# Growth of curvature and perimeter of temperature patches in the 2D
Boussinesq equations

Jaemin Park

###### Abstract.

In this paper, we construct an example of temperature patch solutions for the
two-dimensional, incompressible Boussinesq system with kinematic viscosity
such that both the curvature and perimeter grow to infinity over time. The
presented example consists of two disjoint, simply connected patches. The
rates of growth for both curvature and perimeter in this example are at least
algebraic.

Department of Mathematics and Computer Science, University of Basel,
Spiegelgasse 1, 4051 Basel, Switzerland<EMAIL_ADDRESS>
##  1\. Introduction and the main results

In this paper, we investigate the long-time behavior of the two-dimensional
incompressible Boussinesq equations in the absence of thermal diffusivity:

$\displaystyle\rho_{t}+u\cdot\nabla\rho$ $\displaystyle=0,$ (1.1)
$\displaystyle u_{t}+u\cdot\nabla u$ $\displaystyle=-\nabla p-\rho
e_{2}+\nu\Delta u,\quad t>0,\ x\in\mathbb{R}^{2}$ $\displaystyle\nabla\cdot u$
$\displaystyle=0,$ $\displaystyle(\rho(t,x),u(t,x))|_{t=0}$
$\displaystyle=(\rho_{0}(x),u_{0}(x)),$

where $e_{2}=(0,1)^{T}$ and $\nu>0$ is the kinematic viscosity coefficient.
The system (1.1) describes the evolution of the temperature distribution
$\rho$, of a viscous, heat-conducting fluid moving in an external
gravitational force field, assuming that the Boussinesq approximation is valid
[3, 13, 22, 25]. The primary goal of this paper is to provide an example of a
patch-type temperature distribution whose curvature and perimeter grow as time
approaches infinity.

###  1.1. Overview of long-time behavior in the Boussinesq equations

Before presenting the precise statement of the main theorem, let us provide a
brief review of the relevant literature.

####  1.1.1. Global well-poseness

In the presence of kinematic viscosity as in (1.1), Hou–Li [15] and Chae [5]
obtained global-in-time regularity results in $(\rho,u)\in H^{m}\times
H^{m-1}$ and $(\rho,u)\in H^{m}\times H^{m}$, respectively, with $m\geq 3$. In
essence, when the initial data are sufficiently smooth and decay rapidly,
these results ensure the existence of a global-in-time strong solution. For
cases with rough initial data, Abidi–Hmidi [1] and Hmidi–Keraani [14] proved
the existence of global weak solutions in $(\rho,u)\in
B^{0}_{2,1}\times\left(L^{2}\cap B^{-1}_{\infty,1}\right)$ and $(\rho,u)\in
L^{2}\times\left(L^{2}\cap H^{s}\right)$, where $s\in[0,2)$ and $B^{s}_{p,q}$
represents the Besov spaces.

####  1.1.2. Long-time behavior of classical solutions

In the study of long-time behavior of solutions, stability analysis, by
itself, provides qualitative information about their long-term behavior,
moreover it also plays a crucial role in proving various solution features, as
exploited in [6, 9]. Denoting

$\rho_{s}^{\alpha}:=\alpha y,\quad u_{s}^{\beta}:=(\beta
y,0)^{T},\quad\alpha,\beta\in\mathbb{R},$

one can easily see that any pair $(\rho_{s}^{\alpha},u_{s}^{\beta})$ is a
steady solution (time-independent) for the system (1.1). In [24], the authors
established stability under perturbations in a Gevrey class near the Couette
flow ($(\rho_{s}^{\alpha},u_{s}^{1})$ with $\alpha\leq 0$), when considering
the Boussinesq system in the spatial domain $\mathbb{T}\times\mathbb{R}$. Near
the hydrostatic equilibria ($\rho_{s}^{\alpha},0)$, under perturbations in a
Sobolev space, Doering–Wu–Zhao–Zheng [8] established stability (when
$\alpha<0$) and instability (when $\alpha>0$), considering the Boussinesq
system in a general Lipschitz domain. Also Tao–Wu–Zhao–Zheng [27] conducted
another stability analysis with relaxed assumptions on the initial data in a
spatially periodic domain.

Regarding a long-time behavior of solutions without an assumption on the
smallness of the initial data, several quantitative results are available in
the literature. Since the density $\rho$ is transported by an incompressible
flow, one cannot expect any growth or decay of $\rVert\rho(t)\rVert_{L^{p}}$.
However, a creation of small scale by the flow might induce the growth of
finer norms of $\rho$ (or vorticity $\omega:=\nabla\times u$) over time.
Indeed, considering (1.1) in a bounded domain, Ju [16] showed
$\rVert\rho\rVert_{H^{1}}\lesssim e^{ct^{2}}$, which was further improved to
an exponential bound $e^{ct}$ in $\mathbb{T}^{2}$ by Kukavica–Wang [21].
Subsequently, Kukavica–Massatt–Ziane [20] achieved a slightly better upper
bound $\rVert\rho\rVert_{H^{2}}\lesssim C_{\epsilon}e^{\epsilon t}$ for any
small $\epsilon>0$. In addition to these upper bounds on growth rates, an
interesting lower bound was obtained by Brandolese–Schonbek [4] proving that
in $\mathbb{R}^{2}$, the kinetic energy $\rVert u\rVert_{L^{2}}$ must grow
faster than $c(1+t)^{1/4}$ as $t\to\infty$, provided that the initial density
$\rho_{0}$ does not have a zero average. Recently, Kiselev–Park–Yao [18]
showed that for a large class of initial data, the Sobolev norms
$\rVert\rho\rVert_{H^{m}}$, for $m\geq 1$, must grow at least at some
algebraic rate in $\mathbb{T}^{2}$ and $\mathbb{R}^{2}$. Besides these
quantitative analyses of norm growth, one might expect some asymptotic
behavior due to the damping effect induced by viscosity. In this direction,
Kukavica–Massatt–Ziane [20] and Aydin–Kukavica–Ziane [2] showed that
$\rVert\nabla u\rVert_{L^{2}}$ and $\rVert\nu\Delta u-\mathbb{P}(\rho
e_{2})\rVert_{L^{2}}$, where $\mathbb{P}$ denotes the Leray projection,
converge to $0$ as $t\to\infty$ in a bounded domain.

