On the rate of convergence of the $p$-curve shortening flow

TL;DR

This paper establishes improved and likely sharp rates of convergence for the p-curve shortening flow when p is an integer greater than or equal to 1, advancing understanding of the flow's behavior.

Contribution

It provides new, sharper convergence rate estimates for the p-curve shortening flow, enhancing previous results and likely representing the optimal rates.

Findings

Improved convergence rate estimates for p-curve shortening flow.

Results are probably sharp, indicating optimality.

Advances understanding of flow dynamics for p ≥ 1.

Abstract

In this paper we give rates of convergence for the $p$-curve shortening flow for $p\geq 1$ an integer, which improves on the known estimates and which are probably sharp.

Equations

$$x:{\mathbb{S}}^1\times \left[0,T\right)⟶{\mathbb{R}}^2$$

$$\frac{\partial x}{\partial t}=-\frac{1}{p}{k}^{p}N,$$

$${\Vert {\tilde{k}}^{\left(n\right)}\Vert }_{\infty }\le C{e}^{-\delta t},$$

$$\hat{\psi }(0)\ge c_p{\Vert \psi \Vert }_2,$$

$${\Vert \tilde{k}-1\Vert }_{C^l\left({\mathbb{S}}^1\right)}\le C_{p,l}{e}^{-\left(3p-1\right)\tau },$$

$${\Vert \tilde{k}-1\Vert }_{C^k\left({\mathbb{S}}^1\right)}\le C_{p,k}{e}^{-\left(3p-1\right)\tau },$$

$$\{\begin{array}{l}\frac{\partial k}{\partial t}=k^2\left(k^{p-1}\frac{{\partial }^{2}k}{\partial {\theta }^2}+(p-1)k^{p-2}{\left(\frac{\partial k}{\partial \theta }\right)}^2+\frac{1}{p}{k}^p\right)\text{in}\left[0,2\pi \right]\times \left(0,T\right)\\ k(\theta ,0)=\psi (\theta )\text{on}\left[0,2\pi \right],\end{array}$$

$$\sum _{n\in \mathbb{Z}}\hat{f}\left(n\right)e^{in\theta }$$

$$\hat{f}(n)=\frac{1}{2\pi }{\int }_0^{2\pi }f(\theta )e^{-in\theta }d\theta .$$

$${\hat{u}}^{\ast (m)}(q_1,q_2,\dots ,q_m,t)=\hat{u}(q_1,t)\hat{u}(q_2,t)\cdots \hat{u}(q_m,t),$$

$$H(p,q_1,q_2)=\frac{1}{p}-(p-1)q_1{q}_2-q_1^2,$$

$${\mathcal{B}}_n=\left\{\left(q_1,\dots ,q_{p+2}\right)\in {\mathbb{Z}}^{p+2}:q_{p+2}=n-q_1-\cdots -q_{p+1}\right\},$$

$${\mathcal{A}}_n=\left\{\mathbf{q}\in {\mathcal{B}}_n:\text{there are }1\le i<j\le p+2\text{ such that }{b}_i\ne 0\text{ and }{b}_j\ne 0\right\},$$

$${\mathcal{C}}_n=\left\{\mathbf{q}\in {\mathcal{A}}_n:q_j\ne 0,\pm 1,\text{for all }1\le j\le p+2\right\}.$$

$$\{\begin{array}{ll}\frac{d}{dt}\hat{k}(0,t)& =\frac{1}{p}\hat{k}{\left(0,t\right)}^{p+2}+{\sum }_{\mathbf{q}\in {\mathcal{A}}_0\cap {\mathcal{Z}}^{p+2}}H\left(p,q_1,q_2\right){\hat{k}}^{\ast (p+2)}\left(\mathbf{q},t\right),\\ & \\ \frac{d}{dt}\hat{k}(n,t)& =\left(\frac{p+2}{p}-n^2\right)\hat{k}{\left(0,t\right)}^{p+1}\hat{k}\left(n,t\right)\\ & +{\sum }_{\mathbf{q}\in {\mathcal{A}}_n\cap {\mathcal{Z}}^{p+2}}H\left(p,q_1,q_2\right){\hat{k}}^{\ast (p+2)}\left(\mathbf{q},t\right),\text{ if }n\ne 0,n\in \mathcal{Z}\end{array}$$

$$\hat{k}(n,0)=\hat{\psi }(n),\text{ if }n\in \mathcal{Z}.$$

$${\Vert f\Vert }_{\beta }=\max \left\{\underset{\xi \in \mathbb{Z}}{\sup }{\left|\xi \right|}^{\beta }\left|Re\left(\hat{f}\left(\xi \right)\right)\right|,\underset{\xi \in \mathbb{Z}}{\sup }{\left|\xi \right|}^{\beta }\left|Im\left(\hat{f}\left(\xi \right)\right)\right|\right\}.$$

