\section{Introduction}

We consider the 2-D surface quasi-geostrophic (QGS) equations
$$
\frac{\partial \theta}{\partial t}+u\cdot\nabla\theta +
\kappa (-\Delta)^{\alpha}\theta=0,\quad 0\le \alpha\le 1
$$
where $\kappa>0$ in
the case of the dissipative equations and $\kappa=0$ for the non-dissipative
equations.
Here the velocity $u=(u_1,
u_2)$ is determined from $\theta$ by a stream function $\psi$:
$$
(u_1,u_2)=\left (-\frac{\partial \psi}{\partial x_2},\frac{
\partial \psi}{\partial x_1}\right)
$$
where $\psi$ satisfies 
$$
{(-\Delta)^{\frac{1}{2}}}\psi=-\theta
$$

\vspace{.1in}
These equations have been under intensive investigations because of 
mathematical importance and potential applications in 
meteorology and oceanography (\cite{CMT},\cite{HPGS},\cite{Pe}). As pointed out in \cite{CMT}, 
the 2-D dissipative and non-dissipative QGS equations are strikingly
analogous to the 3-D Navier-Stokes and the Euler equations. 
Inviscid limit results for the Navier-Stokes equations with smooth
and rough initial data have been established (\cite{bm},\cite{c},
\cite{CW1},\cite{CW2},\cite{CW3}).
Naturally the inviscid limit problem for the solutions of the
dissipative QGS equations rises and so far we have seen no work
in this direction.  

\vspace{.15in}
We know from \cite{CMT} that the QGS equations 
(defined on $\Bbb R^2$ or $\Bbb T^2$)  with smooth initial 
data admit unique classical solutions for short times and the 
quantity 
$$
\int_{0}^{T}\|\nabla\theta(\cdot,s)\|_{L^\infty}ds
$$
is responsible for possible singularity formation. The control 
of this quantity is also important in our proof of the inviscid 
limit results for smooth solutions. 

\vspace{.1in}
Both dissipative and non-dissipative QGS equations on 
$\Bbb T^2$ with $L^2$ initial data have weak solutions in the 
distribution sense (\cite{Re}). Inviscid limits for weak solutions are 
in general hard to obtain and in this case we only obtain a weak $L^2$
result without a rate. We believe that explicit rates can be
given in certain negative Sobolev norms.

\vspace{.15in}
The second part is devoted to regularity estimates for the dissipative
QGS equations.
Although we use the ideas of 
Foias-Temam \cite{FT} and Doering-Titi \cite{DT} developed for the
Navier-Stokes equations, our methods of estimating the non-linear
terms is significantly different from theirs because of the
special structure of the QGS equations. 
We treat the norms
$\|e^{\gamma t\Lambda^\alpha}\Lambda^{\beta}\theta\|_{L^2}$
(see notations in Section 3)
and a special consequence of our estimates for 
$\|e^{\gamma t\Lambda^\alpha}\Lambda^{\beta}\theta\|_{L^2}$
is that as long as $\|\Lambda^{\beta}\theta\|_{L^2}^{2}$ remains
bounded, the Fourier spectrum decays exponentially at high wave numbers.
In a similar fashion as Doering and Titi \cite{DT} argue for the 
Navier-Stokes equations,
these exponential decay estimates 
can be used to obtain bounds on small length scale defined
through the exponential decay rate.

\vspace{.1in}
Finally, I would like to thank Professor Peter Constantin for his 
suggestions and help.

\newpage
\section {Inviscid Limits}

We consider the 2-D dissipative ($\kappa>0$) and the non-dissipative
 ($\kappa=0$) quasi-geostrophic equations on 
 $\Bbb R^2$ or $\Bbb T^2$:
$$
\frac{\partial \theta}{\partial t}+u\cdot\nabla\theta +
\kappa \Lambda^{2\alpha}\theta=0,\qquad 0\le \alpha\le 1
$$
where $\Lambda=(-\Delta)^{\frac{1}{2}}$ is the Riesz potential 
operator and the velocity $u$ is defined from the stream function
$\psi=-\Lambda^{-1}\theta$ by
$$
u=(u_1,u_2)=\left (-\frac{\partial \psi}{\partial x_2},\frac{
\partial \psi}{\partial x_1}\right)
$$

