\documentstyle[twoside,amssymb]{article}
% amssymb is used for R in Real numbers
\pagestyle{myheadings}
\markboth{\hfil Quasi-geostrophic type equations \hfil EJDE--1998/16}%
{EJDE--1998/16\hfil Jiahong Wu \hfil}
\begin{document}
\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
{\sc  Electronic Journal of Differential Equations},
Vol.\ {\bf 1998}(1998), No.~16, pp. 1--10. \newline
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp (login: ftp) 147.26.103.110 or 129.120.3.113}
 \vspace{\bigskipamount} \\
  Quasi-geostrophic type equations  \\ with weak initial data 
\thanks{ {\em 1991 Mathematics Subject Classifications:} 35K22, 35Q35, 76U05.
\hfil\break\indent
{\em Key words and phrases:} Quasi-geostrophic equations, Weak data, Well-posedness.
\hfil\break\indent
\copyright 1998 Southwest Texas State University  and University of
North Texas. \hfil\break\indent
Submitted November 26, 1996. Published June 12, 1998. \hfil\break\indent
Supported by NSF grant DMS 9304580 at IAS.} }
\date{}
\author{Jiahong Wu}
\maketitle

\begin{abstract} 
We study the initial value problem for the quasi-geostrophic type
equations 
$$
\frac{\partial \theta}{\partial t}+u\cdot\nabla\theta +
(-\Delta)^{\lambda}\theta=0,\quad \mbox{on}\quad {\Bbb R}^n
\times (0,\infty),
$$
$$
\theta(x,0)=\theta_0(x), \quad x\in {\Bbb R}^n\,,
$$
where $\lambda(0\le \lambda \le 1)$ is a fixed parameter and
$u=(u_j)$ is divergence free and 
determined from $\theta$ through the Riesz transform
$u_j=\pm {\cal R}_{\pi(j)}\theta$, with $\pi(j)$ a permutation of 
$1,2,\cdots,n$. The initial data $\theta_0$ is taken in the Sobolev 
space  $\dot{L}_{r,p}$ with negative indices.   
We prove local well-posedness when 
$$ \frac{1}{2}<\lambda \le 1,\quad 
1<p<\infty, \quad \frac{n}{p}\le 2\lambda -1, \quad r=\frac{n}{p
}-(2\lambda-1) \le 0\,.
$$
We also prove that the solution is global if $\theta_0$ is sufficiently small.
\end{abstract}

\date{}
\maketitle

\newtheorem{thm}{Theorem}[section]
\newtheorem{prop}[thm]{Proposition}
\newtheorem{define}[thm]{Definition}
\newtheorem{rem}[thm]{Remark}
\newtheorem{lemma}[thm]{Lemma}
\def\theequation{\thesection.\arabic{equation}}

\section{Introduction}\setcounter{equation}{0}
\label{sec:1}
In this paper we study the initial value problem (IVP) of the
dissipative quasi-geostrophic type (QGS) equations
\begin{eqnarray}
&\frac{\partial \theta}{\partial t}+u\cdot\nabla\theta +
(-\Delta)^{\lambda}\theta=0,\quad \mbox{on}\quad {\Bbb R}^n
\times (0,\infty),& \label{eq1} \\
&\theta(x,0)=\theta_0(x), \quad x\in {\Bbb R}^n& \label{eq2}
\end{eqnarray}
where $\lambda (0\le \lambda\le 1)$ is a fixed parameter and
the velocity $u=(u_1,u_2,\cdots,u_n)$
is divergence free and determined from $\theta$ by 
\begin{equation}\label{ndu}
u_j=  \pm {\cal R}_{\pi(j)}\theta,\qquad\mbox{$\pi(j)$ is a permutation of
 $1,2,\cdots,n$} 
\end{equation}
where $u_j$ may take either $+$ or $-$ sign and 
${\cal R}_j=\partial_j(-\Delta)^{-1/2}$ are the Riesz 
transforms. Here Riesz potential operator $(-\Delta)^\alpha$ is defined
through the Fourier transform:
\begin{eqnarray*}
&\widehat{f}(\xi)=\int e^{-2\pi ix\cdot \xi} f(x)\,dx& \\
&\widehat{((-\Delta)^\alpha f)}(\xi) = (2\pi|\xi|)^{2\alpha} \widehat{f}(\xi)&
\end{eqnarray*}