####  1.1.3. Temperature patch problem

An interesting class of solutions to a transport equation,

$\displaystyle\rho_{t}+u\cdot\nabla\rho=0,$ (1.2)

is called patch solutions. These are weak solutions composed of characteristic
functions. For instance, if $\rho_{0}=1_{D}$ for some bounded domain $D$, the
solution remains as a characteristic function $\rho(t)=1_{D_{t}}$ for some
time-dependent domain $D_{t}$, provided that the velocity field $u$ is
suitably regular. In a more general sense, in this paper, we refer to a
solution $\rho$ as a patch solution, if it can be expressed as a linear
combination of characteristic functions defined on some bounded domains.

As transport phenomena are prevalent in fluid dynamics, the long-time behavior
of patch solutions has been a subject of active study in various fluid models,
particularly concerning the behavior of the patch boundary. In certain two-
dimensional models, it has been observed that the curvature of the patch
boundary may grow to infinity ([17] for the 2D Euler), the perimeter also may
grow to infinity ([6, 10, 9] for the 2D Euler) as $t\to\infty$, or even a
singularities can develop in a finite time ([19, 12] for the generalized
surface quasi-geostrophic equations).

In the context of the 2D Boussinesq equations (1.1), the temperature
distribution is also transported by the velocity field, making it natural to
explore patch solution $\rho$. The global existence of patch solutions have
been rigorously proved by Danchin–Zhang [7] and Gancedo–García-Juárez [11].
More precisely, [11, Theorem 3.1] states that if $\rho_{0}=1_{D_{0}}$ for a
simply connected domain $D_{0}\in\mathbb{R}^{2}$ with $\partial D_{0}\in
W^{2,\infty}$, and $u_{0}\in C^{\infty}_{c}(\mathbb{R}^{2})$ is divergence-
free, then

  * •

there is a unique global solution $(\rho(t),u(t))$ such that $u\in
L^{1}((0,T);W^{2,\infty}(\mathbb{R}^{2}))$ and $\rho(t)=1_{D_{t}}$, where
$D_{t}=X_{t}(D_{0})$ with $X_{t}$, a flow map associated to the velocity field
$u$. More precisely $X_{t}$ is the unique map determine by

$\displaystyle\frac{d{X}_{t}}{dt}(x)=u(t,X_{t}(x)),\quad X_{0}(x)=x,\text{ for
all $x\in\mathbb{R}^{2}$.}$ (1.3)

  * •

$\partial D_{t}\in L_{loc}^{\infty}(\mathbb{R}^{+};W^{2,\infty})$,

which ensures that the curvature cannot grow to infinity in a finite time.

Now, let us consider an initial data $(\rho_{0},u_{0})$ such that the initial
temperature density consists of multiple patches with smooth boundaries. The
existence of global weak solution is guaranteed. Indeed, for a set $D_{0}$ as
described in (A1), it is well-known that $1_{D_{0}}\in B^{\alpha}_{2,2}$ for
any $\alpha<\frac{1}{2}$ (e.g., [26, Proposition 3.6]), thus $\rho_{0}\in
B^{\alpha}_{2,2}$. In this case, classical embedding theorems in Besov spaces
(e.g., [28, Subsection 2.7]), yield that $\rho_{0}\in B^{0}_{2,1}\cap
B^{0}_{p,\infty}$ for $p\in(2,4)$. Combining this with $u_{0}\in
C_{c}^{\infty}(\mathbb{R}^{2})$, the well-posedness theorem [14, Theorem 1.2]
ensures that there exists a unique weak solution $(\rho,u)$ to (1.1) in the
class,

$\displaystyle(\rho,u)\in C(\mathbb{R}^{+};B^{0}_{2,1}\cap
B^{0}_{p,\infty})\times C(\mathbb{R}^{+};H^{2}).$ (1.4)

However, technically speaking, the global existence results for patch
solutions in [11] are not directly applicable to the initial data as above,
since $\rho_{0}$ consists of two disjoint patches instead of a single patch.
More precisely, it is not trivial to see whether the temperature distribution
$\rho$ in (1.4) qualifies as a patch solution, and if it does, whether
boundary regularity can persist; A rough velocity $u\in
C(\mathbb{R}^{+};H^{2})$ does not guarantee enough regularity of a flow map
$x\mapsto X_{t}(x)$ to ensure any regularity of the boundary of the set
$X_{t}(D_{0})$. Since our primary focus lies elsewhere and considering that
the main ideas from [11] can be readily applied to the proof, we will only
state a theorem concerning the global existence of patch solutions involving
multiple patches. The detailed proof is left to the interested reader.

######  Theorem 1.1.

[11, Theorem 3.1] Let $N\in\mathbb{N}$. For $i=1,\ldots,N$, let us pick real
numbers $a_{i}\in\mathbb{R}$ and simply connected bounded domains
$D_{i}\subset\mathbb{R}^{2}$ such that $\overline{D}_{i}$ are disjoint and
$\partial D_{i}\in W^{2,\infty}$. Let us consider initial data
$(\rho_{0},u_{0})$ such that $\rho_{0}=\sum_{i=1}^{N}a_{i}1_{D_{i}}$ and
$u_{0}\in H^{2}(\mathbb{R}^{2})$ is divergence-free. Then there exists a
unique weak solution $(\rho,u)$ to (1.1) such that $u\in
C(\mathbb{R}^{+};H^{2})\cap
L^{1}_{loc}(\mathbb{R}^{+};W^{2,\infty}(\mathbb{R}^{2}))$. In addition, for
almost every $t\geq 0$

$\rho(t)=\sum_{i=1}^{N}a_{i}1_{D_{i,t}},\quad D_{i,t}=X_{t}(D_{i}),$

where $X_{t}$ is the flow map generated by the velocity $u$. Lastly, we have
persistence of the curvature, $\partial D_{i,t}\in
L^{\infty}_{loc}(\mathbb{R}^{+};W^{2,\infty})$ for $i=1,\ldots,N$.

###  1.2. Main results

The goal of this paper is to construct an example of initial data with a
patch-type temperature distribution such that under the dynamics (1.1), the
temperature patch exhibits a growth of the curvature and the perimeter. To
this end, we will consider initial data $(\rho_{0},u_{0})$ satisfying the
following assumptions:

  1. (A1)

$\rho_{0}(x)=1_{D_{0}}(x)-1_{D_{0}^{*}}$ for a simply connected domain such
that $\overline{D_{0}}\subset\mathbb{R}\times\mathbb{R}^{+}$, $|D_{0}|=1$ and
$\partial D_{0}\in C^{\infty}$, and

$D_{0}^{*}=\left\\{x\in\mathbb{R}^{2}:(x_{1},-x_{2})\in D_{0}\right\\}.$

See Figure 1 for an illustration.

