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\markboth{\hfil Dissipative quasi-geostrophic equations \hfil EJDE--2001/56}
{EJDE--2001/56\hfil Jiahong Wu \hfil}
\begin{document}
\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
{\sc  Electronic Journal of Differential Equations},
Vol. {\bf 2001}(2001), No. 56, pp. 1--13. \newline
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp  ejde.math.swt.edu  (login: ftp)}
 \vspace{\bigskipamount} \\
 %
  Dissipative quasi-geostrophic equations with \\ $L^p$ data 
 %
\thanks{ {\em Mathematics Subject Classifications:} 35Q35, 76U05, 86A10.
\hfil\break\indent
{\em Key words:} 2D quasi-geostrophic equation, initial-value problem,
existence, uniqueness.
\hfil\break\indent
\copyright 2001 Southwest Texas State University. \hfil\break\indent
Submitted June 18, 2001. Published August 3, 2001.} }
\date{}
%
\author{ Jiahong Wu }
\maketitle

\begin{abstract} 
 We seek solutions of the initial value  problem for the 
 2D dissipative quasi-geostrophic (QG) equation with $L^p$ initial data. 
 The 2D dissipative QG equation is a two dimensional model of the 
 3D incompressible Navier-Stokes equations.
 We prove global existence and uniqueness of regular solutions for 
 the dissipative QG equation with sub-critical powers.
 For the QG equation with critical or super-critical powers, 
 we establish  explicit global $L^p$ bounds for its solutions and 
 conclude that any possible  finite time singularity must occur in 
 the first order derivative. 
\end{abstract}

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\newtheorem{lemma}[thm]{Lemma}

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\section{Introduction} \label{sec:1}

We study in this paper the 2D dissipative 
quasi-geostrophic (QG)
equation 
\begin{equation}\label{qgse}
\partial_t \theta + u\cdot\nabla \theta + \kappa (-\Delta)^\alpha \theta =f,
\quad x\in \mathbb{R}^2,\quad t>0,
\end{equation}
where $\kappa>0$ is the diffusivity coefficient, $\alpha\in [0,1]$ is a
fractional power, and $u=(u_1,u_2)$ is the velocity field determined 
from $\theta$ by a stream function $\psi$ via the auxiliary relations
\begin{equation}\label{u}
u= (u_1,u_2) =\left(-\frac{\partial \psi}{\partial x_2},
\frac{\partial \psi}{\partial 
x_1}\right)\quad \mbox{and}\quad (-\Delta)^{1/2} \psi=-\theta. 
\end{equation}
A fractional power of the Laplacian $(-\Delta)^\beta$ is defined by
$$
\widehat{(-\Delta)^\beta f}(\xi) = (2\pi\,|\xi|)^{2\beta} \widehat{f}(\xi),
$$
where $\widehat{f}$ denotes the Fourier transform of $f$. One may 
consult the book of Stein \cite[p.117]{St} for more details. For notational
convenience, we will denote $(-\Delta)^{1/2}$ by $\Lambda$.
The relation in (\ref{u}) can then be identified with
$$
u=\left(\partial_{x_2} \Lambda^{-1} \theta,\, -\partial_{x_1} 
\Lambda^{-1} \theta\right)  = (-{\cal R}_2 \theta,\, {\cal R}_1\theta),
$$ 
where ${\cal R}_1$ and ${\cal R}_2$ 
are the Riesz transforms \cite[p.57]{St}. 


Equation (\ref{qgse}) is the dissipative version of the inviscid QG
equation derived by reducing the general QG models describing 
atmospheric and oceanic fluid flow under special circumstances
of physical interest (\cite{Pe},\cite{CMT}). Physically, the scalar 
$\theta$ represents the potential 
temperature, $u$ is the fluid velocity and $\psi$ can be identified with
the pressure. Mathematically, the 2D QG equation serves as a 
lower dimensional model of the 3D Navier-Stokes equations because of the 
striking similarity between the behavior of its solution and that of
the potentially singular solutions of the 3D hydrodynamic equations.


Our aim of this paper is to establish global existence and uniqueness
results for the initial-value problem (IVP) for the QG equation (\ref{qgse})
with the initial condition 
\begin{equation}\label{init}
\theta(x,0) =\theta_0(x),\quad x \in \mathbb{R}^2.
\end{equation}


We seek solutions of the IVP (\ref{qgse}) and (\ref{init}) in 
$L^q([0,T];L^p)$ for initial data $\theta_0\in L^r(\mathbb{R}^2)$. 
The notation $L^r$ is standard while $L^q([0,T];L^p)$ stands for the 
space of functions $f$ of $x$ and $t$ satisfying 
$$
\|f\|_{L^q([0,T];L^p)} = \Big(\int_0^T\big(\int_{\mathbb{R}^2}
|f(x,t)|^p dx \big)^{q/p} dt\Big)^{1/q} < \infty.
$$
We distinguish between two cases: $\alpha>1/2$ (the
``sub-critical" case) and $\alpha\le 1/2$ (the ``critical" or ``
super-critical" case).
In the $\alpha>1/2$ case, we establish that the IVP (\ref{qgse})
and (\ref{init}) has a unique global (in time) and regular solution 
in $L^q([0,T];L^p)$. Precise statements are presented in Section \ref{sec:3}.
It is not clear in the $\alpha\le \frac12$ case whether regular solutions 
develop finite time singularities. But We show in Section \ref{sec:4} that 
any singularity must occur in the first derivative if there is a 
singularity. This is achieved by obtaining explicit $L^p$ bounds 
for all high order derivatives of any function solving the IVP
(\ref{qgse}) and (\ref{init}). 

In preparation, we provide in Section
\ref{sec:2} properties of the solution operator for the linear 
QG equation 
and show its boundedness when acting on $L^p$ spaces. 


