) \varphi(T) -
\int_0^T \int_{\mathbb{R}^2} \theta (J(\theta)\nabla \varphi)+\kappa
\int_0^T \int_{\mathbb{R}^2} ((-\Delta)^{\alpha/2} \theta) ((-\Delta)^{\alpha/2} \varphi) \\
&= \int \theta^0\varphi(0,x).
\end{align*}

In his dissertation, \cite{R}, Resnick also
established {\it the maximum principle for $L^p$ norms},
that is for smooth solutions of \eqref{eq:1} and $1\leq  p< \infty$, one has
\begin{equation}
\label{eq:2}
\norm{\theta(t)}{L^p(\mathbb{R}^2)}\leq \norm{\theta^0}{L^p(\mathbb{R}^2)}.
\end{equation}
This was later generalized by Constanin-Wu, \cite{CW1}, \cite{CW2}
 for the case $p=2$ and by
C\'ordoba-C\'ordoba, \cite{CC} in the case $p=2^n$ and N. Ju,
\cite{Ju2} for all $p\geq 2$ to actually imply a power
rate of decay for
$\norm{\theta(t)}{L^p(\mathbb{R}^2)}$ and an exponential rate of decay,
when one considers the equation \eqref{eq:1}
 on the torus ${\mathbf T^2}$. In the sequel, we use primarily
 \eqref{eq:2}, but is nevertheless interesting question to determine the
optimal rates of decay for these norms. Note that Constantin and Wu
have shown in \cite{CW1},  that the optimal
rate for $\norm{\theta(t)}{L^2(\mathbb{R}^2)}$ is $<t>^{-1/2\alpha}$.  Ning Ju
has proved  in \cite{Ju2},  that\footnote{For example, $(p-2)/2p\alpha\to 0$ as $p\to 2$, whereas the optimal rate is $(2\alpha)^{-1}$, as shown by Constantin and Wu. On the other hand, we must note that the rate of $L^p$ decay obtained by Ning Ju holds under the assumption that $\theta_0\in L^2(\mathbb{R}^2)$,
while Constantin-Wu assume that $\theta_0\in L^1(\mathbb{R}^2)$.}  $\norm{\theta(t)}{L^p(\mathbb{R}^2)}\leq C(\norm{\theta^0}{L^p})
(1+t)^{-(p-2)/2p\alpha}$.


\subsection{The uniqueness theorem of Constantin-Wu}
Recall the uniqueness theorem by Constantin-Wu
(Theorem 2.2, in \cite{CW1}).

\begin{theorem}(Constantin-Wu) \label{theo:2}
Assume that $\alpha\in (1/2, 1]$ and $p,q$ satisfy $p\geq 1,
q>1$ and $1/p+\alpha/q=\alpha-1/2$. Then for every $T>0$, there is at most one
weak solution of \eqref{eq:1} in $[0,T]$, satisfying
$$
\theta\in L^\infty([0,T], L^2(\mathbb{R}^2))\cap L^2[[0,T], H^\alpha(\mathbb{R}^2))\cap
L^q([0,T], L^p(\mathbb{R}^2)).
$$
\end{theorem}

In particular, one can take $q=\infty, 1/p=\alpha-1/2$ to
obtain uniqueness for weak solutions satisfying
$\theta\in L^\infty([0,T], L^p(\mathbb{R}^2))$.

\subsection{Some Fourier Analysis}
\label{sec:2.3}
Define the Fourier transform  by
$$
\hat{f}(\xi)=\int_{\mathbb{R}^n} f(x)e^{-i x\cdot  \xi} dx
$$
and its inverse by
$$
 f(x)=(2\pi)^{-n} \int_{\mathbb{R}^n} \hat{f}(\xi)e^{i x\cdot  \xi} d\xi.
$$
For a
positive, smooth and  even function $\chi:\mathbb{R}^2\to \mathbb{R}$, supported in
 $\{\xi:|\xi|\leq 2\}$ and so that $\chi(\xi)=1$ for all
$|\xi|\leq 1$.
 Define $\varphi(\xi)=\chi(\xi)-\chi(2\xi)$,
which is supported in the annulus $1/2\leq |\xi|\leq 2$. Clearly
$\sum_{k\in \mathcal{Z}} \varphi(2^{-k} \xi)=1$ for all $\xi\neq 0$.