\vspace{.1in}
We use $\Bbb D^2$ to denote either $\Bbb R^2$ or $\Bbb T^2$.
We first consider the smooth initial data case: $\theta_0\in H^k(
\Bbb D^2)(k\ge 3)$. As shown
by Constantin, Majda and Tabak \cite{CMT}, the QGS
equations with smooth initial data have local (in time) smooth
solutions and the Beale-Kato-Majda type blowup conditions  
have been obtained. More precisely,

\begin{prop}\label{prop:2.1}
If the initial data $\theta|_{t=0}=\theta_0\in H^k(\Bbb D^2)$
for some $k\ge 3$, then both the QGS and the dissipative QGS 
equations have a unique smooth solution for a small time interval,
respectively. Furthermore, the solution $\theta^{QG}$ of the QGS
equations satisfies
$$\int_{0}^{t}\|\nabla \theta^{QG}(\cdot,s)\|_{L^\infty}ds< \infty,$$
$$ \int_{0}^{t}\|\theta^{QG}(\cdot,s)\|_{k}^{2}ds< \infty$$
for any $t$ belonging to the existence interval $[0,T^*)$.
\end{prop}


This proposition admits the possibility of finite-time singularity
formation and consequently, the inviscid limit results for
the smooth solutions are valid only for the time period before
the possible breakdown. We need further estimates on  
the solutions.

\begin{prop}\label{prop:33.1.2}
Let $\theta^{QG}$ and $\theta^{DQG}$ be the smooth solutions of the 
QGS and the dissipative QGS equations with the same initial data $\theta_0\in
H^k(\Bbb D^2)$$(k\ge 3)$. Respectively,
$u^{QG}$ and $u^{DQG}$ are the corresponding velocities and 
$\psi^{QG}$ and $\psi^{DQG}$ are the stream functions. Then for
any $t$ in the maximal time interval $[0,T^*)$ (when the smooth
solutions exist),
\begin{enumerate}
\item The solution $\theta^{QG}$ of the QGS equation satisfies:
$$\int_{\Bbb D^2}G(\theta^{QG}(x,t))dx=\int_{\Bbb D^2} G(\theta_0)dx,
$$ where $G$ is a continuous function with $G(0)=0$.  Especially,
$$\|\theta^{QG}(\cdot,t)\|_{L^p}=\|\theta_0\|_{L^p},\quad 1\le p\le\infty,$$
furthermore,  
$$
\|u^{QG}(\cdot,t)\|_{L^2}=
\|\theta^{QG}(\cdot,t)\|_{L^2}=\|\theta_0\|_{L^2},$$
$$ \|u^{QG}(\cdot,t)\|_{L^q}\le C_q\|\theta^{QG}(\cdot,t)\|_{L^q},\qquad
1<q<\infty,$$
where $C_q$ is a constant depending on $q$.

\item 
The solution $\theta^{DQG}$ of the dissipative QGS equation obeys
$$
\|\theta^{DQG}(\cdot,t)\|_{L^p}\le \|\theta_0\|_{L^p},
\qquad 1\le p\le \infty, $$
$$\|u^{DQG}(\cdot,t)\|_{L^q}\le C_q\|\theta^{DQG}(\cdot,t)\|_{L^q},
\qquad 1<q<\infty. $$
\end{enumerate}
\end{prop}
{\bf Proof.}\quad 
The proof of this proposition is classical. The $L^q$ estimates 
are the classical inequality for the Calderon-Zygmund singular
integrals while the $L^p$ bounds come from energy estimates.
We omit details.

vspace{.27in}
We now state the inviscid limit theorem for smooth solutions.

\begin{thm}\label{thm:chapter4}
Let $\theta^{QG}$ and $\theta^{DQG}$ be the smooth solutions of the 
QGS equations and the dissipative QGS equations with the same
initial data $\theta_0\in H^{k}(\Bbb D^2)(k\ge 3)$. If $[0,T^*)$ is the
maximal time interval of smooth existence, then for any $t<T^*$,
$$
\|\theta^{QG}(\cdot,t)-\theta^{DQG}(\cdot,t)\|_{L^2(\Bbb D^2)}\le C\kappa,
$$
where $C$ is a constant depending on $\theta_0$ and $T^*$ only.
\end{thm}

The convergence rate $O(\kappa)$ is optimal. 
In the proof we only treat the case $\Bbb R^2=\Bbb D^2$ and the case
$\Bbb R^2=\Bbb T^2$ is easier.

