\documentclass[reqno]{amsart}
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\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2017 (2017), No. 290, pp. 1--6.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2017 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2017/290\hfil Improved lifespan of solutions]
{Improved lifespan of solutions to an inviscid
surface quasi-geostrophic model}

\author[Z. Wen \hfil EJDE-2017/290\hfilneg]
{Zhihong Wen}

\address{Zhihong Wen \newline
Department of Mathematics and Statistics,
Jiangsu Normal University,
101 Shanghai Road,
Xuzhou 221116, Jiangsu,  China}
\email{wenzhihong1989@163.com}


\thanks{Submitted November 10, 2016. Published November 21, 2017.}
\subjclass[2010]{35B35, 35Q35, 76B03}
\keywords{Surface quasi-geostrophic equation; lifespan; decay estimate}

\begin{abstract}
 This article consider the two-dimensional (2D) inviscid surface
 quasi-geostrophic  (SQG) model. By studying the decay estimate of the
 operator $e^{\mathcal{R}_1^{2}t}$, we obtain an improved lifespan
 of the solutions to the corresponding model. More precisely,
 if the initial data is of size $\epsilon$, then the lifespan satisfies
 $T_{\epsilon}\simeq\epsilon^{-4/3}$, which improves the result obtained
 via hyperbolic methods.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{remark}[theorem]{Remark}
\allowdisplaybreaks

\section{Introduction} \label{intro}

The classical 2D inviscid surface  quasi-geostrophic (SQG) equation has
 the form
\begin{equation}\label{SQG3}
\begin{gathered}
\partial_{t}\theta+(u\cdot\nabla)\theta=0, \\
u=(-\mathcal{R}_2\theta,\mathcal{R}_1\theta),\\
\theta(x, 0)=\theta_{0}(x),
\end{gathered}
\end{equation}
where the real scalar function $\theta=\theta(x,t)$ represents the potential
temperatures of the fluid, and
\[
\mathcal{R}_1:=\partial_{x_1}\Lambda^{-1}, \mathcal{R}_2
:=\partial_{x_2}\Lambda^{-1}\quad (\Lambda:=(-\Delta)^{1/2})
\]
 are the standard 2D Riesz transforms. The SQG equation is an important
model in geophysical fluid dynamics. In particular, it is the special
case of the general quasi-geostrophic approximations for atmospheric and
oceanic fluid flow with small Rossby and Ekman numbers, see \cite{CMT,PG}
and the references cited there. Mathematically, as pointed out
by Constantin, Majda and Tabak \cite{CMT},  the inviscid SQG equation
shares many parallel properties with those of the 3D Euler equations
such as the vortex-stretching mechanism and thus
serves as a lower-dimensional model of the 3D Euler equations.

The mathematical study on the SQG equation is divided into two major cases.
The first case is the dissipative SQG equation with fractional Laplacian, namely
\eqref{SQG3} adding $\Lambda^{\alpha}$, which has recently attracted
enormous attention and significant progress has been made on the global
well-posedness issue.
The global regularity problem for the SQG
equation with subcritical ($\alpha>1$) can be found in \cite{CW3111,Resnick}
Constantin, C$\rm \acute{o}$rdoba and Wu in \cite{CCW} first addresses
the global regularity issue for the critical case ($\alpha=1$) and
obtained a small data global existence result. Since then, small data
global existence results have been obtained in various functional
settings (see, e.g., \cite{CCCMP04,CMZ07,Chaelee03}).
Recently, the global regularity without small condition for the SQG
equation with critical dissipation has been successfully resolved
(see, e.g., \cite{CV1,CV,KN2,KNV}). The global regularity issue for
the supercritical case ($0<\alpha<1$) remains outstandingly open
(see \cite{Dabkowski,Silvestre} for the eventual regularity).


The second case is the inviscid case
\eqref{SQG3} which is probably the simplest dynamical scalar equation,
however, the global regularity problem still remains open.
The local well-posedness and blow-up criterion of \eqref{SQG3} were
first established in the Sobolev spaces by Constantin, Majda and Tabak \cite{CMT}.
Subsequently, there are various results available in different function spaces
(see for instance \cite{Wu97,Chae03,Resnick,Xiang11}).
We remark that aside from local well-posedness and breakdown criteria not much
is known about the well-posedness of the inviscid SQG equation.
As a matter of fact, the global small data well-posednes result of
the inviscid case is also an unsolved problem. Recently, Wu-Xu-Ye \cite{WXY}
established the global smooth solutions to the damped SQG equation with small
initial data.


