Asymptotic analysis for non-local curvature flows for plane curves with a general rotation number

1.   Introduction

In this paper, we deal with curvature flows comprising non-local terms for plane curves with a general rotation number. Let ${\boldsymbol{f}}$ be an $\mathbb{R}^2$-valued function on $\mathbb{R} / L(t) \mathbb{Z} \times [ 0, T)$ such that for a fixed $t \in [ 0, T)$, it is an arc-length parametrization of a closed plane curve with total length $L(t)$. In the following text, we simply denote $L(t)$ as $L$ in many cases. To explain the curvature flow that is considering in this work, we introduce a certain geometric quantity. For a fixed $t \in [ 0, T)$, $s \in \mathbb{R} / L \mathbb{Z}$ is an arc-length parameter. Then, ${\boldsymbol{\tau}} = \partial_s {\boldsymbol{f}}$ and ${\boldsymbol{\kappa}} = \partial_s^2 {\boldsymbol{f}}$ are the unit tangent vector and the curvature vector respectively. The vector ${\boldsymbol{\nu}}$ is a unit normal vector given by rotating ${\boldsymbol{\tau}}$ counter-clockwise by $\frac \pi 2$. The curvature $\kappa$ and its deviation $\tilde \kappa$ are given by

Here, $\tilde \kappa$ is a non-local quantity. The equation we consider is of the following form:

Here, we assume that the function $g$ is a scale-invariant non-local quantity determined by ${\boldsymbol{f}}$. That is, set ${\boldsymbol{f}}_\lambda (s) = \frac 1 \lambda {\boldsymbol{f}} (\lambda s)$ ($s \in \mathbb{R} / \lambda^{-1} L \mathbb{Z}$), then,

Here we study three cases of $g$:

(AP) If we set $g \equiv 0$, then our equation represents the area-preserving flow. In fact, we set $A$ as

which is the enclosed area when $\mathrm{Im} {\boldsymbol{f}}$ is a simple curve. Consequently, it holds that

(LP) Let ${ g = L \left(\int_0^L \kappa \, ds \right)^{-1} \int_0^L \tilde \kappa^2 ds }$. Here, the equation represents the length-preserving flow:

(JP) Jiang-Pan considered an equation with ${ g = \frac { L^2 } { 2A } - \int_0^L \kappa \, ds }$ in ^[5]. Here, the isoperimetric ratio does not increase along with the flow:

Let

be the rotation number. For classical solutions, the rotation number $n$ is independent of $t$. There are a multitude of literature available considering the case when $n = 1$ in the above equations. First of all, we should mention Gage's result ^[3]. Assume that $\mathrm{Im} {\boldsymbol{f}} (0)$ is a strictly convex, closed curve with a rotation number equal to 1 in the class of $C^2$. Then, the solution ${\boldsymbol{f}}$ with the initial data ${\boldsymbol{f}} (0)$ exists globally in time, and $\mathrm{Im} {\boldsymbol{f}} (t)$ converges to a circle with a surrounding area $A(0)$ as $t \to \infty$. Similar results for (LP) and (JP) were proved by ^[6] and ^[5] respectively under the convexity condition. The authors considered flows without the convexity condition in ^[7,8]. Instead of convexity, we assume the global existence of the solution. Then the solution of (AP), (LP), or (JP) converges to a circle as $t \to \infty$ exponentially. As a result, the curvature uniformly converges to a positive constant, and thus, the curve becomes convex in finite time. In our previous works, the isoperimetric deficit

played an important role. First, we show the decay of $I_{-1}$. Set

In ^[7], we showed the inequality

for an integer $j \in [ 0, \ell ]$ with a positive constant $C = C (j, \ell)$ independent of the total length of curve. Since $I_{-1}$ is small for a sufficiently large $t$, we can regard this inequality as an embedding with a small embedding constant. We showed the exponential decay of $I_\ell$ using the standard energy method, combining the above inequality. Finally, using the decay of $I_\ell$, we showed the convergence of $\mathrm{Im} {\boldsymbol{f}}$ to a circle.