\vspace{.1in}
\noindent{\bf Proof of Theorem \ref{thm:chapter4}.}\quad
Consider the difference 
$$
\theta(x,t)=\theta^{DQG}(x,t)-\theta^{QG}(x,t)
$$
between the solutions of the QGS and the dissipative QGS equations
and let $u(x,t)$ be the corresponding velocity 
difference. This difference $\theta$ satisfies
$$
\frac{\partial \theta}{\partial t}+u^{DQG}\cdot\nabla\theta
+u\cdot\nabla\theta^{QG}+\kappa \Lambda^{2\alpha}(\theta+
\theta^{QG})=0
$$
where $\Lambda^{2\alpha}=(-\Delta)^{\alpha}$.
Multiplying by $\theta$ and integrating in space
$$
\frac{1}{2}\frac{d}{dt}\int \theta^2dx+\kappa \int (\Lambda^{2\alpha}\theta)
\theta dx=I+II+III
$$
where
$$I=\int (u^{DQG}\cdot\nabla\theta)\theta dx,\quad II=\int (u\cdot
\nabla\theta^{QG})\theta dx $$
$$ III=\kappa\int (\Lambda^{2\alpha}\theta^{QG})\theta dx$$
We estimate these three terms and start with the first one. 
Clearly the estimates in Proposition \ref{prop:33.1.2} guarantee that $I$ is
integrable. We show that it is actually zero. Let $\chi$ be a 
smooth cut-off function: $\chi(x)=1$ if $|x|<1$ and $\chi(x)=0$ if
$|x|>2$ and $\chi_r=\chi(\frac{x}{r})$ for $r>0$. Using
the Dominated Convergence Theorem and the divergence theorem,
$$
I=\lim_{r\to \infty}\int (u^{DQG}\cdot\nabla\theta)\theta\chi_r(x)dx
=-\lim_{r\to \infty}\frac{1}{2r}\int\chi'\cdot u^{DQG}\theta^2 dx
$$
since the last integral is bounded,
$$
|\int \chi'\cdot u^{DQG}\theta^2 dx|\le \int |u^{DQG}|\theta^2 dx
\le \|\theta^{DQG}\|_{L^2}\|\theta\|^{2}_{L^4}\le 4\|\theta_0\|_{L^2}
\|\theta_0\|^{2}_{L^4}
$$
we obtain $I=0$. $II$ and $III$ can be estimated by using Proposition 
\ref{prop:33.1.2},
$$
|II|\le \|\nabla\theta^{QG}\|_{L^\infty}\|u\|_{L^2}\|\theta\|_{L^2}
=\|\nabla\theta^{QG}\|_{L^\infty}\|\theta\|_{L^2}^{2}
$$ $$
|III|\le \frac{\kappa^2}{2}\int (\Lambda^{2\alpha}\theta^{QG})^2dx
+\frac{1}{2}\int\theta^2 dx
$$

\vspace{.15in}
Collecting these estimates,
$$
\frac{d}{dt}\int \theta^2dx+\kappa \|\Lambda^{\alpha} \theta\|_{L^2}^{2} dx
\le \cal{P}(t)\|\theta\|_{L^2}^{2}
+\kappa^2 \|\theta^{QG}\|_{2\alpha}^{2}
$$
where 
$$
\cal{P}(t)=2\|\nabla\theta^{QG}(\cdot,t)\|_{L^\infty}+1
$$
By Gronwall's inequality,
$$
\|\theta\|_{L^2}^{2}\le e^{\int_{0}^{t}\cal{P}(s)ds}
\|\theta_0\|_{L^2}^{2}+\kappa^2 \int_{0}^{t}
e^{\int_{\tau}^{t}\cal{P}(s)ds}
\|\theta^{QG}\|^{2}_{2\alpha} d\tau
$$
Noting that 
$\theta_0=0$ and using the result of Proposition \ref{prop:2.1}, especially,
$$
\int_{0}^{t}\|\nabla\theta^{QG}(\cdot,s)\|_{L^\infty}ds< \infty,\quad \int_{0}^{t}\|\theta^{QG}(\cdot,s)
\|_{k}^{2}ds <\infty
$$
we obtain
$$
\|\theta\|_{L^2}\le C\kappa
,$$
which completes the proof of Theorem \ref{thm:chapter4}.