\section{The solution operator for the linear equation}
\label{sec:2}


Consider the solution operator for the linear QG equation 
$$
\partial_t \theta + \kappa\, \Lambda^{2\alpha} \theta =0, \quad 
x\in \mathbb{R}^2,\quad t> 0,
$$ 
where $\kappa >0$, $\Lambda$ denotes $(-\Delta)^{1/2}$ 
and $\alpha\in [0,1]$. For a given initial data $\theta_0$, the solution of
this equation is given by 
$$
\theta = G_\alpha(t) \, \theta_0 
= e^{-\kappa \,\Lambda^{2\alpha} t}\, \theta_0,
$$
where $G_\alpha(t)\equiv e^{-\kappa \Lambda^{2\alpha} t}$ 
is a convolution operator 
with its kernel $g_\alpha$ being defined through the Fourier transform 
$$
\widehat{g_\alpha}(\xi,t) = e^{-\kappa |\xi|^{2\alpha} t}.
$$


The kernel $g_\alpha$ possesses similar properties as the heat kernel
does. For 
example, for $\alpha\in [0,1]$ and $t>0$, $g_\alpha(x,t)$ is a 
nonnegative and non-increasing radial function, and satisfies the dilation
relation 
\begin{equation}\label{dila}
g_\alpha(x,t) = t^{-1/\alpha} g_\alpha(x t^{-1/(2\alpha)},1). 
\end{equation}

Furthermore, the operators $G_\alpha$ and $\nabla G_\alpha$ are bounded 
on $L^p$. To prove this fact, we need the following lemma.  

\begin{lemma} \label{first}
For $t>0$, $\|g_\alpha(\cdot,t)\|_{L^1} =1$ and for $1\le p<\infty$ 
$$
|g_\alpha(\cdot,t)* f|^p \le g_\alpha(\cdot,t) * |f|^p.
$$
\end{lemma}

\paragraph{Proof.}  For any $t>0$, $\|g_\alpha(\cdot,t)\|_{L^1}
=\widehat{g_\alpha}(0,t)=1$.  By H\"{o}lder's inequality,
\begin{eqnarray*}
|g_\alpha(\cdot,t)* f|^p &=& \left|\int_{\mathbb{R}^2} g_\alpha^{1/q}(x-y,t)
\cdot g_\alpha^{1/p}(x-y,t) f(y) dy \right|^p \\
&\le& \|g_\alpha(\cdot,t)\|_{L^1}^\frac{p}q \int_{\mathbb{R}^2}g_\alpha(x-y,t)
|f(y)|^p dy =g_\alpha(\cdot,t) * |f|^p,
\end{eqnarray*}
where $(1/q)+(1/p)=1$. 


\begin{prop}\label{pq}
Let $1\le p\le q\le \infty$.
For any $t>0$, the operators $G_\alpha(t)$ and $\nabla G_\alpha(t)$ 
are bounded operators from $L^{p}$ to $L^{q}$. Furthermore, we have for 
any $f\in L^{p}$,
\begin{equation}\label{gin}
\|G_\alpha(t) f\|_{L^{q}} \le C t^{-\frac1{\alpha}\left(\frac1{p}-
\frac1{q}\right)} \|f\|_{L^{p}}
\end{equation}
and 
\begin{equation}\label{pgin}
\|\nabla G_\alpha(t) f\|_{L^{q}} \le C 
t^{-\left(\frac1{2\alpha} + \frac1{\alpha}\left(\frac1{p}-
\frac1{q}\right)\right)} \|f\|_{L^{p}}
\end{equation}
where $C$ is a constant depending on $\alpha$, $p$ and $q$ only.
\end{prop}

\paragraph{Proof.}  We first prove (\ref{gin}). For $p=q=\infty$, we
have 
$$
\|G_\alpha(t) f\|_{L^\infty} \le \|g_\alpha(\cdot,t)\|_{L^1} \|f\|_{L^\infty}
=\|f\|_{L^\infty}.
$$
For $p=q<\infty$, we combine Lemma \ref{first} and Young's inequality 
to obtain  
\begin{eqnarray*}
\|G_\alpha(t) f\|_{L^p}^p &=&\|g_\alpha(\cdot,t) * f\|_{L^p}^p  
\le \int_{\mathbb{R}^2} g_\alpha(\cdot,t)* |f|^{p} dx \\
&\le& \|g_\alpha(\cdot,t)\|_{L^1}\|f\|_{L^{p}}^{p}
=\|f\|_{L^{p}}^{p}
\end{eqnarray*}
To prove the general case, we first estimate $\|G_\alpha(t) f\|_{L^\infty}$.
Without loss of generality, we consider $G_\alpha(t) f$ at $x=0$. 
\begin{eqnarray}
|(G_\alpha(t) f)(0)|^p &\le& \int_{\mathbb{R}^2} g_\alpha(|x|,t) |f(x)|^p dx
=\int_0^\infty g_\alpha(\rho,t) dr(\rho) \nonumber \\
&\le& \int_0^\infty |g_\alpha'(\rho,t)| r(\rho) d\rho 
\le \|f\|_{L^p}^{p} \cdot \int_0^\infty |g_\kappa'(\rho,t)| d \rho  \label{med}
\end{eqnarray}
where $r(\rho) =\int_{|y|\le \rho} |f(y)|^{p} dy$ and $g_\alpha' 
=\frac{\partial g_\alpha}{\partial \rho}$. Using (\ref{dila}), one 
easily sees that for some constant $C$  
$$
\int_0^\infty |g_\kappa'(\rho,t)| d \rho = C\, t^{-1/\alpha}
$$
and therefore (\ref{med}) becomes (since $x=0$ is not special !) 
$$
\|G_\alpha(t) f\|_{L^\infty} \le C\, t^{-\frac1{p \alpha}} \|f\|_{L^p}. 
$$
We now estimate $\|G_\alpha(t)f\|_{L^q}$ in terms of $\|f\|_{L^{p}}$
for $1\le p\le q<\infty$. 
$$
\|G_\alpha(t)f\|_{L^{q}}^{q} \le C \|G_\alpha(t)f\|_{L^\infty}^{q-p} 
\|G_\alpha(t)f\|_{L^p}^p  
\le C t^{-\frac1{p \alpha}(q-p)} \|f\|_{L^p}^{q-p} \cdot \|f\|_{L^p}^p.  
$$
That is, $\|G_\alpha(t)f\|_{L^{q}} \le C t^{-\frac1\alpha
\left(\frac1p-\frac1q\right)} \|f\|_{L^p}$.


