# Well-posedness for Ohkitani model and long-time existence for surface quasi-
geostrophic equations

Dongho Chae Department of Mathematics, Chung-Ang University University.
E-mail<EMAIL_ADDRESS>In-Jee Jeong Department of Mathematical Sciences and
RIM, Seoul National University. E-mail<EMAIL_ADDRESS>Jungkyoung Na
Department of Mathematical Sciences, Seoul National University. E-mail:
<EMAIL_ADDRESS>Sung-Jin Oh Department of Mathematics, UC Berkeley and
School of Mathematics, Korea Institute for Advanced Study. E-mail:
<EMAIL_ADDRESS>
###### Abstract

We consider the Cauchy problem for the logarithmically singular surface quasi-
geostrophic (SQG) equation, introduced by Ohkitani,

$\begin{split}\partial_{t}\theta-\nabla^{\perp}\log(10+(-\Delta)^{\frac{1}{2}})\theta\cdot\nabla\theta=0,\end{split}$

and establish local existence and uniqueness of smooth solutions in the scale
of Sobolev spaces with exponent decreasing with time. Such a decrease of the
Sobolev exponent is necessary, as we have shown in the companion paper [7]
that the problem is strongly ill-posed in any fixed Sobolev spaces. The time
dependence of the Sobolev exponent can be removed when there is a dissipation
term strictly stronger than log. These results improve wellposedness
statements by Chae, Constantin, Córdoba, Gancedo, and Wu in [4]. This well-
posedness result can be applied to describe the long-time dynamics of the
$\delta$-SQG equations, defined by

$\begin{split}\partial_{t}\theta+\nabla^{\perp}(10+(-\Delta)^{\frac{1}{2}})^{-\delta}\theta\cdot\nabla\theta=0,\end{split}$

for all sufficiently small $\delta>0$ depending on the size of the initial
data. For the same range of $\delta$, we establish global well-posedness of
smooth solutions to the logarithmically dissipative counterpart:

$\begin{split}\partial_{t}\theta+\nabla^{\perp}(10+(-\Delta)^{\frac{1}{2}})^{-\delta}\theta\cdot\nabla\theta+\log(10+(-\Delta)^{\frac{1}{2}})\theta=0.\end{split}$

##  1 Introduction

###  1.1 Main results

In this paper, we are concerned with the logarithmically singular surface
quasi-geostrophic (SQG) equations:

$\left\\{\begin{aligned} &\partial_{t}\theta+u\cdot\nabla\theta=0,\\\
&u=-\nabla^{\perp}\log(10+\Lambda)\theta,\end{aligned}\right.$ (1.1)

where $\nabla^{\perp}=(-\partial_{x_{2}},\partial_{x_{1}})^{\top}$,
$\Lambda=(-\Delta)^{\frac{1}{2}}$,
$\theta(t,\cdot):\Omega\rightarrow\mathbb{R}$ and
$u(t,\cdot):\Omega\rightarrow\mathbb{R}^{2}$ with $\Omega=\mathbb{T}^{2}$,
$\mathbb{R}^{2}$, or $\mathbb{T}\times\mathbb{R}$. Note that (1.1) is an
inviscid nonlinear transport equation and the velocity is strictly more
singular than the advected scalar. The system (1.1) has been introduced by
Ohkitani ([39, 40]) and was referred to as the Ohkitani model later in [4].
Our first main result gives a local well-posedness result in the scale of
Sobolev spaces with an exponent decreasing with time.

######  Theorem A.

For any $s_{0}>4$ and $\theta_{0}\in H^{s_{0}}(\Omega)$, there exist
$T=T(s_{0},\|{\theta_{0}}\|_{H^{s_{0}}})>0$, a continuous decreasing function
$s(t)>4$ with $s(0)=s_{0}$ defined in $t\in[0,T]$, and a solution $\theta\in
L^{\infty}([0,T];H^{s(t)}(\Omega))$ to (1.1) with initial data $\theta_{0}$
satisfying

$\begin{split}\|{\theta(t,\cdot)}\|_{H^{s(t)}(\Omega)}\leq
C\|{\theta_{0}}\|_{H^{s_{0}}(\Omega)}.\end{split}$

The solution is unique in the class $L^{\infty}([0,T];H^{4}(\Omega))$.

######  Remark 1.1.

One may extend the above well-posedness result to the cases where the
multiplier $\log(10+\Lambda)$ in the Biot–Savart law
$u=\nabla^{\perp}\log(10+\Lambda)\theta$ is replaced with less singular radial
multipliers, i.e. $\gamma=\gamma(|\xi|)$ satisfying
$\gamma(|\xi|)\lesssim\log(10+|\xi|)$ together with a few reasonable
assumptions, e.g.
$|\partial^{\alpha}_{\xi}\gamma(\xi)|\lesssim_{\alpha}\langle{\xi}\rangle^{-|\alpha|}|\gamma(\xi)|$
for any multi-index $\alpha$. Some natural examples are $\log(10-\Delta)$,
$\log^{\beta}(10+\Lambda)$ ($\beta<1$), $\log^{\alpha}(10+\log(10+\Lambda))$
($\alpha>0$) etc.

As we shall explain below, the logarithmic Biot–Savart law featured in (1.1)
represents the natural borderline in which we can expect a well-posedness
theory in Sobolev spaces. Interestingly, Ohkitani arrived at the same model in
[39, 40] by investigating the collective behavior of the solutions for the
generalized SQG equations:

$\left\\{\begin{aligned}
&\partial_{t}\theta^{\delta}+u^{\delta}\cdot\nabla\theta^{\delta}=0,\\\
&u^{\delta}=\nabla^{\perp}(10+\Lambda)^{-\delta}\theta^{\delta},\end{aligned}\right.$
(1.2)

when $\delta>0$ is a parameter. He numerically observed that the sequence
$\\{\theta^{\delta}\\}_{\delta>0}$ is convergent after a time rescaling.
Formally, this can be seen as follows: introducing the new time variable
$\tau=\delta t$, (1.2) can be written as

$\begin{split}\partial_{\tau}\theta^{\delta}+\nabla^{\perp}\left(\frac{(10+\Lambda)^{-\delta}-1}{\delta}\right)\theta^{\delta}\cdot\nabla\theta^{\delta}=0,\end{split}$

and then from the Taylor expansion,

$\begin{split}\frac{(10+\Lambda)^{-\delta}-1}{\delta}\longrightarrow-\log(10+\Lambda)\end{split}$
(1.3)

as $\delta\to 0$. That is, the time-rescaled Biot–Savart law for (1.2)
converges to that for (1.1). This seems to suggest that the solutions to (1.2)
may converge to those to (1.1). We make this precise in the following result.

######  Theorem B.

Given any $\delta\in(0,1)$ and initial data $\theta_{0}\in H^{s}(\Omega)$ with
$s>5$, there exists a unique solution $\theta^{\delta}\in
L^{\infty}([0,C^{*}\delta^{-1}];H^{s}(\Omega))$ to (1.2) for some constant
$C^{*}>0$ depending only on $s$ and $\left\|{\theta_{0}}\right\|_{H^{s}}$.
Moreover, there exists a continuous decreasing function $s(t)>4$ defined in
$t\in[0,C^{*}\delta^{-1}]$ satisfying

$\left\|{\theta^{\delta}(t,\cdot)-\theta(\delta
t,\cdot)}\right\|_{H^{s(t)}(\Omega)}\leq\delta,$ (1.4)

where $\theta$ is the unique local solution to (1.1) with the same initial
data $\theta_{0}$.

######  Remark 1.2.

Observe that the maximal existence time $T_{max}^{\delta}$ of the smooth
solution of (1.2) satisfies
$T_{max}^{\delta}\gtrsim\delta^{-1}\rightarrow\infty$ as $\delta\rightarrow
0^{+}$. Furthermore, (1.4) shows that the long-time behavior of the solution
sequence for (1.2) is given by the solution of the Ohkitani model.

Moreover, the convergence (1.3) yields the global existence of smooth
solutions for the following dissipative counterpart of (1.2):

$\left\\{\begin{aligned}
&\partial_{t}\theta^{\delta}+u^{\delta}\cdot\nabla\theta^{\delta}+\kappa\Psi\theta^{\delta}=0,\\\
&u^{\delta}=\nabla^{\perp}(10+\Lambda)^{-\delta}\theta^{\delta},\end{aligned}\right.$
(1.5)

when $\delta>0$ is sufficiently small. Here $\Psi$ can be any super-
logarithmic Fourier multiplier in the sense that its symbol $\psi$ satisfies
the lower bound

$\begin{split}\psi(|\xi|)\geq\log(10+|\xi|),\quad|\xi|>\Xi_{0}\end{split}$
(1.6)

for some $\Xi_{0}>0$.

######  Theorem C.

Given any $\kappa>0$ and initial data $\theta_{0}\in H^{s}(\Omega)$ with
$s>4$, there exists a unique global solution $\theta^{\delta}\in
L^{\infty}([0,\infty);H^{s}(\Omega))$ to (1.5) for all
$\delta\in(0,\delta^{*})$, where $\delta^{*}\in(0,1)$ is a constant depending
only on $\kappa$, $s$, and $\left\|{\theta_{0}}\right\|_{H^{s}}$, and
satisfies

$\lim_{\left\|{\theta_{0}}\right\|_{H^{s}}\rightarrow\infty}\delta^{*}=0.$

Lastly, we consider the following dissipative counterpart of (1.1):

$\left\\{\begin{aligned}
&\partial_{t}\theta+u\cdot\nabla\theta+\kappa\log^{\beta}(10+\Lambda)\theta=0,\\\
&u=-\nabla^{\perp}\log(10+\Lambda)\theta.\end{aligned}\right.$ (1.7)

The following result shows that when there is a dissipative term stronger than
log, there is no need to consider time-dependent Sobolev spaces.

######  Theorem D.

For $\beta>1$, the dissipative system (1.7) is locally well-posed in
$L^{\infty}_{t}H^{s}_{x}$ for any $s>4$, That is, given any initial data
$\theta_{0}\in H^{s}(\Omega)$, $\kappa>0$, there exist $T>0$ and a unique
solution $\theta\in L^{\infty}([0,T];H^{s}(\Omega))$ to (1.7) with initial
data $\theta_{0}$.

######  Remark 1.3.

This result extends to more general systems of the form

$\left\\{\begin{aligned}
&\partial_{t}\theta+u\cdot\nabla\theta+\kappa\Upsilon\theta=0,\\\
&u=-\nabla^{\perp}\log(10+\Lambda)\theta,\end{aligned}\right.$ (1.8)

where the dissipation is given by a Fourier multiplier $\Upsilon$ with symbol
$\upsilon$ satisfying

$\begin{split}\frac{\upsilon(|\xi|)}{\log(10+|\xi|)}\geq\frac{C_{s}}{\kappa}\|{\theta_{0}}\|_{H^{s}(\Omega)},\quad|\xi|>\Xi_{0}\end{split}$
(1.9)

for some $\Xi_{0}>0$ and $C_{s}>0$. Moreover, the requirement $\beta>1$ is
sharp in view of the illposedness result from [7] in the case $\beta\leq 1$.

###  1.2 Previous works

Generalized SQG equations. The generalized SQG equation (gSQG) takes the form

$\begin{split}&\partial_{t}\theta+u\cdot\nabla\theta=0,\\\
&u=\nabla^{\perp}\Gamma\theta,\end{split}$ (1.10)

in a two-dimensional physical domain without a boundary, where $\Gamma$ is
some Fourier multiplier. The special cases $\Gamma=\Lambda^{-2}$ and
$\Lambda^{-1}$ correspond respectively to the two-dimensional incompressible
Euler equation and the (usual) SQG equation, derived in [14, 13]. Furthermore,
the equation in the case $\Gamma=\Lambda$ governs the dynamics of the so-
called electron MHD equation under certain symmetric ansatz (see [7, 38]). It
seems that the generalized model (1.10) was first considered in [5], and
subsequently several authors investigated the question of well-posedness for
various multipliers $\Gamma$ ([6, 4, 24, 26, 25]). In the particular case of
$\Gamma=\Lambda^{\beta}$, standard energy estimates give local well-posedness
for $\beta\leq-1$, and an additional cancellation arising from the even
symmetry of $\Gamma$ is necessary to obtain local well-posedness in the case
$\beta\leq 0$ ([4]).111This even symmetry is essential; in the works [26, 25],
ill-posedness was established for odd $\Gamma$ with scaling
$|\gamma(\xi)|\sim|\xi|^{\beta}$ with $-1<\beta<0$. The latter argument can be
adapted to multipliers $\Gamma$ with even symbols $\gamma$ satisfying
$|\gamma(\xi)|\lesssim 1$, together with a few natural regularity assumptions
on the derivatives of $\gamma$. It turns out that this boundedness assumption
is crucial for local well-posedness ([7]), and for this reason, we shall refer
to $\Gamma$ as a singular multiplier if its symbol $\gamma$ satisfies
$|\gamma(\xi)|\rightarrow\infty$ as $|\xi|\rightarrow\infty$.

Global well-posedness for dissipative SQG equations. The dissipative SQG
equations takes the form

$\left\\{\begin{aligned}
&\partial_{t}\theta+u\cdot\nabla\theta+\kappa(-\Delta)^{\alpha}\theta=0,\\\
&u=\nabla^{\perp}\Lambda^{-1}\theta\end{aligned}\right.$ (1.11)

for $\kappa>0$ and $\alpha\in(0,1)$. The cases $\alpha>\frac{1}{2}$,
$\alpha=\frac{1}{2}$, and $\alpha<\frac{1}{2}$ are referred to as subcritical,
critical, and supercritical, respectively. Global existence and uniqueness of
smooth solutions corresponding to arbitrary smooth initial data were
established for the subcritical regime in [17], and for the critical regime in
[3, 16, 15, 33] (see also [20] for the logarithmically supercritical case). In
the supercritical regime, the question of global regularity was settled only
for sufficiently small initial data ([8, 19, 41, 10]). Global well-posedness
or finite-time blow-up for arbitrary initial data in the supercritical case
remains an outstanding open problem. In this direction, the authors of [18]
proved that the supremum of sizes of initial data leading to global well-
posedness goes to infinity as ${\alpha\rightarrow\frac{1}{2}}$. Comparing this
result with our Theorem C, both of them sacrifice some quantities to obtain
global regularity. Indeed, to absorb the size of large initial data, the order
of dissipation increases to the order corresponding to the critical case in
[18] while the order of velocity goes to the order corresponding to the
trivial steady state in Theorem C. However, recalling (1.5) and (1.6), Theorem
C only requires logarithmic dissipation $\log(10+\Lambda)$, which is much
weaker than $(-\Delta)^{\alpha}$ with $\alpha\in(0,\frac{1}{2})$ in (1.11).