\vspace{.24in}
We now turn to weak solutions of these equations corresponding
to $L^2$ initial data. We restrict ourselves to the periodic
domain $\Bbb T^2=[0,L]\times[0,L]$. We quote the result of 
Resnick \cite{Re} on the existence of weak solutions.


\begin{prop}
Let $\theta_0\in L^2(\Bbb T^2)$ and $T>0$ be arbitrarily fixed.
Then there exist weak solutions $\theta^{QG}\in L^\infty([0,T];L^2(\Bbb T^2))$
and $\theta^{DQG}\in L^\infty([0,T];L^2(\Bbb T^2))\cap \newline L^2([0,T];H^\alpha(
\Bbb T^2))$ of the QGS and the dissipative QGS equations, respectively.
That is, for each test function $\phi\in C^\infty(\Bbb T^2)$,
$$
\int \theta^{QG}\phi dx-\int\theta_0\phi dx-\int_{0}^{T}
\int_{\Bbb T^2}\theta^{QG}(u^{QG}\cdot\nabla \phi)dxdt=0,
$$ $$
\int \theta^{DQG}\phi dx -\int \theta_0\phi dx-
\int_{0}^{T}\int_{\Bbb T^2}\theta^{DQG}(u^{DQG}\cdot\nabla \phi)dxdt=0,
$$
where $u^{QG}$ and $u^{DQG}$ are the velocities corresponding to
$\theta^{QG}$ and $\theta^{DQG}$.
\end{prop}

\vspace{.1in}
These weak solutions are constructed by using classical Galerkin 
approximations. The weak $L^2$ inviscid limit result is an easy
consequence of this construction method.
\begin{thm}
Let $\theta_0\in L^2(\Bbb T^2)$ and $\theta^{QG}$ and $\theta^{DQG}$
be the weak solutions of the QGS and the dissipative QGS equations
with the same initial data $\theta_0$. Then for any arbitrarily 
fixed $T>0$ and any $\phi\in L^2(\Bbb T^2)$,
\begin{equation}\label{eq:33.2.1}
\limsup_{\kappa\to 0}\left(\theta^{DQG}(\cdot,t)-\theta^{QG}(\cdot,t),\phi\right)=0,
\qquad \mbox{for any $t\le T$},
\end{equation}
\end{thm}
{\bf Proof}\quad Consider the $n$-th Galerkin approximation  $\{\theta^{QG}_{n}\}$ and $\{\theta^{DQG}_{n}\}$,
which are in the space $S_n$ spanned by the Fourier modes $e^{imx}$ with 
$0<|m|\le n$ and satisfy
$$
\frac{\partial \theta_n}{\partial t}+P_n(u_n\cdot\nabla\theta_n)+
\kappa\Lambda^{2\alpha}\theta_n=0,
$$ $$
\theta_n|_{t=0}=P_n\theta_0,
$$
where $P_n$ is the orthogonal projection from $L^2$ onto $S_n$ and $\kappa=0$
in the case of $\theta^{QG}$. 
As we know from \cite{Re}, for some subsequences
$$
\theta^{DQG}_{n}\rightharpoonup\theta^{DQG},\quad\theta^{QG}_{n}
\rightharpoonup\theta^{QG} \quad \mbox{weakly in $L^2(\Bbb T^2)$},
$$
So we have for any $\epsilon>0$ by taking large $n$,
$$
|(\theta^{DQG}(\cdot,t)-\theta^{QG}(\cdot,t),\phi)|\le 
\epsilon+ |(\theta^{DQG}_{n}-\theta^{QG}_{n},\phi)|
$$
\begin{equation}\label{eq:2}
\le \epsilon +\|\phi\|_{L^2}\|\theta^{DQG}_{n}-\theta^{QG}_{n}\|_{L^2}
\le \epsilon+C_n\kappa
\end{equation}
which implies (\ref{eq:33.2.1}). Here we've applied the inviscid limit result
for smooth solutions to $\theta^{DQG}_{n}-\theta^{QG}_{n}$.