Estimate (\ref{pgin}) can be proved similarly by using the identity
$$
\partial_x g_\alpha(x,t) = t^{-1/(2\alpha)} \tilde{g}_\alpha(x,t)
$$
where $\tilde{g}_\alpha$ is another radial function enjoying the 
same properties as $g_\alpha$ does.


The following lemma provides point-wise bounds for $\nabla\, g_\alpha$.

\begin{lemma}Let $\alpha\in (0,1]$. Then
for any $x\in \mathbb{R}^2\setminus\{0\}$, $t> 0$, $j=1$ or $2$, 
\begin{equation}\label{decay}
|\partial_{x_j} \, g_\alpha(x,t) | 
\le \left\{ \begin{array}{l} \frac{C}{|x|\, t^{\frac1\alpha}},\\[3pt] 
                             \frac{C}{|x|^2 \, t^{\frac1{2\alpha}}},\\[3pt]
                             \frac{C}{|x|^3\, t},
            \end{array} \right.
\end{equation}
where $C$ is an explicit constant depending on $\alpha$ only.
\end{lemma}

\paragraph{ Proof.} 
Consider the Fourier transform of 
$F(x,t) = x_i \partial_{x_j}\, g_\alpha(x,t)$:
\begin{eqnarray*}
\widehat{F} (\xi,t) &=& i\frac{\partial}{\partial \xi_i} 
\left(i\xi_j \widehat{g_\alpha}
(\xi,t)\right) = (-1) \frac{\partial}{\partial \xi_i} \left(\xi_j \, e^{-\kappa
|\xi|^{2\alpha} t} \right) \\
&=& \left(-\delta_{ij} + 2\kappa \alpha\, t\, \xi_i\,\xi_j |\xi|^{2\alpha-2}
\right)\, e^{-\kappa |\xi|^{2\alpha} t},
\end{eqnarray*}
where $\delta_{ij}$ is the Kronecker delta. Therefore, for $x\in\mathbb{R}^2$ 
and $t>0$, 
\begin{eqnarray*}
|x_i \partial_{x_j}\, g_\alpha(x,t)| &=& |F(x,t)| 
\le \|\widehat{F}(\cdot,t)\|_{L^1} 
\le \int_{\mathbb{R}^2} (1\, + 2\kappa \alpha |\xi|^{2\alpha} t) 
e^{-\kappa |\xi|^{2\alpha} t} d\xi \\
&=&2\pi \int_0^\infty (1+ 2\kappa \alpha \rho^{2\alpha} t ) 
e^{-\kappa \rho^{2\alpha} t} \rho\, d \rho 
=C\, t^{-1/\alpha}. 
\end{eqnarray*}
where $C= \frac{\pi}{\alpha} \int_0^\infty(1\, + 2\kappa \alpha\, r) \,
r^{\frac1\alpha-1}\, e^{-\kappa r} dr$. This proves the first inequality 
in (\ref{decay}). 
The next two inequalities can be established in a similar fashion by 
considering $F(x,t) = x_i\,x_k\,\partial_{x_j}\, g_\alpha(x,t)$ and 
$F(x,t)= x_l\,x_i\,x_k\,\partial_{x_j}\, g_\alpha(x,t)$, respectively, where
the indices $i,j,k,l=1$ or $2$. 


We will need the Hardy-Littlewood-Sobolev inequality, which we now recall.
It states that the fractional integral 
$$
Tf(x) = \int_{\mathbb{R}^2}\frac{f(y)}{|x-y|^{n-\gamma}} dy, \quad 0< \gamma < n
$$
is a bounded operator from $L^p$ to $L^q$ if $p$ and $q$ satisfies
$$
1\le p<q<\infty,\quad \frac1q + \frac{\gamma}{n} =\frac1p.
$$
One can find the Hardy-Littlewood-Sobolev inequality in \cite[p.119]{St}.



\section{Global existence and uniqueness in \\the $\alpha>1/2$ case}
\label{sec:3}

In this section we consider the IVP for the dissipative QG equation
\begin{eqnarray}\label{dqg}
&\theta_t  + u\cdot \nabla \theta 
+ \kappa \Lambda^{2\alpha} \theta=f,\quad 
 (x,t)\in \mathbb{R}^2\times [0,\infty),&\nonumber \\
& u=(u_1,u_2) = (-{\cal R}_2 \theta,\, {\cal R}_1\theta),\quad
 (x,t)\in \mathbb{R}^2\times [0,\infty), & \\
& \theta(x,0) =\theta_0(x),\quad x\in \mathbb{R}^2 ,&\nonumber 
\end{eqnarray}
where $\kappa>0$ and $\alpha\in [0,1]$.
Our major result is that the IVP (\ref{dqg}) with $\alpha>1/2$,
$\theta_0\in L^r$ and $f\in L^{q'}([0,T];L^{r_1})$ has a unique global 
(in time) solution in $L^q([0,T];L^p)$ for proper $p,q,q',r$ and $r_1$.
Furthermore, the solution is shown to be smooth if $\theta_0$ and $f$ 
are sufficiently smooth. Precise statements will be presented in  
Theorem \ref{2main} and Theorem \ref{regular}.