In this paper we study the case of $n > 1$, when the isomerimetric deficit is

The isoperimetric inequality shows $I_{-1}\geqq 0$ when $n = 1$. However, $I_{-1}$ is not necessarily non-negative for $n > 1$. This implies the technique used in ^[7,8] is not applicable for $n > 1$. In spite of this, $I_{-1}$ gives us some useful information. For example, we can show that if $I_{-1}$ is negative for $t = 0$, then the solution blows up in finite time. See our first main result, Theorem 3.1. This implies $I_{-1} \geqq 0$ for global solutions, and that sounds a good information. However, the inequality (1.1) does not hold for $n > 1$. There are at least two approaches for dealing with this difficulty. One is to give a proof without using (1.1), and another is to show an alternative inequality to (1.1). In this paper, we show that both are in success. For the second approach, we use a geometric quantity which has never been considered before, given as follows:

Then we can show

We prepare several inequalities and estimates for closed curves with a rotation number $n$, in § 2.1. And we describe some basic properties of the flows (AP), (LP) and (JP), in § 2.2. Using these, in § 3, we discuss blow-up solutions with blow-up time estimates, blow-up quantities, and blow-up rates. In § 4, the convergence to an $n$-fold circle of global solutions is proved without using (1.2). Finally, we show (1.2) in the final section.

2.   Preliminaries

In this section, we provide several estimates and inequalities for plane curves. Those in § 2.1 hold for curves which are not necessarily solutions of the flows. We derive the basic properties of flows in § 2.2.

2.1.   Estimates for plane curves

Let ${\boldsymbol{f}} = (f_1, f_2)$ be an arc-length parametrization of a plane curve with the rotation number $n \geqq 1$. Set

The functions ${ \varphi_k = \frac 1 { \sqrt L } \exp \left(\frac { 2 \pi i k s } L \right) }$ for $k \in \mathbb{Z}$ generate the standard complete orthogonal system of $L^2 (\mathbb{R} / L \mathbb{Z})$. Let $\hat f (k)$ be the Fourier coefficient of $f$. Subsequently, we can derive the following relations in a manner similar to [7,Corollary 2.1], where we dealt with the case of $n = 1$. The difference is just "$n$" in (2.3) which comes exactly from the definition of the rotation number. We can find similar argument in ^[1,10]

Lemma 2.1.

Note that we have

from (2.2) and (2.3). The above is very useful for our analysis.

Lemma 2.2. We have

Proof. We obtain (2.8) as

from (2.4) and (2.3). Combining this with (2.7), we obtain (2.9).

Though $I_0$ must be non-negative by the definition, it is not obvious to see that from the first expression (2.8). However, it can be seen from the second one (2.9). Furthermore, we see from (2.9) that $I_0 = 0$ if and only if $\mathrm{Im} {\boldsymbol{f}}$ is an $n$-fold circle.

The isoperimetric inequality holds even if $n$ is not $1$.

Lemma 2.3. We have $L^2 - 4 \pi A \geqq 0$.

Proof. It follows from (2.2) and (2.1) that

Similarly, $I_{-1}$ has two expressions.

Lemma 2.4. We have

Proof. It follows from (2.2) and (2.1) that

The second expression of $I_{-1}$ is obtained from the above and (2.7).

Since $k (k - n)$ is not necessarily non-negative when $n > 1$, we know the same holds for $I_{-1}$. However, the modulus of $I_{-1}$ can be estimated by $I_0$ for $n \geqq 1$ as follows. This is Wirtinger's inequality when $n = 1$.

Lemma 2.5. It holds that $4 \pi^2 n | I_{-1} | \leqq I_0$.

Proof. From Lemmas 2.2–2.4 we obtain

Here, we use $k (k \mp 1) \geqq 0$ for $k \in \mathbb{Z}$.

2.2.   Estimates for flows

In this subsection, we derive the basic properties of the flows, which we use in following sections. Let ${\boldsymbol{f}}$ be a classical solution of one of (AP), (LP), or (JP) on $[ 0, T)$, and let $T$ be the maximum existence time. Since ${ \frac { dL } { dt } = - \int_0^L \partial_t {\boldsymbol{f}} \cdot {\boldsymbol{\kappa}} \, ds }$, we have

that is,

Similarly, we have

It follows from the above that

From these, we summarize the basic properties of each solution as follows.