\vspace{.12in}
\begin{rem}
Since the constants $C_n$ in the inequality (\ref{eq:2})
depends on $n$, we obtain no convergence rate. An explicit rate may 
exist in negative Sobolev norms.
\end{rem}





\newpage
\section {Regularity Estimates}

We consider the 2-D dissipative QGS equations with smooth initial
data on the torus $\Bbb T^2=[0,L]\times[0,L]$, which admits a unique
classical local solution. Let $\theta$ be this solution. We will 
estimate the quantity
$$
\|e^{\gamma t\Lambda^{\alpha}}\Lambda^\beta\theta\|_{L^2}^{2}
$$
where the operators $\Lambda^\beta$ and $e^{\lambda
\Lambda^\alpha}$ are defined through the Fourier transform
$$
\Lambda^\beta f=L^{-2}\sum_{k}e^{ik\cdot x}|k|^{\beta}\widehat{f}(k)
$$
$$
e^{\lambda\Lambda^\alpha}f=L^{-2}\sum_{k}e^{ik\cdot x+\lambda|k|^\alpha}
\widehat{f}(k)
$$
with $\widehat{f}(k)$ being the k-th Fourier mode of $f$,
$$
\widehat{f}(k)=\int_{\Bbb T^2}e^{-ik\cdot x}f(x)dx.
$$

It is easy to see from these notations that $e^{\lambda\Lambda^\alpha}$
commutes with $\Lambda^\beta$ and partial derivatives for periodic
boundary conditions considered here.

\vspace{.14in}
We obtain bounds for the quantity $
\|e^{\gamma t\Lambda^{\alpha}}\Lambda^\beta\theta\|_{L^2}^{2}
$, which lead to the exponential decay of the Fourier spectrum of $\theta$.
The precise estimates are 

\begin{thm}
Consider the 2-D dissipative QGS equations
\begin{equation}\label{eq:3.1}
\frac{\partial \theta}{\partial t}+u\cdot\nabla \theta
+\kappa\Lambda^{2\alpha}\theta=0,\qquad \kappa>0,\quad \frac{1}{2}<\alpha\le 1
\end{equation}
on the 2-D torus $\Bbb T^2=[0,L]\times[0,L]$. Let the initial 
data $\theta_0\in H^k(\Bbb T^2)$ and $\theta$ be the unique
smooth solution. We take $\beta:$
$$
\beta>0, \qquad\beta+2\alpha> 2.
$$
Then 
$\theta$ satisfies for any $\gamma>0$
\begin{equation}\label{eq:3.2}
\|e^{\gamma t\Lambda^{\alpha}}\Lambda^\beta\theta
\|_{L^2}^{2}\le \frac{e^{\frac{2\gamma^2t}{\kappa}}\|\Lambda^\beta\theta_0\|_{
L^2}^{2}}
{\left(1-C(\|\Lambda^\beta\theta_0\|_{L^2}^{2})^{N-1}\kappa^{-(M-1)}\gamma^{-2}
(e^{\frac{2(N-1)\gamma^2t}{\kappa}}-1)\right)^{\frac{1}{N-1}}},
\end{equation}
which is finite for $t\in[0,t^*)$,
$$
t^*=\frac{\kappa}{2(N-1)\gamma^2}\log\left(1+\frac{\kappa^{M-1}\gamma^2}{
C\|\Lambda^\beta\theta_0\|^{2(N-1)}_{L^2}}\right)
$$
Here $C$ is a constant and $M,N$ are given by
$$
M=\frac{\alpha+\sigma}{\alpha-\sigma},\quad N=1+\frac{\alpha}{\alpha-\sigma},
\quad \sigma=\left\{\begin{array}{ll}1-\alpha,&\quad \mbox{if $\beta\ge 1$}
\\2-\beta-\alpha,&\quad \mbox{if $\beta\le 1$}\end{array}\right.
$$
\end{thm}