The theorems of this section are proved by the method of integral
equations and the contraction mapping argument. To proceed, we write 
the QG equation into the integral form
\begin{equation}\label{integ}
\theta(t)=G_\alpha(t)\,\theta_0 + \int_0^t G_\alpha(t-\tau)\,(f-u\cdot
\nabla \theta)(\tau) d\tau,
\end{equation}

We observe that $u\cdot\nabla \theta=\nabla\cdot(u\theta)$ because 
$\nabla\cdot u =0$. The nonlinear term can then be alternatively written as 
$$
B(u,\theta)(t) \equiv \int_0^t \nabla G_\alpha (t-\tau) (u\theta)(\tau) d\tau.
$$ 
We will solve (\ref{integ}) in $L^p([0,T];L^q)$ and the following estimates 
for the operator $B$ acting on this type of spaces will be used.

\begin{prop}\label{bpq}
Let $\alpha>1/2$ and $T>0$. Assume that $u$ and $\theta$ are in
$L^q([0,T];L^p)$ with $p$ and $q$ satisfying 
$$
p> \frac{2}{2\alpha-1},\quad 
\frac1p + \frac{\alpha}{q} = \alpha-\frac12.
$$
Then the operator $B$ is bounded in $L^q([0,T];L^p)$ with 
$$
\|B(u,\theta)\|_{L^q([0,T]; L^p)}
\le C \|u\|_{L^q([0,T];L^p(\mathbb{R}^2))}\cdot
\|\theta\|_{L^q([0,T];L^p(\mathbb{R}^2))}.
$$
where $C$ is a constant depending on $\alpha$, $p$ and $q$ only. 
\end{prop}

\paragraph{Proof.}  For $p> \frac2{2\alpha-1}\ge 2$, 
we obtain after applying (\ref{pgin}) of Proposition \ref{pq} 
\begin{eqnarray}
\|B(u,\theta)\|_{L^p} &\le&  \int_0^t 
\|\nabla G_\alpha(t-\tau)(u\, \theta)(\tau)\|_{L^p} d \tau \nonumber\\
&\le& C \int_0^t \frac{1}{|t-\tau|^{\frac1{2\alpha} + \frac1\alpha\, \left(
\frac2p-\frac1p\right)}}\,\, \|u\, \theta(\cdot,\tau)\|_{L^{p/2}} d\tau  
\label{best} \\
&\le& C \int_0^t \frac{1}{|t-\tau|^{\frac1{2\alpha} + \frac1{p \,\alpha}}}
\|u(\cdot,\tau)\|_{L^p}\,\|\theta(\cdot,\tau)\|_{L^p} d\tau \nonumber
\end{eqnarray}
for some constant $C$ depending on $\alpha$ and $p$ only. 
For $\alpha>1/2$ and $p>\frac2{2\alpha-1}$, we have 
$$
0<\frac1{2\alpha} + \frac1{p \,\alpha} <1. 
$$
Applying the Hardy-Littlewood-Sobolev inequality to (\ref{best}) with
$$
\frac1q + \, \frac{1-\frac1{2\alpha}-\frac1{p \,\alpha}}{1} =\frac2q,
\quad i.e., \quad \frac1p + \frac{\alpha}{q} = \alpha-\frac12, 
$$
we obtain
\begin{eqnarray*}
\|B(u,\theta)\|_{L^q([0,T]; L^p)} 
&\le& C \|\left(\|u(\cdot,t)\|_{L^p}
\|\theta(\cdot,t)\|_{L^p}\right)\|_{L^{q/2}([0,T])} \\
&\le& C \|u\|_{L^q([0,T];L^p(\mathbb{R}^2))}\cdot
\|\theta\|_{L^q([0,T];L^p(\mathbb{R}^2))}.
\end{eqnarray*}

The next two lemmas detail how $G_\alpha$ behaves when acting on $\theta_0$ 
and $f$.

\begin{lemma}\label{theta0}
Let $1/2<\alpha\le 1$,\, $T>0$,\, and $p$ and $q$ satisfy 
$$
p> \frac{2}{2\alpha-1},\quad
\frac1p + \frac{\alpha}{q} = \alpha-\frac12.
$$
Assume that $\theta_0\in L^r(\mathbb{R}^2)$ with $\frac{2}{2\alpha-1}
<r\le p$. Then we have 
$$
\|G_\alpha(t)\theta_0\|_{L^q([0,T];L^p)} 
\le C\, T^{1-\frac1\alpha\left(\frac12+\frac1r\right)} \,\|\theta_0\|_{L^r},
$$
where $C$ is a constant depending on $\alpha$, $p$, $q$ and $r$ only.
\end{lemma}

\paragraph{Proof.}  By (\ref{gin}),
\begin{eqnarray*}
\|G_\alpha(t)\theta_0\|_{L^q([0,T];L^p)} &=&\Big[\int_0^T
\|G_\alpha(t)\theta_0\|_{L^p}^q dt\Big]^{1/q} \\
&\le& \Big[\int_0^T t^{-\frac1\alpha\left(\frac1r-\frac1p\right)
\cdot q} \|\theta_0\|_{L^r}^q \,dt \Big]^{1/q} \\
&=&C T^{1-\frac1\alpha\left(\frac12+\frac1r\right)} \,\|\theta_0\|_{L^r}.
\end{eqnarray*}


\begin{lemma}\label{force}
Let $1/2<\alpha\le 1$,\, $T>0$,\, and $p$ and $q$ satisfy
$$
p> \frac{2}{2\alpha-1},\quad
\frac1p + \frac{\alpha}{q} = \alpha-\frac12.
$$
Assume $f\in L^{q'}([0,T];L^{r_1})$ with 
$q'$ being the conjugate of $q$ (i.e., 1/q'+1/q=1) and $r_1$ satisfying
$\frac{2}{2\alpha-1}<r_1\le p$. Then 
$$
\left\|\int_0^t G_\alpha(t-\tau)f(\tau) d\tau\right\|_{L^q([0,T];L^p)} 
\le C T^{1+\frac1q-\frac1\alpha\left(\frac12+\frac{1}{r_1}\right)}\,
\|f\|_{L^{q'}([0,T];L^{r_1})},
$$
where $C$ is a constant depending on $\alpha$, $p$, $q$ and $r_1$ only.
\end{lemma}