Proposition 2.1. Assume that the initial curve is smooth, and that $A(0)$ is positive. Let ${\boldsymbol{f}}$ be a classical solution of one of (AP), (LP), or (JP) on $[ 0, T)$ and let $T$ be the maximum existence time. Then, the following holds for $t \in (0, T)$.

1). For solutions of (AP),

2). For solutions of (LP),

3). For solutions of (JP),

4). For solutions of (AP), (LP), (JP),

In other words,

Proof. In the cases of (AP) and (LP), the signs of ${ \frac { dA } { dt } }$ and ${ \frac { d L^2 } { dt } }$ immediately follow from (2.11) and (2.10). Therefore, $A > 0$ and

In the case of (JP), we prove the positivity of $A$ by applying the contradiction argument. In this case,

It follows from (2.11) that

Assume that $A(t_0)^2 = 0$ for some first time $t_0 \in (0, T)$. Since $A^2 \geqq 0$, we have

Since $A (0)^2 > 0$, there exists $t_1 \in (0, t_0)$ such that

It follows from (2.14) and (2.12) that

Therefore, by (2.16)

This contradicts (2.15). Hence, $A > 0$ on $(0, T)$. Using (2.12), (2.10), $I_{-1} - 1 = - \frac { 4 \pi n A } { L^2 }$, and (2.13), we have

Since $I_{-1}$ is non-increasing, we have $I_{-1} \leqq I_{-1} (0)$. Lemma 2.3 gives us

3.   Blow-up solutions

The non-positivity of $I_{-1} (0)$ implies that the blow-up phenomena occurs in finite time.

Theorem 3.1. Let ${\boldsymbol{f}}$ be a classical solution of one of (AP), (LP), or (JP) on $[ 0, T)$ and let $T$ be the maximum existence time. Assume that the initial curve is smooth, and satisfies $A(0) > 0$, $I_{-1} (0) < 0$. Then, the solution blows up in finite time. The blow-up time $T$ is estimated from above as follows:

${\rm (AP)} \; { T \leqq \frac { L(0)^2 - 4 \pi A (0) } { - 8 \pi^2 n I_{-1} (0) } }$,

${\rm (LP)} \; { T \leqq \frac { L(0)^2 - 4 \pi A (0) } { - 8 \pi^2 I_{-1} (0) } }$,

${\rm (JP)} \; { T \leqq \frac { L(0)^2 } { - 8 \pi^2 nI_{-1} (0) } }$.

Proof. In the case of (AP), $g \equiv 0$. It follows from Proposition 2.1 that $I_{-1} (t) \leqq I_{-1} (0) < 0$. By (2.10) and Lemma 2.5, we have

Integrating this from $0$ to $t \in (0, T)$, and using Lemma 2.3, we obtain

Since the first side is non-positive by the isoperimetric inequality (Lemma 2.3), $t$ must satisfy

In the case of (LP), ${ g = \frac { I_0 } { 2 \pi n } \geqq 0 }$. Proposition 2.1 shows $I_{-1} (t) \leqq I_{-1} (0) < 0$. From (2.11) and Lemma 2.5, we have

We integrate this from $0$ to $t \in (0, T)$. Using Lemma 2.3, we obtain

Consequently, $t$ must satisfy

In the case of (JP), ${ g = \frac { L^2 I_{-1} } { 2A } }$. It follows from (2.10), Proposition 2.1, and Lemma 2.5 that

We integrate this from $0$ to $t \in (0, T)$. Using Lemma 2.3, we obtain

Consequently $t$ must satisfy

Corollary 3.1. Let ${\boldsymbol{f}}$ be a classical solution of one of (AP), (LP), or (JP) on $[ 0, T)$ and let $T$ be the maximum existence time. Assume that the initial curve is smooth, and that satisfies $A(0) > 0$, and $I_{-1} (0) = 0$, but it is not an $n$-fold circle. Then, $T < \infty$.