  2. (A2)

$u_{0}\in C_{c}^{\infty}(\mathbb{R}^{2})$ and $u_{0}$ is divergence-free.
Moreover, denoting $u_{0}=(u_{01},u_{02})$, we assume that $u_{02}$ is odd in
$x_{2}$ and $u_{01}$ is even in $x_{2}$.

Figure 1. Illustration of the initial patch $\rho_{0}$

When considering $(\rho_{0},u_{0})$ satisfying (A1) and (A2), the uniqueness
part of Theorem 1.1 ensures the preservation of the $x_{2}$-odd symmetry in
the solution $\rho(t)$ obtained in Theorem 1.1:
$\rho(t,x_{1},-x_{2})=-\rho(t,x_{1},x_{2})$. In other words, $\rho(t)$ takes
the form

$\rho(t)=1_{D_{t}}-1_{D_{t}^{*}},\text{ where $D_{t}=X_{t}(D_{0})$ and
$D_{t}^{*}=\left\\{x\in\mathbb{R}^{2}:(x_{1},-x_{2})\in D_{t}\right\\}.$}$

Now, we are ready to state the paper’s main theorem:

######  Theorem 1.2.

Let $(\rho_{0},u_{0})$ satisfy (A1) and (A2). Then, the global patch-type
solution $\rho(t)=1_{D_{t}}-1_{D_{t}^{*}}$, where $D_{t}^{*}$ denotes the
$x_{2}$ symmetric copy of $D_{t}$, satisfies the following:

  1. (a)

Infinite growth of curvature: We have

$\limsup_{t\to\infty}t^{-\frac{1}{6}}|\kappa(t)|=\infty,$

where $\kappa(t)$ is the maximum curvature of $\partial D_{t}$.

  2. (b)

Infinite growth of perimeter: Denoting $L_{t}$ be the distance between a far-
left and a far-right point on $\partial D_{t}$, we have

$\limsup_{t\to\infty}t^{-\frac{1}{6}}L_{t}=\infty.$

Since $D_{t}$ is simply connected, we have infinite growth of perimeter.

##  2\. prelimineries

In this section, we collect several well-known conserved properties and some
useful uniform estimates for the solutions.

Due to the incompressibility of the flow, the conservation $L^{p}$ norms of
the density follows immediately,

$\displaystyle\rVert\rho(t)\rVert_{L^{p}}=\rVert\rho_{0}\rVert_{L^{p}},\text{
for all $p\in[1,\infty]$.}$ (2.1)

Another well-known conserved quantity is the total energy of the system. We
define the potential energy $E_{P}(t)$ and the kinetic energy $E_{K}(t)$ as
follows:

$\displaystyle E_{P}(t):=\int_{\mathbb{R}^{2}}\rho(t,x)x_{2}dx,\quad
E_{K}(t):=\frac{1}{2}\int_{\mathbb{R}^{2}}|u(t,x)|^{2}dx.$

Then it follows straightforwardly from (1.1) that

$\displaystyle\frac{d}{dt}E_{P}(t)$
$\displaystyle=\int_{\mathbb{R}^{2}}\rho_{t}x_{2}dx=\int_{\mathbb{R}^{2}}-u\cdot\nabla\rho
x_{2}dx=\int_{\mathbb{R}^{2}}\rho u_{2}dx,$ (2.2)
$\displaystyle\frac{d}{dt}E_{K}(t)$ $\displaystyle=\int_{\mathbb{R}^{2}}u\cdot
u_{t}dx=\int_{\mathbb{R}^{2}}-\rho e_{2}\cdot u+\nu\Delta u\cdot
udx=\int_{\mathbb{R}^{2}}-\rho u_{2}dx-\nu\int_{\mathbb{R}^{2}}|\nabla
u|^{2}dx.$ (2.3)

By summing up the above two quantities and integrating over time, we obtain

$\displaystyle E_{P}(0)+E_{K}(0)=E_{P}(t)+E_{K}(t)+\nu\int_{0}^{t}\rVert\nabla
u(t)\rVert_{L^{2}}^{2}dt=:E_{T}(t)+\nu\int_{0}^{t}\rVert\nabla
u(t)\rVert_{L^{2}}^{2}dt.$ (2.4)

Under the assumptions (A1) and (A2), both $E_{P}(t)$ and $E_{K}(t)$ are always
nonnegative. Thus the above energy equality gives us a uniform bound for the
vorticity $\omega:=\nabla\times u$,

$\displaystyle\int_{0}^{t}\rVert\omega(t)\rVert_{L^{2}}^{2}dt\leq\int_{0}^{t}\rVert\nabla
u(t)\rVert_{L^{2}}^{2}dt\leq C(\rho_{0},u_{0})(1+{\nu^{-1}})\text{ for all
$t>0$}.$ (2.5)

##  3\. Uniform in time estimates

In this section, let us derive another uniform time estimate that is simple
but will play a crucial role in proving the main theorem. Roughly speaking,
the estimate in (2.5) tells us that time-averaged vorticity dissipates
eventually, for instance,
$\frac{1}{T}\int_{T}^{2T}\rVert\omega(t)\rVert_{L^{2}}^{2}dt\to 0$, as
$T\to\infty$. Now, let us consider the vorticity equation,

$\displaystyle\omega_{t}+u\cdot\nabla\omega=-\partial_{1}\rho+\nu\Delta\omega,$
(3.1)

which can be easily derived by taking the curl operator in the second equation
of (1.1). Assuming $\omega$ is sufficiently small in some sense, the quadratic
term $u\cdot\nabla\omega$ is comparatively less dominant (in a weak sense)
when compared to the linear terms in (3.1). Consequently, considering the
dissipation of vorticity, we may anticipate a convergence towards zero for the
quantity $-\partial_{1}\rho+\nu\Delta\omega$. Establishing such asymptotic
behavior in a rigorous sense may be nontrivial. However, in the next lemma, we
will derive a uniform estimate which exhibits a convergence towards zero of a
time-averaged quantity of $-\partial_{1}\rho+\nu\Delta\omega$.