Wellposedness in the Ohkitani model. Chae, Constantin, Cordoba, Gancedo, and
Wu have established a local existence and uniqueness result for the
logarithmically singular SQG equations in [4]. Precisely, in [4, Theorem 1.3],
the authors proved that the system

$\left\\{\begin{aligned}
&\partial_{t}\theta+u\cdot\nabla\theta+\kappa(-\Delta)^{\alpha}\theta=0,\\\
&u=\nabla^{\perp}(\log(10-\Delta))^{\mu}\theta\end{aligned}\right.$ (1.12)

for any $\kappa,\mu,\alpha>0$ is locally well-posed in
$L^{\infty}([0,T];H^{4}(\mathbb{R}^{2}))$. On the other hand, our Theorem A
says that when $\mu\leq 1$, there is no need for a dissipation term for local
well-posedness. Furthermore, Theorem D states that a logarithmic dissipation
suffices; although we omit the proof, Theorem D can be generalized to any
$\mu>0$, by simply replacing the condition (1.9) with

$\begin{split}\frac{\upsilon(|\xi|)}{(\log(10+|\xi|))^{\mu}}\geq\frac{C_{s}}{\kappa}\|{\theta_{0}}\|_{H^{s}}.\end{split}$
(1.13)

Convergence to the Ohkitani model. As we shall explain below, the key idea in
the well-posedness of (1.1) is the use of losing estimate in the scale of
Sobolev spaces. Interestingly, the same strategy can be applied to the family
of generalized SQG equations (1.2) for $\delta>0$, this time giving long-time
existence for smooth solutions as $\delta\to 0$. Furthermore, the solutions to
(1.2) as $\delta\to 0$ converges to the solution of (1.1), after a time
rescaling. This confirms the expectations of Ohkitani [39, 40]. We have been
also inspired by the incompressible Euler limit of (1.2) (namely, $\delta\to
2$) recently considered in [43]. This limit is interesting as well since while
the two-dimensional Euler equation is known to be globally well-posed. The
authors of [43] actually show the long-time existence of solutions to (1.2) as
$\delta\to 2$.

Illposedness in the Ohkitani model. The above main theorems nicely complement
the illposedness results from our recent work [7]; among other things, one can
prove a nonexistence result for (1.1) and (1.7) in a fixed Sobolev space
$H^{s}$ with $s$ arbitrarily large, in the sense that there exists a smooth
initial data $\theta_{0}$ for which there cannot exist a corresponding
solution in $L^{\infty}([0,T];H^{s})$ for $s>3$ with any $T>0$. The
dissipative version requires a condition of the form (roughly)

$\begin{split}\frac{\upsilon(|\xi|)}{\log(10+|\xi|)}\ll 1,\end{split}$ (1.14)

see [7] for the precise statements. This shows that _a decrease in the Sobolev
exponent_ (as in Theorem A) _is unavoidable if one insists in having
wellposedness in the $H^{s}$-spaces_. The paper [7] actually proves
nonexistence for a large class of singular even multipliers $\Gamma$.

###  1.3 Ideas of the proof

To understand how such a losing estimate in the scale of time-dependent
Sobolev spaces is possible even when the time-independent estimate fails, it
is instructive to make an analogy with the well-known Cauchy–Kovalevskaya
theorem, which gives local well-posedness of the Cauchy problem in the
analytic class when the loss of derivative is less than or equal to one222We
do not expect the analytic regularity to propagate in time for the gSQG
equation (1.10) with singular $\Gamma$; that is, Cauchy–Kovalevskaya theorem
fails. Indeed, one may prove some analytic (and even Gevrey) illposedness
results for linear and nonlinear SQG equations near quadratically degenerate
shear steady states, closely following the arguments of [7, 29]. Note that
formally the loss of derivative in the gSQG equation with singular $\Gamma$ is
always strictly larger than one..

To be concrete, the key point in the proof of Cauchy–Kovalevskaya theorem for
the model equation $\partial_{t}b=\Lambda b$ on $\mathbb{R}$ with initial data
$b(t=0)=b_{0}$ is that if one considers the time-dependent multiplier
$m(t,\xi):=\exp((c_{0}-Ct)|\xi|),$ (with $c_{0}>0$ depending on the radius of
analyticity of $b_{0}$) then we have

$\begin{split}\frac{\mathrm{d}}{\mathrm{d}t}\|{m(t,\xi)\widehat{b}(\xi)}\|_{L^{2}}^{2}+C\|{|\xi|^{\frac{1}{2}}m(t,\xi)\widehat{b}(\xi)}\|_{L^{2}}^{2}=\int
m^{2}(t,\xi)\overline{\widehat{b}(\xi)}\partial_{t}\widehat{b}(\xi)\,\mathrm{d}\xi\end{split}$

so that for $C>0$ large, we can absorb the right hand side into the positive
term on the left hand side. Now, as a reasonable linear model of the
logarithmically singular SQG equation whose initial value problem is illposed
in a _fixed_ Sobolev space [7, 29], we consider the degenerate dispersive
equation

$\begin{split}\partial_{t}b=x\partial_{x}\ln(10+\Lambda)b+\frac{1}{2}\ln(10+\Lambda)b\end{split}$
(1.15)

on $\mathbb{R}$. More generally, we can consider

$\begin{split}\partial_{t}b=x\partial_{x}\Gamma(\Lambda)b+\frac{1}{2}\Gamma(\Lambda)b,\end{split}$
(1.16)

where $\Gamma$ is a multiplier with the symbol $\gamma=\gamma(|\xi|)\geq 0$
singular in the sense that $\gamma(|\xi|)\rightarrow\infty$ as
$|\xi|\rightarrow\infty$. When $\gamma$ is not singular, one may prove a
Cauchy-Kovaleskaya type theorem under some natural assumptions on the
derivatives of $\gamma$. The presence of last terms in (1.15) and (1.16) is
not essential; they were inserted to just guarantee the conservation

$\begin{split}\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\|{\gamma^{\frac{1}{2}}(\Lambda)b}\|_{L^{2}}^{2}=-\frac{1}{2}\|{\gamma^{\frac{1}{2}}(\Lambda)b}\|_{L^{2}}^{2}+\frac{1}{2}\|{\gamma^{\frac{1}{2}}(\Lambda)b}\|_{L^{2}}^{2}=0\end{split}$

which models conservation of $\|{\Gamma^{\frac{1}{2}}\theta}\|_{L^{2}}$ in
(1.10). To explore possible function spaces in which (1.16) is well-posed, we
consider a time-dependent multiplier $m(t,\xi)$ (positive and increasing with
$|\xi|$). Then, we compute that from

$\begin{split}\partial_{t}\widehat{b}(\xi)=-\partial_{\xi}(\xi\gamma(|\xi|)\widehat{b})+\frac{1}{2}\gamma(|\xi|)\widehat{b},\end{split}$
$\begin{split}\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\int
m^{2}(t,\xi)|\widehat{b}(t,\xi)|^{2}=\int
m\dot{m}|\widehat{b}|^{2}+\operatorname{Re}\int
m^{2}\,\overline{\widehat{b}}\left(-\partial_{\xi}(\xi\gamma(|\xi|)\widehat{b})+\frac{1}{2}\gamma(|\xi|)\widehat{b}\right)\end{split}$

and

$\begin{split}-\operatorname{Re}\int
m^{2}\,\overline{\widehat{b}}\partial_{\xi}(\xi\gamma(|\xi|)\widehat{b})&=\operatorname{Re}\int\partial_{\xi}(m^{2}\,\overline{\widehat{b}})\xi\gamma(|\xi|)\widehat{b}=\operatorname{Re}\int(2m\partial_{\xi}m\overline{\widehat{b}}+m^{2}\partial_{\xi}\overline{\widehat{b}})\xi\gamma(|\xi|)\widehat{b}\\\
&=\operatorname{Re}\int
2m\partial_{\xi}m\xi\gamma(|\xi|)|\widehat{b}|^{2}-\frac{1}{2}\operatorname{Re}\int\partial_{\xi}(m^{2}\gamma(|\xi|)\xi)|\widehat{b}|^{2}\end{split}$

that

$\begin{split}\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\int
m^{2}(t,\xi)|\widehat{b}(t,\xi)|^{2}=\int
m(\xi)\left(\dot{m}+\xi\gamma(|\xi|)\partial_{\xi}m-m\xi\partial_{\xi}\gamma(|\xi|)\right)|\widehat{b}|^{2}\leq
0\end{split}$

if $m$ is chosen that

$\begin{split}\dot{m}+\xi\gamma(|\xi|)\partial_{\xi}m-m\xi\partial_{\xi}\gamma(|\xi|)\leq
0.\end{split}$ (1.17)

Define the characteristics $\dot{\Xi}=\Xi\gamma(|\Xi|),\Xi(0)=\xi$ so that the
above is equivalent to

$\begin{split}\frac{\mathrm{d}}{\mathrm{d}t}m(t,\Xi(t,\xi))\leq\Xi\gamma^{\prime}(|\Xi|)m(t,\Xi(t,\xi)).\end{split}$

Hence if the characteristics $\Xi$ are well-defined for $O(1)$-time uniformly
for all $\xi$, then we can take

$\begin{split}m(t,\Xi(t,\xi))=m_{0}(\xi)\exp\left(\int_{0}^{t}\Xi(s,\xi)\gamma^{\prime}(|\Xi(s,\xi)|)ds\right).\end{split}$

In the case of $\gamma(|\xi|)=\ln(1+|\xi|)$, we have that (1.17) is given by
$\dot{m}+\xi\ln(1+|\xi|)\partial_{\xi}m-m\leq 0,$ and for large $|\xi|$,

$\begin{split}\Xi(t,\xi)\simeq\xi^{\exp(t)},\qquad m(t,\xi^{\exp(t)})\simeq
m_{0}(\xi)\exp(t).\end{split}$

Hence, in the Sobolev case where $m_{0}(\xi)\simeq|\xi|^{s}$,
$m(t,\xi)\simeq|\xi|^{s\exp(-t)}\exp(t)$. That is, a losing estimate in the
scale of $H^{s(t)}$-spaces is available.333In the Gevrey case where
$m_{0}(\xi)\simeq\exp(|\xi|^{\frac{1}{\sigma}})$ for some $\sigma>0$, we have
that $m(t,\xi)\simeq\exp(|\xi|^{\frac{1}{e^{t}\sigma}})$. We see that a losing
estimate in the Gevrey class can be obtained, but unlike the usual case, the
index $\sigma$ should be sacrificed and not the radius of Gevrey regularity.

In parallel with the above argument, if one considers the time-dependent
multiplier of the form

$\begin{split}\widetilde{m}(t,\xi):=|\xi|^{s_{0}-Ct}\end{split}$

(with $s_{0}>0$ depending on $\theta_{0}$ and $C>0$ large) for a solution
$\theta$ of (1.1), then we have

$\begin{split}\frac{\mathrm{d}}{\mathrm{d}t}\|{\widetilde{m}(t,\xi)\widehat{\theta}(t,\xi)}\|_{L^{2}_{\xi}}^{2}+C\|{(\log|\xi|)^{\frac{1}{2}}\widetilde{m}(t,\xi)\widehat{\theta}(t,\xi)}\|_{L^{2}_{\xi}}^{2}=\int\widetilde{m}^{2}(t,\xi)\overline{\widehat{\theta}(t,\xi)}\partial_{t}\widehat{\theta}(t,\xi)\,\mathrm{d}\xi\end{split}$

so that if the loss of regularity in the right hand side is sub-logarithmic,
then we may deduce that

$\begin{split}\frac{\mathrm{d}}{\mathrm{d}t}\|{\widetilde{m}(t,\xi)\widehat{\theta}(t,\xi)}\|_{L^{2}_{\xi}}^{2}\leq
0\end{split}$

which results in an a priori estimate. This type of observation (that is,
handling a logarithmic loss of derivative with a losing estimate in the scale
of Sobolev spaces) is not new. For instance, one may consider the task of
solving the 2D incompressible Euler equation (SQG with
$\Gamma=(-\Delta)^{-1}$) with Sobolev critical initial data:

$\begin{split}\partial_{t}\omega+\nabla^{\perp}(-\Delta)^{-1}\omega\cdot\nabla\omega=0.\end{split}$

When $\omega_{0}\in W^{s,p}$ with $sp=2$ and $1<p<\infty$, we do not have in
general $\nabla\nabla^{\perp}(-\Delta)^{-1}\omega\in L^{\infty}$, which is
necessary to propagate $W^{s,p}$-regularity. Indeed, Bourgain and Li in [1]
has shown that (see [23, 30, 32] for simplified proofs) there exist initial
data $\omega_{0}\in W^{s,p}\cap L^{\infty}\cap L^{1}$ whose unique
$L^{\infty}_{t}(L^{\infty}\cap L^{1})$ solution444For the 2D Euler equations
with $\omega_{0}\in L^{\infty}\cap L^{1}$ initial data, Yudovich’s theorem
gives a unique global in time solution $\omega\in
L^{\infty}_{t}(L^{\infty}\cap L^{1})$ ([44]). instantaneously escapes
$W^{s,p}$. That is, 2D Euler is strongly illposed in critical Sobolev spaces.
On the other hand, it is not difficult to see that such an illposedness
behavior is logarithmic in nature; one way to quantify it is to consider the
logarithmically regularized Euler equation ([5]), defined by

$\begin{split}\partial_{t}\omega+\nabla^{\perp}(-\Delta)^{-1}(\log(10-\Delta))^{-\gamma}\omega\cdot\nabla\omega=0\end{split}$

for some $\gamma>0$. It has been shown in [9] that the above is locally
wellposed in $W^{s,p}$ for $\gamma>\frac{1}{2}$. (On the other hand, there is
still strong illposedness for $\gamma\leq\frac{1}{2}$; see [37].) Since the
loss is only logarithmic, one can consider a losing estimate in the scale of
Sobolev spaces; indeed, with $H^{1}\cap L^{\infty}\cap L^{1}$ initial data, it
can be shown that the unique $L^{\infty}_{t}(L^{\infty}\cap L^{1})$ solution
at time $t$ belongs to $H^{s(t)}\cap W^{1,p(t)}$ where $s,p$ are continuously
decreasing functions of time with $s(0)=p(0)=2$ ([23, 28, 2]).

###  1.4 Open problems

Given local well-posedness, the natural question is to ask whether local
solutions could blow up in finite time or not. For the inviscid equation
(1.1), one may also ask whether there exists a solution with Sobolev
regularity actually decreasing continuously with time. We plan to address
these issues in the future.

In general, not much is known regarding the long-time dynamics and the
possibility of singularity formation for the gSQG equations. While the 2D
Euler equation ((1.10) with $\Gamma=(-\Delta)^{-1}$) is globally well-posed,
this global regularity result so far only extends to models slightly more
singular than 2D Euler (by loglog in [5]). Several constructions of smooth
solutions to gSQG equations with Sobolev norms growing with time have been
obtained in [21, 22, 34, 27, 45, 42, 11]. Finite-time singularity formation
for a class of patch-type solutions touching the boundary was established for
a range of gSQG equations in [35, 36].

##  2 Proofs

We give the proofs of Theorems A and D in §2.1, and Theorems B and C in §2.2.
In the proofs, we shall assume that $\Omega=\mathbb{R}^{2}$, but the case of
$\mathbb{R}\times\mathbb{T}$ and $\mathbb{T}^{2}$ can be treated in the same
manner.

###  2.1 Local well-posedness for (1.1) and (1.7)

We need an elementary lemma.

######  Lemma 2.1.

For any $s\geq 3$, we have the inequality

$\begin{split}\left||\xi|^{s}-|\eta|^{s}-|\xi-\eta|^{s}-s(\xi-\eta)\cdot\eta|\xi-\eta|^{s-2}\right|\leq
C_{s}(|\eta|^{2}|\xi-\eta|^{s-2}+|\xi-\eta||\eta|^{s-1})\end{split}$

for any $\xi,\eta\in\mathbb{R}^{2}$. Here $C_{s}>0$ is a constant depending
only on $s$.