The $k^{th}$ Littlewood-Paley projection is
$\widehat{P_k f}(\xi)=\varphi(2^{-k}\xi) \hat{f}(\xi)$.  Similarly
$P_{<k}=\sum_{l\leq k}P_l$ given by the multiplier $\chi(2^{-k}\xi)$.
Note that the kernels of $P_k,P_{<k}  $ are  uniformly  integrable and thus
$P_k, P_{<k}:L^p\to L^p$ for $1\leq p\leq \infty$ and
$\norm{P_k}{L^p\to L^p}\leq
C \norm{\hat{\chi}}{L^1}$. In particular, the  bounds are
independent of $k$.

The  kernels  of $P_k$ are smooth and real-valued\footnote{Thus for
a real valued function $\psi$, $P_k \psi$ is a real-valued function as well.}
and $P_k$ commutes with differential operators with constant coefficients.  We will frequently use the notation
$\psi_k(x)$ instead of  $P_k \psi$, when this will not create confusion.

It is convenient to define the (homogeneous and inhomogeneous)
Sobolev norms in terms of
the Littlewood-Paley operators. Namely for any   $s\geq 0$,
define for every Schwartz
function $\psi$ th norms
\begin{gather*}
 \norm{\psi}{\dot{H}^s}:=\Big(\sum_{k=-\infty}^{\infty}
 2^{2ks} \norm{\psi_k}{L^2}^2\Big)^{1/2}\\
 \norm{\psi}{H^s}:=\Big(\norm{\psi}{L^2}^2+ \sum_{k=0}^{\infty}
 2^{2ks} \norm{\psi_k}{L^2}^2\Big)^{1/2}
\end{gather*}
and the corresponding spaces are then obtained as the closure of
the set of all Schwartz functions in these norms.
 Clearly $H^s=L^2\cap \dot{H}^s$.

Introduce the operator $\Lambda$ acting via
$\widehat{\Lambda \psi}(\xi):=|\xi| \hat{\psi}(\xi)$. Clearly, by the
uniform boundedness
of $P_k$ in the scale of  $L^p$ spaces, $\norm{\Lambda^s \psi_k}{L^p}\sim
2^{ks} \norm{\psi_k}{L^p}$.


Next, we introduce some basic facts from the
theory of the paraproducts, which will be useful for us,
when estimating the contribution of the nonlinearity. \\
Write for any two Schwartz functions $f, g$ and any integer $k$,
\[
P_k(f g)=P_k(\sum_{l_1, l_2} f_{l_1} g_{l_2})=
P_k(\sum_{l_1, l_2: |l_1-l_2|\leq 3} f_{l_1} g_{l_2})+
P_k(\sum_{l_1, l_2: |l_1-l_2|> 3} f_{l_1} g_{l_2})
\]
But
$$
P_k(\sum_{l_1, l_2: |l_1-l_2|\leq 3} f_{l_1} g_{l_2})
= P_k(\sum_{l_1, l_2: |l_1-l_2|\leq 3, \min(l_1, l_2)>k-3} f_{l_1} g_{l_2})
$$
since by the properties of the convolution
$2^{l_1+1}+2^{l_2+1}$ must be at least $2^{k-1}$ and
$$
P_k(\sum_{l_1, l_2: |l_1-l_2|> 3} f_{l_1} g_{l_2})=
P_k(\sum_{l_1, l_2: |l_1-l_2|> 3, |\max(l_1, l_2)-k|\leq 3} f_{l_1} g_{l_2})
$$
since otherwise $supp \widehat{f_{l_1} g_{l_2}}\subset
\{\xi: |\xi|\sim 2^{\max(l_1, l_2)}\}$, which would be away from the set
$\{\xi: |\xi|\sim 2^{k}\}$ and thus $P_k(f_{l_1} g_{l_2})=0$.