######  Lemma 3.1.

Let $(\rho,u)$ be a solution to (1.1) with initial data $(\rho_{0},u_{0})$
satisfying (A1) and (A2). Then,

$\displaystyle\int_{0}^{t}\rVert\partial_{1}\Delta^{-1}\rho-\nu\omega\rVert_{\dot{H}^{1}}^{2}dt\leq(1+\nu^{-1})C(\rho_{0},u_{0}),\text{
for all $t>0$.}$ (3.2)

######  Remark 3.2.

When considering (1.1) in a bounded domain, in [20, 2], it was shown that for
a general initial data, $\rVert\nu\Delta u-\mathbb{P}(\rho
e_{2})\rVert_{L^{2}}$ converges to $0$ as $t\to\infty$, where $\mathbb{P}$ is
the Leray projection, which is equivalent to
$\rVert\nu\omega-\partial_{1}\Delta^{-1}\rho\rVert_{\dot{H}^{1}}\to 0$. When
considering the Boussinesq system in an unbounded domain, obtaining such a
result for general initial data becomes nontrivial. The challenge arises from
the fact that while the proof provided in [20, 2] relies on a uniform bound of
the kinetic energy, the kinetic energy in an unbounded domain may not be
uniformly bounded in time in general. Although the total energy is inferred to
be bounded from (2.4), it remains possible for the kinetic energy to increase
indefinitely throughout the evolution without a lower bound of the potential
energy. In our case, this issue is overcome by the assumptions (A1) and (A2),
which ensure a uniform lower bound of the potential energy.

###### Proof.

The proof relies on the second derivative of the potential energy. To begin,
we recall the expression of $E^{\prime\prime}_{P}(t)$ from [18]:

######  Lemma 3.3.

[18, Lemma 2.1, Lemma 2.3] Let $(\rho,u)$ be a solution to (1.1) with initial
data $(\rho_{0},u_{0})$ satisfying (A1) and (A2). Then the potential energy
$E_{P}(t)$ satisfies

$E_{P}^{\prime\prime}(t)=A(t)+B(t)-\int|\nabla\partial_{1}\Delta^{-1}\rho(t)|^{2}dx\quad\text{
for all }t\geq 0,$ (3.3)

where

$A(t):=\sum_{i,j=1}^{2}\int_{\Omega}((-\Delta)^{-1}\partial_{2}\rho)\partial_{i}u_{j}\partial_{j}u_{i}\,dx,~{}~{}\text{and}~{}~{}B(t):=\nu\int_{\Omega}\rho\Delta
u_{2}dx.$ (3.4)

Furthermore, $A(t)$ satisfies

$\displaystyle\int_{0}^{t}|A(t)|dt\leq C(\rho_{0})\int_{0}^{t}\rVert\nabla
u(t)\rVert_{L^{2}}^{2}dt,\text{ for all $t>0$.}$ (3.5)

Towards the proof of (3.2), we rewrite $B(t)$, using the Biot-Savart law
($u_{2}=\partial_{1}\Delta^{-1}\omega$), as

$B(t)=\int\rho\Delta(\partial_{1}\Delta^{-1}\omega)dx=-\int\partial_{1}\rho\omega
dx.$

Therefore (3.3) can be also rewritten as

$\displaystyle E_{P}^{\prime\prime}(t)=A(t)-\nu\int\partial_{1}\rho\omega
dx-\int|\nabla\partial_{1}\Delta^{-1}\rho|^{2}dx.$ (3.6)

From the vorticity equation (3.1), we compute

$\displaystyle\nu\frac{d}{dt}\left(\frac{1}{2}\rVert\omega(t)\rVert_{L^{2}}^{2}\right)$
$\displaystyle=\nu\int\omega\omega_{t}dx=\nu\int\omega(-\partial_{1}\rho+\nu\Delta\omega)dx=-\int\nu\omega\partial_{1}\rho
dx-\int\nu^{2}|\nabla\omega|^{2}dx.$ (3.7)

Combining this with (3.6), we obtain

$\displaystyle\frac{d}{dt}\left(E^{\prime}_{P}(t)+\frac{\nu}{2}\rVert\omega(t)\rVert_{L^{2}}^{2}\right)$
$\displaystyle=A(t)-2\int\nu\omega\partial_{1}\rho
dx-\int|\nabla\partial_{1}\Delta^{-1}\rho|^{2}dx-\int\nu^{2}|\nabla\omega|^{2}dx$
$\displaystyle=A(t)-2\int\nabla\nu\omega\cdot\nabla\partial_{1}\Delta^{-1}\rho
dx-\int|\nabla\partial_{1}\Delta^{-1}\rho|^{2}dx-\int\nu^{2}|\nabla\omega|^{2}dx$
$\displaystyle=A(t)-\int|\nabla(\partial_{1}\Delta^{-1}\rho-\nu\omega)|^{2}dx.$

Thus integrating this over time, we obtain

$\displaystyle
E^{\prime}_{P}(t)+\frac{\nu}{2}\rVert\omega(t)\rVert_{L^{2}}^{2}+\int_{0}^{t}\rVert\partial_{1}\Delta^{-1}\rho-\nu\omega\rVert_{\dot{H}^{1}}^{2}dt=E_{P}^{\prime}(0)+\frac{\nu}{2}\rVert\omega_{0}\rVert_{L^{2}}^{2}+\int_{0}^{t}A(t)dt,\text{
for all $t>0$.}$ (3.8)

Finally, sending $E^{\prime}_{P}(t)$ on the left-hand side to the other side,

$\int_{0}^{t}\rVert\partial_{1}\Delta^{-1}\rho-\nu\omega\rVert_{\dot{H}^{1}}^{2}dt\leq
C(\rho_{0},u_{0})+\int_{0}^{t}A(t)dt-
E_{P}^{\prime}(t)\leq(1+\nu^{-1})C(\rho_{0},u_{0})-E_{P}^{\prime}(t),$

where the last inequality follows from (2.5) and (3.5). Recalling
$E_{P}^{\prime}(t)=\int_{\mathbb{R}^{2}}\rho u_{2}dx$ from (2.2), and using
the Cauchy-Schwarz inequality, we have $|E_{P}^{\prime}(t)|\leq
C\rVert\rho(t)\rVert_{L^{2}}\rVert u\rVert_{L^{2}}\leq C(\rho_{0},u_{0})$.
This gives the desired estimate (3.2). ∎

######  Remark 3.4.

In the case $\Omega=\mathbb{T}^{2}$ (or in a suitable bounded domains), the
Sobolev inequality allows us to derive a more concise estimate:

$\displaystyle\int_{0}^{t}\rVert\partial_{1}\rho\rVert_{\dot{H}^{-2}}^{2}dt\leq
C(\rho_{0},u_{0})(1+\nu^{-1}).$ (3.9)

Indeed, the triangular inequality and the Sobolev inequality give

$\rVert\partial_{1}\Delta^{-1}\rho\rVert_{L^{2}(\mathbb{T}^{2})}^{2}\leq\rVert\nu\omega\rVert_{L^{2}(\mathbb{T}^{2})}^{2}+\rVert\nu\omega-\partial_{1}\Delta^{-1}\rho\rVert_{L^{2}(\mathbb{T}^{2})}^{2}\leq\rVert\nu\omega\rVert_{L^{2}(\mathbb{T}^{2})}^{2}+\rVert\nu\omega-\partial_{1}\Delta^{-1}\rho\rVert_{\dot{H}^{1}(\mathbb{T}^{2})}^{2}.$

Thus, integrating over time and combining it with (3.2) and (2.5), we obtain
(3.9).