Cannone, Miao, and Xue \cite{Cannone} proved the existence of global
strong solutions to the following dispersive SQG equation with supercritical
 dissipation
$$
\partial_{t}\theta+(u\cdot\nabla)\theta+\Lambda^{\alpha}\theta
+K\mathcal{R}_1\theta=0,\quad 0<\alpha<1.
$$
More precisely, they show that for given initial data $\theta_{0}$
there exists $K$ large enough such that the
solution is global.
Very recently, Elgindi and Widmayer \cite{Elgindi} considered the
 following dispersive SQG equation (without any dissipation)
 \begin{equation}\label{dispersiveSQG}
\begin{gathered}
\partial_{t}\theta+(u\cdot\nabla)\theta=\mathcal{R}_1\theta, \\
u=(-\mathcal{R}_2\theta,\mathcal{R}_1\theta),\\
\theta(x, 0)=\theta_{0}(x).
\end{gathered}
\end{equation}
By studying the anisotropic
linear semigroup $e^{\mathcal{R}_1t}$ and using the stationary phase lemma,
 the lifespan of the above system \eqref{dispersiveSQG} was given 
in \cite{Elgindi} as follows
\begin{equation}\label{ok88}
T_{\epsilon}\simeq\epsilon^{-4/3}.
\end{equation}
Let us also point out that Elgindi \cite{Elgindi11} considered
the following inviscid porous medium equation
\begin{equation}\label{PPM}
\begin{gathered}
\partial_{t}\theta+( {u}\cdot\nabla)\theta=0, \\
 {u}=-\nabla p-\theta e_2,\\
\nabla\cdot {u}=0,\\
\theta(x, 0)=\theta_{0}(x).
\end{gathered}
\end{equation}
By classifying all stationary
solutions of the inviscid  porous
medium equation under mild conditions, he proved that
sufficiently regular perturbations which are also small must
 be globally regular
and strongly converge to a steady state.

In this article, we consider system \eqref{dispersiveSQG} when
replacing $\mathcal{R}_1\theta$ by $\mathcal{R}_1^{2}\theta$.
 More precisely, we consider
\begin{equation}\label{SQG}
\begin{gathered}
\partial_{t}\theta+(u\cdot\nabla)\theta=\mathcal{R}_1^{2}\theta, \\
u=(-\mathcal{R}_2\theta,\mathcal{R}_1\theta),\\
\theta(x, 0)=\theta_{0}(x).
\end{gathered}
\end{equation}
By establishing the decay estimate of the operator $e^{\mathcal{R}_1^{2}t}$
(see Lemma \ref{Lem23}), we also obtain the lifespan \eqref{ok88} for
the above system \eqref{SQG}. Precisely, the main result can be stated as follows.

\begin{theorem}\label{ThSQG}
Let $s_1>4$, $s_2>3$ and $\epsilon$ be a sufficiently small positive constant.
If $\|\theta_{0}\|_{H^{s_1}}+\|\theta_{0}\|_{W^{s_2,1}}\leq\epsilon$,
then there exists a unique solution 
$\theta\in C([0,\,T_{\epsilon}]; H^{s_1}(\mathbb{R}^{2}))$ of the system
 \eqref{SQG}, where $T_{\epsilon}$ satisfies
$$
T_{\epsilon}\simeq\epsilon^{-4/3}.
$$
Moreover, 
$$
\|\theta(t)\|_{H^{s_1}}\lesssim\epsilon, \quad \forall t\in [0,T].
$$
\end{theorem}

It is well-known that by using hyperbolic methods, it is easy to get the 
maximal existence time $T_{\epsilon}$ with $T_{\epsilon}\geq \frac{C}{\epsilon}$.
Thus, it is clear that Theorem \ref{ThSQG} improves this result.