\paragraph{Proof.}  The result is a consequence of direct computation.
By (\ref{gin}) and then H\"{o}lder's inequality, 
\begin{eqnarray*}
\lefteqn{\big\|\int_0^t G_\alpha(t-\tau)f(\tau) d\tau\big\|_{L^q([0,T];L^p)} }\\ 
&\le& \Big[\int_0^T \Big(\int_0^t (t-\tau)^{-\frac1\alpha
\left(\frac1{r_1}-\frac1p\right)}\|f(\cdot,\tau)\|_{L^{r_1}} d\tau\Big)^q dt
\Big]^{1/q} \\
&\le& \Big[\int_0^T \int_0^t (t-\tau)^{-\frac1\alpha 
\left(\frac1{r_1}-\frac1p\right) \cdot q}\, d\tau \cdot \Big(\int_0^t 
\|f(\cdot,\tau)\|_{L^{r_1}}^{q'} d\tau\Big)^{q/q'}\, dt\Big]^{1/q} \\
&\le& C T^{1+\frac1q-\frac1\alpha\left(\frac12+\frac1{r_1}\right)} 
\|f\|_{L^{q'}([0,T];L^{r_1})}.
\end{eqnarray*}

Now we state and prove the main theorem.

\begin{thm}\label{2main}
Let $1/2<\alpha\le 1$, $T>0$, and $p$ and $q$ satisfy 
$$
p> \frac{2}{2\alpha-1},\quad 
\frac1p + \frac{\alpha}{q} = \alpha-\frac12.
$$
Assume that $\theta_0\in L^r(\mathbb{R}^2)$ with $\frac{2}{2\alpha-1}
<r\le p$  and $f\in L^{q'}([0,T];L^{r_1})$ with $\frac{2}{2\alpha-1}<r_1\le p$,
where $q'$ denotes the conjugate of $q$ (i.e., 1/q'+1/q=1).
Then there exists a constant $C$ such that 
for any $\theta_0$ and $f$ satisfying 
$$
T^{1-\frac1\alpha\left(\frac12+\frac1r\right)} \,\|\theta_0\|_{L^r} + \,
T^{1+\frac1q-\frac1\alpha\left(\frac12+\frac{1}{r_1}\right)}\,
\|f\|_{L^{q'}([0,T];L^{r_1})} \le C, 
$$
there exists a unique strong solution $\theta\in L^q([0,T];L^p)$
for the IVP (\ref{dqg}) in the sense of (\ref{integ}).
\end{thm}

\paragraph{Proof.} We write the integral equation (\ref{integ}) 
symbolically as $\theta=A\theta$. The operator $A$ is seen as  a mapping of 
the space $E\equiv L^q([0,T];L^p)$ into itself. 
Let 
$$
b= T^{1-\frac1\alpha\left(\frac12+\frac1r\right)} \,\|\theta_0\|_{L^r} + \,
T^{1+\frac1q-\frac1\alpha\left(\frac12+\frac{1}{r_1}\right)}\,
\|f\|_{L^{q'}([0,T];L^{r_1})}
$$
and set $R=2b$. Define $B_R$ to be the closed ball with radius $R$ 
centered at the origin in $E$. We now show that if $b$ is bounded by  
an appropriate constant, then $A$ is a contraction map on $B_R$.
Let $\theta$ and $\bar{\theta}$ be any two elements of $B_R$. Then we have 
$$
\|A\,\theta- A\,\bar{\theta}\|_E = \Big\|\int_0^t G_\alpha(t-\tau) (u\cdot
\nabla \theta) d\tau - \int_0^t G_\alpha(t-\tau) (\bar{u}\cdot
\nabla\bar{\theta}) d\tau \Big\|_E,
$$
where $u$ and $\bar{u}$ are determined by $\theta$ and $\bar{\theta}$, 
respectively, through the second relation in (\ref{dqg}). 
Recalling the notation $B$, we have
\begin{eqnarray*}
\|A\,\theta- A\,\bar{\theta}\|_E 
&=&\left\| B(u-\bar{u}, \theta) + B(\bar{u}, \theta-\bar{\theta}) \right\|_E \\
&\le& \|B(u-\bar{u}, \theta)\|_E + \|B(\bar{u}, \theta-\bar{\theta})\|_E.
\end{eqnarray*}
It then follows from applying Proposition \ref{bpq} that 
$$
\|A\,\theta- A\,\bar{\theta}\|_E
\le C \|u-\bar{u}\|_E \|\theta\|_E 
+ C \|\bar{u}\|_E \|\theta-\bar{\theta}\|_E,
$$
where $C$ is a constant depending on $\alpha$, $p$ and $q$ only. 
Since $u$ and $\bar{u}$
are Riesz transforms of $\theta$ and $\bar{\theta}$, respectively, the 
classical Calderon-Zygmund singular integral estimates imply that 
$$
\|u\|_E \le C \|\theta\|_E,\quad \|\bar{u}\|_E \le C \|\bar{\theta}\|_E.
$$
One can consult the book of Stein \cite{St}
for more details on Riesz transforms.
Therefore,
$$
\|A\,\theta- A\,\bar{\theta}\|_E \le C (\|\theta\|_E + \|\bar{\theta}\|_E)\,
\|\theta-\bar{\theta}\|_E \le C\,R\,\|\theta-\bar{\theta}\|_E.
$$
We now estimate $\|A\theta\|_E$. By Lemma \ref{theta0} and Lemma \ref{force},
the norm of 
$$
A\,0 = G_\alpha(t)\theta_0 + \int_0^t G_\alpha (t-\tau)f(\tau) d\tau
$$
in $E$ can be bounded by
$$
\|A\,0\|_E \le C\, T^{1-\frac1\alpha\left(\frac12+\frac1r\right)} 
\,\|\theta_0\|_{L^r} + \, C
T^{1+\frac1q-\frac1\alpha\left(\frac12+\frac{1}{r_1}\right)}\,
\|f\|_{L^{q'}([0,T];L^{r_1})}= b
$$
Therefore, 
$$
\|A\,\theta\|_E =\|A\,\theta -A\, 0 + A\,0\|_E
\le \|A\,\theta -A\, 0\|_E + \|A\,0\|_E \le C\,R\,\|\theta\|_E + b.
$$
If $2C\, b \le \frac12$, then $C\,R =2C\, b\le \frac12$ and we have
$$
\|A\,\theta- A\,\bar{\theta}\|_E \le \frac12 \|\theta-\bar{\theta}\|_E,\quad
\mbox{and}\quad  \|A\,\theta\|_E \le R.
$$
It follows from the contraction mapping principle that there exists a 
unique $\theta\in E=L^q([0,T];L^p)$ solving (\ref{integ}).
This finishes the proof of Theorem \ref{2main}.