\vspace{.12in}
A special consequence of this theorem is that each Fourier mode 
amplitude can be individually controlled. In fact, a rough estimate
gives
$$
e^{2\gamma t |k|^{\alpha}}|k|^{2\beta}|\theta(k,t)|^2
\le \sum_{k}e^{2\gamma t |k|^{\alpha}}|k|^{2\beta}|\theta(k,t)|^2
=L^{2}\|e^{\gamma t\Lambda^{\alpha}}\Lambda^\beta\theta
\|_{L^2}^{2}
$$
Thus the k-th mode is bounded by 
$$
|\theta(k,t)|^2\le \frac{L^2}{|k|^{2\beta}}\frac{e^{\frac{\gamma^2 t}{\kappa}
-2\gamma t|k|^{\alpha}}\|\Lambda^\beta\theta_0\|_{L^2}^{2}}
{\left(1-C(\|\Lambda^\beta\theta_0\|_{L^2}^{2})^{N-1}\kappa^{-(M-1)}\gamma^{-2}
(e^{\frac{2(N-1)\gamma^2t}{\kappa}}-1)\right)^{\frac{1}{N-1}}}
$$
for $t\in[0,t^*).$

\vspace{.1in}
Doering and Titi \cite{DT} establish the exponential decay of power spectrum 
for the flow field of the 3-D Navier-Stokes equations.
Similar analysis based on these bounds can be made to conclude that
if $\|\Lambda^\beta\theta\|^{2}_{L^2}$ is bounded uniformly in time, then
after a transient time of length $t^*/2$ the Fourier spectrum of $\theta$ 
decays exponentially at high wave numbers. Furthermore, the associated
decay length can be defined and estimated in terms of the dissipation
rates.

\vspace{.2in}
The main difficulty in proving the estimate (\ref{eq:3.2}) is how to bound 
the non-linear term properly. We need the inequalities for the 
Calderon-Zygmund type singular integrals. We'll also use the
following lemma concerning the operator $\Lambda^s$, which is proved
in \cite{Re},\cite{Ta}.

\begin{lemma}
For $s>0$ and $1<r<p\le\infty$,
$$
\|\Lambda^s(uv)\|_{L^r}\le C(\|u\|_{L^p}\|\Lambda^sv\|_{L^q}
+\|v\|_{L^p}\|\Lambda^su\|_{L^q})
$$
where $\frac{1}{r}=\frac{1}{p}+\frac{1}{q}$ and $C$ is a constant.
\end{lemma}

\vspace{.16in}
\noindent {\bf Proof of Theorem 3.1}.\quad
Using the equation (\ref{eq:3.1}),
$$
\frac{1}{2}\frac{d}{dt}\int|e^{\gamma t\Lambda^\alpha}\Lambda^\beta
\theta|^2dx =\int \left(e^{\gamma t\Lambda^\alpha}\Lambda^\beta\theta\right)
\left(\gamma e^{\gamma t\Lambda^\alpha}\Lambda^\alpha\Lambda^\beta\theta
+e^{\gamma t\Lambda^\alpha}\Lambda^\beta\frac{\partial \theta}
{\partial t}\right)dx
$$ $$
=I+II+III\qquad\qquad
$$
where
$$
I= \gamma\int \left(e^{\gamma t\Lambda^\alpha}\Lambda^\beta\theta\right)
\left(e^{\gamma t\Lambda^\alpha}\Lambda^\alpha\Lambda^\beta\theta\right)dx
$$ $$
\quad \qquad \le \frac{\gamma^2}{\kappa}\|
e^{\gamma t\Lambda^\alpha}\Lambda^\beta\theta\|_{L^2}^{2} +
\frac{\kappa}{4}\|e^{\gamma t\Lambda^\alpha}\Lambda^{\alpha
+\beta}\theta\|_{L^2}^{2}
,$$ $$
II=-\int \left(e^{\gamma t\Lambda^\alpha}\Lambda^\beta\theta\right)
\left(e^{\gamma t\Lambda^\alpha}\Lambda^\beta(u\cdot\nabla \theta)\right)dx
,$$
$$
III=-\kappa \int\left(e^{\gamma t\Lambda^\alpha}\Lambda^\beta\theta\right)\left(e^{\gamma t\Lambda^\alpha}\Lambda^{2\alpha+\beta}\theta\right) dx
=-\kappa\|e^{\gamma t\Lambda^\alpha}\Lambda^{\alpha
+\beta}\theta\|_{L^2}^{2}
$$