##  4\. Lemmas for curvature and perimeter

In this section, we study relations between the curvature/perimeter of a patch
and the (negative) Sobolev norms. Throughout the section, $D$ is always
assumed to be a simply connected bounded domain such that
$\overline{D}\subset\mathbb{R}\times\mathbb{R}^{+}$, $|D|=1$ and $\partial
D\in C^{\infty}$. Also, we will denote the disk centered at
$x\in\mathbb{R}^{2}$ by $B_{r}(x)\subset\mathbb{R}^{2}$. A constant $C>0$ will
denote a universal constant that does not depend on any variables, while it
might vary from line to line.

Let us recall the Pestov–Ionin theorem, which asserts that every simple-closed
curve with a curvature of at most one encloses a unit disk. In other words, it
holds that

$\displaystyle\sup\left\\{r:B_{r}(x)\subset D,\text{ for some
$x\in\mathbb{R}^{2}$}\right\\}\leq\frac{1}{\max_{x\in\partial D}|\kappa(x)|},$
(4.1)

where $\kappa(x)$ is the signed curvature at $x\in\partial D$. In the next
lemma, we explore how the radius of a maximal disk within $D$ can be
constrained in terms of the negative Sobolev norms of $1_{D}$.

######  Lemma 4.1.

Suppose $D$ contains a disk with radius $r>0$. Then there exists a universal
constant $C>0$ such that for any $\Omega\in C^{\infty}_{c}(\mathbb{R}^{2})$,

$\displaystyle r^{3}\leq
C\left(\rVert\partial_{1}\Delta^{-1}(1_{D}-1_{D^{*}})-\Omega\rVert_{\dot{H}^{1}(\mathbb{R}^{2})}+\rVert\Omega\rVert_{L^{2}(\mathbb{R}^{2})}\right),$
(4.2)

where
$D^{*}:=\left\\{(x_{1},x_{2})\subset\mathbb{R}\times\mathbb{R}^{-}:(x_{1},-x_{2})\in
D\right\\}$.

######  Remark 4.2.

Since $1_{D}-1_{D^{*}}\in L^{2}(\mathbb{R}^{2})$ and it has a zero average, we
have $\partial_{1}\Delta^{-1}(1_{D}-1_{D^{*}})\in L^{2}(\mathbb{R}^{2})$ [23,
Proposition3.3]. Therefore, if we simply take
$\Omega:=\partial_{1}\Delta^{-1}(1_{D}-1_{D^{*}})$, then Lemma 4.1 tells us
that the radius $r$ of a maximal disk contained in $D$ must satisfy

$\displaystyle
r^{3}\leq\rVert\partial_{1}\Delta^{-1}(1_{D}-1_{D^{*}})\rVert_{L^{2}(\mathbb{R}^{2})}.$

In our proof of the main theorem, we do not have any smallness of
$\rVert\partial_{1}\Delta^{-1}\rho(t)\rVert_{L^{2}(\mathbb{R}^{2})}$,
therefore, we will make a use of carefully chosen $\Omega$ in the application
of the lemma.

###### Proof.

Let $B_{0}$ be a disk contained in $D$. Without loss of generality, we assume
that the center of $B_{0}$ lies on the $x_{2}$-axis, that is,
$B_{0}=B_{r}((0,b))$ for some $b>0$. Clearly, we must have $r<1$, due to the
assumption $|D|=1$. Next, let us consider a sequence of horizontally
translated disks
$B_{n}:=\left\\{(x_{1},x_{2})\in\mathbb{R}^{2}:(x_{1}-2rn,x_{2})\in
B_{r}(b)\right\\}$ for $n\in\mathbb{N}$ and choose
$N^{*}:=\inf\left\\{n\in\mathbb{N}:|D\cap
B_{n}|\leq\frac{r^{2}}{16}\right\\}.$ We claim that

$\displaystyle|D\cap B_{N^{*}}|\leq\frac{r^{2}}{16}\text{ and
}N^{*}\leq\frac{32}{r^{2}}.$ (4.3)

The first statement is clear by the definition of $N^{*}$. To see the upper
bound of $N^{*}$, let us suppose, to the contrary, that
$N^{*}>\frac{32}{r^{2}}$ and denote $n^{*}$:=$\lfloor\frac{32}{r^{2}}\rfloor$,
where $\lfloor a\rfloor$ denotes the largest integer not exceeding $a$. Since
$r<1$, we have $n^{*}>\frac{16}{r^{2}}$. This implies that $|D\cap
B_{n}|\geq\frac{r^{2}}{16}$ for all $n=1,...n^{*}$. However, in this case, we
must have

$1=|D|\geq\sum_{n=1}^{n^{*}}|D\cap B_{n}|\geq n^{*}\frac{r^{2}}{16}>1,$

which is a contradiction.

Towards the proof of the lemma, we define a function $x_{1}\mapsto g(x_{1})$
and $x_{2}\mapsto h(x_{2})$ as

$\displaystyle g(x_{1}):=\begin{cases}0,&\text{ if $x_{1}\leq 0$},\\\
\frac{1-\cos(\pi x_{1}/r)}{2}&\text{ if $x_{1}\in(0,r]$},\\\ 1&\text{ if
$x_{1}\in(r,2rN^{*}]$},\\\ \frac{1+\cos(\pi(x_{1}-2rN^{*})/r)}{2}&\text{ if
$x_{1}\in(2rN^{*},2rN^{*}+r]$},\\\ 0&\text{ if $x_{1}>2rN^{*}+r$},\end{cases}$

and

$\displaystyle
h(x_{2}):=\begin{cases}\frac{1+\cos(\frac{\pi(x_{2}-b)}{r})}{2}&\text{ if
$x_{2}\in(b-r,b+r)$},\\\ 0&\text{ otherwise}.\end{cases}$

And we define $f=f(x_{1},x_{2})$ as

$\displaystyle\begin{cases}f(x_{1},x_{2}):=g(x_{1})h(x_{2}),&\text{ if
$x_{2}\geq 0$,}\\\ f(x_{1},x_{2}):=f(x_{1},-x_{2}),&\text{ if
$x_{2}<0$}.\end{cases}$