We now show that the solution obtained in the previous theorem is actually
smooth. We introduce a notation. 
For a non-negative multi-index $k=(k_1, k_2)$, we define 
$$
D^k=\left(\frac\partial {\partial_{x_1}}\right)^{k_1}\, 
\left(\frac\partial {\partial_{x_2}}\right)^{k_2}.
$$
and $|k|=k_1+ \,k_2$.

\begin{thm}\label{regular}
Let $1/2<\alpha\le 1$, $T>0$,
and $p$ and $q$ satisfy
$$
p> \frac{2}{2\alpha-1},\quad
\frac1p + \frac{\alpha}{q} = \alpha-\frac12.
$$
Assume that for a non-negative multi-index $k$
\begin{equation}\label{as}
D^k\,\theta_0\in L^r(\mathbb{R}^2)
\quad \mbox{and}\quad 
 D^k \,f\in L^{q'}([0,T];L^{r_1}(\mathbb{R}^2)),
\end{equation}
where $\frac{2}{2\alpha-1}<r\le p$,\, $\frac{2}{2\alpha-1}<r_1\le p$ and 
$q'$ denotes the conjugate of $q$. Then for any non-negative multi-index
$j$ with $|j|\le |k|$ 
\begin{equation}\label{reg}
D^j \, \theta \in  L^q([0,T];L^p).
\end{equation}
Furthermore, for each $j$ with $0\le |j|\le |k|-2$ and almost every $t
\in [0,T]$ 
\begin{equation}\label{pt}
\partial_t D^j \, \theta \in L^p(\mathbb{R}^2).
\end{equation}
\end{thm}

\paragraph{Proof.}  The basic tool of establishing (\ref{reg}) is still 
the contraction mapping argument. We first consider the case  
$|j| =1$. Taking $D$ of (\ref{integ}), we obtain
\begin{equation}\label{nabla}
D\,\theta(t) =\,G_\alpha(t)(D \theta_0) + \int_0^t 
 G_\alpha(t-\tau)(D f(\tau)) d\tau 
+ B(D\, u, \theta) + B(u, D\, \theta).
\end{equation}
This integral equation can then be viewed as 
$(D \theta) =\tilde{A}(D\theta)$
and $\tilde{A}$ is seen as a mapping of the space $E$ consisting of 
functions $\theta$ such that 
$$
\theta\in L^q([0,T];L^p) \quad \mbox{and}\quad D\theta \in L^q([0,T];L^p).
$$  
The norm in $E$ is given by 
$$
\|\theta\|_E = \|\theta\|_{L^q([0,T];L^p)} + \|D\theta\|_{L^q([0,T];L^p)}.
$$
For $\theta_0$ and $f$ satisfying (\ref{as}), the first two terms are bounded
in $E$. The two nonlinear terms acting on $E$ have similar bounds as stated
in Proposition \ref{bpq}. As in the proof of Theorem \ref{2main}, we can then 
show that $\tilde{A}$ is a contraction mapping of $E$ into itself. Therefore
$\tilde{A}$ has a fixed point in $E$. The uniqueness result of 
Theorem \ref{2main} indicates that this $\theta$ is 
just the original $\theta$. Thus we have shown that $D \theta\in 
L^q([0,T];L^p)$. The proof of (\ref{reg}) for $|j|=2,3,\cdots, |k|$ 
is similar and
we thus omit details.  
  