\vspace{.12in}
Now we deal with the second term II. Since the operators commute,
$$
II=-\int\left(\Lambda^{(\alpha+\beta)}e^{\gamma t\Lambda^\alpha}\theta
\right)\left(R\Lambda^{\beta-\alpha+1}e^{\gamma t\Lambda^\alpha}
(u\cdot\theta)\right)dx
$$
where $R=(\partial_{x_1}\Lambda^{-1},\partial_{x_2}\Lambda^{-1})$ are  
Riesz transforms. For brevity, we'll use the notations
$$
\tilde {u}=e^{\gamma t\Lambda^\alpha}u,\qquad 
\tilde {\theta}=e^{\gamma t\Lambda^\alpha}\theta
$$
To obtain further estimates, we break up the term $e^{
\gamma t\Lambda^\alpha}(u\cdot\theta)$. Its structure
can be better seen from the Fourier transform.
$$
e^{\gamma t\Lambda^\alpha}(u\cdot\theta)= L^{-2}
\sum_{k'+k''=k}e^{\gamma t|k|^{\alpha}}\widehat{u}(k')
\widehat{\theta}(k'')
$$
For $0\le\alpha\le 1$, 
$$
|k|^{\alpha}=|k'+k''|^{\alpha}\le |k'|^{\alpha}+|k''|^{\alpha}
$$
So this term is bounded by
$$
L^{-2}\sum_{k'}e^{\gamma t|k'|^{\alpha}}|u(k')|\cdot
\sum_{k''}e^{\gamma t|k''|^{\alpha}}|\theta(k'')|
$$

Now using H\"{o}lder's inequality and the boundedness of Riesz transform,
$$
|II|\le C\|\tilde\theta\|_{H^{\alpha+\beta}}\|
\Lambda^{\beta+1-\alpha}(\tilde{u}\cdot\tilde{\theta})\|_{L^2}
$$
where $C$ is a constant.


\vspace{.1in}
Now we take $q=2$ if $\beta>1$ and 
$q=\frac{2}{\beta}$ if $\beta\le 1$. Choose $p$ such that 
$$
\frac{2}{p}+\frac{2}{q}=1
$$
and $\sigma=2-\frac{2}{q}-\alpha$. The condition that $\beta+2\alpha>2$
implies that $0<\sigma<\alpha$. We apply Lemma 2.2 to obtain
$$
|II|\le C\|\tilde{\theta}\|_{H^{\alpha+\beta}}\left(\|\tilde{u}\|_{L^p}
\|\Lambda^{\beta+1-\alpha}\tilde{\theta}\|_{L^q}
+\|\tilde{\theta}\|_{L^p}\|\Lambda^{\beta+1-\alpha}\tilde{u}\|_{L^q}\right)
=II_1+II_2
$$

\vspace{.1in}
Using Sobolev imbeddings
$$
H^{\beta}\hookrightarrow H^{\frac{2}{q}}\hookrightarrow L^p,\qquad
H^{\beta+\sigma}\hookrightarrow L^{p}_{\beta+1-\alpha}
$$
and the Gagliado-Nirenberg interpolation (since $\sigma<\alpha$):
$$
\beta+\sigma=\frac{\sigma}{\alpha}(\beta+\alpha)+\left(1-\frac{\sigma}
{\alpha}\right)\beta
$$
we have
$$
II_1\le C\|\tilde{\theta}\|_{H^{\beta+\alpha}}\|\tilde{u}\|_{H^{\beta}}
\|\tilde{\theta}\|_{H^{\beta+\sigma}}\le C\|\tilde{\theta}\|_{H^{\beta}}
\|\tilde{\theta}\|_{H^{\beta+\alpha}}\|\tilde{\theta}\|_{H^{\beta+\sigma}}
$$
$$
\le C\|\tilde{\theta}\|_{H^{\beta+\alpha}}^{1+\frac{\sigma}{\alpha}}
\|\tilde{\theta}\|_{H^{\beta}}^{2-\frac{\sigma}{\alpha}}
$$
where $C$ is constant depending on $\alpha$ and $\beta$.
By Young's inequality
$$
II_1\le \frac{\kappa}{8}\|\tilde{\theta}\|_{H^{\beta+\alpha}}^{2}+
\frac{C}{\kappa^{M}}\left(\|\tilde{\theta}\|_{\beta}^{2}\right)^{N}
$$
where $N=\frac{2\alpha-\sigma}{\alpha-\sigma}$
and $M=\frac{\alpha+\sigma}{\alpha-\sigma}$.