All in all,
\begin{equation}\label{eq:10}
\begin{gathered}
 P_k(f g) = P_k(\sum_{l=k-3}^\infty P_{l}f  P_{l-3\leq \cdot\leq l+3} g) +\\
   P_k(\sum_{j=-3}^3 P_{k+j} f P_{<k+j-3} g)+ P_k(\sum_{j=-3}^3 P_{k+j} g P_{<k+j-3} f).
   \end{gathered}
\end{equation}
We will refer to the first term as ``high-high interaction'' term, while
the second and the third terms represent  the ``high-low interaction'' term.
We have the following lemma, which is an application of the representation formula
\eqref{eq:10}.

\begin{lemma} \label{le:1}
For every $0<s\leq 1$, $2<p,q<\infty: 1/p+1/q=1/2$, there is the estimate
$$
|\int P_k \psi_k  [J(\psi)\cdot  \nabla\psi ] dx|
\leq C 2^{k(1-s)}\norm{\psi_k}{L^p}
(\sum_{l\geq k-3} 2^{-s(l-k)} \norm{\Lambda^s \psi_l}{L^2}) \norm{\psi}{L^q}.
$$
for some absolute constant $C$.
\end{lemma}

\begin{proof}
Integration by parts and $\mathop{\rm div}(J(\theta))=0$ yield
$$
\int P_k \psi_k  [J(\psi)\cdot  \nabla\psi ] dx=-
\int \nabla \psi_k  \cdot  P_k[J(\psi) \psi ] dx
$$
At this point, by the boundedness of the Riesz transform on $L^p$, we  treat
$J(\psi)$ as $T\psi$, where $T:L^r\to L^r$ for all $1<r<\infty$
 and ignore the vector structure.
By H\"older's inequality,
$$
\big|\int \nabla \psi_k  P_k [T(\psi)\cdot \psi ] dx\big|
\lesssim 2^k \norm{\psi_k}{L^p}
\norm{P_k [T(\psi) \psi ]}{L^{p'}}.
$$
By  \eqref{eq:10},
\begin{align*}
&\norm{P_k [T(\psi) \psi ]}{L^{p'}} \\
&\leq \norm{\sum_{l=k-3}^\infty P_{l}T \psi  P_{l-3\leq \cdot\leq l+3} \psi}{L^{p'}}
+ \big\|\sum_{j=-3}^3 P_{k+j} (T \psi)  P_{<k+j-3} \psi\big\|_{L^{p'}} \\
&\quad + \big\|\sum_{j=-3}^3 P_{k+j} (\psi)  P_{<k+j-3} T \psi
\big\|_{L^{p'}}\\
& \leq \sum_{l=k-3}^\infty \norm{P_l \psi}{L^2}
\norm{P_{l-3\leq \cdot\leq l+3} \psi}{L^q}
+ \sum_{j=-3}^3 \norm{P_{k+j}\psi}{L^2}\norm{P_{<k+j-3} \psi}{L^q} \\
&\leq   C(\sum_{l\geq k-3} \norm{\psi_l}{L^2}) \norm{\psi}{L^q}.
\end{align*}
The Lemma follows by the observation  $\norm{\psi_l}{L^2}\sim 2^{-ls}
\norm{\Lambda^s \psi_l}{L^2}$ and by reshufling the $2^{ks}$.
\end{proof}

\section{Proof of Theorem \ref{theo:1}}
The main step of the proof of Theorem \ref{theo:1}
is the energy estimate \eqref{eq:3}.

We start with the assumption that we are given a smooth solution
$\theta(t,x)$, corresponding to an initial data $\theta^0$ up to time $T$ and
we will prove \eqref{eq:3} based on it. Assume \eqref{eq:3} for a moment for
such smooth solutions. We will show that the global existence and
uniqueness follows in a standard way from
an approximation argument and  the Constantin-Wu
uniqueness result, Theorem \ref{theo:2}.

Indeed, for a given initial data $\theta^0$, take an approximating sequence in
$\dot{B}^{2-2\alpha}_{2, \infty}\cap L^{2/(2\alpha-1)}$,
 $\{\theta^0_l\}$ of
smooth functions (say in the Schwartz class $\mathcal{S}$). By the Constan\-tin-Wu existence result for
data in $H^s: s>(2-2\alpha)$, we have  global and smooth solutions $\theta_l(t)$. In addition,
they will satisfy the energy estimate \eqref{eq:3}.  Moreover, by the $L^p$ maximum principle, $\norm{\theta_l(t)}{L^q}\leq \norm{\theta_l(0)}{L^q}$ for all $1<q<\infty$, in particular for
$q=2, q=2/(2\alpha-1)$.

Taking weak limits will produce a weak solution
$\theta(t)$ of \eqref{eq:1}, corresponding to initial data $\theta^0$,
so that it  satisfies the energy estimate
\eqref{eq:3} and
$\norm{\theta}{L^\infty_t L^{2/(2\alpha-1)}}\leq \norm{\theta^0}{L^{2/(2\alpha-1)}}$.  This shows the existence of a weak solution with the required smoothness of the initial data.