A particularly important special case of (\ref{eq1}) is the 2-D 
dissipative quasi-geostrophic equations in which the velocity $u=
(u_1,u_2)$ can also be defined  through the
stream function $\psi$:
\begin{equation}\label{2du}
u=(u_1, u_2)=\left(-\frac{\partial \psi}{\partial x_2},\frac{
\partial \psi}{\partial x_1}\right), \quad
(-\Delta)^{1/2}\psi=-\theta
\end{equation}
The 2-D QGS equations are derived from more general quasi-geostrophic 
approximations for flow in rapidly rotating 3-D half space, which
in some important cases
reduce to the evolution equation for the temperature  on the 2-D
boundary given in (\ref{eq1}), 
(\ref{eq2}),(\ref{2du}) (\cite{P,CMT}).
The scalar $\theta$ represents the potential temperature and $u$
is the fluid velocity. These equations have been under active 
investigation because of mathematical importance and potential 
applications in meteorology and oceanography (\cite{P,CMT,CP,HPGS}).
As pointed out in \cite{CMT}, the non-dissipative 2-D QGS equations are 
strikingly analogous to the 3-D Euler equations and thus 
serve as a simple model in seeking possible singular solutions. 

We are interested mainly in the well-posedness result for initial data
$\theta_0$ in homogeneous Lebesgue spaces, $\theta_0\in \dot{L}_{r,p}({\Bbb R}
^n)$ (defined below).  By well-posedness we
mean existence, uniqueness and persistence (i.e. the solution describes
a continuous curve belonging to the same space as does the initial data)
and continuous dependence on the data.
 
Here the homogeneous Lebesgue space $\dot{L}_{s,q}({\Bbb R}^n)$ consists of all
$v$ such that 
$$
(-\Delta)^{\frac{s}{2}} v \in L^q, \quad s\in {\Bbb R},\quad 1\le q<\infty\,,
$$
and the standard norm is given by 
$$
\|v\|_{s,q}=\|(-\Delta)^{s/2}v\|_{L^q}\,.
$$
These spaces are also called the spaces of Riesz potentials.
Kato and Ponce \cite{kp1} consider the Navier-Stokes 
equations with initial data in this type of spaces. 

We prove that 
if $\frac{1}{2}< \lambda\le 1$ and $\theta_0\in \dot{L}_{r,p}$ with 
$r,p$ satisfying 
$$
1<p<\infty, \quad \frac{n}{p}\le 2\lambda -1, \quad r=\frac{n}{p
}-(2\lambda-1)\le 0\,,
$$
then the IVP (\ref{eq1}), (\ref{ndu}), (\ref{eq2}) is locally well-posed. 
The solution is global if $\theta_0$ is sufficiently small. 
The detailed statements
are given in Theorem \ref{thm:2.1} of the next section. 
 

Although there is a large body of literature on quasi-geostrophic equations (
\cite{P,CP,HPGS,CMT}), not 
many rigorous mathematical results concerning the solutions 
have been obtained. 
In \cite{CMT} Constantin-Majda-Tabak proved finite time 
existence results for smooth data  and developed mathematical criteria
characterizing blowup  for the 2-D non-dissipative QGS equation. 
In \cite{Re} 
Resnick obtained solutions of 2-D QGS equations with $L^2$ data on  
periodic domain by using Galerkin approximation. 
In a previous paper \cite{Wu},
the vanishing dissipation limits and Gevrey class regularity \cite{FT}
  for the 
2-D dissipative QGS equations are obtained. 
In this paper we consider the IVP of  the general $n$-D QGS type equations 
(defined by (\ref{eq1}), (\ref{ndu}, 
(\ref{eq2})) with initial data in Sobolev 
spaces of negative indices and establish local well-posedness 
results. For sufficiently 
small initial data, the solution is global. 
By taking $n=2$ and $p=2$, the well-posedness reduces to the $L^2$ 
results in 2-D. 

The main result is presented in the next section, and it is proven using 
the contraction-mapping principle.