\vspace{.11in}
A similar estimate results in the same bound for $II_2$.

\vspace{.17in}
Collecting the estimates for $I,II$ and $III$ and reintroducing 
$\tilde{\theta}=e^{\gamma t\Lambda^{\alpha}}\theta$,
$$
\frac{d}{dt}\|e^{\gamma t\Lambda^{\alpha}}\Lambda^{\beta}\theta\|_{L^2}
^{2}\le -\kappa \|e^{\gamma t\Lambda^{\alpha}}\Lambda^{\beta+\alpha}
\theta\|_{L^2}^{2} 
$$
$$
\qquad\qquad \qquad+\frac{2\gamma^2}{\kappa}\|
e^{\gamma t\Lambda^\alpha}\Lambda^\beta\theta\|_{L^2}^{2}
+\frac{C}{\kappa^M}\|e^{\gamma t\Lambda^\alpha}\Lambda^\beta\theta\|_{L^2}^{2N}
$$
Now we let
$$
Z(t)= \|e^{\gamma t\Lambda^{\alpha}}\Lambda^{\beta}\theta\|_{L^2}
^{2}
$$
the differential inequality becomes
$$
\frac{dZ}{dt}\le \frac{2\gamma^2}{\kappa}Z+\frac{C}{\kappa^M}Z^N
$$
An elementary algebra results
$$
\frac{dY}{dt}\le \frac{Ce^{2(N-1)\frac{\gamma^2t}{\kappa}}}{\kappa^M}Y^{N}
$$
where $Y=e^\frac{-2\gamma^2t}{\kappa}Z$. After a simple calculation,
$$
Y\le \frac{Y_0}{\left(1-CY_{0}^{N-1}\kappa^{-(M-1)}\gamma^{-2}(
e^{\frac{2(N-1)\gamma^2t}{\kappa}}-1)\right)^{\frac{1}{N-1}}}
$$
Reintroducing $Z(t)=\|e^{\gamma t\Lambda^{\alpha}}\Lambda^\beta\theta
\|_{2}^{2}$ and noting that $Z(0)=\|\Lambda^\beta\theta_0\|_{2}^{2}$
that
$$
\|e^{\gamma t\Lambda^{\alpha}}\Lambda^\beta\theta
\|_{2}^{2}\le \frac{e^\frac{2\gamma^2t}{\kappa}\|\Lambda^\beta\theta_0\|_{2}^{2}}
{\left(1-C(\|\Lambda^\beta\theta_0\|_{2}^{2})^{N-1}\kappa^{-(M-1)}\gamma^{-2}
(e^{\frac{2(N-1)\gamma^2t}{\kappa}}-1)\right)^{\frac{1}{N-1}}}
$$
This means that $\|e^{\gamma t\Lambda^{\alpha}}\Lambda^\beta\theta
\|_{2}^{2}$ is finite on the interval $[0,t^*)$, where
$$
t^*=\frac{\kappa}{2(N-1)\gamma^2}\log\left(1+\frac{\kappa^{M-1}\gamma^2}{
C\|\Lambda^\beta\theta_0\|^{2(N-1)}_{2}}\right)
$$
The smaller the initial decay rate $\|\Lambda^\beta\theta_0\|_{2}$
is , the larger $t^*$. Similarly, the larger the parameter $\gamma$
, the shorter $t^*$.




\newpage
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