For the uniqueness part, we should require in addition that
$\theta^0\in L^2(\mathbb{R}^2)$. Then, we will show
$\norm{\theta}{L^2_t H^\alpha_x}<\norm{\theta^0}{L^2}$, which allows us to
apply  the  Constantin-Wu
uniqueness result (Theorem \ref{theo:2}). That is, $\theta$  is the
 unique solution in the class
 $L^\infty([0,T], L^2(\mathbb{R}^2))\cap L^2[[0,T], H^\alpha(\mathbb{R}^2))\cap
L^\infty([0,T], L^{2/(2\alpha-1)}(\mathbb{R}^2))$. Thus, it remains to
prove \eqref{eq:3} for smooth solutions
and \eqref{eq:890}. Since, \eqref{eq:890} is relatively easy,
we start with \eqref{eq:3}.

\subsection{Proof of the energy estimate \eqref{eq:3}}
 Let  $s_0=2-2\alpha$.
Take a Littlewood-Paley operator on both sides of \eqref{eq:1}
\[
\partial_t \theta_k+\kappa (-\Delta)^{\alpha} \theta_k +P_k(J(\theta) \nabla \theta)=0.
\]
Taking a dot product with $\theta_k$ (which is real-valued!) yields
\[
\partial_t \norm{\theta_k}{L^2}^2  +2 \kappa \norm{(-\Delta)^{\alpha/2}\theta_k}{L^2}^2
 +2 \int P_k\theta_k J(\theta) \nabla \theta=0.
\]
By the properties of the Littlewood-Paley operators,
$\norm{(-\Delta)^{\alpha/2}\theta_k}{L^2}^2\sim 2^{2\alpha k} \norm{\theta_k}{L^2}^2$.
For the integral term, use Lemma \ref{le:1}  with $1/p=1/2-s_0/2, 1/q=s_0/2$. We have
\begin{align*}
\big|\int P_k\theta_k J(\theta) \nabla \theta dx\big|
&\leq C 2^{k(1-s_0)}
\norm{\theta_k(t)}{L^q}\big(\sum_{l\geq  k-3}2^{-s_0(l-k)}
\norm{\Lambda^s\theta_l}{L^2}\big) \norm{\theta(t)}{L^p} \\
& \leq C 2^{k(1-s_0)} \norm{\theta_k(t)}{L^q} \sup_{l}
\norm{\Lambda^s\theta_l}{L^2} \norm{\theta(t)}{L^p}.
\end{align*}
By the $L^p$ maximum principle, \eqref{eq:2}, we have
$\norm{\theta(t)}{L^p}\leq \norm{\theta^0}{L^p}$.
Substituting everything in the equation allows us to conclude
\begin{equation} \label{eq:15}
\partial_t \norm{\theta_k}{L^2}^2  +c \kappa 2^{2k\alpha}\norm{\theta_k}{L^2}^2\leq
C 2^{k(1-s_0)} \norm{\theta^0}{L^p}
\norm{\theta_k(t)}{L^q} \sup_{l} \norm{\Lambda^{s_0}\theta_l}{L^2}
\end{equation}
At this point,  the argument splits in two cases with a threshold value of
$\alpha=3/4$. As expected, the case $3/4\leq \alpha<1$ proves out to be slightly
 simpler, so we start with it.