\section{Well-posedness}\setcounter{equation}{0}
\label{sec:well}
We need to use the spaces of weighted continuous functions in time, which
have been introduced by Kato, Ponce and others in solving
the Navier-Stokes equations (\cite{k2,kp1,kp2}). 

\begin{define}
Suppose $T>0$ and $\alpha\ge 0$ are real numbers. The spaces 
$ C_{\alpha,s,q}$ and $\dot{C}_{\alpha,s,q}$ are defined as
$$
C_{\alpha,s,q} \equiv \{f \in C((0,T), \dot{L}_{s,q}), \quad
\|f\|_{\alpha,s,q} <\infty\}\,,
$$
where the norm is given by
$$
\|f\|_{\alpha,s,q}=\sup \{t^\alpha \|f\|_{s,q}, \quad t\in (0,T)\}\,.
$$
Note that $\dot{C}_{\alpha,s,q}$ is a subspace of $C_{\alpha,s,q}$:
$$
\dot{C}_{\alpha,s,q}\equiv\{f\in C_{\alpha,s,q}, \quad \lim_{t\to 0}
t^\alpha\|f(t)\|_{s,q}= 0\}\,.
$$
When $\alpha=0$, the spaces $\bar{C}_{s,q}$ are used for $BC([0,T),\dot{L}_{s,q})$.
\end{define}

These spaces are important in uniqueness and local existence problems 
(\cite{k2,kp1,kp2}).  Notice that
$f\in C_{\alpha,s,q}$ (resp. $f\in \dot{C}_{\alpha,s,q}$) implies
that $\|f(t)\|_{s,q}=O(t^{-\alpha})$ (resp. $o(t^{-\alpha})$).  

The main result of this section is the well-posedness theorem that states
\begin{thm}\label{thm:2.1}
Assume that $\lambda>1/2$ and $\theta_0\in \dot{L}_{r,p}$ with $r,p$ satisfying
\begin{equation}\label{res}
1<p<\infty, \quad \frac{n}{p}\le 2\lambda -1, \quad r=\frac{n}{p
}-(2\lambda-1) (\le 0)
\end{equation}
Then there exists $T=T(\theta_0)$ and a unique solution $\theta(t)$ of 
the IVP (\ref{eq1}),(\ref{ndu}),
(\ref{eq2}) in the time interval $[0,T)$ satisfying
$$
\theta\in Y_T\equiv (\cap_{p\le q<\infty}\bar{C}_{\frac{n}{q}-(2\lambda-1)
, q})\cap (\cap
_{p\le q<\infty}\cap_{s>\frac{n}{q}-(2\lambda-1)}
\dot{C}_{\left(s-\frac{n}{q}+(2\lambda-1)\right)
/(2\lambda),s,q})
$$
In particular,
$$
\theta\in BC([0,T), \dot{L}_{r,p})\cap (\cap_{s>r} C((0,T), \dot{L}_{s,p}))\,.
$$
Furthermore, for some neighborhood $V$ of 
$\theta_0$,  the mapping  
$$
{\frak P}: V\longmapsto Y_{T}:\quad \theta_0\longmapsto \theta
$$
is Lipschitz.
\end{thm}

\begin{rem}
If $\|\theta_0\|_{r,p}$ is small enough, then we can take $T=\infty$.
\end{rem}

We prove this theorem by the method of integral equations and 
contraction-mapping arguments. Following standard practice (\cite{G, 
GMO, k1,kp1}), 
we write the QGS equation (\ref{eq1}) into the integral form:
\begin{equation}\label{int}
\theta=K\theta_0(t) -G(u,\theta)(t)\equiv e^{-\Lambda^{2\lambda}t}\theta_0
-\int_{0}^{t} e^{-\Lambda^{2\lambda}(t-\tau)}(u\cdot\nabla\theta)(\tau)d\tau\,,
\end{equation}
where $K(t)=e^{-\Lambda^{2\lambda}t}$ is the solution operator of the 
linear equation 
$$
\partial_t\theta +\Lambda^{2\lambda}\theta =0,\quad\mbox{with}\quad 
\Lambda=(-\Delta)^{1/2}\,.
$$
We observe that $u\cdot\nabla \theta=\sum_{j}u_j\partial_j\theta=
\nabla\cdot(u\theta)$ provided that $\nabla\cdot u=0$. This provides an
alternative expression for $G$:
$$
G(u,\theta)(t)=G(u\theta)(t)=\int_{0}^{t}\nabla\cdot e^{-\Lambda^{2\lambda}(t-\tau)}
(u\theta)(\tau)d\tau\,.
$$

We shall solve (\ref{int}) in the spaces of weighted continuous functions 
in time introduced in the beginning of this section. To this end we need
estimates for the operators $K$ and $G$ acting between these spaces. These
are established in the two propositions that follow.

\begin{prop}\label{U}
\begin{description}
\item{(i)} For $1\le q<\infty$ and $s\in {\Bbb R}$, 
the operator $K$ maps continuously from $\dot{L}_{s,q}$ into $\bar{C}_{s,q}
\equiv BC([0,\infty), \dot{L}_{s,q})$.