We note that $C_{s}$ could be taken to be independent of $s$ if we restrict
$s$ to a closed interval.

###### Proof.

To establish the inequality, we introduce

$\begin{split}A(\rho)=(1-\rho)\xi+\rho\eta,\quad
B(\rho)=(1-\rho)(\xi-\eta)\end{split}$

so that we have

$\begin{split}|\xi|^{s}-|\eta|^{s}=\int_{0}^{1}\partial_{\rho}|A(\rho)|^{s}\,\mathrm{d}\rho=-s(\eta-\xi)\cdot\int_{0}^{1}|A(\rho)|^{s-2}A(\rho)\,\mathrm{d}\rho,\end{split}$
$\begin{split}|\xi|^{s}-|\eta|^{s}-|\xi-\eta|^{s}=-s(\eta-\xi)\int_{0}^{1}\left(|A(\rho)|^{s-2}A(\rho)-|B(\rho)|^{s-2}B(\rho)\right)\,\mathrm{d}\rho.\end{split}$

We write

$\begin{split}|A(\rho)|^{s-2}A(\rho)-|B(\rho)|^{s-2}B(\rho)=|B(\rho)|^{s-2}(A(\rho)-B(\rho))+(|A(\rho)|^{s-2}-|B(\rho)|^{s-2})A(\rho)=I+II\end{split}$

and observe the following formulas:

$\begin{split}-s(\eta-\xi)\int_{0}^{1}I\,\mathrm{d}\rho=\frac{s}{s-1}(\xi-\eta)\cdot\eta|\xi-\eta|^{s-2},\end{split}$
$\begin{split}|A(\rho)|^{s-2}-|B(\rho)|^{s-2}=\int_{0}^{1}\partial_{\sigma}|E(\rho,\sigma)|^{s-2}\,\mathrm{d}\sigma=-(s-2)\eta\cdot\int_{0}^{1}|E|^{s-4}E\,\mathrm{d}\sigma,\end{split}$

and

$\begin{split}-s(\eta-\xi)\int_{0}^{1}II\,\mathrm{d}\rho=-s(s-2)(\eta-\xi)\cdot\int_{0}^{1}\int_{0}^{1}A(\rho)|E|^{s-4}\eta\cdot
E\,\mathrm{d}\sigma\,\mathrm{d}\rho,\end{split}$

where $E(\rho,\sigma)=(1-\rho)\xi+(\rho-\sigma)\eta.$ We may therefore write

$\begin{split}|\xi|^{s}-|\eta|^{s}-|\xi-\eta|^{s}-s(\xi-\eta)\cdot\eta|\xi-\eta|^{s-2}&=-s(\eta-\xi)\int_{0}^{1}II\,\mathrm{d}\rho-\frac{s(s-2)}{s-1}(\xi-\eta)\cdot\eta|\xi-\eta|^{s-2}\\\
&=s(s-2)(\xi-\eta)\left[\int_{0}^{1}\int_{0}^{1}A|E|^{s-4}\eta\cdot
E-\frac{1}{s-1}\eta|\xi-\eta|^{s-2}\mathrm{d}\sigma\mathrm{d}\rho\right].\end{split}$

Noticing that

$\begin{split}\left|A|E|^{s-4}\eta\cdot
E-\frac{1}{s-1}\eta|\xi-\eta|^{s-2}\right|\leq
C_{s}|\eta|^{2}\left(|\xi-\eta|^{s-3}+|\eta|^{s-3}\right)\end{split}$

holds, the proof is complete. ∎

###### Proof of Theorem A.

We consider the inviscid case. We shall perform an $H^{s}$ estimate, with
$s=s(t)>4$ depending on time. For convenience, we re-define the $H^{s}$ norm
by

$\begin{split}\|{\theta}\|_{H^{s}(\mathbb{R}^{2})}^{2}=\int_{\mathbb{R}^{2}}(10+|\xi|)^{2s}|\widehat{\theta}(\xi)|^{2}\,\mathrm{d}\xi.\end{split}$
(2.1)

Then, we compute that

$\begin{split}\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\|{\theta}\|_{{H}^{s(t)}}^{2}&=\int\dot{s}(t)(10+|\xi|)^{2s(t)}\log(10+|\xi|)|\widehat{\theta}(\xi)|^{2}\,\mathrm{d}\xi\\\
&\qquad-\operatorname{Re}\int(10+|\xi|)^{2s(t)}\overline{\widehat{\theta}}(\xi)\int\left(\widehat{u}(\xi-\eta)\cdot
i\eta\widehat{\theta}(\eta)\right)\,\mathrm{d}\eta\,\mathrm{d}\xi.\end{split}$

Neglecting the constant 10 for now, we shall estimate the following quantity
with $s=s(t)$:

$\begin{split}\operatorname{Re}\int|\xi|^{s}\overline{\widehat{\theta}}(\xi)\int|\xi|^{s}\left(\widehat{u}(\xi-\eta)\cdot
i\eta\widehat{\theta}(\eta)\right)\,\mathrm{d}\eta\,\mathrm{d}\xi=:I+II+III+IV,\end{split}$
(2.2)

where the four terms are given by the decomposition

$\begin{split}|\xi|^{s}=|\eta|^{s}+|\xi-\eta|^{s}+s(\xi-\eta)\cdot\eta|\xi-\eta|^{s-2}+R(\eta,\xi).\end{split}$
(2.3)

That is, we define

$\begin{split}I=\operatorname{Re}\int|\xi|^{s}\overline{\widehat{\theta}}(\xi)\int|\eta|^{s}\left(\widehat{u}(\xi-\eta)\cdot
i\eta\widehat{\theta}(\eta)\right)\,\mathrm{d}\eta\,\mathrm{d}\xi,\end{split}$
$\begin{split}II=\operatorname{Re}\int|\xi|^{s}\overline{\widehat{\theta}}(\xi)\int|\xi-\eta|^{s}\left(\widehat{u}(\xi-\eta)\cdot
i\eta\widehat{\theta}(\eta)\right)\,\mathrm{d}\eta\,\mathrm{d}\xi,\end{split}$
$\begin{split}III=\operatorname{Re}\int|\xi|^{s}\overline{\widehat{\theta}}(\xi)\int
s(\xi-\eta)\cdot\eta|\xi-\eta|^{s-2}\left(\widehat{u}(\xi-\eta)\cdot
i\eta\widehat{\theta}(\eta)\right)\,\mathrm{d}\eta\,\mathrm{d}\xi,\end{split}$

and

$\begin{split}IV=\operatorname{Re}\int|\xi|^{s}\overline{\widehat{\theta}}(\xi)\int
R(\xi,\eta)\left(\widehat{u}(\xi-\eta)\cdot
i\eta\widehat{\theta}(\eta)\right)\,\mathrm{d}\eta\,\mathrm{d}\xi.\end{split}$

To begin with, we claim that $I=0$. By symmetrizing555This argument is
equivalent to using the anti-symmetry of $u\cdot\nabla$. with respect to the
change of variables $(\xi,\eta)=(\eta^{\prime},\xi^{\prime})$, and also using
$\overline{\widehat{u}}(\xi-\eta)=\widehat{u}(-\xi+\eta)$ (since each
component of $u$ is real-valued), we see that

$I=\frac{1}{2}\operatorname{Re}\int|\xi|^{s}\overline{\widehat{\theta}}(\xi)\int|\eta|^{s}\left(\widehat{u}(\xi-\eta)\cdot
i(\eta-\xi)\widehat{\theta}(\eta)\right)\,\mathrm{d}\eta\,\mathrm{d}\xi,$

which vanishes since $u$ is divergence-free (hence $\widehat{u}(\xi-\eta)\cdot
i(\eta-\xi)=0$). Next, we claim that

$\begin{split}\left|IV\right|\leq C_{s}\|{\theta}\|_{H^{s}}^{3}\end{split}$

for $s>4$. To see this, we recall the definition of $u$ and the form of $R$
from Lemma 2.1: with Young’s convolution inequality,

$\begin{split}IV&\leq\iint|\xi|^{s}|\overline{\widehat{\theta}}(\xi)|(|\eta|^{2}|\xi-\eta|^{s-2}+|\xi-\eta||\eta|^{s-1})\log(10+|\xi-\eta|)|\xi-\eta||\widehat{\theta}(\xi-\eta)||\eta||\widehat{\theta}(\eta)|\,\mathrm{d}\eta\,\mathrm{d}\xi\\\
&\leq\iint|\xi|^{s}|\overline{\widehat{\theta}}(\xi)||\xi-\eta|^{s-1}\log(10+|\xi-\eta|)|\widehat{\theta}(\xi-\eta)||\eta|^{3}|\widehat{\theta}(\eta)|\,\mathrm{d}\eta\,\mathrm{d}\xi\\\
&\qquad+\iint|\xi|^{s}|\overline{\widehat{\theta}}(\xi)|\log(10+|\xi-\eta|)|\xi-\eta|^{2}|\widehat{\theta}(\xi-\eta)||\eta|^{s}|\widehat{\theta}(\eta)|\,\mathrm{d}\eta\,\mathrm{d}\xi\\\
&\leq
C\|{{\langle{\xi}\rangle^{3}}\widehat{\theta}(\xi)}\|_{L^{1}_{\xi}}\|{\theta}\|_{H^{s}}^{2}\leq
C_{s}\|{\theta}\|_{H^{s}}^{3}.\end{split}$

(Observe that $s>4$ is required in the last inequality to apply Sobolev
embedding in two dimensions.) It remains to handle the terms $II$ and $III$.
To estimate $II$, we may rewrite

$\begin{split}II&=-\operatorname{Re}\iint|\xi|^{s}\overline{\widehat{\theta}}(\xi)\log^{\frac{1}{2}}(10+|\xi-\eta|)\\\
&\phantom{=-\operatorname{Re}\iint}\times(\log^{\frac{1}{2}}(10+|\xi-\eta|)-\log^{\frac{1}{2}}(10+|\xi|))\widehat{\theta}(\xi-\eta)|\xi-\eta|^{s}(\xi-\eta)^{\perp}\cdot\eta\widehat{\theta}(\eta)\,\mathrm{d}\eta\,\mathrm{d}\xi.\end{split}$

Here, we used

$\begin{split}\operatorname{Re}\iint|\xi|^{s}\overline{\widehat{\theta}}(\xi)\log^{\frac{1}{2}}(10+|\xi|)\log^{\frac{1}{2}}(10+|\xi-\eta|)\widehat{\theta}(\xi-\eta)|\xi-\eta|^{s}(\xi-\eta)^{\perp}\cdot\eta\widehat{\theta}(\eta)\,\mathrm{d}\eta\,\mathrm{d}\xi=0,\end{split}$

which follows by symmetrizing666This argument is equivalent to using the anti-
symmetry of
$\log^{\frac{1}{2}}(10+\Lambda)\nabla\theta\cdot\nabla^{\perp}\log^{\frac{1}{2}}(10+\Lambda)$.
with respect to the change of variables
$(\xi,\eta)=(\xi^{\prime}-\eta^{\prime},-\eta^{\prime})$ and using
$\overline{\widehat{\theta}}(\eta)=\widehat{\theta}(-\eta)$ (since $\theta$ is
real-valued). We write

$\begin{split}\log^{\frac{1}{2}}(10+|\xi-\eta|)-\log^{\frac{1}{2}}(10+|\xi|)=\frac{\log(10+|\xi-\eta|)-\log(10+|\xi|)}{\log^{\frac{1}{2}}(10+|\xi-\eta|)+\log^{\frac{1}{2}}(10+|\xi|)}\end{split}$

and deduce a rough estimate

$\begin{split}\left|\log^{\frac{1}{2}}(10+|\xi-\eta|)-\log^{\frac{1}{2}}(10+|\xi|)\right|\leq
C\left|\log\frac{10+|\xi-\eta|}{10+|\xi|}\right|\leq
C|\eta|\left(\frac{1}{10+|\xi|}+\frac{1}{10+|\xi-\eta|}\right).\end{split}$

With this in hand, we can bound

$\begin{split}\left|II\right|&\leq
C\iint|\xi|^{s}|\widehat{\theta}(\xi)|\log^{\frac{1}{2}}(10+|\xi-\eta|)|\xi-\eta|^{s}|\widehat{\theta}(\xi-\eta)||\eta|^{2}|\widehat{\theta}(\eta)|\,\mathrm{d}\eta\,\mathrm{d}\xi\\\
&\leq
C\|{|\eta|^{2}\widehat{\theta}(\eta)}\|_{L^{1}_{\eta}}\int\log(10+|\xi|)|\xi|^{2s}|\widehat{\theta}(\xi)|^{2}\,\mathrm{d}\xi.\end{split}$

Here, we used $(\xi-\eta)^{\perp}\cdot\eta=\xi^{\perp}\cdot\eta$ to estimate
$((10+|\xi|)^{-1}+(10+|\xi-\eta|)^{-1})|{(\xi-\eta)^{\perp}\cdot\eta}|\lesssim|{\eta}|$.
For $III$, it would be convenient to make a change of variables
$\eta\mapsto\mu$ by $\mu=\xi-\eta$. Then, using the definition of $u$, we have
that

$\begin{split}III&=-\operatorname{Re}\int|\xi|^{s}\overline{\widehat{\theta}}(\xi)\int
s(\mu\cdot(\xi-\mu))(\mu^{\perp}\cdot(\xi-\mu))|\mu|^{s-2}\log(10+|\mu|)\widehat{\theta}(\mu)\widehat{\theta}(\xi-\mu)\,\mathrm{d}\mu\,\mathrm{d}\xi\\\
&=-\operatorname{Re}\iint|\xi|^{2}\log(10+|\mu|)G(\xi,\mu)\overline{\widehat{\theta}(\xi)}{\widehat{\theta}}(\mu)\widehat{\theta}(\xi-\mu)\,\mathrm{d}\mu\,\mathrm{d}\xi\end{split}$

where
$G(\xi,\mu):=s|\xi|^{s-2}|\mu|^{s-2}(\mu\cdot(\xi-\mu))(\mu^{\perp}\cdot(\xi-\mu)).$
It encourages us to consider the symmetrization

$\begin{split}III^{\prime}:=-\operatorname{Re}\iint|\xi||\mu|\log^{\frac{1}{2}}(10+|\mu|)\log^{\frac{1}{2}}(10+|\xi|)G(\xi,\mu)\overline{\widehat{\theta}(\xi)}{\widehat{\theta}}(\mu)\widehat{\theta}(\xi-\mu)\,\mathrm{d}\mu\,\mathrm{d}\xi.\end{split}$

With Young’s convolution inequality and the simple identity
$\mu^{\perp}\cdot(\xi-\mu)=(\mu-\xi)^{\perp}\cdot\xi$, we see that

$\begin{split}\left|III^{\prime}\right|&\leq
C\|{|\eta|^{2}\widehat{\theta}(\eta)}\|_{L^{1}_{\eta}}\int\log(10+|\xi|)|\xi|^{2s}|\widehat{\theta}(\xi)|^{2}\,\mathrm{d}\xi.\end{split}$