Proof. Assume $T = \infty$. Then, Theorem 3.1 implies that $I_{-1} (t) \geqq 0$ for all $t \in [ 0, \infty)$. On the other hand, (2.12) with $I_{-1} (0) = 0$ shows that $I_{-1} (t) \leqq 0$. Hence, $I_{-1} (t) \equiv 0$. When $t > 0$,

Combining this with the rotation number $n$, we find that $\mathrm{Im} {\boldsymbol{f}} (t)$ is an $n$-fold circle. However, this does not satisfy the initial condition.

Corollary 3.2. ${\boldsymbol{f}}$ is a classical stationary solution of one of (AP), (LP), or (JP), if and only of it is an $n$-fold circle.

Proof. Assume that $\mathrm{Im} {\boldsymbol{f}}$ is an $n$-fold circle. Then, $\tilde \kappa \equiv 0$. Since $f = \hat f (0) \varphi_0 + \hat f (n) \varphi_n$, we see $I_0 = I_{-1} = 0$ by Lemmas 2.2 and 2.4. Hence, ${ \tilde \kappa - \frac gL \equiv 0 }$ for each case. Consequently, it is a stationary solution.

Conversely, assume that ${\boldsymbol{f}}$ is a stationary solution. It follows from (2.12) that $I_0 (t) \equiv 0$. Hence, we can conclude that $\mathrm{Im} {\boldsymbol{f}} (t)$ is an $n$-fold circle in a manner similar to the proof of the previous corollary.

Suppose now ${\boldsymbol{f}}$ blows up as $t \nearrow T < \infty$. Then, we have

Indeed, if ${ \limsup\limits_{ t \nearrow T } I_0 (t) < \infty }$, then ${ \sup\limits_{ t \in (0, T) } I_0 (t) }$ is bounded. We can show the boundedness of ${ \sup\limits_{ t \in (0, T) } I_\ell (t) }$ by the standard energy method. Using this and the equation of the flow, we can see that ${\boldsymbol{f}} (t)$ converges to a smooth function as $t \nearrow T$. Consequently, the solution can be expanded beyond $T$. This is a contradiction.

Set

We will show the blow-up of $W$ and its blow-up rate. Firstly, we consider the limit supremum of $W$.

Lemma 3.1. It holds that ${ \limsup\limits_{ t \nearrow T } W(t) = \infty }$.

Proof. Set

and we have

Hence,

Therefore, the assertion immediately follows in the case of (LP).

In the case of (AP), $L$ is non-increasing by Proposition 2.1. Lemma 2.3 implies that $L \geqq \sqrt{ 4 \pi A } = \sqrt{ 4 \pi A_0 }$. Consequently, $L(t)$ converges to a positive constant as $t \nearrow T$, and the assertion follows.

We show that $L(t)$ converges to a positive constant in the case of (JP) as well. We assume that ${ \liminf\limits_{ t \nearrow T } A (t) = 0 }$. $I_{-1}$ is monotone by Proposition 2.1. Therefore, it follows from

that $A$ does not oscillate near $t = T$. Hence, we may assume ${ \lim\limits_{ t \nearrow T } A (t) = 0 }$. From the above relation and Proposition 2.1, we find that ${ \frac { dA } { dt } }$ is bounded. Consequently, the estimate

holds. Thus, we have

and therefore,

This implies that the left derivative of $A^2$ at $T$ vanishes:

However, $A(0)^2 > 0$ and $A(T - 0)^2 = 0$ show the existence of $t_\ast \in (0, T)$ such that

Since

we have

for $t \in (t_\ast, T)$. This contradicts (3.1). Now, we prove ${ \liminf\limits_{ t \nearrow T } A(t) > 0 }$. Since

has a constant sign near $T$, we conclude that ${ \lim\limits_{ t \nearrow T } A(t) > 0 }$. The convergence of ${ \lim\limits_{ t \nearrow T } L(t) }$ follows from the convergence of $A$, and the monotonicity and boundedness of $I_{-1}$. Since ${ \frac { L^2 } A }$ is strictly positive by Proposition 2.1, the limit of $L$ is positive.

Next, we derive the time derivative of $W$. Set

which are scale-invariant quantities. Note that $I_0 = J_2$.

Lemma 3.2. It holds that

Proof. The proof is a direct calculation:

Thirdly, we estimate $\frac { dW } { dt }$ from above.

Lemma 3.3. We have

Here,

with the constant $C$ being independent of the initial curve and the rotation number.