We now prove (\ref{pt}) and start with the case $|j|=0$. 
Because $\theta$ satisfies  
$$
\partial_t \theta =\, f -u\cdot\nabla \theta - \kappa \Lambda^{2\alpha}
\theta,
$$
it suffices to show that the terms on the right are in $L^p$ for almost 
every $t$. Since for almost every $t$
$$
f\in L^{r_1},\quad Df\in L^{r_1},\quad u\in L^p, \quad Du\in L^p,\quad
D\theta\in L^p, \quad D^2\theta\in L^p,
$$ 
we obtain by applying the Gagliardo-Nirenberg inequality 
$$
\|f(\cdot,t)\|_{L^p} \le C \|f(\cdot,t)\|_{L^{r_1}}^{1-\sigma} 
\,\,\| Df(\cdot,t)\|_{L^{r_1}}^\sigma,\quad \sigma=\frac2{r_1}-\frac{2}{p}. 
$$
\begin{eqnarray*}
\|(u\cdot\nabla\theta)(\cdot,t)\|_{L^p}
&\le& C \|u(\cdot,t)\|_{L^{2p}}\,\,\|D\theta(\cdot,t)\|_{L^{2p}} \\
&\le& C \|u(\cdot,t)\|_{L^p}^{1-\epsilon}\,\|Du(\cdot,t)\|_{L^p}^{\epsilon}
\,\,\|D\theta(\cdot,t)\|_{L^p}^{1-\epsilon}\,\,
\|D^2\theta(\cdot,t)\|_{L^p}^{\epsilon},
\end{eqnarray*}
where $\epsilon=1/p$. Therefore, for almost every $t$ 
$$
f(\cdot,t)\in L^p,\quad u\cdot\nabla\in L^p, \quad 
\mbox{and}\quad \Lambda^{2\alpha}\theta\in L^p
$$
and this in turn implies $\partial_t \theta\in L^p$. The proof for 
$\partial_t D^j \theta\in L^p$ with $|j|>0$ is similar. 
This completes the proof of (\ref{pt}). 



\section{$L^p$ bounds in the $\alpha\le 1/2$ case}
\label{sec:4}

For $\alpha>1/2$, the issue of global existence, uniqueness and 
regularity concerning the IVP (\ref{dqg}) with $L^r$ initial data is 
resolved in Section \ref{sec:3}.  Our major interest of this section 
is in the $\alpha\le \frac12$ case although all theorems to be presented 
hold for any $\alpha \in [0,1]$. We conclude that any possible finite 
time singularity must occur in the first derivative. This is achieved
by bounding the $L^p$ norms of all high order derivatives of 
$\theta$ by the initial $L^p$ norms and a magic quantity. 

\begin{lemma}\label{useful}
Let $\alpha \in [0,1]$, $p\in (1,\infty)$ and $k$ be a nonnegative multi-index.
Then for any sufficiently smooth $\theta$, we have for any $t\ge 0$
$$
\int_{\mathbb{R}^2} |D^k\theta|^{p-2}(x,t)\,\, (D^k\theta(x,t))\,\,
\Lambda^{2\alpha} D^k\theta (x,t) dx \ge 0.
$$
\end{lemma}

\paragraph{Proof.}  Let $g_\alpha(x,s)$ be the kernel of the solution operator
for the linear QG equation, as defined in the previous section. Then 
$\Theta(x,s)\equiv g_\alpha(\cdot,s)*(D^k\theta)$ satisfies the equation 
\begin{equation}\label{Th}
\partial_s \Theta + \kappa \Lambda^{2\alpha}\, \Theta =0
\end{equation}
and $\Theta(x,s) \to D^k\theta$ as $s\to 0$. Multiplying both sides of 
(\ref{Th}) by $p|\Theta|^{p-2} \Theta$ and integrate over $\mathbb{R}^2$, 
we obtain
$$
\frac{d}{ds} \int_{\mathbb{R}^2} |\Theta|^p dx 
+ \,p\,\kappa \,\int_{\mathbb{R}^2} |\Theta|^{p-2} \,\Theta 
\Lambda^{2\alpha}\, \Theta dx =0.
$$ 
Integrating the above over $[s_1,s_2]$ with respect to $s$, we have 
\begin{equation}\label{s1s2}
\int_{\mathbb{R}^2} |\Theta|^p(x,s_2) dx - \int_{\mathbb{R}^2} |\Theta|^p(x,s_1)dx
=-\,p\,\kappa \,\int_{s_1}^{s_2} \int_{\mathbb{R}^2} |\Theta|^{p-2} \Theta
\Lambda^{2\alpha}\, \Theta dx ds,
\end{equation}
where $s_1$ and $s_2$ are arbitrarily fixed. Applying (\ref{gin}) of 
Proposition \ref{pq}, we have
\begin{eqnarray*}
\int_{\mathbb{R}^2} |\Theta|^p(x,s_2) dx 
&=&\|g_\alpha(\cdot,s_2)*(D^k\theta)\|_{L^p}^p \\
&=&\|g_\alpha(\cdot,s_2-s_1)*[g_\alpha(\cdot,s_1)*(D^k\theta)]\|_{L^p}^p \\
&\le& \|g_\alpha(\cdot,s_1)*(D^k\theta)\|_{L^p}^p
=\int_{\mathbb{R}^2} |\Theta|^p(x,s_1)dx.
\end{eqnarray*}
That is, the left hand side of (\ref{s1s2}) is not positive. Therefore
$$
\int_{s_1}^{s_2} \int_{\mathbb{R}^2} |\Theta|^{p-2} \Theta
\Lambda^{2\alpha}\, \Theta dx ds \ge 0.
$$
The arbitrariness of $s_1$ and $s_2$ then implies that for any $s> 0$, 
\begin{equation}\label{>0}
\int_{\mathbb{R}^2} |\Theta|^{p-2}(x,s)\, \Theta(x,s)\, 
\Lambda^{2\alpha}\, \Theta(x,s) dx ds \ge 0.
\end{equation}
Letting $s\to 0$ and recalling the definition of $\Theta$, we obtain 
for any $t\ge 0$
$$
\int_{\mathbb{R}^2} |D^k\theta|^{p-2}(x,t)\, (D^k\theta(x,t))\,
\Lambda^{2\alpha} D^k\theta (x,t) dx \ge 0.
$$


One consequence of the previous lemma is that the 
$L^p$-norm ($p\in(1,\infty]$) 
of any solution $\theta$ of the IVP (\ref{dqg}) is uniformly bounded by 
the $L^p$ norm of the initial data. Thus finite-time singularity 
is only possible in the derivatives of $\theta$. The following result was
shown in \cite{Re} and we now briefly describe it.  

\begin{thm}
Let $\alpha\in [0,1]$ and $p\in (1,\infty]$. Then any solution $\theta$ of the
IVP (\ref{dqg}) satisfies for $t\ge 0$ 
$$
\|\theta(\cdot,t)\|_{L^p(\mathbb{R}^2)} \le \|\theta\|_{L^p(\mathbb{R}^2)}.
$$
\end{thm}
A sketch of the proof for this theorem is given in \cite{CCW}.