Next, $III-III^{\prime}$ is given by

$\begin{split}-\operatorname{Re}\iint
K(\xi,\mu)G(\xi,\mu)\overline{\widehat{\theta}(\xi)}{\widehat{\theta}}(\mu)\widehat{\theta}(\xi-\mu)\,\mathrm{d}\mu\,\mathrm{d}\xi\end{split}$

where

$\begin{split}|K(\xi,\mu)|&=|\xi|\log(10+|\mu|)\log^{\frac{1}{2}}(10+|\xi|)\left|\frac{|\xi|}{\log^{\frac{1}{2}}(10+|\xi|)}-\frac{|\mu|}{\log^{\frac{1}{2}}(10+|\mu|)}\right|\\\
&\leq C|\xi||\xi-\mu|\log(10+|\mu|)\log^{\frac{1}{2}}(10+|\xi|)\\\ &\leq
C|\xi||\xi-\mu|(\log(10+|\xi|)+\log(10+|\xi-\mu|))\log^{\frac{1}{2}}(10+|\xi|).\end{split}$

Observing
$\|{\log(10+|\eta|)|\eta|^{3}\widehat{\theta}(\eta)}\|_{L^{1}_{\eta}}\leq
C\|{{\theta}}\|_{H^{s}},$ we obtain from the above bound of $K$ that

$\begin{split}|III-III^{\prime}|\leq C\|{\theta}\|_{H^{s}}^{3}.\end{split}$

Together with the previous bound for $III^{\prime}$, we conclude

$\begin{split}\left|III\right|&\leq
C\|{\theta}\|_{H^{s}}\int\log(10+|\xi|)|\xi|^{2s}|\widehat{\theta}(\xi)|^{2}\,\mathrm{d}\xi+C_{s}\|{\theta}\|_{H^{s}}^{3}.\end{split}$

Combining the estimates for $I$, $II$, $III$, and $IV$, we have

$\begin{split}&\left|\operatorname{Re}\int|\xi|^{s}\overline{\widehat{\theta}}(\xi)\int|\xi|^{s}\left(\widehat{u}(\xi-\eta)\cdot
i\eta\widehat{\theta}(\eta)\right)\,\mathrm{d}\eta\,\mathrm{d}\xi\right|\leq
C_{s}\|{\theta}\|_{H^{s}}\int\log(10+|\xi|)|\xi|^{2s}|\widehat{\theta}(\xi)|^{2}\,\mathrm{d}\xi+C_{s}\|{\theta}\|_{H^{s}}^{3}.\end{split}$
(2.4)

We now claim that the expression

$\begin{split}V:=\operatorname{Re}\iint((10+|\xi|)^{2s}-|\xi|^{2s})\overline{\widehat{\theta}}(\xi)\left(\widehat{u}(\xi-\eta)\cdot
i\eta\widehat{\theta}(\eta)\right)\,\mathrm{d}\eta\,\mathrm{d}\xi\end{split}$

in absolute value is bounded by the right hand side of (2.4). We only sketch
the argument; setting

$\begin{split}Z(\xi):=\sqrt{(10+|\xi|)^{2s}-|\xi|^{2s}},\end{split}$

we consider

$\begin{split}V^{\prime}:=\operatorname{Re}\iint
Z(\xi)Z(\xi-\eta)\overline{\widehat{\theta}(\xi)}\left(\widehat{u}(\xi-\eta)\cdot
i\eta\widehat{\theta}(\eta)\right)\,\mathrm{d}\eta\,\mathrm{d}\xi\end{split}$

There is a highest order cancellation in $V^{\prime}$, which can be obtained
by symmetrizing the log in $|\xi|$ and $|\xi-\eta|$. Furthermore, the highest
order term in $|\xi-\eta|$ cancels in the expression $V-V^{\prime}$, giving
rise to a factor of $|\eta|^{3}$ in the kernel. We omit the details.

Therefore, we have arrived at the following:

$\begin{split}\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\|{\theta}\|_{{H}^{s}}^{2}\leq\left(\dot{s}+C_{s}\|{\theta}\|_{H^{s}}\right)\int\log(10+|\xi|)|\xi|^{2s}|\widehat{\theta}(\xi)|^{2}\,\mathrm{d}\xi+C_{s}\|{\theta}\|_{H^{s}}^{3}.\end{split}$

We set $F(t)=\|{\theta}\|_{H^{s(t)}}$ and define $s(t)$ to be a smooth
function of time such that the system of differential inequalities777 We note
that the dependence of $C_{s}$ on $s$ is minor; it can be assumed to be
independent of $s$ if we work on a closed interval of values of $s$.

$\begin{split}&\dot{s}+C_{s}F\leq 0,\qquad\dot{F}\leq C_{s}F^{2},\end{split}$

are satisfied for some small interval of time. (One can simply take
$s=s_{0}-Mt$ for some large $M>0$.) Then, this allows us to prove $F(t)\leq
CF_{0}$ for a possibly smaller interval of time. Hence, we have proved the a
priori estimate $\|{\theta(t,\cdot)}\|_{H^{s(t)}}\leq
C\|{\theta_{0}}\|_{H^{s_{0}}}$ in $t\in[0,T]$ for some
$T=T(s_{0},\|{\theta_{0}}\|_{H^{s_{0}}})>0$, as long as $s_{0}>4$.

Proving existence and uniqueness of the solution satisfying this a priori
estimate is straightforward and we shall only sketch the argument. For the
proof of existence, one can consider the dissipative system with a Laplacian,
namely

$\begin{split}\partial_{t}\theta+u\cdot\nabla\theta=\kappa\Delta\theta.\end{split}$

There exists a unique local in time smooth solution $\theta^{(\kappa)}$ with
initial data $\theta_{0}\in H^{s_{0}}$ for any $\kappa>0$ ([4]) which clearly
satisfies the $H^{s(t)}$ a priori estimate above. Therefore, there exists
$T=T(s_{0},\|{\theta_{0}}\|_{H^{s_{0}}})>0$ such that the solutions
$\theta^{(\kappa)}$ are uniformly bounded in
$L^{\infty}([0,T];H^{4+\varepsilon})$ for some $\varepsilon>0$. Furthermore,
since $\theta^{(\kappa)}$ are uniformly bounded in
$L^{\infty}([0,T];L^{\infty})$ as well. An interpolation gives that there
exists (by taking a subsequence if necessary) a strong limit
$\theta^{(\kappa)}\to\theta$ in $L^{\infty}([0,T];H^{4})$ as
$\kappa\rightarrow 0$. It is straightforward to verify that $\theta$ is a
solution to the inviscid equation with initial data $\theta_{0}$. To prove
uniqueness, we assume that there exist two solutions $\theta$ and
$\widetilde{\theta}$ to (1.1) with the same initial data $\theta_{0}$,
belonging to $L^{\infty}([0,T];H^{4})$ for some $T>0$. We set
$\phi=\theta-\widetilde{\theta}$ and compute, for some $M>0$ to be chosen
later,

$\begin{split}&\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\|{(10+\Lambda)^{-Mt}\phi}\|_{L^{2}}^{2}+M\|{(10+\Lambda)^{-Mt}\log^{\frac{1}{2}}(10+\Lambda)\phi}\|_{L^{2}}^{2}\\\
&\qquad=-\langle{\nabla^{\perp}\log(10+\Lambda)\theta\cdot\nabla\phi,(10+\Lambda)^{-2Mt}\phi}\rangle+\langle{\nabla^{\perp}\log(10+\Lambda)\phi\cdot\nabla\widetilde{\theta},(10+\Lambda)^{-2Mt}\phi}\rangle\\\
&\qquad=:I+II.\end{split}$

Note that

$\begin{split}I&=-\langle{(10+\Lambda)^{-Mt}(\nabla^{\perp}\log(10+\Lambda)\theta\cdot\nabla\phi),(10+\Lambda)^{-Mt}\phi}\rangle\\\
&=-\langle{[(10+\Lambda)^{-Mt},\nabla^{\perp}\log(10+\Lambda)\theta\cdot\nabla]\phi,(10+\Lambda)^{-Mt}\phi}\rangle\\\
\end{split}$

and with a commutator estimate for
$[(10+\Lambda)^{-Mt},\nabla^{\perp}\log(10+\Lambda)\theta\cdot\nabla]$,

$\begin{split}\left|I\right|\leq
C\|{\theta}\|_{H^{4}}\|{(10+\Lambda)^{-Mt}\phi}\|_{L^{2}}^{2}.\end{split}$

Similarly, for $II$ we can derive

$\begin{split}\left|II\right|\leq
C\|{\widetilde{\theta}}\|_{H^{4}}\|{(10+\Lambda)^{-Mt}\log^{\frac{1}{2}}(10+\Lambda)\phi}\|_{L^{2}}^{2}.\end{split}$

Taking $M=2C\|{\widetilde{\theta}}\|_{H^{4}}$ gives

$\begin{split}\frac{\mathrm{d}}{\mathrm{d}t}\|{(10+\Lambda)^{-Mt}\phi}\|_{L^{2}}^{2}\leq
C\|{\theta}\|_{H^{4}}\|{(10+\Lambda)^{-Mt}\phi}\|_{L^{2}}^{2},\end{split}$

or

$\begin{split}\|{(10+\Lambda)^{-Mt}\phi(t)}\|_{L^{2}}\leq\exp(C\|{\theta}\|_{H^{4}}t)\|{\phi_{0}}\|_{L^{2}}=0.\end{split}$

This shows $\phi(t)\equiv 0$, which finishes the proof of uniqueness. ∎

######  Remark 2.2.

This proof does not say that $C^{\infty}$-smooth data remains
$C^{\infty}$-smooth for some interval of time. It is only guaranteed that for
any $k$, there exists some time interval $[0,t_{k}]$ with $t_{k}>0$ depending
on $k$ such that the solution is $C^{k}$-smooth in space for $t\in[0,t_{k}]$.

###### Proof of Theorem D.

To establish local well-posedness in the dissipative case, one can derive an
estimate of the form

$\begin{split}\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\|{\theta}\|_{{H}^{s}}^{2}+\kappa\int\upsilon(|\xi|)(10+|\xi|)^{2s}|\widehat{\theta}(\xi)|^{2}\,\mathrm{d}\xi\leq
C_{s}\|{\theta}\|_{H^{s}}\int\log(10+|\xi|)|\xi|^{2s}|\widehat{\theta}(\xi)|^{2}\,\mathrm{d}\xi+C_{s}\|{\theta}\|_{H^{s}}^{3}\end{split}$

by arguing as in the proof of Theorem A above. Under the assumption (1.9) on
$\upsilon$, the first term on the right hand side can be absorbed to the left
hand side, giving

$\begin{split}\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\|{\theta}\|_{{H}^{s}}^{2}+\frac{\kappa}{2}\int\upsilon(|\xi|)(10+|\xi|)^{2s}|\widehat{\theta}(\xi)|^{2}\,\mathrm{d}\xi\leq
C_{s}\|{\theta}\|_{H^{s}}^{3}.\end{split}$

This gives the a priori estimate $\|{\theta(t)}\|_{H^{s}}\leq
C_{s}\|{\theta_{0}}\|_{H^{s}}$ for $0\leq t\leq T$ with
$T=T(\|{\theta_{0}}\|_{H^{s}})>0$. We omit the proof of existence and
uniqueness. ∎

###  2.2 The relation between solutions of (1.2) and (1.1)

This subsection is devoted to the proofs of Theorem B and C. Hereafter, we
denote the identity operator by $I$. Before starting the proof, we present
some lemmas. The first one is the well-known Kato–Ponce commutator estimate.

######  Lemma 2.3 ([31]).

For any $\epsilon>0$ and $s>0$, there exists a constant $C=C_{\epsilon,s}>0$
depending only on $\epsilon$ and $s$ satisfying

${\left\|\left[\Lambda^{s},f\right]g\right\|}_{L^{2}\left(\mathbb{R}^{2}\right)}\leq
C\left(\|{f}\|_{H^{2+\epsilon}(\mathbb{R}^{2})}{\left\|\Lambda^{s-1}g\right\|}_{L^{2}\left(\mathbb{R}^{2}\right)}+{\left\|\Lambda^{s}f\right\|}_{L^{2}\left(\mathbb{R}^{2}\right)}\|{g}\|_{H^{1+\epsilon}(\mathbb{R}^{2})}\right).$

The following estimates are concerned with velocities of (1.2) and (1.1).

######  Lemma 2.4.

For any $\epsilon>0$, $0<\delta<1$, and $s\geq 0$, there exists a constant
$C=C_{\epsilon}>0$ depending only on $\epsilon$ satisfying

$\displaystyle{\left\|\nabla^{\perp}\left(\delta\log(10+\Lambda)-I+(10+\Lambda)^{-\delta}\right)f\right\|}_{H^{s}(\mathbb{R}^{2})}$
$\displaystyle\leq C\delta^{2}\|{f}\|_{H^{s+1+\epsilon}(\mathbb{R}^{2})},$
(2.5a)
$\displaystyle{\left\|\nabla^{\perp}\left((10+\Lambda)^{-\delta}-I\right)f\right\|}_{H^{s}(\mathbb{R}^{2})}$
$\displaystyle\leq C\delta\|{f}\|_{H^{s+1+\epsilon}(\mathbb{R}^{2})}.$ (2.5b)

###### Proof.

Plancherel’s theorem gives us that

$\begin{split}&{\left\|\nabla^{\perp}\left(\delta\log(10+\Lambda)-I+(10+\Lambda)^{-\delta}\right)f\right\|}_{H^{s}}\\\
&\qquad={\left\|(1+|\xi|^{2})^{\frac{s}{2}}|\xi|\left|\delta\log(10+|\xi|)-1+(10+|\xi|)^{-\delta}\right|\hat{f}(\xi)\right\|}_{L^{2}}.\end{split}$
(2.6)

Applying the Taylor’s theorem to $(10+|\xi|)^{-\delta}$ at $0$, we have

$\begin{split}\left|\delta\log(10+|\xi|)-1+(10+|\xi|)^{-\delta}\right|&\leq\delta^{2}\left|\log(10+|\xi|)\right|^{2}=\delta^{2}(10+|\xi|)^{\epsilon}(10+|\xi|)^{-\epsilon}\left|\log(10+|\xi|)\right|^{2}\\\
&\leq C\delta^{2}(10+|\xi|)^{\epsilon},\end{split}$

where in the last line, we used the fact that the maximum of
$h(x):=x^{-\epsilon}(\log x)^{2}$ with $x\geq 1$ is $\left(\frac{2}{\epsilon
e}\right)^{2}$. Plugging this into (2.6), we obtain (2.5a). With a similar
argument, we obtain the estimate:

$\left|(10+|\xi|)^{-\delta}-1\right|\leq\delta\left|\log(10+|\xi|)\right|\leq
C\delta|\xi|(10+|\xi|)^{\epsilon},$

which yields (2.5b). ∎

The next one is another type of commutator estimate.