Proof. Here, we use Lemma 3.2. In the case of (AP), since $g = 0$, we have

Set $\theta = \frac 12 - \frac 1p$. Then, Gagliardo-Nirenberg's inequality yields

Hence,

Since $0 \leqq I_0 \leqq LW$ and $R^2 \leqq LW$, we obtain

Furthermore,

Consequently, we conclude that

In the case of (LP), since ${ g = \frac { I_0 } R }$, we have

In the case of (JP), since ${ g = \frac { L^2 } { 2A } - R }$, we have

By Proposition 2.1, we have

Consequently, we can conclude that

Now, we prove the following theorem.

Theorem 3.2. Let $T \in (0, T)$ be the blow-up time for a solution of one of (AP), (LP), or (JP). Then, $W(t)$ blows up as

where

with a constant $C$ that is independent of the initial curve and the rotation number.

Proof. It follows from Lemma 3.3 that

Due to Lemma 3.1, there exists a sequence $\{ t_n \}$ such that $t_n t \nearrow T$ and $W(t_n)^{-2} \to 0$ as $n \to \infty$. Integrating the differential inequality from $t$ to $t_n$, we have

Therefore, we obtain the theorem as $n \to \infty$.

The curve $\mathrm{Im} {\boldsymbol{f}}$ may have several loops. When the orientation of a loop is counter-clockwise as $s$ increases, it is called a positive loop. Otherwise, it is called a negative loop. It has already been shown that $L(t)$ converges to a positive constant as $t \to \infty$. Therefore, from the above theorem we know that

or

If a positive/negative loop of $\mathrm{Im} {\boldsymbol{f}}$ shrinks as $t \nearrow T$, the maximum/minimum value of the curvature may not remain bounded. On the other hand, there is a possibility of the maximum or minimum remaining bounded as $t \nearrow T$. For example, if a negative loop shrinks as $t \nearrow T$ before the positive loops shrink, the minimum value of the curvature goes to $- \infty$, but the maximum remains bounded. In the last part of this section, we discuss the blow-up of the maximum and minimum.

Theorem 3.3. Let $T \in (0, \infty)$ be the blow-up time for a solution of one of (AP), (LP), or (JP). Assume that

then it satisfies

Proof. Set

Define the set $S_t$ by $S_t = \{ s \in \mathbb{R} / L(t) \mathbb{Z} \, | \, \kappa (s, t) = K(t) \}$. After re-parametrizing ${\boldsymbol{f}} (\cdot, t)$ by a new parameter that is independent of $t$, we apply [2,Lemma B.40]. Consequently, we can conclude that $K$ is a continuous function of $t$, and that

$\kappa$ satisfies the equation

For the cases of (AP) and (LP), $R + g > 0$ as $R > 0$ and $g \geqq 0$. In the case of (JP),

$\partial_s^2 \kappa \leqq 0$ holds for $s \in S_t$. Hence, we have

for $s \in S_t$, and

We calculate Dini's derivative of $K^{-2}$ as

According to the assumption of the theorem, there exists a sequence $\{ t_k \}_{ k \in \mathbb{N} }$ such that $t_k \nearrow T$ and $K(t_k)^{-2} \to 0$ as $k \to \infty$. Using [4,Theorem 3], we have

for $t_k \in (t, T)$. Therefore, we can conclude that

by $k \to \infty$

Theorem 3.4. Let $T \in (0, \infty)$ be the blow-up time for a solution of one of (AP), (LP), or (JP). Assume that

For the solution of (AP),

holds.

For the solution of (LP),

holds.

For the solution of (JP), there exists a time $T_\ast \in [ 0, T)$ such that

holds for $t \in [ T_\ast, T)$. Additionally, it holds that

where

Remark 3.1. The time $T_\ast$ above exists for all cases. And for the proof, it does not need to be the first or last such time.