$${\Vert f\Vert }_{C^l\left([0,2\pi ]\right)}=\underset{j=0,\dots ,l}{\max }\underset{\theta \in [0,2\pi ]}{\sup }\left|\frac{d^{j}f(\theta )}{d{\theta }^j}\right|$$

$$Q(k)={\int }_0^{2\pi }\frac{e^{i\theta }}{k(\theta ,t)}d\theta =0,$$

$$2\delta \cdot \psi (0)\ge q^2|\hat{k}(q,t)|,$$

$$\hat{k}(0,t)\ge \left(1-\delta \right)\hat{\psi }(0),$$

$$U:=\sum _{\mathbf{q}\in {\mathcal{A}}_0\cap {\mathcal{Z}}^{p+2}}H\left(p,q_1,q_2\right){\hat{k}}^{\ast (p+2)}\left(\mathbf{q},t\right)=O\left(\delta \hat{k}(0,t{)}^{p+2}\right),$$

$$2\delta \cdot \hat{\psi }(0)\ge |\hat{\psi }(\pm 1)|\text{and}\delta \cdot \hat{\psi }(0)\ge q^2|\hat{\psi }(q)|\text{for}\left|q\right|\ge 2$$

$$2\delta \cdot \hat{\psi }(0)\ge |\hat{k}(\pm 1,t)|,$$

$$\hat{k}(0,t)\ge \left(1-\delta \right)\hat{\psi }(0).$$

$$\delta \cdot \hat{\psi }(0)\ge q^2|\hat{k}(q,t)|.$$

$$\frac{d}{dt}\log R_n=\left(\frac{p+2}{p}-n^2\right)\hat{k}{\left(0,t\right)}^{p+1}+\sum _{i=1}^2{B}_i$$

$$B_1$$

$$B_2$$

$$\left|\hat{k}\left(n,t\right)\right|\left|B_1\right|$$

$$\left|{\hat{k}}^{\ast (p+2)}\left(\mathbf{q},t\right)\right|=\frac{2^{p+2}{\delta }^{p+2}\hat{\psi }(0{)}^{p+2}}{q_1^2\cdots q_{p+1}^2{\left(n-q_1-\cdots -q_{p+1}\right)}^2},$$

Full text

1. Introduction

Let us first introduce the main character of this story, the $p$-curve shortening flow, with $p$ a positive integer. So, we let

[TABLE]

be a family of smooth convex embeddings of $\mathbb{S}^{1}$, the unit circle, into $\mathbb{R}^{2}$. We say that $x$ satisfies the $p$-curve shortening flow, $p\geq 1$, if $x$ satisfies

[TABLE]

where $k$ is the curvature of the embedding and $N$ is the normal vector pointing outwards the region bounded by $x\left(\cdot,t\right)$.

This is just a natural generalisation of the well known and well studied curve shortening flow. A solution to 1 starting from an embedded convex simple curve will contract, via embedded convex curves, towards a round point in finite time: this means that if we start with a simple convex curve, via the $p$-curve shortening, after a convenient normalisation, which includes a time reparametrisation, the embedded curves converge smoothly to a circle (see [1]). It is also known that this convergence is exponential in the following sense (here $\tilde{k}$ denotes the curvature of the embedded curves after normalisation)

[TABLE]

with where $\tilde{k}^{\left(n\right)}$ represents the $n$-derivative of $\tilde{k}$ with respect to the arclength parameter in $\mathbb{S}^{1}$, $n\geq 1$, and $\delta>0$. For the curve shortening flow, we can use as $\delta$ any $2\alpha$ for $0<\alpha<1$, this was proved by Gage and Hamilton in their by now famous (by mathematical standards) paper [6]. For the $p$-curve shortening, Huang in [7] showed that $\delta$ can be taken as $2\alpha p$, with the same restrictions on $\alpha$. Interestingly enough, with the exception of the curve shortening flow ($p=1$), it has not been showed that $\tilde{k}\rightarrow 1$ exponentially! For the curve shortening flow ($p=1$), in the book [3] exponential convergence of the curvature towards 1 is shown, and Andrews and Bryan showed in [2] (although they did not stated explicitly) that $\tilde{k}\rightarrow 1$ as fast as $e^{-2\tau}$.

Related to this problem is the mean curvature flow, and Sesum in [9], using Huisken’s work as a departure point, has given sharp rates of convergence for this flow.

The main goal of this paper to give better rates of convergence for the $p$-curve shortening flow, $p\geq 1$ an integer, than the ones previously known. Our main result, from which the said rates of convergence can be deduced, is the following.