######  Lemma 2.5.

For any $\epsilon>0$ and $s\geq 0$, there exists a constant $C=C_{\epsilon}>0$
depending only on $\epsilon$ satisfying

$\displaystyle{\left\|\left[\partial_{j}\left(I-(10+\Lambda)^{-\delta}\right)^{\frac{1}{2}},f\right]\Lambda^{s}g\right\|}_{L^{2}\left(\mathbb{R}^{2}\right)}\leq
C\delta^{\frac{1}{2}}\left\|{f}\right\|_{H^{2+\epsilon}(\mathbb{R}^{2})}{\left\|\log^{\frac{1}{2}}(10+\Lambda)g\right\|}_{H^{s}(\mathbb{R}^{2})}\quad(j=1,2).$

###### Proof.

We note that

$\begin{split}&{\left\|\left[\partial_{j}\left(I-(10+\Lambda)^{-\delta}\right)^{\frac{1}{2}},f\right]\Lambda^{s}g\right\|}_{L^{2}}\leq{\left\|\int\left|\Phi_{j}(\xi,\eta)\right||\hat{f}(\eta)||\xi-\eta|^{s}|\hat{g}(\xi-\eta)|\,d\eta\right\|}_{L^{2}_{\xi}},\end{split}$

where

$\Phi_{j}(\xi,\eta):=\xi_{j}\left(1-(10+|\xi|)^{-\delta}\right)^{\frac{1}{2}}-(\xi-\eta)_{j}\left(1-(10+|\xi-\eta|)^{-\delta}\right)^{\frac{1}{2}}.$

By the mean value theorem, we obtain $\rho\in(0,1)$ such that

$\begin{split}\left|\Phi_{j}(\xi,\eta)\right|&\leq
C|\eta|\left(\left(1-(10+|\rho\xi+(1-\rho)(\xi-\eta)|)^{-\delta}\right)^{\frac{1}{2}}\right.\\\
&\left.\qquad+\delta(10+|\rho\xi+(1-\rho)(\xi-\eta)|)^{-\delta}\left(1-(10+|\rho\xi+(1-\rho)(\xi-\eta)|)^{-\delta}\right)^{-\frac{1}{2}}\right)\\\
&=C|\eta|\left(\left(1-(10+|(\xi-\eta)+\rho\eta|)^{-\delta}\right)^{\frac{1}{2}}\right.\\\
&\left.\qquad+\delta(10+|(\xi-\eta)+\rho\eta|)^{-\frac{\delta}{2}}\left((10+|(\xi-\eta)+\rho\eta|)^{\delta}-1\right)^{-\frac{1}{2}}\right).\end{split}$

Using

$\begin{split}\left(1-(10+|(\xi-\eta)+\rho\eta|)^{-\delta}\right)^{\frac{1}{2}}&\leq\delta^{\frac{1}{2}}\log^{\frac{1}{2}}(10+|(\xi-\eta)+\rho\eta|)\leq
C\delta^{\frac{1}{2}}\left(\log^{\frac{1}{2}}(10+|\xi-\eta|)+\log^{\frac{1}{2}}(10+|\eta|)\right),\end{split}$
$\begin{split}&\delta(10+|(\xi-\eta)+\rho\eta|)^{-\frac{\delta}{2}}\left((10+|(\xi-\eta)+\rho\eta|)^{\delta}-1\right)^{-\frac{1}{2}}\leq\delta^{\frac{1}{2}},\end{split}$

we have

$\begin{split}\left|\Phi_{j}(\xi,\eta)\right|&\leq
C\delta^{\frac{1}{2}}\log^{\frac{1}{2}}(10+|\xi-\eta|)|\eta|\left(1+\log^{\frac{1}{2}}(10+|\eta|)\right).\end{split}$

Hence, Young’s convolution inequality gives

$\begin{split}&{\left\|\left[\partial_{j}\log^{\frac{1}{2}}(10+\Lambda),f\right]\Lambda^{s}g\right\|}_{L^{2}}\\\
&\qquad\leq
C\delta^{\frac{1}{2}}{\left\|\int|\eta|\left(1+\log^{\frac{1}{2}}(10+|\eta|)\right)|\hat{f}(\eta)||\xi-\eta|^{s}\log^{\frac{1}{2}}(10+|\xi-\eta|)|\hat{g}(\xi-\eta)|d\eta\right\|}_{L^{2}_{\xi}}\\\
&\qquad\leq
C\delta^{\frac{1}{2}}\left\|{|\eta|\left(1+\log^{\frac{1}{2}}(10+|\eta|)\right)|\hat{f}(\eta)|}\right\|_{L^{1}_{\eta}}{\left\|\log^{\frac{1}{2}}(10+\Lambda)g\right\|}_{H^{s}}\leq
C\delta^{\frac{1}{2}}\left\|{f}\right\|_{H^{2+\epsilon}}{\left\|\log^{\frac{1}{2}}(10+\Lambda)g\right\|}_{H^{s}}.\qed\end{split}$

Finally, we present an elementary lemma from [12].

######  Lemma 2.6.

Let $T>0$, $G>0$ be given constants and let $F(t)$ be a nonnegative continuous
function on $[0,T]$. Assume further that $\nu$ be a constant satisfying

$0<8\nu TG\int_{0}^{T}F(t)dt\leq 1.$ (2.7)

Then every solution $y(t)\geq 0$ of

$\left\\{\begin{aligned} &\frac{\mathrm{d}}{\mathrm{d}t}y(t)\leq\nu
F(t)+Gy^{2}(t),\\\ &y(0)=0,\end{aligned}\right.$ (2.8)

is uniformly bounded on $[0,T]$ and satisfies

$y(t)\leq\min\left\\{\frac{3}{2TG},12\nu\int_{0}^{T}F(t)dt\right\\}.$ (2.9)

###### Proof of Theorem B.

By Theorem A, $\theta_{0}\in H^{s}$ with $s>5$ guarantees a time $T^{\log}>0$
and a unique solution $\theta\in
L^{\infty}([0,T^{\log}];H^{s_{0}+1+\epsilon})$ to (1.1) for some $s_{0}>4$ and
$\epsilon>0$. Since $\theta^{\delta}\in L^{\infty}_{t}H^{s}_{x}$ with $s>3$
implies $\theta^{\delta}\in L^{\infty}_{t}H^{\infty}_{x}$ (see [4]), it
suffices to show the existence of a continuous decreasing function $s(t)>4$
with $s(0)=s_{0}$ defined on $[0,C^{*}\delta^{-1}]$ satisfying

$\left\|{\theta^{\delta}(t)-\theta(\delta t)}\right\|_{H^{s(t)}}\leq\delta,$

where $C^{*}>0$ is a constant to be determined later. For simplicity, we
denote

$\tilde{\theta}^{\delta}(x,t):=\theta(x,\delta
t),\qquad\text{and}\qquad\theta^{D}(x,t):=\theta^{\delta}(x,t)-\tilde{\theta}^{\delta}(x,t).$

Noticing
$\nabla^{\perp}(10+\Lambda)^{-\delta}\theta^{\delta}\cdot\nabla\theta^{\delta}=\nabla^{\perp}\left((10+\Lambda)^{-\delta}-I\right)\theta^{\delta}\cdot\nabla\theta^{\delta},$
we have that

$\partial_{t}\theta^{\delta}+\nabla^{\perp}\left((10+\Lambda)^{-\delta}-I\right)\theta^{\delta}\cdot\nabla\theta^{\delta}=0.$

Moreover, observing
$\partial_{t}\tilde{\theta}^{\delta}-\nabla^{\perp}\delta\log(10+\Lambda)\tilde{\theta}^{\delta}\cdot\nabla\tilde{\theta}^{\delta}=0,$
we obtain the equation of $\theta^{D}$:

$\begin{split}\partial_{t}&\theta^{D}+\nabla^{\perp}\left((10+\Lambda)^{-\delta}-I\right)\theta^{D}\cdot\nabla\tilde{\theta}^{\delta}+\nabla^{\perp}\left((10+\Lambda)^{-\delta}-I\right)\theta^{D}\cdot\nabla\theta^{D}\\\
&+\nabla^{\perp}\left((10+\Lambda)^{-\delta}-I\right)\tilde{\theta}^{\delta}\cdot\nabla\theta^{D}+\nabla^{\perp}\left(\delta\log(10+\Lambda)-I+(10+\Lambda)^{-\delta}\right)\tilde{\theta}^{\delta}\cdot\nabla\tilde{\theta}^{\delta}=0.\end{split}$

Re-defining the $H^{s}$ norm by (2.1), we have

$\left|\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\left\|{\theta^{D}}\right\|_{H^{s(t)}}^{2}-\dot{s}(t)\left\|{\log^{\frac{1}{2}}(10+\Lambda)\theta^{D}}\right\|^{2}_{H^{s(t)}}\right|\leq\left|\int\theta^{D}\partial_{t}\theta^{D}\right|+\left|\int\Lambda^{s(t)}\theta^{D}\Lambda^{s(t)}\partial_{t}\theta^{D}\right|.$

To begin with, we estimate $\int\theta^{D}\partial_{t}\theta^{D}$. We compute
that

$\displaystyle\int\theta^{D}\partial_{t}\theta^{D}$
$\displaystyle=-\int\theta^{D}\left(\nabla^{\perp}\left((10+\Lambda)^{-\delta}-I\right)\theta^{D}\cdot\nabla\tilde{\theta}^{\delta}\right)-\int\theta^{D}\left(\nabla^{\perp}\left((10+\Lambda)^{-\delta}-I\right)\theta^{D}\cdot\nabla\theta^{D}\right)$
$\displaystyle\quad-\int\theta^{D}\left(\nabla^{\perp}\left((10+\Lambda)^{-\delta}-I\right)\tilde{\theta}^{\delta}\cdot\nabla\theta^{D}\right)-\int\theta^{D}\left(\nabla^{\perp}\left(\delta\log(10+\Lambda)-I+\Lambda^{-\delta}\right)\cdot\tilde{\theta}^{\delta}\nabla\tilde{\theta}^{\delta}\right)$
$\displaystyle=:L_{1}+L_{2}+L_{3}+L_{4}.$

Using (2.5b) and the Sobolev embedding $H^{3}(\mathbb{R}^{2})\hookrightarrow
W^{1,\infty}(\mathbb{R}^{2})$, we have

$|L_{1}|\leq{\left\|\theta^{D}\right\|}_{L^{2}}{\left\|\nabla^{\perp}\left((10+\Lambda)^{-\delta}-I\right)\theta^{D}\right\|}_{L^{2}}{\left\|\nabla\tilde{\theta}^{\delta}\right\|}_{L^{\infty}}\leq
C\delta\left\|{\theta^{D}}\right\|_{H^{2}}^{2}\left\|{\tilde{\theta}^{\delta}}\right\|_{H^{3}}.$
(2.10)

Since $\nabla^{\perp}\left((10+\Lambda)^{-\delta}-I\right)\theta^{\delta}$ is
divergence-free, integrating by parts gives

$L_{2}+L_{3}=-\int\theta^{D}\left(\nabla^{\perp}\left((10+\Lambda)^{-\delta}-I\right)\theta^{\delta}\cdot\nabla\theta^{D}\right)=0.$
(2.11)

By (2.5a), we obtain

$\begin{split}|L_{4}|&\leq{\left\|\theta^{D}\right\|}_{L^{2}}{\left\|\nabla^{\perp}\left(\delta\log(10+\Lambda)-I+(10+\Lambda)^{-\delta}\right)\tilde{\theta}^{\delta}\right\|}_{L^{2}}{\left\|\nabla\tilde{\theta}^{\delta}\right\|}_{L^{\infty}}\leq
C\delta^{2}{\left\|\theta^{D}\right\|}_{L^{2}}\left\|{\tilde{\theta}^{\delta}}\right\|_{H^{3}}^{2}.\end{split}$
(2.12)

Next, we estimate
$\int\Lambda^{s(t)}\theta^{D}\Lambda^{s(t)}\partial_{t}\theta^{D}$. We compute
that

$\displaystyle\int\Lambda^{s}\theta^{D}\Lambda^{s}\partial_{t}\theta^{D}$
$\displaystyle=-\int\Lambda^{s}\theta^{D}\,\Lambda^{s}\left(\nabla^{\perp}\left((10+\Lambda)^{-\delta}-I\right)\theta^{D}\cdot\nabla\tilde{\theta}^{\delta}\right)$
$\displaystyle\quad-\int\Lambda^{s}\theta^{D}\,\Lambda^{s}\left(\nabla^{\perp}\left((10+\Lambda)^{-\delta}-I\right)\theta^{D}\cdot\nabla\theta^{D}\right)$
$\displaystyle\quad-\int\Lambda^{s}\theta^{D}\,\Lambda^{s}\left(\nabla^{\perp}\left((10+\Lambda)^{-\delta}-I\right)\tilde{\theta}^{\delta}\cdot\nabla\theta^{D}\right)$
$\displaystyle\quad-\int\Lambda^{s}\theta^{D}\,\Lambda^{s}\left(\nabla^{\perp}\left(\delta\log(10+\Lambda)-I+(10+\Lambda)^{-\delta}\right)\tilde{\theta}^{\delta}\cdot\nabla\tilde{\theta}^{\delta}\right)$
$\displaystyle=:H_{1}+H_{2}+H_{3}+H_{4}.$

We claim that

$\begin{split}|H_{1}|\leq
C\delta{\left\|\tilde{\theta}^{\delta}\right\|}_{H^{s+1}}\left(\left\|{\log^{\frac{1}{2}}(10+\Lambda)\theta^{D}}\right\|^{2}_{H^{s}}+{\left\|\theta^{D}\right\|}_{H^{s}}^{2}\right).\end{split}$
(2.13)

As in (2.2), we use (2.3) to decompose

$\begin{split}H_{1}&=-\operatorname{Re}\int|\xi|^{s}\overline{\widehat{\theta^{D}}}(\xi)\int|\eta|^{s}\left((\xi-\eta)^{\perp}\left((10+|\xi-\eta|)^{-\delta}-1\right)\widehat{\theta^{D}}(\xi-\eta)\cdot
i\eta\widehat{\tilde{\theta}^{\delta}}(\eta)\right)\,\mathrm{d}\eta\,\mathrm{d}\xi\\\
&\quad-\operatorname{Re}\int|\xi|^{s}\overline{\widehat{\theta^{D}}}(\xi)\int|\xi-\eta|^{s}\left((\xi-\eta)^{\perp}\left((10+|\xi-\eta|)^{-\delta}-1\right)\widehat{\theta^{D}}(\xi-\eta)\cdot
i\eta\widehat{\tilde{\theta}^{\delta}}(\eta)\right)\,\mathrm{d}\eta\,\mathrm{d}\xi\\\
&\quad-\operatorname{Re}\int|\xi|^{s}\overline{\widehat{\theta^{D}}}(\xi)\int
s(\xi-\eta)\cdot\eta|\xi-\eta|^{s-2}\\\
&\qquad\qquad\qquad\qquad\qquad\qquad\times\left((\xi-\eta)^{\perp}\left((10+|\xi-\eta|)^{-\delta}-1\right)\widehat{\theta^{D}}(\xi-\eta)\cdot
i\eta\widehat{\tilde{\theta}^{\delta}}(\eta)\right)\,\mathrm{d}\eta\,\mathrm{d}\xi\\\
&\quad-\operatorname{Re}\int|\xi|^{s}\overline{\widehat{\theta^{D}}}(\xi)\int
R(\xi,\eta)\left((\xi-\eta)^{\perp}\left((10+|\xi-\eta|)^{-\delta}-1\right)\widehat{\theta^{D}}(\xi-\eta)\cdot
i\eta\widehat{\tilde{\theta}^{\delta}}(\eta)\right)\,\mathrm{d}\eta\,\mathrm{d}\xi\\\
&=:H_{11}+H_{12}+H_{13}+H_{14}.\end{split}$

With (2.5b) and the Sobolev embedding $H^{s-2}(\mathbb{R}^{2})\hookrightarrow
L^{\infty}(\mathbb{R}^{2})$, we estimate