\item{(ii)} If $ q_1,q_2,s_1,s_2$ and $\alpha_2$ satisfy $
q_1\le q_2, \quad s_1\le s_2$, and 
$$
\alpha_2=\frac{1}{2\lambda}(s_2-s_1)+\frac{1}{2\lambda}\left(\frac{n}{q_1}
-\frac{n}{q_2}\right)\,,
$$
then $K$ maps continuously from $\dot{L}_{s_1,q_1}$ to $\dot{C}_{\alpha_2,
s_2,q_2}$ (When $\alpha_2=0$, $\dot{C}$ should be replaced by $\bar{C}$).
\end{description}
\end{prop}

\noindent{\bf Proof.}  To prove Assertion (i),
it suffices to prove that for some constant $C$, 
$$
\|K\phi (t)\|_{L^q} \le C\|\phi\|_{L^q},\quad\mbox{for any $t\in[0,\infty)$}\,,
$$
which can be established  using the Young's inequality
$$
\|K\phi(t)\|_{L^q}\le \|K(t)\|_{L^1}\|\phi\|_{L^q}
$$
and the fact that
$$
\widehat{K}(t)(\xi)=e^{-|2\pi \xi|^{2\lambda}t},\quad \|K(t)\|_{L^1}=
\widehat{K}(t)(0)=1\,.
$$

To prove Assertion (ii), we first note that the operator $
(-\Delta)^{s_0/2}K(t)$
has the property
\begin{equation}\label{ppp}
\|(-\Delta)^{s_0/2}K(t)\|_{L^q({\Bbb R}^n)} 
\le C t^{\frac{1}{2\lambda}\left(-s_0- n(1-\frac{1}{q}
)\right)}\,,
\end{equation}
where $s_0\ge 0$, $q\in [1,\infty)$ and $C$ is a constant. The proof of this
 property is similar to that for the heat operator (\cite{G,GMO,kp1}). 
 To show (ii),it suffices show that for some constant $C$, 
$$
\sup_{t\in[0,T)} t^{\alpha_2}\|(-\Delta)^{\frac{s_0}{2}}K\phi(t)\|_{L^
{q_2}} \le C\|\phi\|_{L^{q_1}}
$$
with $s_0=s_2-s_1\ge 0$. 
This can be proved using the property (\ref{ppp}) and 
Young's inequality
$$
\|(-\Delta)^{\frac{s_0}{2}}K\phi(t)\|_{L^{q_2}} \le C 
\|(-\Delta)^{\frac{s_0}{2}}K(t)\|_{L^q} \|\phi\|_{L^{q_1}}
$$
with $\frac{1}{q}=1-\left(\frac{1}{q_1}-\frac{1}{q_2}\right)$. \qquad$\Box$
\bigskip
 
Now we give estimates for the operator
$$
G(g)(t)=\int_{0}^{t}\nabla\cdot K(t-\tau)g(\tau)d\tau
$$

\begin{prop}\label{G}
If $q_1,q_2,s_1,s_2, \alpha_1$ and $\alpha_2$ satisfy $q_1\le q_2$,
\begin{eqnarray*}
&s_1-1 \le s_2< s_1 +2\lambda -1 -\left(\frac{n}{q_1}-\frac{n}{q_2}\right)&\\
&\alpha_1<1,\quad\mbox{and}\quad  \alpha_2=\alpha_1-1 +\frac{1}{2\lambda}\left[
s_2-s_1 +1+\frac{n}{q_1}-\frac{n}{q_2}\right]\,,&
\end{eqnarray*}
then $G$ is a continuous mapping from $\dot{C}_{\alpha_1,s_1,q_1}$
to $\dot{C}_{\alpha_2,s_2,q_2}$.
\end{prop}