From the properties of $g$ and $h$, it is clear that the support of $f$ is
contained in the vertical strip bounded by
$\left\\{x_{1}=0\right\\}\cup\left\\{x_{1}=2rN^{*}+r\right\\}$ whose width is
$2rN^{*}+r\leq\frac{C}{r}$ (see (4.3)). At the same time, the support of $f$
in $\mathbb{R}\times\mathbb{R}^{+}$ lies in the horizontal strip whose width
is less than $2r$. Consequently, we have

$\displaystyle|\text{supp}(f)|,\ |\text{supp}(\nabla f)|,\ |\text{supp}(\Delta
f)|\leq C.$ (4.4)

Now, denoting $\mu=1_{D}-1_{D^{*}}$, we see that for any $\Omega\in
C^{\infty}_{c}(\mathbb{R}^{2})$,

$\displaystyle{\int_{\mathbb{R}^{2}}\mu(x)\partial_{1}f(x)dx}$
$\displaystyle={\int_{\mathbb{R}^{2}}\mu(x)\Delta^{-1}\Delta\partial_{1}f(x)dx}=-{\int_{\mathbb{R}^{2}}\partial_{1}\Delta^{-1}\mu(x)\Delta
f(x)dx}$
$\displaystyle=\int_{\mathbb{R}^{2}}\left(\partial_{1}\Delta^{-1}\mu(x)-\Omega(x)\right)\Delta
f(x)dx+\int_{\mathbb{R}^{2}}\Omega(x)\Delta f(x)dx$
$\displaystyle\leq\rVert\partial_{1}\Delta^{-1}\mu-\Omega\rVert_{\dot{H}^{1}}\rVert\nabla
f\rVert_{L^{2}}+\rVert\Omega\rVert_{L^{2}}\rVert\Delta f\rVert_{L^{2}},$ (4.5)

where we used the integration by parts and the Cauchy-Schwarz inequality to
get the last inequality.

We will estimate the left/right-hand side of the inequality (4) separately. To
get a lower bound of the left-hand side, we notice that $\mu$ and
$\partial_{1}f$ are both odd in $x_{2}$, therefore,

$\displaystyle\int_{\mathbb{R}^{2}}\mu(x)\partial_{1}f(x)dx$
$\displaystyle=2\int_{\mathbb{R}\times\mathbb{R}^{+}}\mu(x)\partial_{1}f(x)dx=2\int_{D}\partial_{1}f(x)dx$
$\displaystyle=2\int_{D\cap B_{0}}\partial_{1}f(x)dx+2\int_{D\cap
B_{N^{*}}}\partial_{1}f(x)dx,$ (4.6)

where the last equality follows from the fact that, by the definition of $f$
(especially the definition of $g$),
$\partial_{1}f(x_{1},x_{2})=g^{\prime}(x_{1})h(x_{2})=0$ if $x\in(B_{0}\cup
B_{N^{*}})^{c}$. Using $B_{0}\subset D$ and the definition of $g,h$, we can
estimate the first integral as

$\displaystyle\int_{D\cap B_{0}}\partial_{1}f(x)dx$
$\displaystyle=\int_{B_{0}}g^{\prime}(x_{1})h(x_{2})dx\geq\int_{B_{0},\
\left\\{\frac{r}{4}<x_{1}<\frac{3r}{4}\right\\}}g^{\prime}(x_{1})h(x_{2})dx$
$\displaystyle\geq\int_{\frac{r}{4}}^{\frac{3r}{4}}\int_{b-\frac{\sqrt{7}}{4}r}^{b+\frac{\sqrt{7}}{4}r}g^{\prime}(x_{1})h(x_{2})dx_{2}dx_{1}$
$\displaystyle\geq\int_{\frac{r}{4}}^{\frac{3r}{4}}\int_{-\frac{\sqrt{7}}{4}r}^{\frac{\sqrt{7}}{4}r}\frac{\pi}{2r}\sin\left(\frac{\pi
x_{1}}{r}\right)\frac{(1+\cos(\pi x_{2}/r))}{2}dx_{2}dx_{1}$
$\displaystyle=\pi r\int_{1/4}^{1/2}\int_{0}^{\frac{\sqrt{7}}{4}}\sin(\pi
x_{1})(1+\cos(\pi x_{2}))dx_{2}dx_{1}$ $\displaystyle=\pi
r\int_{1/4}^{1/2}\sin(\pi x_{1})dx_{1}\int_{0}^{\sqrt{7/}4}(1+\cos(\pi
x_{2}))dx_{2}$ $\displaystyle\geq\pi
r\frac{\sqrt{2}}{2}\frac{\sqrt{7}}{4}\geq\pi r\frac{\sqrt{14}}{8}.$

By using $|\partial_{1}f|_{L^{\infty}}\leq\frac{\pi}{2r}$, the second integral
can be estimated as

$\int_{D\cap B_{N^{*}}}\partial_{1}f(x)dx\leq|D\cap
B_{N^{*}}|\frac{\pi}{2r}\leq r^{2}/16\cdot\frac{\pi}{2r}=\frac{\pi r}{32},$

where the second inequality follows from (4.3). Thus, in (4), we see that

$\displaystyle\int_{\mathbb{R}^{2}}\mu(x)\partial_{1}f(x)dx\geq
2\left(\frac{\pi r\sqrt{14}}{8}-\frac{\pi r}{32}\right)\geq Cr.$ (4.7)

Let us estimate the right-hand side of (4) Again, using the properties of
$g,h$, we have that $\rVert\Delta f\rVert_{L^{\infty}}\leq\rVert
g^{\prime\prime}\rVert_{L^{\infty}}+\rVert
h^{\prime\prime}\rVert_{L^{\infty}}\leq\frac{C}{r^{2}}$ and $\rVert\nabla
f\rVert_{L^{\infty}}\leq\rVert g^{\prime}\rVert_{L^{\infty}}+\rVert
h^{\prime}\rVert_{L^{\infty}}\leq\frac{C}{r}$. Combining this with (4.4), we
get

$\displaystyle\rVert\Delta f\rVert_{L^{2}}\leq\frac{C}{r^{2}},\text{ and
}\rVert\nabla f\rVert_{L^{2}}\leq\frac{C}{r}\leq\frac{C}{r^{2}},$ (4.8)

where the last inequality follows from $r<1$. Plugging this and (4.7) into
(4), we obtain the desired estimate (4.2). ∎

Now, we make a lemma to estimate the parameter of the domain $D$.