\begin{remark}\rm
Theorem \ref{ThSQG} still holds for the system
\begin{equation}\label{SQG111}
\begin{gathered}
\partial_{t}\theta+(u\cdot\nabla)\theta=\mathcal{R}_2^{2}\theta, \\
u=(-\mathcal{R}_2\theta,\mathcal{R}_1\theta),\\
\theta(x, 0)=\theta_{0}(x).
\end{gathered}
\end{equation}
The proof is the same as that of Theorem \ref{ThSQG}.
\end{remark}

\section{Proof of Theorem \ref{ThSQG}} \label{proofofThm1}

We first apply
$(I+\Lambda)^{s}$ ($s>0$) to the equation \eqref{SQG} and multiply the resultant by
$(I+\Lambda)^{s}\theta$, add them up to to conclude that
\begin{equation}\label{t71901}
 \frac{d}{dt}\|\theta(t)\|_{H^{s}}^{2}
+\|\mathcal{R}_1 \theta\|_{H^{s}}^{2}
=-\int {(I+\Lambda)^{s}(u\cdot\nabla\theta)(I+\Lambda)^{s}\theta\,dx},
\end{equation}
where we have used
$$
\int {(I+\Lambda)^{s}\mathcal{R}_1^{2}\theta  (I+\Lambda)^{s}\theta\,dx}
=-\int {(I+\Lambda)^{s}\mathcal{R}_1 \theta \,(I+\Lambda)^{s}\mathcal{R}_1 \theta\,dx}
=\|\mathcal{R}_1 \theta\|_{H^{s}}^{2} .
$$
Thanks to the Kato-Ponce inequality and the divergence condition, we have
\begin{equation}\label{t71902}
\begin{aligned}
\int {(I+\Lambda)^{s}(u\cdot\nabla\theta)(I+\Lambda)^{s}\theta\,dx}
&=\int {[(I+\Lambda)^{s},\,u\cdot\nabla]\theta (I+\Lambda)^{s}\theta\,dx} \\
&\leq C ( \| \nabla u \|_{L^{\infty}}
 +\|\nabla \theta \|_{L^{\infty}})\|\theta \|_{H^{s}}^{2}.
\end{aligned}
\end{equation}
Putting \eqref{t71902} into \eqref{t71901} yields
$$
\frac{d}{dt}\|\theta(t)\|_{H^{s}}^{2}
+\|\mathcal{R}_1 \theta\|_{H^{s}}^{2} 
\leq  C ( \| \nabla u \|_{L^{\infty}}
+\|\nabla \theta \|_{L^{\infty}})\|\theta \|_{H^{s}}^{2}.
$$
By integrating the above inequality in time, we obtain
\begin{equation}\label{t71903}
\|\theta(t)\|_{H^{s}}^{2}+\int_{0}^{t}{\|\mathcal{R}_1 
\theta(\tau)\|_{H^{s}}^{2}\,d\tau}
\leq \|\theta_{0}\|_{H^{s}}^{2}e^{C \int_{0}^{t}( \| \nabla u (\tau)\|_{L^{\infty}}
+\|\nabla \theta (\tau)\|_{L^{\infty}})\,d\tau}.
\end{equation}
Next, our goal is to estimate $\| \nabla u\|_{L^{\infty}}$ and 
$\| \nabla \theta\|_{L^{\infty}}$ at the right-hand side of \eqref{t71903}.
To this end, we apply the Duhamel principle to the first equation of the 
system \eqref{SQG} to show
\begin{equation}\label{t71904}
\theta(x,t)=e^{\mathcal{R}_1^{2}t}\theta_{0}
-\int_{0}^{t}{e^{\mathcal{R}_1^{2}(t-\tau)}(u\cdot\nabla\theta)(\tau)\,d\tau}.
\end{equation}
The following lemma concerns the decay estimate of the operator 
$e^{\mathcal{R}_1^{2}t}$, which is a key component in proving our main result.