\noindent{\bf Proof.} 
Let $g\in \dot{C}_{\alpha_1, s_1, q_1}$. Then clearly, 
$$
\|G(g)\|_{\alpha_2,s_2,q_2}=\sup_{t\in [0,T)} t^{\alpha_2}
\int_{0}^{t}\|(-\Delta)^{\frac{(1+s_0)}{2}}
K(t-\tau)\left((-\Delta)^{\frac{s_1}{2}}g(\tau)
\right)\|_{L^{q_2}}d\tau
$$
where $s_0=s_2-s_1$. Using Young's inequality, 
$$
\|G(g)\|_{\alpha_2,s_2,q_2}\le \sup_{t\in [0,T)} t^{\alpha_2}
\int_{0}^{t}\|(-\Delta)^{\frac{(1+s_0)}{2}}K(t-\tau)\|_{L^q}
\|\left((-\Delta)^{\frac{s_1}{2}}g(\tau)\right)\|_{L^{q_1}}d\tau
$$
with $\frac{1}{q}=1-\left(\frac{1}{q_1}-\frac{1}{q_2}\right)$. 
If $s_0+1\ge 0$, we can use
the property (\ref{ppp}) of operator $K$ and obtain
\begin{eqnarray*}
\|G(g)\|_{\alpha_2,s_2,q_2}&\leq& C \|g\|_{\alpha_1,s_1,
q_1} \sup_{t\in [0,T)} t^{\alpha_2}\int_{0}^{t}(t-\tau)^{-\frac{1}{2\lambda}
\left(s_0+1 +n(1-\frac{1}{q})\right)}\tau^{-\alpha_1}d\tau \\
&\leq&  C \|g\|_{\alpha_1,s_1,q_1} 
\sup_{t\in [0,T)} t^{\alpha_2-\alpha_1 +1 -\frac{1}{2\lambda}\left(
s_0+1 +n(1-\frac{1}{q})\right)} \times \\
&& B\left(1-\frac{1}{2\lambda}\left[s_0+1+n(1-\frac
{1}{q})\right], 1-\alpha_1\right)\,,
\end{eqnarray*}
where $C$ is a constant and $B(a,b)$ is the Beta function 
$$
B(a,b)=\int_{0}^{1}(1-x)^{a-1} x^{b-1}\,dx\,.
$$
By noticing that $B(a,b)$ is finite when $a>0$, $b>0$ and that
$$
s_0=s_2-s_1,\quad 1-\frac{1}{q}=\frac{1}{q_1}-\frac{1}{q_2}
$$
we obtain 
$$
\|G(g)\|_{\alpha_2,s_2,q_2}\le C \|g\|_{\alpha_1,s_1,q_1}\,, 
$$
if the indices satisfy $
0\le s_2-s_1+1< 2\lambda -\frac{n}{q_1}-\frac{n}{q_2}$, 
$\alpha_1<1$,  and 
$$\alpha_2=\alpha_1-1 +\frac{1}{2\lambda}\left[
s_2-s_1+1+\frac{n}{q_1}-\frac{n}{q_2}\right]\,.
$$
\hfill$\Box$

To prove Theorem \ref{thm:2.1}, we also need the following 
singular integral operator estimate whose proof can be found in \cite{St}.
\begin{lemma}\label{uo}
For $u=(u_j)$ with $u_j=\pm{\cal R}_{\pi(j)}\theta$( $j=1,2,\cdots,n)$,
where ${\cal R}_j$ are the Riesz transforms, we have the estimate
$$
\|u\|_{L^q}\le C_q \|\theta\|_{L^q}, \quad 1<q<\infty
$$
with $C_q$ a constant depending on $q$. 
\end{lemma}

\noindent{\bf Proof of Theorem \ref{thm:2.1}.} We distinguish between two cases:
$r<0$, and $r=0$. For $r<0$, we define
$$
X= \bar{C}_{r,p}\cap \dot{C}_{-\frac{r}{2\lambda},0,p}
$$
with norm for $\theta\in X$ given by 
$$
\|\theta\|_{X}=\|\theta-K\theta_0\|_{0,r,p} +\|\theta\|_{-\frac{r}
{2\lambda},0,p}\,,
$$
and the complete metric space $X_R$ to be the closed ball in $X$ of radius
$R$. 
Consider the operator ${\cal A}(\theta,\theta_0):X_R\times V\longmapsto X$
$$
{\cal A}(\theta,\theta_0)(t)=K\theta_0(t) - G(u\theta)(t),\quad  0<t<T \,,
$$
where $V$ is some neighborhood of $\theta_0$ in $\dot{L}_{r,p}$ and $T$ 
will be chosen.
Using Proposition \ref{U} by substituting $s=r, q=p$ in (i) and 
$$
q_1=q_2 =p, \quad s_1=r, \quad s_2=0, \quad \alpha_2=-\frac{r}{2\lambda}
$$
in (ii), we find that $K\tilde{\theta}_0(t)\in X_R$ for 
$\tilde{\theta}_0\in V$  if $T$ is taken small enough and $V$ is chosen 
properly. 