$\begin{split}|H_{11}|&\leq{\left\|\Lambda^{s}\theta^{D}\right\|}_{L^{2}}{\left\|\nabla^{\perp}\left((10+\Lambda)^{-\delta}-I\right)\theta^{D}\right\|}_{L^{\infty}}{\left\|\nabla\Lambda^{s}\tilde{\theta}^{\delta}\right\|}_{L^{2}}\\\
&\leq
C{\left\|\theta^{D}\right\|}_{H^{s}}\left\|{\nabla^{\perp}\left((10+\Lambda)^{-\delta}-I\right)\theta^{D}}\right\|_{H^{s-2}}{\left\|\tilde{\theta}^{\delta}\right\|}_{H^{s+1}}\leq
C\delta{\left\|\tilde{\theta}^{\delta}\right\|}_{H^{s+1}}{\left\|\theta^{D}\right\|}_{H^{s}}^{2}.\end{split}$

We integrate by parts to decompose $H_{12}$ into

$\begin{split}H_{12}&=-\int\Lambda^{s}\theta^{D}\left(\nabla^{\perp}\Lambda^{s}\left(I-(10+\Lambda)^{-\delta}\right)\theta^{D}\cdot\nabla\tilde{\theta}^{\delta}\right)\\\
&=\int\Lambda^{s}\left(I-(10+\Lambda)^{-\delta}\right)^{\frac{1}{2}}\theta^{D}\,\nabla^{\perp}\left(I-(10+\Lambda)^{-\delta}\right)^{\frac{1}{2}}\left(\cdot\nabla\tilde{\theta}^{\delta}\Lambda^{s}\theta^{D}\right)\\\
&=\int\Lambda^{s}\left(I-(10+\Lambda)^{-\delta}\right)^{\frac{1}{2}}\theta^{D}\left[\nabla^{\perp}\left(I-(10+\Lambda)^{-\delta}\right)^{\frac{1}{2}},\cdot\nabla\tilde{\theta}^{\delta}\right]\Lambda^{s}\theta^{D}\\\
&\quad+\int\Lambda^{s}\left(I-(10+\Lambda)^{-\delta}\right)^{\frac{1}{2}}\theta^{D}\,\nabla\tilde{\theta}^{\delta}\cdot\nabla^{\perp}\Lambda^{s}\left(I-(10+\Lambda)^{-\delta}\right)^{\frac{1}{2}}\theta^{D}\\\
&=H_{121}+H_{122}.\end{split}$

Using Lemma 2.5 and
$\left|1-(10+|\xi|)^{-\delta}\right|^{\frac{1}{2}}\leq\delta^{\frac{1}{2}}\log^{\frac{1}{2}}(10+|\xi|)$,
we estimate

$\begin{split}|H_{121}|&\leq{\left\|\Lambda^{s}\left(I-(10+\Lambda)^{-\delta}\right)^{\frac{1}{2}}\theta^{D}\right\|}_{L^{2}}{\left\|\left[\nabla^{\perp}\left(I-(10+\Lambda)^{-\delta}\right)^{-\frac{1}{2}},\cdot\nabla\tilde{\theta}^{\delta}\right]\Lambda^{s}\theta^{D}\right\|}_{L^{2}}\\\
&\leq
C\delta{\left\|\tilde{\theta}^{\delta}\right\|}_{H^{s}}\left\|{\log^{\frac{1}{2}}(10+\Lambda)\theta^{D}}\right\|^{2}_{H^{s}}.\end{split}$

It follows from $\nabla^{\perp}\cdot\nabla\tilde{\theta}^{\delta}=0$ that
$H_{122}=0.$ For $H_{13}$, we similarly further decompose

$\begin{split}H_{13}&=-s\int\Lambda^{s}\theta^{D}\left(\nabla\nabla^{\perp}\Lambda^{s-2}\left(I-(10+\Lambda)^{-\delta}\right)\theta^{D}\cdot\nabla^{2}\tilde{\theta}^{\delta}\right)\\\
&=s\int\nabla\Lambda^{s-2}\left(I-(10+\Lambda)^{-\delta}\right)^{\frac{1}{2}}\theta^{D}\left[\nabla^{\perp}\left(I-(10+\Lambda)^{-\delta}\right)^{\frac{1}{2}},\cdot\nabla^{2}\tilde{\theta}^{\delta}\right]\Lambda^{s}\theta^{D}\\\
&\quad+s\int\nabla\Lambda^{s-2}\left(I-(10+\Lambda)^{-\delta}\right)^{\frac{1}{2}}\theta^{D}\,\nabla^{2}\tilde{\theta}^{\delta}\cdot\nabla^{\perp}\Lambda^{s}\left(I-(10+\Lambda)^{-\delta}\right)^{\frac{1}{2}}\theta^{D}\\\
&=H_{131}+H_{132}.\end{split}$

With Lemma 2.5 and
$\left|1-(10+|\xi|)^{-\delta}\right|^{\frac{1}{2}}\leq\delta^{\frac{1}{2}}\log^{\frac{1}{2}}(10+|\xi|)$,
we estimate

$\begin{split}|H_{131}|&\leq
C\delta{\left\|\tilde{\theta}^{\delta}\right\|}_{H^{s}}\left\|{\log^{\frac{1}{2}}(10+\Lambda)\theta^{D}}\right\|^{2}_{H^{s}}.\end{split}$

The integration by parts yields

$H_{132}=s\int\nabla^{\perp}\nabla\Lambda^{s-2}\left(I-(10+\Lambda)^{-\delta}\right)^{\frac{1}{2}}\theta^{D}\cdot\nabla^{2}\tilde{\theta}^{\delta}\Lambda^{s}\left(I-(10+\Lambda)^{-\delta}\right)^{\frac{1}{2}}\theta^{D},$

so that we have

$\begin{split}|H_{132}|&\leq{\left\|\nabla^{\perp}\nabla\Lambda^{s-2}\left(I-(10+\Lambda)^{-\delta}\right)^{\frac{1}{2}}\theta^{D}\right\|}_{L^{2}}{\left\|\nabla^{2}\tilde{\theta}^{\delta}\right\|}_{L^{\infty}}{\left\|\Lambda^{s}\left(I-(10+\Lambda)^{-\delta}\right)^{\frac{1}{2}}\theta^{D}\right\|}_{L^{2}}\\\
&\leq
C\delta{\left\|\tilde{\theta}^{\delta}\right\|}_{H^{s}}\left\|{\log^{\frac{1}{2}}(10+\Lambda)\theta^{D}}\right\|^{2}_{H^{s}}.\end{split}$

Using $\left|(10+|\xi|)^{-\delta}-1\right|\leq\delta\log(10+|\xi|)$, we can
slightly modify the proof of the estimate of $IV$ in Theorem A to obtain

$|H_{14}|\leq
C\delta{\left\|\tilde{\theta}^{\delta}\right\|}_{H^{s}}{\left\|\theta^{D}\right\|}_{H^{s}}^{2}.$

Combining all the estimates, we have (2.13). Next, we claim that

$\begin{split}|H_{2}|\leq
C\delta{\left\|\theta^{D}\right\|}_{H^{s}}\left(\left\|{\log^{\frac{1}{2}}(10+\Lambda)\theta^{D}}\right\|^{2}_{H^{s}}+{\left\|\theta^{D}\right\|}_{H^{s}}^{2}\right).\end{split}$
(2.14)

Using (2.3) again, we decompose

$\begin{split}H_{2}&=-\operatorname{Re}\int|\xi|^{s}\overline{\widehat{\theta^{D}}}(\xi)\int|\eta|^{s}\left((\xi-\eta)^{\perp}\left((10+|\xi-\eta|)^{-\delta}-1\right)\widehat{\theta^{D}}(\xi-\eta)\cdot
i\eta\widehat{\theta^{D}}(\eta)\right)\,\mathrm{d}\eta\,\mathrm{d}\xi\\\
&\quad-\operatorname{Re}\int|\xi|^{s}\overline{\widehat{\theta^{D}}}(\xi)\int|\xi-\eta|^{s}\left((\xi-\eta)^{\perp}\left((10+|\xi-\eta|)^{-\delta}-1\right)\widehat{\theta^{D}}(\xi-\eta)\cdot
i\eta\widehat{\theta^{D}}(\eta)\right)\,\mathrm{d}\eta\,\mathrm{d}\xi\\\
&\quad-\operatorname{Re}\int|\xi|^{s}\overline{\widehat{\theta^{D}}}(\xi)\int
s(\xi-\eta)\cdot\eta|\xi-\eta|^{s-2}\\\
&\qquad\qquad\qquad\qquad\qquad\qquad\times\left((\xi-\eta)^{\perp}\left((10+|\xi-\eta|)^{-\delta}-1\right)\widehat{\theta^{D}}(\xi-\eta)\cdot
i\eta\widehat{\theta^{D}}(\eta)\right)\,\mathrm{d}\eta\,\mathrm{d}\xi\\\
&\quad-\operatorname{Re}\int|\xi|^{s}\overline{\widehat{\theta^{D}}}(\xi)\int
R(\xi,\eta)\left((\xi-\eta)^{\perp}\left((10+|\xi-\eta|)^{-\delta}-1\right)\widehat{\theta^{D}}(\xi-\eta)\cdot
i\eta\widehat{\theta^{D}}(\eta)\right)\,\mathrm{d}\eta\,\mathrm{d}\xi\\\
&=:H_{21}+H_{22}+H_{23}+H_{24}.\end{split}$

It is clear that $H_{21}=0$, and for the estimates of $H_{22}$, $H_{23}$, and
$H_{24}$, we replace $\tilde{\theta}^{\delta}$ in the proofs of the estimates
of $H_{12}$, $H_{13}$, and $H_{14}$ with $\theta^{D}$, respectively, which
yields (2.14). Now we handle $H_{3}$. Since
$\nabla^{\perp}\left((10+\Lambda)^{-\delta}-I\right)\tilde{\theta}^{\delta}$
is divergence-free, we have

$H_{3}=-\int\Lambda^{s}\theta^{D}\left[\Lambda^{s},\nabla^{\perp}\left((10+\Lambda)^{-\delta}-I\right)\tilde{\theta}^{\delta}\cdot\right]\nabla\theta^{D},$

and therefore, Lemma 2.3 and (2.5b) give

$\begin{split}|H_{3}|&\leq{\left\|\Lambda^{s}\theta^{D}\right\|}_{L^{2}}{\left\|\left[\Lambda^{s},\nabla^{\perp}\left((10+\Lambda)^{-\delta}-I\right)\tilde{\theta}^{\delta}\cdot\right]\nabla\theta^{D}\right\|}_{L^{2}}\\\
&\leq
C{\left\|\theta^{D}\right\|}_{H^{s}}\left(\left\|{\nabla^{\perp}\left((10+\Lambda)^{-\delta}-I\right)\tilde{\theta}^{\delta}}\right\|_{H^{2+\epsilon}}\left\|{\nabla\theta^{D}}\right\|_{H^{s-1}}\right.\\\
&\qquad\qquad\qquad\left.+{\left\|\nabla^{\perp}\left((10+\Lambda)^{-\delta}-I\right)\tilde{\theta}^{\delta}\right\|}_{H^{s}}\left\|{\nabla\theta^{D}}\right\|_{H^{1+\epsilon}}\right)\\\
&\leq
C\delta\left\|{\tilde{\theta}^{\delta}}\right\|_{H^{s+1+\epsilon}}{\left\|\theta^{D}\right\|}_{H^{s}}^{2}.\end{split}$
(2.15)

For $H_{4}$, noticing that $H^{s}(\mathbb{R}^{2})$ is a Banach algebra, (2.5a)
implies that

$\begin{split}|H_{4}|&\leq{\left\|\Lambda^{s}\theta^{D}\right\|}_{L^{2}}{\left\|\Lambda^{s}\left(\nabla^{\perp}\left(\delta\log(10+\Lambda)-I+(10+\Lambda)^{-\delta}\right)\tilde{\theta}^{\delta}\cdot\nabla\tilde{\theta}^{\delta}\right)\right\|}_{L^{2}}\\\
&\leq
C{\left\|\theta^{D}\right\|}_{H^{s}}{\left\|\nabla^{\perp}\left(\delta\log(10+\Lambda)-I+(10+\Lambda)^{-\delta}\right)\tilde{\theta}^{\delta}\right\|}_{H^{s}}{\left\|\nabla\tilde{\theta}^{\delta}\right\|}_{H^{s}}\\\
&\leq
C\delta^{2}\left\|{\tilde{\theta}^{\delta}}\right\|_{H^{s+1+\epsilon}}^{2}{\left\|\theta^{D}\right\|}_{H^{s}}.\end{split}$
(2.16)

Summing from (2.10) to (2.16), we have arrived at

$\begin{split}\frac{\mathrm{d}}{\mathrm{d}t}\left\|{\theta^{D}}\right\|^{2}_{H^{s}}&\leq
2\left(\dot{s}+C\delta\left(\left\|{\tilde{\theta}^{\delta}}\right\|_{H^{s+1+\epsilon}}+{\left\|\theta^{D}\right\|}_{H^{s}}\right)\right)\left\|{\log^{\frac{1}{2}}(10+\Lambda)\theta^{D}}\right\|^{2}_{H^{s}}\\\
&\quad+C\delta\left(\left\|{\tilde{\theta}^{\delta}}\right\|_{H^{s+1+\epsilon}}{\left\|\theta^{D}\right\|}_{H^{s}}^{2}+{\left\|\theta^{D}\right\|}_{H^{s}}^{3}+\delta\left\|{\tilde{\theta}^{\delta}}\right\|_{H^{s+1+\epsilon}}^{2}{\left\|\theta^{D}\right\|}_{H^{s}}\right).\end{split}$
(2.17)

We now define

$s(t):=s_{0}-C\delta\left(1+\sup_{t\in\left[0,T^{log}\right]}\left\|{\theta(t)}\right\|_{H^{s+1+\epsilon}}\right)t,$

and

$C^{*}:=\frac{\min\left\\{{1,\,T^{log}},\,s_{0}-4\right\\}}{100C\left(1+\sup_{t\in[0,{T^{log}}]}\left\|{\theta(t)}\right\|_{H^{s+1+\epsilon}}\right)^{2}}$

We claim that for any $\delta\in(0,1)$,

$s(t)>4,$ (2.18)

and

$\left\|{\theta^{D}(t)}\right\|_{H^{s(t)}}\leq\delta$ (2.19)

on $\left[0,C^{*}\delta^{-1}\right]$. We can check (2.18) with the definitions
of $s(t)$ and $C^{*}$. To obtain (2.19), suppose, on the contrary, that

$\bar{T}:=\inf{\left\\{t\,:\;\left\|{\theta^{D}(t)}\right\|_{H^{s(t)}}>\delta\right\\}}<C^{*}\delta^{-1}.$

Then since

$\dot{s}+C\delta\left(\left\|{\tilde{\theta}^{\delta}}\right\|_{H^{s+1+\epsilon}}+{\left\|\theta^{D}\right\|}_{H^{s}}\right)\leq
0$

on $[0,\bar{T}]$, (2.17) implies that

$\frac{\mathrm{d}}{\mathrm{d}t}\left\|{\theta^{D}}\right\|_{H^{s}}\leq
C\delta\left(\left\|{\tilde{\theta}^{\delta}}\right\|_{H^{s+1+\epsilon}}{\left\|\theta^{D}\right\|}_{H^{s}}+{\left\|\theta^{D}\right\|}_{H^{s}}^{2}+\delta\left\|{\tilde{\theta}^{\delta}}\right\|_{H^{s+1+\epsilon}}^{2}\right)$

on $[0,\bar{T}]$. In order to make use of Lemma 2.6, we multiply this by
$\exp{\left(-C\delta\int_{0}^{t}\left\|{\tilde{\theta}^{\delta}(\tau)}\right\|_{H^{s+1+\epsilon}}d\tau\right)}$
and consider the quantity