Proof. Here, we set

Define the set $S_t$ by $S_t = \{ s \in \mathbb{R} / L(t) \mathbb{Z} \, | \, - \kappa (s, t) = K(t) \}$. As shown before, it holds that

$- \kappa$ satisfies

Since $\partial_s^2 (- \kappa) \leqq 0$ and $- \kappa = K$ for $s \in S_t$,

If $\kappa \leqq C < \infty$ holds on $[ 0, T)$, then,

by Theorem 3.2. Since $L$ is bounded, we conclude that $K \to \infty$ as $t \nearrow T$. Therefore, $| \kappa | \leqq \max\{ C, K \} \leqq K$ near $T$. Hence, there exists $T_\ast \in [ 0, T)$ as mentioned in the statement. Considering $t \geqq T_\ast$, we may assume that $| \kappa | \leqq K$.

In the case of (AP), since $g = 0$,

Using this and

we have $\partial_t (- \kappa) \leqq 2 K^3$ on $S_t$, i.e.,

Consequently, we obtain the assertion as before.

In the case of (LP),

The estimate ${ \frac RL \leqq K }$ holds for all cases. Hence,

Consequently, we have

Here, we use $L \equiv L(0)$. The statement follows from the above, as shown before.

In the case of (JP), using ${ R + g = \frac { L^2 } { 2A } }$ and Lemma 2.1, we have

Hence, it holds that

which leads to the required conclusion and ends the proof.

Remark 3.2. At a glance, the power $\frac 13$ of blow-up rate in (LP) seems to be curious. The difference with other cases is that there is the length $L(0)$ in the braces. If an estimate

holds, then the power $p$ must be $\frac 13$. To see this, assume that ${\boldsymbol{f}}$ is a solution of (LP) which blows up at $T < \infty$. For a positive constant $\lambda$, set

We denote quantities of ${\boldsymbol{f}}_\lambda$ the notation with the suffix $\lambda$; for example $\kappa_\lambda$ is its curvature. Then, ${\boldsymbol{f}}_\lambda$ satisfies (LP) with the length $L_\lambda = \lambda^{-1} L (0)$, and blows up at $T_\lambda = \lambda^{-2} T$. The minimum of curvature is

Using $L(0) = \lambda L_\lambda (0)$ and $T = \lambda^2 T_\lambda$, we have

Hence, $p$ must be $\frac 13$. The $L(0)$ in braces comes from the estimate $\frac { K^2 g } L \leqq \frac { L K^4 } R$ in the proof. If we can improve this as $\frac { K^2 g } L \leqq C K^3$, then the blow-up rate coincides with other cases.

4.   Convergence of global solutions

In this section, we assume that ${\boldsymbol{f}}$ is a classical global solution of one of (AP), (LP), or (JP), and that the initial curve satisfies $A(0) > 0$. We prove that $\mathrm{Im} {\boldsymbol{f}}$ converges to an $n$-fold circle exponentially as $t \to \infty$.

Remark 4.1. However, this conclusion is meaningless if $n$-fold circles are only global solutions. At least, in the case of (AP), under suitable assumptions on the initial curve, regarding symmetry and convexity, solutions exist globally in time even if $n > 1$. See ^[9].

Firstly we prove the decay of $I_{-1}$.

Lemma 4.1. For the global solution above, $I_{-1} (t)$ fulfills

In particular, the estimate

is satisfied with respect to the global solution for (AP); the estimate

for the global solution of (LP). In the case of (JP), setting ${ \bar L = \sup\limits_{ t \in [ 0, \infty) } L(t) }$, we have $\bar L < \infty$, and

Proof. For global solutions, we know, from Theorem 3.1, that $I_{-1} (t) \geqq 0$. Hence, we have

by Lemma 2.5. Consequently, (2.12) becomes

Solving this differential inequality, we obtain the first assertion.

We use $\sqrt{ 4 n \pi A(0) } \leqq L(t) \leqq L(0)$ for (AP), and $L(t) \equiv L(0)$ for (LP). Then, the second assertion follows for these two cases.

Now, we consider the case of (JP). Integrating (2.12), we have

${ \frac { L^2 } A }$ is uniformly positive and bounded by Proposition 2.1. From this, (2.10) with ${ g = \frac { L^2 I_{-1} } { 2A } }$ and (4.1), we have

Integrating this, we have

Hence, $\bar L < \infty$. It follows from (2.11) and ${ g = \frac { L^2 I_{-1} } { 2A } \geqq 0 }$ that

Therefore, the lower bound $L$ follows from $L(t)^4 \geqq (4 \pi n A(t))^2 \geqq (4 \pi n A(0))^2$. Consequently, we obtain the second assertion for (JP).