\begin{lemma}\label{Lem23}
For any $\rho>2$, it holds
\begin{align}\label{t71905}
\|e^{\mathcal{R}_1^{2}t}f\|_{L^{\infty}}\leq
C(1+t)^{-1/2}\|f\|_{W^{\rho,1}},\end{align}
\begin{gather}\label{addt71905}
\|e^{\mathcal{R}_1^{2}t}\mathcal{R}_1f\|_{L^{\infty}}\leq
C(1+t)^{-1/2}\|f\|_{W^{\rho,1}}, \\
\label{addddt71905}
\|e^{\mathcal{R}_1^{2}t}\mathcal{R}_2f\|_{L^{\infty}}\leq
C(1+t)^{-1/2}\|f\|_{W^{\rho,1}}.
\end{gather}
\end{lemma}

\begin{proof}[Proof of Lemma \ref{Lem23}]
We first prove \eqref{t71905}. Using the polar coordinates $\xi_1=r\cos\alpha$,
 $\xi_2=r\sin\alpha$,  we get the  estimate
\begin{align*}
\|e^{\mathcal{R}_1^{2}t}f\|_{L^{\infty}}
&\leq \|e^{-\frac{\xi_1^{2}}{|\xi|^{2}}t}\widehat{f}(\xi)\|_{L^{1}} \\
&\leq C\int_{0}^{2\pi}e^{-t\cos^{2}\alpha }
\underbrace{\int_{0}^{\infty}{|\widehat{f}(\xi)|}r\,dr}_{N}d\alpha.
\end{align*}
Now we show that $N$ can be bounded by
$$
N\leq C\|f\|_{W^{\rho,1}}.
$$
As a matter of fact, it is not hard to see that
\begin{align*}
N&= \int_{0}^{\infty}{\frac{|(1+|\xi|)^{\rho}\widehat{f}(\xi)|r}{(1+|\xi|)^{\rho}}
\,dr} \\
&= \int_{0}^{\infty}{\frac{|\widehat{(1+\Lambda)^{\rho}f}(\xi)|r}{(1+r)^{\rho}}
\,dr} \\
&\leq C \int_{0}^{\infty}{\frac{\|(1+\Lambda)^{\rho}f\|_{L^{1}}\,r}{(1+r)^{\rho}}
\,dr} \\
&\leq C \|f\|_{W^{\rho,1}} \int_{0}^{\infty}{\frac{ r}{(1+r)^{\rho}}
\,dr} 
\leq C \|f\|_{W^{\rho,1}}.
\end{align*}
It directly gives
\begin{equation}\label{t71906}
\|e^{\mathcal{R}_1^{2}t}f\|_{L^{\infty}}
\leq C\|f\|_{W^{\rho,1}}\int_{0}^{2\pi}e^{-t\cos^{2}\alpha }d\alpha.
\end{equation}
By the simple calculation, we get
\begin{equation} \label{t71907}
\begin{aligned}
\int_{0}^{2\pi}e^{-t\cos^{2}\alpha }d\alpha
&= 4\int_{0}^{\pi/2}e^{-t\cos^{2}\alpha }d\alpha \\
&= 4\int_{0}^{\pi/4}e^{-t\cos^{2}\alpha }d\alpha
 +4\int_{\pi/4}^{\pi/2}e^{-t\cos^{2}\alpha }d\alpha \\
&\leq 4\int_{0}^{\pi/4}e^{-t(\frac{\sqrt{2}}{2})^{2} }d\alpha
 +4\sqrt{2}\int_{\pi/4}^{\pi/2}e^{-t\cos^{2}\alpha }
 \sin\alpha d\alpha \\
&\leq \pi e^{-t/2}-4\sqrt{2}t^{-1/2}
 \int_{\pi/4}^{\pi/2}e^{-t\cos^{2}\alpha }
 d (t^{1/2}\cos\alpha)\\
&\leq C(1+t)^{-1/2}.
\end{aligned}
\end{equation}
Combining \eqref{t71906} and \eqref{t71907} implies the desired estimate
 \eqref{t71905}. Noting the simple fact that $|\sin\alpha|,\,|\cos\alpha|\leq1$, 
the estimates \eqref{addt71905} and \eqref{addddt71905} follow directly 
from the proof of \eqref{addt71905}.
Thus, the proof is complete.
\end{proof}

The estimate \eqref{addt71905} can be improved as
$$
\|e^{\mathcal{R}_1^{2}t}\mathcal{R}_1f\|_{L^{\infty}}\leq
C(1+t)^{-1}\|f\|_{W^{\rho,1}}.
$$
The estimate \eqref{addt71905} itself would suffice our purpose.

