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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 $00$, 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 $^{-1/2\alpha}$. 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_{ 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_{(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