We denote the relevant statement of Lemma 4.1 as

Corollary 4.1. For the global solution above, there exists $L_\infty > 0$ and $A_\infty > 0$ such that

Proof. In the case of (AP), by Proposition 2.1, we have ${ \frac { dL } { dt } \leqq 0 }$. Hence, we conclude the convergence of ${ \lim\limits_{ t \to \infty } L(t) }$. Set the limit value as $L_\infty$. Since $A(t) \equiv A (0)$, and since ${ \lim\limits_{ t \to \infty } I_{-1} (t) = 0 }$, it holds that

and $L_\infty \leqq L \leqq L(0)$. Therefore,

In the case of (LP), since ${ \frac { dA } { dt } \geqq 0 }$ and since $4 \pi A \leqq L^2 = L(0)^2$, we conclude the convergence of ${ \lim\limits_{ t \to \infty } A(t) }$. Set the limit value as $A_\infty$. Since $L(t) \equiv L(0)$, and ${ \lim\limits_{ t \to \infty } I_{-1} (t) = 0 }$, it holds that $4 \pi n A_\infty = L(0)^2$. Consequently, (2.11) with ${ g = \frac { I_0 } { 2 \pi n } }$ yields

Here, we use (2.12) and Lemma 4.1.

In the case of (JP), ${ \frac { dA } { dt } = \frac { L^2 I_{-1} } { 2A } \geqq 0 }$. By Proposition 2.1, ${ \frac A { L^2 } }$ is uniformly positive and bounded. Combining the above two statements with Lemma 4.1, we conclude

Furthermore, we estimate that

Corollary 4.2. For the global solution above, it holds that

Proof. We know that $L$ is uniformly bounded for all cases. Therefore, (2.12) implies that

Lemma 4.2. For the global solution above, there exists $\lambda_0 > 0$ such that

Proof. As in Section 3, we set

As we know that $L \to L_\infty > 0$ as $t \to \infty$, it is enough to show that

Since $I_0 = J_2 = LW - R^2$, we have from (2.10) and Lemma 3.2

We obtain

in a manner similar to the proof of Lemma 3.3.

Since $g = 0$ in (AP), and ${ g = \frac { L^2 I_{-1} } { 2A } }$ in (JP), $| g |$ is uniformly bounded for these cases. In (LP), $g = R^{-1} I_0$. Hence, it holds for every case that

This can be presented as

By Corollary 4.2, there exists $t_0 > 0$ such that

Set

If $t_1 < \infty$, then,

For $t \in (t_0, t_1)$, we have

and therefore,

Letting $t \nearrow t_1$, we obtain a contradiction. Consequently, $t_1 = \infty$, that is, ${ I_0 (t) < \frac 3C }$ for $t \in [ t_0, \infty)$. Since we know that $I_0$ is uniformly bounded, we obtain

It follows from Wirtinger's inequality and the uniform estimate of $L^2$ that

for some constant $\lambda > 0$. Multiplying both sides by $e^{ 2 \lambda t }$, and integrating from ${ \frac t2 }$ to $t$, we have

That is, we have

Using the uniform estimate of $L$ and the exponential decay of ${ \int_{ \frac t2 }^\infty I_0 dt }$, we finally obtain the exponential decay of $I_0$.

Once we obtain the exponential decay of $\widetilde I_{-1}$ and $I_0$, we can obtain the convergence of $\mathrm{Im} {\boldsymbol{f}}$ to an $n$-fold circle as $t \to \infty$.