To estimate $G$, we use Proposition \ref{G} with 
$$
q_1=\frac{p}{2},\quad q_2=p,\quad s_1=0, \quad s_2=l+r,\quad \alpha_1=-\frac
{r}{\lambda}, 
\quad \alpha_2=\frac{l}{2\lambda}
$$
to obtain for a constant $c$  such that
$$
\|G(u\theta)\|_{\frac{l}{2\lambda}, l+r, p}\le c\|u\theta\|_{-\frac{r}{\lambda}, 0,\frac{p}
{2}}\le c\|u\|_{-\frac{r}{2\lambda},0,p}\|\theta\|_{-\frac{r}{2\lambda},0,p}
$$
for $l\in [0,-2r)$.  To estimate $u$ in terms of $\theta$, we use 
Lemma \ref{uo}, i.e. for $1<p<\infty$, 
$$
\|u\|_{L^p}\le C_p \|\theta\|_{L^p}
$$
and eventually we obtain
$$
\|G(u\theta)\|_{\frac{l}{2\lambda}, l+r, p}\le cC_p\|\theta\|_{-\frac{r}{2
\lambda},0,p}
^{2} \le cC_p R^2\,.
$$
Notice that the restrictions (\ref{res})
 on $r,p$ are necessary 
in order to apply Propositions \ref{U}, \ref{G} and Lemma \ref{uo}.
 
Furthermore,
$$
\|{\cal A}(\theta,\theta_0)-{\cal A}(\tilde{\theta},\theta_0)\|_{X}
= \|G(u\theta)-G(\tilde{u}\tilde{\theta})\|_{X}\,,
$$
where $\tilde{u}=(\tilde{u}_j)$ with $\tilde{u}_j=\pm {\cal R}_{\pi(j)}\tilde{\theta}
(j=1,2,\cdots,n)$.
Using Proposition \ref{G} again, 
\begin{eqnarray*}
\|{\cal A}(\theta,\theta_0)-{\cal A}(\tilde{\theta},\theta_0)\|_{X}
&\le& \|G((\tilde{u}-u)\tilde{\theta})\|_X + \|G(u(\theta-\tilde{\theta}))\|_X\\
&\le& c\left(\|\tilde{u}-u\|_{X}\|\tilde{\theta}\|_X + \|\theta-\tilde{\theta}\|_X
\|u\|_{X}\right)\,.
\end{eqnarray*}
Since $(\tilde{u}-u)_j=\pm {\cal R}_{\pi(j)}(\tilde{\theta}-\theta)$, 
Lemma \ref{uo}  implies 
$$
\|u\|_X\le C_p\|\theta \|_X,\quad 
\|\tilde{u}-u\|_{X} \le C_p \|\tilde{\theta}-\theta\|_{X}\,.
$$
Therefore, for constant satisfies $C=cC_p$ and 
$$
\|{\cal A}(\theta,\theta_0)-{\cal A}(\tilde{\theta},\theta_0)\|_{X}
\le C(\|\tilde{\theta}\|_X + \|\theta\|_X)\|\tilde{\theta}-\theta\|_X\,.
$$ 

Our above estimates show that if we choose $T$ small and $R$ 
appropriately, then ${\cal A}$ maps $X_R$ into itself and is a contraction.
Consequently there exists a unique fixed point $\theta\in X_R$: $
\theta ={\frak P}(\theta_0)$ satisfying $\theta= {\cal A}(\theta,
\theta_0)$. It is easy to see from these estimates that the uniqueness 
can be extended to all $R'$ by further reducing the the time interval and
thus to the whole $X$.