$y(t):={\left\|\theta^{D}(t)\right\|}_{H^{s}}\exp{\left(-C\delta\int_{0}^{t}\left\|{\tilde{\theta}^{\delta}(\tau)}\right\|_{H^{s+1+\epsilon}}d\tau\right)}.$

Denoting

$G:=C\delta\exp{\left(C\delta\int_{0}^{\bar{T}}\left\|{\tilde{\theta}^{\delta}(\tau)}\right\|_{H^{s+1+\epsilon}}d\tau\right)},$

we then obtain

$\left\\{\begin{aligned} &\frac{\mathrm{d}}{\mathrm{d}t}y(t)\leq
C\delta^{2}\left\|{\tilde{\theta}^{\delta}(t)}\right\|_{H^{s+1+\epsilon}}^{2}+Gy^{2}(t),\\\
&y(0)=0\end{aligned}\right.$

on $[0,\bar{T}]$. Replacing $\nu$, $T$, and $F(t)$ in Lemma 2.6 by
$C\delta^{2}$, $\bar{T}$, and
$\left\|{\tilde{\theta}^{\delta}(t)}\right\|_{H^{s+1+\epsilon}}^{2}$,
respectively, we have

$\begin{split}8\nu TG&\int_{0}^{T}F(t)dt\leq
8\delta\left(C\delta\bar{T}\sup_{t\in\left[0,T^{log}\right]}\left\|{\theta(t)}\right\|_{H^{s+1+\epsilon}}\right)^{2}\exp{\left(C\delta\bar{T}\sup_{t\in\left[0,T^{log}\right]}\left\|{\theta(t)}\right\|_{H^{s+1+\epsilon}}\right)}\\\
&<8\delta\left(CC^{*}\sup_{t\in\left[0,T^{log}\right]}\left\|{\theta(t)}\right\|_{H^{s+1+\epsilon}}\right)^{2}\exp{\left(CC^{*}\sup_{t\in\left[0,T^{log}\right]}\left\|{\theta(t)}\right\|_{H^{s+1+\epsilon}}\right)}\leq
1\end{split}$

for any $\delta\in(0,1)$. Then Lemma 2.6 implies that

$\begin{split}{\left\|\theta^{D}(t)\right\|}_{H^{s}}&\leq
12C\delta^{2}\bar{T}\left(\sup_{t\in\left[0,T^{log}\right]}\left\|{\theta(t)}\right\|_{H^{s+1+\epsilon}}\right)^{2}\exp{\left(C\delta\bar{T}\sup_{t\in\left[0,T^{log}\right]}\left\|{\theta(t)}\right\|_{H^{s+1+\epsilon}}\right)}\leq\frac{\delta}{2}\end{split}$

for any $\delta\in(0,1)$ and $t\in\left[0,\bar{T}\right]$, which is a
contradiction to the definition of $\bar{T}$. ∎

###### Proof of Theorem C.

By the local well-posedness theory of (1.5), there exist a maximal existence
time $T_{max}^{\delta}>0$ and a unique solution $\theta^{\delta}\in
L^{\infty}([0,T_{max}^{\delta});H^{s})\cap
L^{2}([0,T_{max}^{\delta});\Psi^{-1}H^{s})$ to (1.5), where $\Psi^{-1}H^{s}$
is a function space defined by

$\Psi^{-1}H^{s}:=\left\\{f:\Psi f\in H^{s}\right\\}.$

In the same manner as the proof of (2.14), we estimate

$\begin{split}\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}{\left\|\theta^{\delta}\right\|}_{H^{s}}^{2}+\kappa{\left\|\Psi\theta\right\|}_{H^{s}}^{2}&\leq
C\delta{\left\|\theta^{\delta}\right\|}_{H^{s}}\left({\left\|\log(10+\Lambda)\theta^{\delta}\right\|}_{H^{s}}^{2}+{\left\|\theta^{\delta}\right\|}_{H^{s}}^{2}\right)\\\
&\leq
C\delta{\left\|\theta^{\delta}\right\|}_{H^{s}}{\left\|\log(10+\Lambda)\theta^{\delta}\right\|}_{H^{s}}^{2}\end{split}$

on $[0,T_{max}^{\delta})$. Defining
$\delta^{*}:=\frac{\kappa}{10C{\left\|\theta_{0}\right\|}_{H^{s}}},$ the
assumption (1.6) implies that for any $\delta\in(0,\delta^{*})$,

$\begin{split}\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}{\left\|\theta^{\delta}\right\|}_{H^{s}}^{2}&\leq\left(C\delta{\left\|\theta^{\delta}\right\|}_{H^{s}}-\kappa\right){\left\|\log(10+\Lambda)\theta^{\delta}\right\|}_{H^{s}}^{2}\leq\kappa\left(\frac{{\left\|\theta^{\delta}\right\|}_{H^{s}}}{10{\left\|\theta_{0}\right\|}_{H^{s}}}-1\right){\left\|\log(10+\Lambda)\theta^{\delta}\right\|}_{H^{s}}^{2}\end{split}$

on $[0,T_{max}^{\delta})$. To obtain $T_{max}^{\delta}=\infty$ for any
$\delta\in(0,\delta^{*})$, it suffices to show that

$\bar{T}:=\inf{\left\\{t\,:\;\left\|{\theta^{\delta}(t)}\right\|_{H^{s}}>2{\left\|\theta_{0}\right\|}_{H^{s}}\right\\}}\geq
T_{max}^{\delta}$

for any $\delta\in(0,\delta^{*})$. Suppose, on the contrary, that
$\bar{T}<T_{max}^{\delta}$. Then on $[0,\bar{T}]$, we have

$\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}{\left\|\theta^{\delta}\right\|}_{H^{s}}^{2}\leq-\frac{4\kappa}{5}{\left\|\log(10+\Lambda)\theta^{\delta}\right\|}_{H^{s}}^{2}\leq
0,$

and therefore,

${\left\|\theta^{\delta}(\bar{T})\right\|}_{H^{s}}\leq{\left\|\theta_{0}\right\|}_{H^{s}},$

which is a contradiction to the definition of $\bar{T}$. As in the case of
(1.2), we can check that $\theta^{\delta}\in L^{\infty}_{t}H^{s}_{x}$ with
$s>3$ implies $\theta^{\delta}\in L^{\infty}_{t}H^{\infty}_{x}$ (see [4]). ∎

### Acknowledgments

D. Chae was supported partially by NRF grant 2021R1A2C1003234. I.-J. Jeong and
J. Na has been supported by the Samsung Science and Technology Foundation
under Project Number SSTF-BA2002-04. S.-J. Oh was supported by the Samsung
Science and Technology Foundation under Project Number SSTF-BA1702-02, a Sloan
Research Fellowship and a National Science Foundation CAREER Grant under NSF-
DMS-1945615.

## References

  * [1] Jean Bourgain and Dong Li, _Strong ill-posedness of the incompressible Euler equation in borderline Sobolev spaces_ , Invent. Math. 201 (2015), no. 1, 97–157. MR 3359050
  * [2] Elia Brue and Quoc-Hung Nguyen, _Sobolev estimates for solutions of the transport equation and ODE flows associated to non-Lipschitz drifts_ , Math. Ann. (2020).
  * [3] Luis A. Caffarelli and Alexis Vasseur, _Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation_ , Ann. of Math. (2) 171 (2010), no. 3, 1903–1930. MR 2680400
  * [4] Dongho Chae, Peter Constantin, Diego Córdoba, Francisco Gancedo, and Jiahong Wu, _Generalized surface quasi-geostrophic equations with singular velocities_ , Comm. Pure Appl. Math. 65 (2012), no. 8, 1037–1066. MR 2928091
  * [5] Dongho Chae, Peter Constantin, and Jiahong Wu, _Inviscid models generalizing the two-dimensional Euler and the surface quasi-geostrophic equations_ , Arch. Ration. Mech. Anal. 202 (2011), no. 1, 35–62. MR 2835862
  * [6] , _Deformation and symmetry in the inviscid SQG and the 3D Euler equations_ , J. Nonlinear Sci. 22 (2012), no. 5, 665–688. MR 2982049
  * [7] Dongho Chae, In-Jee Jeong, and Sung-Jin Oh, _Illposedness via degenerate dispersion for generalized surface quasi-geostrophic equations with singular velocities_.
  * [8] Dongho Chae and Jihoon Lee, _Global well-posedness in the super-critical dissipative quasi-geostrophic equations_ , Comm. Math. Phys. 223 (2003), no. 2, 297–311. MR 1962043
  * [9] Dongho Chae and Jiahong Wu, _Logarithmically regularized inviscid models in borderline Sobolev spaces_ , J. Math. Phys. 53 (2012), no. 11, 115601, 15. MR 3026546
  * [10] Qionglei Chen, Changxing Miao, and Zhifei Zhang, _A new Bernstein’s inequality and the 2D dissipative quasi-geostrophic equation._ , Comm. Math. Phys. 271 (2007), no. 3, 821–838. MR 2291797
  * [11] Kyudong Choi and In-Jee Jeong, _Infinite growth in vorticity gradient of compactly supported planar vorticity near Lamb dipole_ , Nonlinear Anal. Real World Appl. 65 (2022), Paper No. 103470, 20. MR 4350517
  * [12] Peter Constantin, _Note on loss of regularity for solutions of the 3-D incompressible Euler and related equations_ , Comm. Math. Phys. 104 (1986), no. 2, 311–326. MR 0836008
  * [13] Peter Constantin, Andrew J. Majda, and Esteban Tabak, _Formation of strong fronts in the $2$-D quasigeostrophic thermal active scalar_, Nonlinearity 7 (1994), no. 6, 1495–1533. MR 1304437
  * [14] Peter Constantin, Andrew J. Majda, and Esteban G. Tabak, _Singular front formation in a model for quasigeostrophic flow_ , Phys. Fluids 6 (1994), no. 1, 9–11. MR 1252829
  * [15] Peter Constantin, Andrei Tarfulea, and Vlad Vicol, _Long time dynamics of forced critical SQG_ , Comm. Math. Phys. 335 (2015), no. 1, 93–141. MR 3314501
  * [16] Peter Constantin and Vlad Vicol, _Nonlinear maximum principles for dissipative linear nonlocal operators and applications_ , Geom. Funct. Anal. 22 (2012), no. 5, 1289–1321. MR 2989434
  * [17] Peter Constantin and Jiahong Wu, _Behavior of solutions of 2D quasi-geostrophic equations_ , SIAM J. Math. Anal. 30 (1999), no. 5, 937–948. MR 1709781
  * [18] Michele Coti Zelati and Vlad Vicol, _On the global regularity for the supercritical sqg equation_ , Indiana Univ. Math. J. 65 (2016), no. 2, 535–552. MR 3498176
  * [19] Diego Córdoba, Antonio; Córdoba, _A maximum principle applied to quasi-geostrophic equations_ , Comm. Math. Phys. 249 (2004), no. 3, 511–528. MR 2084005
  * [20] Michael Dabkowski, Alexander Kiselev, Luis Silvestre, and Vlad Vicol, _Global well-posedness of slightly supercritical active scalar equations_ , Anal. PDE 7 (2014), 43–72. MR 3219499
  * [21] Sergey A. Denisov, _Infinite superlinear growth of the gradient for the two-dimensional Euler equation_ , Discrete Contin. Dyn. Syst. 23 (2009), no. 3, 755–764. MR 2461825
  * [22] , _Double exponential growth of the vorticity gradient for the two-dimensional Euler equation_ , Proc. Amer. Math. Soc. 143 (2015), no. 3, 1199–1210. MR 3293735
  * [23] Tarek M. Elgindi and In-Jee Jeong, _Ill-posedness for the Incompressible Euler Equations in Critical Sobolev Spaces_ , Ann. PDE 3 (2017), no. 1, 3:7. MR 3625192
  * [24] Tarek Mohamed Elgindi, _Osgood’s lemma and some results for the slightly supercritical 2D Euler equations for incompressible flow_ , Arch. Ration. Mech. Anal. 211 (2014), no. 3, 965–990. MR 3158812
  * [25] Susan Friedlander, Francisco Gancedo, Weiran Sun, and Vlad Vicol, _On a singular incompressible porous media equation_ , J. Math. Phys. 53 (2012), no. 11, 115602, 20. MR 3026547
  * [26] Susan Friedlander and Vlad Vicol, _On the ill/well-posedness and nonlinear instability of the magneto-geostrophic equations_ , Nonlinearity 24 (2011), no. 11, 3019–3042. MR 2844825
  * [27] Siming He and Alexander Kiselev, _Small scale creation for solutions of the SQG equation_ , arXiv:1903.07485.
  * [28] In-Jee Jeong, _Dynamics of the incompressible Euler equations at critical regularity_ , Ph.D. thesis, Princeton University, 2017.
  * [29] In-Jee Jeong and Sung-Jin Oh, _On the Cauchy problem for the Hall and electron magnetohydrodynamic equations without resistivity I: Illposedness near degenerate stationary solutions_ , Ann. PDE 8 (2022), no. 2, Paper No. 15, 106. MR 4456288
  * [30] In-Jee Jeong and Tsuyoshi Yoneda, _Enstrophy dissipation and vortex thinning for the incompressible 2D Navier-Stokes equations_ , Nonlinearity 34 (2021), no. 4, 1837–1853. MR 4246445
  * [31] Tosio Kato and Gustavo Ponce, _Commutator estimates and the euler and navier-stokes equations_ , Comm. Pure Appl. Math. 41 (1988), no. 7, 891–907. MR 0951744
  * [32] Junha Kim and In-Jee Jeong, _A simple ill-posedness proof for incompressible Euler equations in critical Sobolev spaces_ , J. Funct. Anal. 283 (2022), no. 10, Paper No. 109673, 34. MR 4470735
  * [33] A. Kiselev, F. Nazarov, and A. Volberg, _Global well-posedness for the critical 2D dissipative quasi-geostrophic equation_ , Invent. Math. 167 (2007), no. 3, 445–453. MR 2276260
  * [34] Alexander Kiselev and Fedor Nazarov, _A simple energy pump for the surface quasi-geostrophic equation_ , Nonlinear partial differential equations, Abel Symp., vol. 7, Springer, Heidelberg, 2012, pp. 175–179. MR 3289364
  * [35] Alexander Kiselev, Lenya Ryzhik, Yao Yao, and Andrej Zlatoš, _Finite time singularity for the modified SQG patch equation_ , Ann. of Math. (2) 184 (2016), no. 3, 909–948. MR 3549626
  * [36] Alexander Kiselev, Yao Yao, and Andrej Zlatoˇs, _Local regularity for the modified SQG patch equation_ , Comm. Pure Appl. Math. 70 (2017), no. 7, 1253–1315. MR 3666567
  * [37] Hyunju Kwon, _Strong ill-posedness of logarithmically regularized 2D Euler equations in the borderline Sobolev space_ , J. Funct. Anal. 280 (2021), no. 7, 108822. MR 4211020
  * [38] Yuri E. Litvinenko and Liam C. McMahon, _Finite-time singularity formation at a magnetic neutral line in Hall magnetohydrodynamics_ , Appl. Math. Lett. 45 (2015), 76–80. MR 3316965
  * [39] Koji Ohkitani, _Growth rate analysis of scalar gradients in generalized surface quasigeostrophic equations of ideal fluids_ , Phys. Rev. E (3) 83 (2011), no. 3, 036317, 8. MR 2788267
  * [40] , _Asymptotics and numerics of a family of two-dimensional generalized surface quasi-geostrophic equations_ , Physics of Fluids 24 (2012), no. 095101.
  * [41] Jiahong Wu, _Global solutions of the 2d dissipative quasi-geostrophic equation in Besov spaces_ , SIAM J. Math. Anal. 36 (2004/05), no. 3, 1014–1030. MR 2111923
  * [42] Xiaoqian Xu, _Fast growth of the vorticity gradient in symmetric smooth domains for 2D incompressible ideal flow_ , J. Math. Anal. Appl. 439 (2016), no. 2, 594–607. MR 3475939
  * [43] Huan Yu, Xiaoxin Zheng, and Quansen Jiu, _Remarks on well-posedness of the generalized surface quasi-geostrophic equation_ , Arch. Ration. Mech. Anal. 232 (2019), no. 1, 265–301. MR 3916975
  * [44] V. I. Yudovich, _Non-stationary flows of an ideal incompressible fluid_ , Z. Vycisl. Mat. i Mat. Fiz. 3 (1963), 1032–1066. MR 0158189
  * [45] Andrej Zlatoš, _Exponential growth of the vorticity gradient for the Euler equation on the torus_ , Adv. Math. 268 (2015), 396–403. MR 3276599