Theorem 4.1. Let ${\boldsymbol{f}}$ be a classical global solution of one of (AP), (LP), or (JP), with the smooth initial curve satisfying $A(0) > 0$. Then, $\mathrm{Im} {\boldsymbol{f}}$ converges to an $n$-fold circle with centre ${\boldsymbol{c}}_\infty$, and radius ${ r_\infty = \frac { L_\infty } { 2 \pi n } }$ in the following sense. Set

with the $\mathbb{R} / L(t) \mathbb{Z}$-valued function $\sigma$ defined by

Then, there exist ${\boldsymbol{c}}_\infty \in \mathbb{R}^2$, ${ r_\infty = \frac { L_\infty } { 2 \pi n } > 0 }$, $\sigma_\infty \in \mathbb{R} / L_\infty \mathbb{Z}$, $\lambda > 0$, and $C > 0$ such that

Furthermore, for $k \in \{ 0 \} \cup \mathbb{N}$, there exist $\gamma_k > 0$ and $C_k > 0$ such that

When $n = 1$, we used (1.1) for the proof of this theorem in [7,§ 4], and [8,§ 2.2]. The most crucial part is to show the decay of $I_0$. As above, we have already obtained a decay estimate of $I_0$ without using (1.1) for $n \geqq 1$. Once we obtain it, to show the theorem, we can perform the standard energy method with help of usual Gagliardo-Nirenberg's inequality rather than (1.1) as the previous papers. In this sense, (1.1) is not absolutely necessary, however, we need several modification of argument. Using (1.2) which is an alternative inequality to (1.1), we can develop the argument almost word to word as the previous papers. Thus, we deal with (1.2) in the next section.

5.   Interpolation inequalities

We discuss (1.2) in this section. Set

Proposition 5.1. We have

$\widetilde I_{-1}$ vanishes if and only if $\mathrm{Im} {\boldsymbol{f}}$ is an $n$-fold circle.

Proof. Setting

we have

The squared $L^2$-norm of $\nu$ is

On the other hand, we have

Since the last right-hand side expression is a real number, it holds that

Consequently, we obtain

$\widetilde I_{-1}$ vanishes if and only if

Hence, $\mathrm{Im} {\boldsymbol{f}}$ is an $n$-fold circle.

An estimate similar to Lemma 2.5 holds for $\widetilde I_{-1}$ as well.

Lemma 5.1. It holds that $4 \pi^2 \widetilde I_{-1} \leqq I_0$.

Proof. Since $k^2 - 1 \geqq 0$ for $k \in \mathbb{Z} \setminus \{ 0 \}$, we have

The next proposition corresponds to [7,Theorem 2.2].

Proposition 5.2. It holds that

Proof. It follows from Lemma 2.2, Schwarz' inequality, and (2.6) that

Using this proposition, we can prove the following estimates.

Theorem 5.1. Let $j \in [ 0, \ell ]$ be an integer. Then, there exists a positive constant $C = C (j, \ell)$ independent of $L$ such that

Proof. Since the assertion can be proven in a manner similar to the proof of [7,Theorem 3.1], we give only the sketch. Firstly, we derive

from Proposition 5.2 and Gagliardo-Nirenberg's inequality

for $p \geqq 2$ and $j \leqq m$. Here $C (j, m, p)$ is independent of $L$. It follows from (5.2) that

Combining this together with (5.1), we obtain the assertion for $j = 0$. It gives also the assertion for $j \geqq 1$ with help of (5.3).

For the proof of convergence of global flow to a circle, we use in ^[7] the following properties of $I_{-1}$:

(ⅰ) $I_{-1} \geqq 0$,

(ⅱ) $I_{-1} = 0$ holds if and if the image of ${\boldsymbol{f}}$ is a circle,

(ⅲ) $C^{-1} I_{-1} \leqq I_0$ (an inequality of Wirtinger's type).

These are satisfied when $n = 1$, but not when $n > 1$. The quantity $\widetilde I_{-1}$ satisfies

(ⅰ) $\widetilde I_{-1} \geqq 0$,

(ⅱ) $\widetilde I_{-1} = 0$ holds if and if the image of ${\boldsymbol{f}}$ is an $n$-fold circle,

(ⅲ) $C^{-1} \widetilde I_{-1} \leqq I_0$ (an inequality of Wirtinger's type).

Hence, it is an alternative quantity to $I_{-1}$.

Acknowledgments

The first author is partly supported by Grant-in-Aid for Scientific Research (C) (17K05310), and (B) (20H01813), Japan Society for the Promotion Science. The authors express their appreciation to the anonymous referee for his/her suggestive comments and information of related articles ^[1,10].

Conflict of interest

The authors declare no conflict of interest.