If we denote $L_1=\nabla\nabla^{\perp}\Lambda^{-1}$ and $L_2=\nabla$, 
then $L_1\theta=\nabla u$ and $L_2\theta=\nabla \theta$.
Applying $L_{i}$ ($i=1,2$) to the first equation of the system \eqref{SQG} 
and using the Duhamel principle yield
\begin{equation}\label{t71908}
L_{i}\theta(x,t)=e^{\mathcal{R}_1^{2}t}L_{i}\theta_{0}
-\int_{0}^{t}{e^{\mathcal{R}_1^{2}(t-\tau)}L_{i}(u\cdot\nabla\theta)(\tau)\,d\tau}.
\end{equation}
Recalling the above mentioned estimates of Lemma \ref{Lem23} and invoking some 
simple embedding allows us to deduce
\begin{equation}\label{t71909}
\begin{aligned}
&\|L_{i}\theta\|_{L^{\infty}} \\
&\leq \|e^{\mathcal{R}_1^{2}t}L_{i}\theta_{0}\|_{L^{\infty}}
 +\int_{0}^{t}{\|e^{\mathcal{R}_1^{2}(t-\tau)}L_{i}(u\cdot\nabla\theta)(\tau)
 \|_{L^{\infty}}\,d\tau} \\
&\leq C(1+t)^{-1/2}\|\nabla \theta_{0}\|_{W^{s_2-1,1}}
+C\int_{0}^{t}{(1+t)^{-1/2}\|\nabla(u\cdot\nabla\theta)(\tau)
\|_{W^{s_1-2,1}}\,d\tau} \\
&\leq C(1+t)^{-1/2}\|\theta_{0}\|_{W^{s_2,1}}
+C\int_{0}^{t}{(1+t)^{-1/2}\|(u\cdot\nabla\theta)(\tau)
\|_{W^{s_1-1,1}}\,d\tau} \\
&\leq C(1+t)^{-1/2}\|\theta_{0}\|_{W^{s_2,1}}
+C\int_{0}^{t}{(1+t)^{-1/2}\|\theta(\tau)
\|_{H^{s_1}}^{2}\,d\tau}.
\end{aligned}
\end{equation}
If we assume for any $t\in [0,T]$ that 
$\|\theta(t)\|_{H^{s_1}}\leq2\epsilon$,
then  from \eqref{t71909} we deduce that
\begin{equation}\label{t71912}
\|L_{i}\theta\|_{L^{\infty}}\leq
\widetilde{C}\epsilon t^{-1/2}
+\widetilde{C}\epsilon^{2}t^{1/2},
\end{equation}
where $\widetilde{C}>0$ is an absolute constant. 
The above estimate together with the estimate \eqref{t71903} yields for any 
$t\in [0,T]$ that
\begin{equation}
\begin{aligned}
\|\theta(t)\|_{H^{s_1}}
&\leq \|\theta_{0}\|_{H^{s_1}}e^{\frac{C}{2} \int_{0}^{t}
 ( \| \nabla u (\tau)\|_{L^{\infty}}+\|\nabla \theta (\tau)\|_{L^{\infty}})\,d\tau} \\
&\leq \|\theta_{0}\|_{H^{s_1}}e^{\frac{C}{2} \int_{0}^{t}
 ( \widetilde{C}\epsilon \tau^{-1/2}
+\widetilde{C}\epsilon^{2}\tau^{1/2})\,d\tau} \\
&\leq \|\theta_{0}\|_{H^{s_1}}e^{\frac{C}{2} (2\widetilde{C}
 \epsilon t^{1/2}
+\frac{2\widetilde{C}}{3}\epsilon^{2}t^{3/2})}.
\end{aligned}
\end{equation}
Thus, if 
$$
\|\theta(t)\|_{H^{s_1}}\leq2\epsilon, \quad \forall\, t\in [0,T],
$$
then it suffices  that
$$
C\widetilde{C}\epsilon T^{1/2}
+\frac{C\widetilde{C}}{3}\epsilon^{2}T^{3/2}=\ln2,
$$
which further implies
$T\simeq\epsilon^{-4/3}$.
This completes the proof of Theorem \ref{ThSQG}

\subsection*{Acknowledgements}
The author would like to thank the anonymous referee for the careful
 reading and valuable suggestions. 
This research was supported by the Natural
Science Foundation of Colleges of Jiangsu Province (No. 17KJD110002) 
and the Foundation Project of Jiangsu Normal University (No. 16XLR033).

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\end{document}