######  Lemma 4.3.

Let $L>0$ be the distance between a far-left and a far-right points on
$\partial D$, and let $A:=\int_{\mathbb{R}^{2}}1_{D}(x)x_{2}dx$. Then, there
exists a universal constant $C>0$ such that for any $\Omega\in
C^{\infty}_{c}(\mathbb{R}^{2})$,

$\displaystyle 1\leq
C(A+1)(1+L^{3})\left(\rVert\partial_{1}\Delta^{-1}(1_{D}-1_{D^{*}})-\Omega\rVert_{\dot{H}^{1}}+\rVert\Omega\rVert_{L^{2}}\right),$
(4.9)

where
$D^{*}:=\left\\{(x_{1},x_{2})\subset\mathbb{R}\times\mathbb{R}^{-}:(x_{1},-x_{2})\in
D\right\\}$.

######  Remark 4.4.

As explained in Remark 4.2, if we simply choose
$\Omega=\partial_{1}\Delta^{-1}(1_{D}-1_{D^{*}})$ in (4.9), then we get

$1\leq
C(A+1)(1+L^{3})\rVert\partial_{1}\Delta^{-1}(1_{D}-1_{D^{*}})\rVert_{L^{2}}.$

This inequality indeed tells us that assuming
$\int_{\mathbb{R}^{2}}1_{D}(x)x_{2}dx$ is bounded, the perimeter of $\partial
D$ grows to infinity, as
$\rVert\partial_{1}(1_{D}-1_{D^{*}})\rVert_{\dot{H}^{-2}}$ goes to zero. In
our proof of the main theorem, we will use the slightly finer estimate stated
in the lemma, due to the lack of smallness of
$\rVert\partial_{1}\rho(t)\rVert_{\dot{H}^{-2}}$.

###### Proof.

Without loss of generality, let us assume that
$\inf\left\\{x_{1}:(x_{1},x_{2})\in D\right\\}=0$ so that a far-right point of
$\partial D$ can be denoted by $(L,x_{2}^{r})$ for some $x_{2}^{r}>0$. Using
that $|D|=1$, we see

$\displaystyle\int_{\mathbb{R}^{2},\ \left\\{x_{2}>4A\right\\}}1_{D}dx$
$\displaystyle\leq\frac{1}{4A}\int_{\mathbb{R}^{2},\
\left\\{x_{2}>4A\right\\}}1_{D}(x)x_{2}dx\leq\frac{1}{4},$ (4.10)
$\displaystyle\int_{\mathbb{R}^{2},\
\left\\{x_{2}<\frac{1}{4L}\right\\}}1_{D}dx$
$\displaystyle=\int_{-\infty}^{\infty}\int_{0}^{\frac{1}{4L}}1_{D}(x)dx\leq\frac{1}{4}.$
(4.11)

This implies

$\displaystyle 4A>\frac{1}{4L}.$ (4.12)

Indeed, if it were not true, we would have

$1=|D|=\int_{\mathbb{R}^{2}}1_{D}dx\leq\int_{\mathbb{R}^{2},\
\left\\{x_{2}>4A\right\\}}1_{D}dx+\int_{\mathbb{R}^{2},\
\left\\{x_{2}<\frac{1}{4L}\right\\}}1_{D}dx\leq\frac{1}{2},$

which is a contradiction. Moreover, the above estimates (4.10) and (4.11)
imply that at least a half of $D$ is contained in the horizontal strip bounded
by $\left\\{x_{2}=4A\right\\}$ and $\left\\{x_{2}=\frac{1}{4L}\right\\}$.
Therefore, taking away $\left\\{0\leq x_{1}\leq\frac{1}{32A},\ 0\leq x_{2}\leq
4A\right\\}\cup\left\\{L-\frac{1}{32A}\leq x_{1}\leq L,\ 0\leq x_{2}\leq
4A\right\\}$ from $D$, whose total measure is at most $\frac{1}{4}$, we have
that

$\displaystyle\left|\left\\{x\in D:\frac{1}{32A}\leq x_{1}\leq
L-\frac{1}{32A},\quad\frac{1}{4L}\leq x_{2}\leq
4A\right\\}\right|\geq\frac{1}{4}.$ (4.13)

Towards the proof of the lemma, we choose nonnegative smooth functions
$g(x_{1})$ and $h(x_{2})$ satisfying

$\displaystyle\begin{cases}\text{$g(x_{1})=0$ if $x_{1}\leq 0$},\\\
\text{$0\leq g^{\prime}(x_{1})\leq 1$ for $x_{1}\in(0,L)$ and
$g^{\prime}(x_{1})=1$ for $x_{1}\in(\frac{1}{32A},L-\frac{1}{32A})$},\\\
\text{$g(x_{1})$ is symmetric about the axis $\left\\{x_{1}=L\right\\}$, that
is, $g(x_{1})=g(2L-x_{1})$ for $x_{1}\geq L$},\\\ \text{$0\leq g(x_{1})\leq
L$, $|g^{\prime}(x_{1})|\leq 1$ and $|g^{\prime\prime}(x_{1})|\leq 32A$ for
all $x_{1}\in\mathbb{R}$,}\end{cases}$

and

$\displaystyle\begin{cases}\text{$h(x_{2})=0$ for $x_{2}\leq 0$ or $x_{2}\geq
4A+\frac{1}{4L}$},\\\ \text{$h(x_{2})=1$ for $x_{2}\in(\frac{1}{4L},4A)$},\\\
\text{$0\leq h(x_{2})\leq 1$, $|h^{\prime}(x_{2})|\leq 4L$ and
$|h^{\prime\prime}(x_{2})|\leq 32L^{2}$ for all
$x_{2}\in\mathbb{R}$.}\end{cases}$

A construction of $g,h$ satisfying above properties is straightforward. For
such $g,h$, we define

$\displaystyle f(x_{1},x_{2}):=g(x_{1})h(x_{2})\text{ for $x_{2}\geq 0$ and
}f(x_{1},x_{2})=-f(x_{1},-x_{2}),\text{ for $x_{2}<0$.}$

Clearly, $|\text{supp}(f)|\leq C(AL+1)$, thus the above properties of $g,h$
give us that

$\displaystyle|\text{supp}(\nabla f)|\leq C(AL+1)\leq CAL,$ (4.14)

where the last inequality is due to (4.12). Furthermore, noticing that
$g(x_{1})$ is linear for $x_{1}\in(\frac{1}{32A},L-\frac{1}{32A})$, it is not
difficult to see that

$\displaystyle|\text{supp}(\Delta f)|\leq C.$ (4.15)

Next, denoting $\mu:=1_{D}-1_{D^{*}}$ and following the same computations in
(4), we get