* [187] Keyulu Xu, Weihua Hu, Jure Leskovec, and Stefanie Jegelka.  How powerful are graph neural networks?  In ICLR, 2019.
  * [188] Zhiming Xu, Xiao Huang, Yue Zhao, Yushun Dong, and Jundong Li.  Contrastive attributed network anomaly detection with data augmentation.  In PAKDD, 2022.
  * [189] Haotian Xue, Kaixiong Zhou, Tianlong Chen, Kai Guo, Xia Hu, Yi Chang, and Xin Wang.  Cap: Co-adversarial perturbation on weights and features for improving generalization of graph neural networks.  arXiv, 2021.
  * [190] Ling Yang, Jiayi Zheng, Heyuan Wang, Zhongyi Liu, Zhilin Huang, Shenda Hong, Wentao Zhang, and Bin Cui.  Individual and structural graph information bottlenecks for out-of-distribution generalization.  IEEE TKDE, 2023.
  * [191] Wensi Yang, Yuhang Zhang, Kejiang Ye, Li Li, and Cheng-Zhong Xu.  Ffd: A federated learning based method for credit card fraud detection.  In BigData, 2019.
  * [192] Yuhang Yao, Weizhao Jin, Srivatsan Ravi, and Carlee Joe-Wong.  Fedgcn: Convergence and communication tradeoffs in federated training of graph convolutional networks.  arXiv, 2022.
  * [193] Nan Yin, Li Shen, Mengzhu Wang, Xiao Luo, Zhigang Luo, and Dacheng Tao.  Omg: Towards effective graph classification against label noise.  IEEE TKDE, 2023.
  * [194] Junchi Yu, Jian Liang, and Ran He.  Mind the label shift of augmentation-based graph ood generalization.  In CVPR, 2023.
  * [195] Xiangchi Yuan, Chunhui Zhang, Yijun Tian, Yanfang Ye, and Chuxu Zhang.  Mitigating severe robustness degradation on graphs.  In ICLR, 2024.
  * [196] Rahul Yumlembam, Biju Issac, Seibu Mary Jacob, and Longzhi Yang.  Iot-based android malware detection using graph neural network with adversarial defense.  IEEE IoT, 2022.
  * [197] Ziqian Zeng, Rashidul Islam, Kamrun Naher Keya, James Foulds, Yangqiu Song, and Shimei Pan.  Fair representation learning for heterogeneous information networks.  In ICWSM, 2021.
  * [198] Abdelhafid Zeroual, Fouzi Harrou, Abdelkader Dairi, and Ying Sun.  Deep learning methods for forecasting covid-19 time-series data: A comparative study.  Chaos Solit. Fractals., 2020.
  * [199] Binchi Zhang, Yushun Dong, Chen Chen, Yada Zhu, Minnan Luo, and Jundong Li.  Adversarial attacks on fairness of graph neural networks.  In ICLR, 2024.
  * [200] Binchi Zhang, Minnan Luo, Shangbin Feng, Ziqi Liu, Jun Zhou, and Qinghua Zheng.  Ppsgcn: A privacy-preserving subgraph sampling based distributed gcn training method.  arXiv, 2021.
  * [201] Chunhui Zhang, Yijun Tian, Mingxuan Ju, Zheyuan Liu, Yanfang Ye, Nitesh Chawla, and Chuxu Zhang.  Chasing all-round graph representation robustness: Model, training, and optimization.  In ICLR, 2023.
  * [202] He Zhang, Bang Wu, Xiangwen Yang, Chuan Zhou, Shuo Wang, Xingliang Yuan, and Shirui Pan.  Projective ranking: A transferable evasion attack method on graph neural networks.  In CIKM, 2021.
  * [203] He Zhang, Bang Wu, Xingliang Yuan, Shirui Pan, Hanghang Tong, and Jian Pei.  Trustworthy graph neural networks: Aspects, methods and trends.  arXiv, 2022.
  * [204] Jiaqiang Zhang, Senzhang Wang, and Songcan Chen.  Reconstruction enhanced multi-view contrastive learning for anomaly detection on attributed networks.  In IJCAI, 2022.
  * [205] Kai Zhang, Yu Wang, Hongyi Wang, Lifu Huang, Carl Yang, and Lichao Sun.  Efficient federated learning on knowledge graphs via privacy-preserving relation embedding aggregation.  In ACL Workshops, 2022.
  * [206] Ke Zhang, Carl Yang, Xiaoxiao Li, Lichao Sun, and Siu Ming Yiu.  Subgraph federated learning with missing neighbor generation.  NeurIPS, 2021.
  * [207] Sen Zhang and Weiwei Ni.  Graph embedding matrix sharing with differential privacy.  IEEE Access, 2019.
  * [208] Xianren Zhang, Jing Ma, Yushun Dong, Chen Chen, Min Gao, and Jundong Li.  Sd-attack: Targeted spectral attacks on graphs.  In PAKDD, 2024.
  * [209] Y Zhang, G Song, L Du, S Yang, and Y Jin.  Dane: Domain adaptive network embedding.  In IJCAI, 2019.
  * [210] Yi Zhang, Yuying Zhao, Zhaoqing Li, Xueqi Cheng, Yu Wang, Olivera Kotevska, Philip S Yu, and Tyler Derr.  A survey on privacy in graph neural networks: Attacks, preservation, and applications.  arXiv, 2023.
  * [211] Yingxue Zhang, Soumyasundar Pal, Mark Coates, and Deniz Ustebay.  Bayesian graph convolutional neural networks for semi-supervised classification.  In AAAI, 2019.
  * [212] Yingxue Zhang, Florence Regol, Soumyasundar Pal, Sakif Khan, Liheng Ma, and Mark Coates.  Detection and defense of topological adversarial attacks on graphs.  In AISTATS, 2021.
  * [213] Yuan Zhang, Elizaveta Levina, and Ji Zhu.  Estimating network edge probabilities by neighborhood smoothing.  Biometrika, 2017.
  * [214] Ziwei Zhang, Peng Cui, and Wenwu Zhu.  Deep learning on graphs: A survey.  IEEE TKDE, 2020.
  * [215] Lingxiao Zhao and Leman Akoglu.  On using classification datasets to evaluate graph outlier detection: Peculiar observations and new insights.  Big Data, 2023.
  * [216] Wentao Zhao, Qitian Wu, Chenxiao Yang, and Junchi Yan.  Graphglow: Universal and generalizable structure learning for graph neural networks.  KDD, 2023.
  * [217] Xiangyu Zhao, Hanzhou Wu, and Xinpeng Zhang.  Watermarking graph neural networks by random graphs.  In ISDFS, 2021.
  * [218] Cheng Zheng, Bo Zong, Wei Cheng, Dongjin Song, Jingchao Ni, Wenchao Yu, Haifeng Chen, and Wei Wang.  Robust graph representation learning via neural sparsification.  In ICML, 2020.
  * [219] Yu Zheng, Ming Jin, Yixin Liu, Lianhua Chi, Khoa T Phan, and Yi-Ping Phoebe Chen.  Generative and contrastive self-supervised learning for graph anomaly detection.  IEEE TKDE, 2021.
  * [220] Mengxiao Zhu and Haogang Zhu.  Mixedad: A scalable algorithm for detecting mixed anomalies in attributed graphs.  In AAAI, 2020.
  * [221] Qi Zhu, Natalia Ponomareva, Jiawei Han, and Bryan Perozzi.  Shift-robust gnns: Overcoming the limitations of localized graph training data.  NeurIPS, 2021.
  * [222] Qi Zhu, Carl Yang, Yidan Xu, Haonan Wang, Chao Zhang, and Jiawei Han.  Transfer learning of graph neural networks with ego-graph information maximization.  NeurIPS, 2021.
  * [223] Wei Zhu, Jiebo Luo, and Andrew D White.  Federated learning of molecular properties with graph neural networks in a heterogeneous setting.  Patterns, 2022.
  * [224] Yongchun Zhu, Qiang Sheng, Juan Cao, Qiong Nan, Kai Shu, Minghui Wu, Jindong Wang, and Fuzhen Zhuang.  Memory-guided multi-view multi-domain fake news detection.  IEEE TKDE, 2022.
  * [225] Yun Zhu, Haizhou Shi, Zhenshuo Zhang, and Siliang Tang.  Mario: Model agnostic recipe for improving ood generalization of graph contrastive learning.  arXiv, 2023.
  * [226] Dingyi Zhuang, Shenhao Wang, Haris Koutsopoulos, and Jinhua Zhao.  Uncertainty quantification of sparse travel demand prediction with spatial-temporal graph neural networks.  In SIGKDD, 2022.
  * [227] Daniel Zügner, Amir Akbarnejad, and Stephan Günnemann.  Adversarial attacks on neural networks for graph data.  In SIGKDD, 2018.
  * [228] Daniel Zügner and Stephan Günnemann.  Adversarial attacks on graph neural networks via meta learning.  In ICLR, 2019.
  * [229] Daniel Zügner and Stephan Günnemann.  Certifiable robustness and robust training for graph convolutional networks.  In SIGKDD, 2019.

|  Song Wang is a Ph.D. student at the Department of Electrical and Computer
Engineering at the University of Virginia, advised by Professor Jundong Li.
Previously, he received a B.E. degree in Electronic Engineering from Tsinghua
University in 2020. His research interests include knowledge graphs and few-
shot learning on graphs. His works have been published in top conferences such
as IJCAI, NeurIPS, WSDM, SIGIR, and SIGKDD.
---|---
|  Yushun Dong is a Ph.D. candidate in the Department of Electrical and
Computer Engineering at the University of Virginia. He received a B.S. degree
in Telecommunications from Beijing University of Posts and Telecommunications
in 2019. His research interests are broadly in data mining and machine
learning, with a particular focus on graph learning. In the past few years,
his works have been published in top-tier venues including SIGKDD, WWW, and
AAAI.
---|---
|  Binchi Zhang is a Ph.D. student in the Department of Electrical and
Computer Engineering at the University of Virginia, advised by Professor
Jundong Li. He received his B.E. degree in Electrical Engineering at Xi’an
Jiaotong University in 2022. His research interests include graph mining,
trustworthy machine learning, and federated learning. His works have been
published in top venues including ICLR, SDM, and NeurIPS Datasets and
Benchmarks Track.
---|---
|  Zihan Chen is a Ph.D. student in the Department of Electrical and Computer
Engineering at the University of Virginia, advised by Professor Jundong Li and
Professor Cong Shen. Previously he received his bachelor’s degree in
mathematics and master’s degree in data science from the University of Science
and Technology of China. His research interests include federated learning,
graph learning, and trustworthy machine learning.
---|---
|  Xingbo Fu is a Ph.D. student in the Department of Electrical and Computer
Engineering at the University of Virginia. He received his B.S. degree and
M.S. degree in Automation from Xi’an Jiaotong University in 2017 and 2020,
respectively. His research interests include graph learning, federated
learning, and AI for healthcare. His recent studies have been published in
SIGKDD Explorations, ICHI, and SIGKDD.
---|---
|  Yinhan He is a Ph.D. student in the Department of Electrical and Computer
Engineering at the University of Virginia. He received the B.S. degree in
Mathematics and Applied Mathematics from University of Chinese Academy of
Sciences in 2022. His research interest is graph machine learning, explainable
AI, and machine learning for healthcare.
---|---
|  Cong Shen received the B.S. and M.S. degrees from the Department of
Electronic Engineering, Tsinghua University, China, in 2002 and 2004,
respectively, and the Ph.D. degree from the Electrical Engineering Department,
UCLA, in 2009. From 2009 to 2014, he worked for Qualcomm Research, San Diego,
CA, USA. From 2015 to 2019, he was a specially-appointed Professor with the
School of Information Science and Technology, University of Science and
Technology of China (USTC). He is currently an Assistant Professor with the
Charles L. Brown Department of Electrical and Computer Engineering, University
of Virginia. His research interests span a number of interdisciplinary areas
in wireless communications, networking, and machine learning. He currently
serves as an Editor for IEEE Transactions On Wireless Communications and IEEE
Wireless Communications Letters.
---|---
|  Chuxu Zhang is an Assistant Professor of Computer Science at the Brandeis
University. His research lies in the intersection of artificial intelligence,
graph machine learning, and societal applications. Recently, he has focused on
developing effective, efficient, safe, and generative machine learning models
and algorithms on graph and multi-modality data. Besides, he applies machine
learning to solve societal challenges in healthcare, social media, science,
and others. His research has led to over 100 papers in major AI venues such as
ICML, NeurIPS, ICLR, KDD, WWW, and EMNLP. He is the recipient of the NSF
CAREER Award (2024) and several best paper (candidate) awards including CIKM
2021 and WWW 2022.
---|---
|  Nitesh V. Chawla (Fellow, IEEE) received the PhD degree in computer science
and engineering from the University of South Florida, Tampa, Florida, in
2002\. He is the Frank M. Freimann professor of computer science and
engineering with the University of Notre Dame. He is the founding director of
the Lucy Family Institute for Data and Society. His research interests focus
on machine learning, data science, and network science, and is motivated by
the question of how technology can advance the common good through
interdisciplinary research.
---|---
|  Jundong Li is an Assistant Professor in the Charles L. Brown Department of
Electrical and Computer Engineering, with a joint appointment in the
Department of Computer Science, and the School of Data Science. He received
his Ph.D. degree in Computer Science at Arizona State University in 2019. His
research interests are in data mining, machine learning, and causal inference.
He has published over 150 articles in high-impact venues (e.g., KDD, WWW,
NeurIPS, ICLR, AAAI, IJCAI, WSDM, EMNLP, CSUR, TPAMI, TKDE, TKDD, and TIST).
He has won several prestigious awards, including NSF CAREER Award, KDD Best
Research Paper Award, JP Morgan Chase Faculty Research Award, Cisco Faculty
Research Award, among others.
---|---