\subsubsection*{The case $3/4\leq \alpha<1$}
The significance of the restriction $\alpha\geq 3/4$ is in the fact that
$s_0=2-2\alpha\in (0,1/2]$. Therefore
$1/q=s_0/2\leq 1/2-s_0/2=1/p$, implying $p\leq q$.  Thus, by
 the Sobolev embedding\footnote{or more
appropriately the Bernstein inequality}, the boundedness of $P_k$ on $L^p$ and the $L^p$ maximum principle  imply
$$
\norm{\theta_k(t)}{L^q}\lesssim
2^{2k(1/p-1/q)}\norm{\theta_k(t)}{L^p}\lesssim
 2^{k(1-2s_0)}\norm{\theta_k(t)}{L^p}\lesssim 2^{k(1-2s_0)}\norm{\theta^0}{L^p}
$$
By \eqref{eq:15}, we infer
\begin{equation}
\label{eq:316}
\partial_t \norm{\theta_k}{L^2}^2  +  c \kappa 2^{2k\alpha}\norm{\theta_k}{L^2}^2\leq
C 2^{k(2-3 s_0)} \norm{\theta^0}{L^p}^2
 \sup_{l} \norm{\Lambda^{s_0}\theta_l(t)}{L^2}
\end{equation}
It is a standard step now to make use of the Gronwal's inequality,
namely rewrite \eqref{eq:316} as
$$
\partial_t (\norm{\theta_k}{L^2}^2 e^{c \kappa 2^{2k\alpha}t})\leq C 2^{k(2-3 s_0)}
e^{ c \kappa 2^{2k\alpha}t} \norm{\theta^0}{L^p}^2
 \sup_{l} \norm{\Lambda^{s_0}\theta_l(t)}{L^2}
$$
and estimate after integration
\begin{equation} \label{eq:17}
\norm{\theta_k(t)}{L^2}^2\leq C_\kappa 2^{k(2-3 s_0-2\alpha)}
\norm{\theta^0}{L^p}^2 \sup_{0\leq z\leq t}  \sup_l \norm{\Lambda^{s_0}\theta_l(z)}{L^2}+
\norm{\theta^0_k}{L^2}^2 e^{- c \kappa 2^{2 k\alpha}t}.
\end{equation}
Note that in the formula above $C_k\sim 1/\kappa$ and $2-3 s_0-2\alpha=-2s_0$. \\
Introduce the functional
$$
E(t)=\sup_{0\leq z\leq t} \sup_k 2^{k s_0} \norm{\theta_k(z)}{L^2}.
$$
Clearly, one may deduce from  \eqref{eq:17} that
$$
E^2(t)\leq E^2(0) + C_\kappa  E(t) \norm{\theta^0}{L^p}^2,
$$
hence
$$
E(t)\leq 2 J(0)+ C_\kappa \norm{\theta^0}{L^p}^2,
$$
which is
\begin{equation} \label{eq:19}
\sup_k 2^{k(2-2\alpha)} \norm{\theta_k(t)}{L^2}\leq
2 \sup_k 2^{k(2-2\alpha)} \norm{\theta^0_k}{L^2}+C_\kappa
\norm{\theta^0}{L^p}^2.
\end{equation}
This is the {\it a priori} estimate of the solution $\theta$,
\eqref{eq:3} for the case $\alpha\in [3/4,1)$.
As we have observed in the beginning of the section,
 it follows that the 2 D quasi-geostrophic
equation \eqref{eq:1} has
global solution with (potentially large)
data in the scale invariant space
$\dot{B}^{2-2\alpha}_{2, \infty}(\mathbb{R}^2)\cap L^{2/(2\alpha-1)}(\mathbb{R}^2)$.