$\displaystyle{\int_{\mathbb{R}^{2}}\mu(x)\partial_{1}f(x)dx}\leq\rVert\partial_{1}\Delta^{-1}\mu-\Omega\rVert_{\dot{H}^{1}}\rVert\nabla
f\rVert_{L^{2}}+\rVert\Omega\rVert_{L^{2}}\rVert\Delta f\rVert_{L^{2}},\text{
for any $\Omega\in C^{\infty}_{c}(\mathbb{R}^{2})$}.$ (4.16)

Using $x_{2}$-odd symmetry of $\mu$ and $f$, we see that
$\int_{\mathbb{R}^{2}}\mu(x)\partial_{1}f(x)dx=2\int_{D}g^{\prime}(x_{1})h(x_{2})dx$,
while the properties of $g,h$ give us that

$\displaystyle\int_{D}g^{\prime}(x_{1})h(x_{2})dx\geq\int_{\frac{1}{32A}}^{L-\frac{1}{32A}}\int_{\frac{1}{4L}}^{4A}1_{D}(x_{1},x_{2})dx_{2}dx_{1}\geq\frac{1}{4},$

where the last inequality follows from (4.13). Thus, we have

$\displaystyle\int_{\mathbb{R}^{2}}\mu(x)\partial_{1}f(x)dx\geq\frac{1}{2}.$
(4.17)

On the other hand, we have that

$\rVert\Delta f\rVert_{L^{\infty}}\leq\rVert
g^{\prime\prime}\rVert_{L^{\infty}}+\rVert
h^{\prime\prime}\rVert_{L^{\infty}}\rVert g\rVert_{L^{\infty}}\leq
32A+32L^{3},\quad\rVert\nabla f\rVert_{L^{\infty}}\leq\rVert
g^{\prime}\rVert_{L^{\infty}}+\rVert h^{\prime}\rVert_{L^{\infty}}\rVert
g\rVert_{L^{\infty}}\leq 1+4L^{2}.$

Combining this with (4.14) and (4.15), we get

$\rVert\Delta f\rVert_{L^{2}}\leq C(A+L^{3})\leq
C(A+1)(1+L^{3}),\quad\rVert\nabla f\rVert_{L^{2}}\leq C(AL^{3}+AL)\leq
C(A+1)(1+L^{3}).$

Plugging this and (4.17) into (4.16), we obtain the desired estimate (4.9). ∎

##  5\. Proof of the main theorem

In this section, we prove the main theorem of the paper. Let
$\rho(t)=1_{D_{t}}-1_{D_{t}^{*}}$ be the global solution with the initial data
$(\rho_{0},u_{0})$ satisfying the assumptions (A1) and (A2). In the rest of
the proof, $C$ denotes some positive constant that depends on only
$(\rho_{0},u_{0},\nu)$ and might vary from line to line.

From (2.5) and Lemma 3.1, we have that for any $n\in\mathbb{N}$, we can find
$T_{n}>0$ such that $T_{n}\mapsto\infty$ and

$\int_{T_{n}}^{2T_{n}}\rVert\omega(t)\rVert_{L^{2}}^{2}+\rVert\partial_{1}\Delta^{-1}\rho(t)-\nu\omega(t)\rVert_{\dot{H}^{1}}^{2}dt\leq\frac{1}{n}.$

Therefore, there exists $t_{n}\in[T_{n},2T_{n}]$ such that

$\displaystyle\rVert\omega(t_{n})\rVert_{L^{2}}+\rVert\partial_{1}\Delta^{-1}\rho(t_{n})-\nu\omega(t_{n})\rVert_{\dot{H}^{1}}\leq\frac{1}{\sqrt{nT_{n}}}.$
(5.1)

We prove the growth of curvature (a) first.

######  Proof of Theorem 1.2, part (a).

Let $B_{r}(x^{*})$ be the largest disk contained in $D_{t_{n}}$, centered at
$x^{*}=(x_{1}^{*},x_{2}^{*})$ with radius $r$. Then the Pestov–Ionin theorem
(see (4.1)) tells us that

$\displaystyle r\geq\frac{1}{\max_{x\in\partial D_{t_{n}}}|\kappa(t_{n})|},$
(5.2)

where $\kappa(t_{n})$ is the signed curvature of $\partial D_{t_{n}}$.
Applying Lemma 4.1 with $D=D_{t_{n}}$ and $\Omega=\omega(t_{n})$, we obtain

$r^{3}\leq{C}\left(\rVert\partial_{1}\Delta^{-1}\rho(t_{n})-\omega(t_{n})\rVert_{\dot{H}^{1}}+\rVert\omega(t_{n})\rVert_{L^{2}}\right)\leq\frac{C}{\sqrt{nT_{n}}},$

where the last inequality follows from (5.1). Combining this with (5.2) yields
that

$\max_{x\in\partial
D_{t_{n}}}|\kappa(t_{n})|\geq(nT_{n})^{\frac{1}{6}}\geq(nt_{n})^{\frac{1}{6}},$

where we used $t_{n}\in[T_{n},2T_{n}]$ for the last inequality. Since
$t_{n}\mapsto\infty$, as $n\to\infty$, we obtain the desired infinite growth
of the curvature. ∎

######  Proof of Theorem 1.2, part (b).

Let $L_{n}>0$ be the distance between a far-left and a far-right points on
$\partial D_{t_{n}}$. From the energy conservation (2.4), we see that
$E_{P}(t)\leq E_{T}(0)<C$ for all $t>0$. Therefore, applying Lemma 4.3 with
$D=D_{t_{n}}$, $\Omega=\omega(t_{n})$ and $A=\frac{1}{2}E_{P}(t_{n})\leq C$,
we obtain

$1\leq
C(L_{n}^{3}+1)\left(\rVert\partial_{1}\Delta^{-1}\rho(t_{n})-\omega(t_{n})\rVert_{\dot{H}^{1}}+\rVert\omega(t_{n})\rVert_{L^{2}}\right)\leq\frac{C}{\sqrt{nT_{n}}}(L_{n}^{3}+1),$

where the last inequality follows from (5.1). Since $T_{n}\to\infty$ as
$n\to\infty$, the above inequalities give us that $L_{n}\geq
C(nT_{n})^{\frac{1}{6}}\geq C(nt_{n})^{\frac{1}{6}}$ for all $n\in\mathbb{N}$,
where $t_{n}\leq 2T_{n}$ was used to justify the second inequality. Since
$D_{t_{n}}$ is simply connected, this proves the desired infinite growth of
perimeter. ∎

### Acknowledgements

The author was partially supported by the SNF grant 212573-FLUTURA and the
Ambizione fellowship project PZ00P2-216083. The author also extends gratitude
to and Eduardo García-Juárez and Yao Yao for their valuable discussions and
insightful suggestions during the course of this research.

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