To prove the Lipschitz continuity  of ${\frak P}$ on $V$, let $\theta=
{\frak P}(\theta_0)$ and $\zeta={\frak P}(\zeta_0)$ for $\theta_0, \zeta_0
\in V$. Then 
\begin{eqnarray*}
\lefteqn{\|\theta-\zeta\|_{X}=\|{\cal A}(\theta, \theta_0)-{\cal A}(\zeta,\zeta_0)\|
_X } && \\
&\le& \|{\cal A}(\theta,\theta_0)-{\cal A}(\zeta, \theta_0)\|_X
+ \|{\cal A}(\zeta,\theta_0)-{\cal A}(\zeta,\zeta_0)\|_{X} \\
&\le& \gamma \|\theta-\zeta\|_X + \|K(\theta_0-\zeta_0)\|_X
\end{eqnarray*}
Since ${\cal A}$ is a contraction, $\gamma<1$. Therefore, the
 asserted property 
is obtained by applying Proposition \ref{U} to the second term of the last 
inequality. 

To show that $\theta$ is in the asserted class $Y_T$ (defined 
in Theorem \ref{thm:2.1}),  we notice that
$$
\theta ={\cal A}(\theta, \theta_0)\equiv K\theta_0 - G(u\theta)\,.
$$
We apply Proposition \ref{U} twice to $K\theta_0$ to show that
$$
K\theta_0\in \bar{C}_{\frac{n}{q}-(2\lambda-1),q}\,,\qquad K\theta_0
\in \dot{C}_{\left(s-\frac{n}{q}+(2\lambda-1)\right)
/(2\lambda),s,q}
$$ 
for any $p \le q <\infty$ and $s>\frac{n}{q}-(2\lambda-1)$. To show the 
second part  
\begin{equation}\label{2nd}
G(u\theta)\in \bar{C}_{\frac{n}{q}-(2\lambda-1),q}, 
\quad p\le q<\infty
\end{equation}
we use 
Proposition \ref{G} with 
$$
q_1=\frac{p}{2},\quad q_2=q,\quad s_1=0,\quad s_2=\frac{n}{q}-(2\lambda-1),
\quad \alpha_1=-\frac{r}{\lambda},\quad \alpha_2=0
$$
and  obtain  
$$
\|G(u\theta)\|_{0, \frac{n}{q}-(2\lambda-1),q}\le C\|u\theta\|_
{-\frac{r}{\lambda},0,\frac{p}{2}}\le C\|u\|_{-\frac{r}{2\lambda},0 ,p}
\|\theta\|_{-\frac{r}{2\lambda},0 ,p}\,.
$$
The asserted property (\ref{2nd})
 is established after we apply Lemma \ref{uo} to $u$. 

Once again, we apply Proposition \ref{G} with 
\begin{eqnarray*}
&q_1=\frac{p}{2},\quad q_2=q,\quad s_1=0,\quad s_2=s, & \\
&\alpha_1=-\frac{r}{\lambda},\quad \alpha_2=\frac{1}{2\lambda}\left[
s-\left(\frac{n}{q}-(2\lambda-1)\right)\right] &
\end{eqnarray*}
to show that 
\begin{equation}\label{3rd}
G(u\theta)\in \dot{C}_{\left(s-\frac{n}{q}+(2\lambda-1)\right)
/(2\lambda),s,q}
,\quad\mbox{for $s>\frac{n}{q}-(2\lambda-1)$},
\end{equation}
but $s$ should also satisfy
$$
s< 2\lambda -1 -\left(\frac{2n}{p} -\frac{n}{q}\right)
$$
as required by Proposition \ref{G}. For large $s$, (\ref{3rd})
 can be shown by 
an induction process (see an analogous argument in \cite{k2}). 

We now deal with the case $r=0$. Define 
$$
X=\bar{C}_{0,p}\cap \dot{C}_{\frac{1}{4}, 0, \frac{4\lambda-2}{3\lambda-2}p}
$$
with the norm 
$$
\|\theta\|_X =\|\theta- K\theta_0\|_{0,0,p} + \|\theta\|_{
\frac{1}{4}, 0, \frac{4\lambda-2}{3\lambda-2}p}\,.
$$
For $\theta\in X_R$, we have by Proposition \ref{G},
\begin{eqnarray*}
\|G(u\theta)\|_X &=&\|G(u\theta)\|_{0,0,p} +\|G(u\theta)\|_{
\frac{1}{4}, 0, \frac{4\lambda-2}{3\lambda-2}p}  \\
& \le& c\|u\theta\|_{\frac{1}{2}, 0, \frac{2\lambda-1}{3\lambda-2}p}\\
&\le & c\|u\|_{\frac{1}{4}, 0, \frac{4\lambda-2}{3\lambda-2}p}\|\theta\|_
{\frac{1}{4}, 0, \frac{4\lambda-2}{3\lambda-2}p}\,.
\end{eqnarray*}
Here $c$ is a constant which may depend on the indices $\lambda$, $p$, and $n$.
Using Lemma \ref{uo} again, we obtain a constant $C$ such that
$$
\|G(u\theta)\|_X \le C \|\theta\|_{X}^{2} \le C R^2\,.
$$
Once the above estimates have been established, the rest of 
the proof in this case is similar to that described in the case $r<0$.
\qquad$\Box$
 