We now state and prove our main theorem, in which we establish estimates
to bound the $L^p$ norms of derivatives of any solution $\theta$ of the 
IVP (\ref{dqg}) in terms of $\nabla u$ ($u$ is related to $\theta$ through 
the second relation in (\ref{dqg})). Roughly speaking, this means that 
no finite time 
singularity in high-order derivatives is possible if $\nabla u$ does not
become infinite first.  The role of the forcing term $f$ is not crucial,
so we set it equal to zero for the sake of clear presentation.


\begin{thm}
Let $\alpha\in [0,1]$. Assume that $\theta$ is a solution of the IVP
(\ref{dqg}). Then for any $p\in (1,\infty]$ and a multi-index 
$k$ with $|k|\ge 1$,  
\begin{equation}\label{major}
\|D^k\theta(\cdot,t)\|_{L^p} \le \|D^k\theta_0\|_{L^p}
\cdot e^{\int_0^t \|\nabla u(\cdot,\tau)\|_{L^\infty} d\tau}
\end{equation}
holds for any $t\ge 0$, where $u$ is determined by $\theta$ through 
the second relation in (\ref{dqg}).
\end{thm}

\paragraph{Proof.} We start with the case $|k|=1$. For $p\in (0,\infty)$, we
take $D$ of the first equation
in (\ref{dqg}), multiply by $p\,|D\theta|^{p-2}\, D\theta$ and 
then integrate over $\mathbb{R}^2$ to obtain
\begin{eqnarray}
\lefteqn{\frac{d}{dt}\int_{\mathbb{R}^2} |D\theta|^p dx  
+ \,p\,\kappa \,\int_{\mathbb{R}^2} |D\theta|^{p-2} \, 
D\theta\cdot \Lambda^{2\alpha} (D\theta) \, dx  }\nonumber\\
\label{dth}
&=&-p \,\int_{\mathbb{R}^2} |D\theta|^{p-2} \, D\theta\, \cdot D(u\cdot\nabla 
\theta) dx  \hspace{25mm}
\end{eqnarray}
The right hand side actually consists of two terms
$$
-p \,\int_{\mathbb{R}^2} |D\theta|^{p-2} \, D\theta\, \cdot Du \cdot 
\nabla \theta dx \quad \mbox{and}\quad
-p \,\int_{\mathbb{R}^2} |D\theta|^{p-2} \, D\theta\, \cdot 
u\cdot \nabla (D\theta) dx,
$$ 
but one of them is zero
$$
\int_{\mathbb{R}^2} |D\theta|^{p-2} \, D\theta\, \cdot
u\cdot \nabla (D\theta) dx = \int_{\mathbb{R}^2} u\cdot 
\nabla \left(|D\theta|^p\right) dx =0 
$$
because $\nabla\cdot u=0$. Therefore, (\ref{dth}) becomes
\begin{eqnarray*}
\lefteqn{ \frac{d}{dt}\int_{\mathbb{R}^2} |D\theta|^p dx
+ p\,\kappa \int_{\mathbb{R}^2} |D\theta|^{p-2} \,
D\theta\cdot \Lambda^{2\alpha} (D\theta) \, dx }\\
&=& -p \,\int_{\mathbb{R}^2} |D\theta|^{p-2} \, D\theta\, \cdot Du \cdot
\nabla \theta dx, 
\end{eqnarray*}
which in turn implies that 
$$
\frac{d}{dt}\int_{\mathbb{R}^2} |D\theta|^p dx
+p\,\kappa \int_{\mathbb{R}^2} |D\theta|^{p-2} \,
D\theta\cdot \Lambda^{2\alpha} (D\theta) \, dx
\le p\|\nabla u(\cdot,t)\|_{L^\infty} \int_{\mathbb{R}^2} |D\theta|^p dx.
$$
By Lemma \ref{useful}, the second term on the left hand side is nonnegative.
So
$$
\frac{d}{dt}\int_{\mathbb{R}^2} |D\theta|^p dx
\le p\, \|\nabla u(\cdot,t)\|_{L^\infty} \int_{\mathbb{R}^2} |D\theta|^p dx.
$$ 
Gronwall's inequality then implies (\ref{major}). Once we have the 
bound (\ref{major}) for any $p<\infty$, we can then take the limit of 
(\ref{major}) as $p\to \infty$ to establish (\ref{major}) 
for $p=\infty$.


The inequality (\ref{major}) for general $k$ can be proved by induction.
One needs the Calderon-Zygmund inequality for Riesz transforms
$$
\|D^j u(\cdot,t)\|_{L^p} \le C \|D^j\theta(\cdot,t)\|_{L^p},
\quad p\in (1,\infty),\quad |j|\le |k|.
$$


\paragraph{Acknowledgments.} 
This research was partially supported by the NSF 
grant DMS 9971926, by the American Mathematical Society Centennial Fellowship, 
and by the ORAU Ralph E. Powe Junior Faculty Enhancement Award. 


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\end{thebibliography}

\noindent\textsc{Jiahong Wu}\\
Department of Mathematics\\
Oklahoma State University \\
401 Mathematical Sciences\\
Stillwater, OK 74078 USA\\ 
e-mail: jiahong@math.okstate.edu

\end{document}