\subsubsection*{The case $1/2< \alpha<3/4$}
In this case, it is clear that $s_0=2-2\alpha\in (1/2, 1)$,
whence $2<q=(1-\alpha)^{-1}<p=(\alpha-1/2)^{-1}$. At this point, we make use of the
Gagliardo-Nirenberg's inequality (see for example \cite{Ni}
or the classical
\cite{Ber}), which  states that whenever $X=(X_0, X_1)_{\theta}$,
say by the complex interpolation
method,  then
$\|\cdot\|_{X}\leq \|\cdot\|_{X_0}^{1-\theta} \|\cdot\|_{X_1}^\theta$. In particular,
applying this to the Sobolev spaces $\dot{W}^{p,k}$, we obtain
$$
\norm{\theta_k}{L^q}\leq C \norm{\Lambda^{2-2\alpha}\theta_k}{L^2}^\gamma
\norm{\Lambda^{-a}\theta_k}{L^p}^{1-\gamma},
$$
with $\gamma=\frac{3-4\alpha}{2-2\alpha}\in (0,1)$ and
$a=\frac{(2-2\alpha)(3-4\alpha)}{2\alpha-1}$. Thus, by $\norm{\Lambda^{-a}\theta_k}{L^p}\sim
2^{-a k}\norm{\theta_k}{L^p}$, whence
it follows
that
$$
\norm{\theta_k}{L^q}\leq C 2^{-k(3-4\alpha)}
\sup_l \norm{\Lambda^{s_0}\theta_l}{L^2}^\gamma
\norm{\theta_k(t)}{L^p}^{1-\gamma}.
$$
Substituting this in \eqref{eq:15} yields
\begin{equation}\label{eq:318}
\partial_t \norm{\theta_k(t)}{L^2}^2 + c  \kappa 2^{2 k\alpha} \norm{\theta_k(t)}{L^2}^2
\leq 2^{k(1-s_0-3+4\alpha)}\sup_l \norm{\Lambda^{s_0}\theta_l}{L^2}^{1+\gamma}\norm{\theta_k(t)}{L^p} \norm{\theta^0}{L^p}^{1-\gamma}.
\end{equation}
Using the maximum principle $\|\theta_k(t)\|_{L^p}\lesssim \|\theta^0\|_{L^p}$,
this reduces to
$$
\partial_t \norm{\theta_k(t)}{L^2}^2 + c  \kappa 2^{2 k\alpha} \norm{\theta_k(t)}{L^2}^2
\leq 2^{k(1-s_0-3+4\alpha)}\sup_l \norm{\Lambda^{s_0}\theta_l}{L^2}^{1+\gamma}\norm{\theta^0}{L^p}^{2-\gamma}.
$$
By the Gronwall's inequality, we deduce
\begin{equation} \label{eq:21}
\norm{\theta_k(t)}{L^2}^2\leq
\norm{\theta^0_k}{L^2}^2e^{-c\kappa 2^{2k\alpha} t}+ C_\kappa 2^{-2k s_0}
\sup_{0\leq z\leq t} \sup_l \norm{\Lambda^{s_0}\theta_l(z)}{L^2}^{1+\gamma}
\norm{\theta^0}{L^p}^{2-\gamma}.
\end{equation}
By using the same energy functional $E(t)$  defined above, we conclude that
$$
E^2(t)\leq E^2(0)+ C_k [E (t)]^{1+\gamma} \norm{\theta^0}{L^p}^{2-\gamma}.
$$
Since $1+\gamma<2$, by Young's inequality
$$
E^2(t)\leq E^2(0)+ \frac{E^2(t)}{2}+ C_{\kappa, \gamma}
\norm{\theta^0}{L^p}^{(4-2\gamma)/(1-\gamma)}.
$$
whence
$$
E(t)\leq 2 E(0)+ C_{\kappa, \gamma}\norm{\theta^0}{L^p}^{(2-\gamma)/(1-\gamma)}.
$$
which is
\begin{equation}\label{eq:25}
\sup_k 2^{k(2-2\alpha)} \norm{\theta_k(t)}{L^2}\leq
\sup_k 2^{k(2-2\alpha)} \norm{\theta^0_k}{L^2}+
C_{\kappa, \gamma}\norm{\theta^0}{L^p}^{(2-\gamma)/(1-\gamma)}.
\end{equation}
Again, this implies \eqref{eq:3} with $M(\alpha)=1/(2\alpha-1)$
and  the problem \eqref{eq:1} has  global solution in
$\dot{B}^{2-2\alpha}_{2, \infty}(\mathbb{R}^2)\cap L^{2/(2\alpha-1)}(\mathbb{R}^2)$,
when the initial data is
taken in the same space.

\subsection{$\theta\in L^\infty([0,\infty), L^2(\mathbb{R}^2))\cap L^2((0,\infty), H^{\alpha}(\mathbb{R}^2))$}
Both of these estimates are classical for smooth solutions, but we
sketch their proofs for completeness.

In fact,  $\theta\in L^\infty([0,\infty), L^2(\mathbb{R}^2))$  follows from the
maximum principle \eqref{eq:2}.
For the second estimate, we multiply the equation by $\theta$ and
integrate in $x$. We get
\[
 \partial_t \norm{\theta(t)}{L^2}^2+ \norm{\Lambda^{\alpha} \theta(t)}{L^2}^2=-\int \theta [J(\theta)\nabla \theta] dx=0
\]
Time integration now yields
$$
\int_0^T \norm{\Lambda^{\alpha}
\theta(t)}{L^2}^2dt\leq \norm{\theta^0}{L^2}^2-\norm{\theta(T)}{L^2}^2 <\norm{\theta^0}{L^2}^2,
$$
whence $\theta\in  L^2((0,\infty), H^{\alpha}(\mathbb{R}^2))$.

\subsection*{Acknowledgement}
 It is a pleasure to thank Ning Ju for several stimulating
discussions on the topic.


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\end{document}