\paragraph{Acknowledgments}
 I would like to thank Professor P. Constantin for teaching me the
quasi-geostrophic equations and Professor C. Kenig for his helpful suggestions.



\begin{thebibliography}{99}

\bibitem{CP} J. Charney, N. Phillips, {\em Numerical integrations of the
quasi-geostrophic equations for barotropic and simple baroclinic
flows}, J. Meteorol., {\bf 10}(1953), 71-99.
\bibitem{CMT} P. Constantin, A. Majda, E. Tabak, {\em 
Formation of strong fronts in the 2-D quasi-geostrophic thermal
active scalar}, Nonlinearity, {\bf 7}(1994), 1495-1533
\bibitem{FT} C. Foias, R. Temam, {\em Gevrey class regularity for 
the solutions.
of the Navier-Stokes equations}, J. Funct. Anal., {\bf 87}(1989), 359-369.
\bibitem{G} Y. Giga, {\em Solutions for semilinear parabolic equations
in $L^p$ and regularity of weak solutions of the Navier-Stokes system},
J. Diff. Eq., {\bf 62}(1986), 186-212.
\bibitem{GMO} Y. Giga, T. Miyakawa, H. Osada, {\em Two dimensional
Navier-Stokes flow with measures as initial vorticity}, Arch. Rational
Mech. Anal., {\bf 104}(1988), 223-250. 
\bibitem{HPGS} I. Held, R. Pierrehumbert, S. Garner, K. Swanson, {\em
Surface quasi-geostrophic dynamics}, J. Fluid Mech., {\bf 282}(1995), 1-20.
\bibitem{k1} T. Kato, {\em Strong $L^p-$ solutions of the Navier-Stokes 
equation in ${\Bbb R}^m$ with applications to weak solutions}, 
Math. Z., {\bf 187}(1984), 471-480.
\bibitem{k2} T. Kato, {\em The Navier-Stokes solutions for an incompressible fluid in ${\Bbb R}^2$ with measure as the initial vorticity}, 
Diff. Integral Eq., {\bf 7}(1994), 949-966.
\bibitem{kf}  T. Kato, H. Fujita, {\em On the nonstationary
Navier-Stokes equations}, Rend. Sem. mat. Univ. Padova, {\bf 32}(1962),
243-260.
\bibitem{kp1} T. Kato, G. Ponce, {\em The Navier-Stokes equations with 
weak initial data}, Int. Math. Research Notices, {\bf 10}(1994), 435-444.
\bibitem{kp2} T. Kato, G. Ponce, {\em Well-posedness of the Euler and the
Navier-Stokes equations in the Lebesgue spaces $L^{p}_{s}({\Bbb R}^2)$},
Rev. Mat. Iber. {\bf 2}(1986), 73-88.
\bibitem{P} J. Pedlosky, ``Geophysical Fluid Dynamics", Springer, New York,
1987.
\bibitem{Re} S. Resnick, ``Dynamical Problems in Non-linear Advective
Partial Differential Equations", Ph.D. thesis, University of Chicago, 1995
\bibitem{St} E.M. Stein, ``Singular Integrals 
and Differentiability Properties
of Functions", NJ: Princeton University Press, 1970
\bibitem{Wu} J. Wu, {\em Inviscid limits and regularity estimates 
for the solutions of the 2-D dissipative quasi-geostrophic equations},
Indiana Univ. Math. J., {\bf 46}(1997), in press.
\end{thebibliography}

{\sc Jiahong Wu}  \\
School of Mathematics, Institute for Advanced Study\\
Princeton, NJ 08540. USA\\
E-mail address: jiahong@math.utexas.edu

\end{document}