Theorem 1**.**

Let $\psi>0$ be the curvature of the initial condition to (1). Then there exists a constant $c_{p}>0$ such that if

[TABLE]

then the solution to the normalised $p$-curve shortening flow ($p$ a positive integer), that is for the curvature $\tilde{k}$ of the curves given by the rescaled embedding $\left(\dfrac{p+1}{p}\right)^{\frac{1}{p+1}}\left(T-t\right)^{-\frac{1}{p+1}}x$, with rescaled time parameter $\tau=-\dfrac{1}{p+1}\log\left(1-\dfrac{t}{T}\right)$, where $0<T<\infty$ is the maximum time of existence for (1), it holds that

[TABLE]

*where $C_{p,l}$ is a constant that only depends on $p,l$ and $\psi$. *

Together with Theorem I1.1 from [1], this gives the following

Theorem 2**.**

For any simple convex curve as initial data, the normalised version of (1), converges towards a circle smoothly and the curvature of the normalised embeddings satisfy

[TABLE]

where $C_{p,k}$ is a constant that only depends on $p,k$ and the curvature of the initial condition.

Indeed, by the theorem of Andrews referred to above, (2) eventually holds if we start (1) with a given convex simple curve as initial data. The rate of convergence given by our main result seems to be the sharpest possible rate of convergence for the $p$-curve shortening flow (see the remark at the end of this paper).

A naive idea for proving Theorem 1 would be to use the Parabolic PDE to which the normalised version of $p$-curve shortening flow is equivalent to (see equation (14) in Section 3.2), and then linearise around the steady solution to obtain exponential convergence. However, if we linearise around the steady solution, the elliptic part of the parabolic operator corresponding to the $p$-curveshortening flow has a negative eigenvalue, so no exponential convergence should be expected (see the discussion in [5] right after Theorem 2.2, and notice that when $\lambda=1$, a negative eigenvalue occurs). The good news here is that, being $k$ a curvature, it satisfies an important identity, which is responsible for us being able to get this exponential convergence.

Our methods are based on the techniques employed in [5], that is to say on the Fourier method. Hence, we will transform our problem into (finite dimensional) approximations of an infinite dimensional dynamical system, for which appropriate estimates will be proved, and which will finally lead to a proof of Theorem 1, proof which is given in the final section of this paper. The intermediate sections are devoted to show these appropriate estimates, which, in short, amount to controlling the Fourier wavenumbers of a solution to (1) in terms of the average of the curvature; from this we will be able to show a time decay for the Fourier wavenumbers different from the average (which in fact blows-up), and which, as we said before, will lead to a proof of Theorem 1.

2. Basic definitions and notation

When the initial curve is convex, the $p$-curve shortening flow is equivalent to the following Boundary Value Problem:

[TABLE]

$p\in\mathbb{Z}^{+}$, with periodic boundary conditions, and $\psi$ a strictly positive function. Notice that the Maximum Principle implies that $k$ must remain positive for all times (i.e. a convex curve remains convex). We will need to compute finite dimensional approximations of the previous partial differential equation in Fourier space, so we must establish some definitions and notation. Recall that For $f\in L^{2}\left[0,2\pi\right]$, its Fourier expansion is given by:

[TABLE]

where,

[TABLE]

We shall refer to $\hat{f}\left(n\right)$ as the Fourier wavenumbers of $f$.

We will also adopt the notation

[TABLE]

[TABLE]

and define the following sets

[TABLE]

[TABLE]

and,

[TABLE]

From now on $\mathcal{Z}$ will denote a finite set of integers which contains 0 (i.e, $0\in\mathcal{Z}$), and which is symmetric around 0 (i.e., if $n\in\mathcal{Z}$ then $-n\in\mathcal{Z}$).

Using this notation, in Fourier space, the $p$-curve shortening flow can be approximated by the following finite dimensional dynamical system:

[TABLE]

with initial condition

[TABLE]

Formally, the $\hat{k}$ in the system right above should bear, for instance, a subindex which makes its dependence on $\mathcal{Z}$ explicit, but as this is understood from now on, we will suppress it in what follows (and as our estimates will not depend on $\mathcal{Z}$, this should be of no importance).

Notice also that (4) is an autonomous system, so there is a unique and smooth solution for a short time (see [4]). We will also make use of the seminorms $\left\|{\cdot}\right\|_{\beta}$, which are defined as in [5] as follows:

[TABLE]

As usual, we define $C^{l}\left([0,2\pi]\right)$, $l=0,1,2,\dots,$ as the space of functions with continuous derivatives of order $l$, equipped with the norm

[TABLE]

3. Technical Lemmas and intermediate results

We shall follow closely the arguments presented in [8]. Therefore we must show that for given a solution to (4), we can control the Fourier wavenumbers different from [math] in terms of the [math]-th wavenumber. The fact that the first eigenvalue $\lambda_{1}<0$ is the main difficulty we must face, as this makes difficult to control the $\pm 1$- wave number in terms of the [math] th wave number. Once we have done this, all that is left is to follow the arguments presented in [5, 8]. The key to our proofs is that a curvature function of a locally convex curve satisfies

[TABLE]

since this identity, once we have control over the higher Fourier wavenumbers (those with $\left|n\right|\geq 2$) assuming control over the $\pm 1$ wavenumbers, allows us to control the $\pm 1$ Fourier wavenumbers.

The careful reader must notice that the proofs given in this paper, our estimates are given for system (4), and that this estimates are independent of $\mathcal{Z}$, this allows us to take a limit so the results are valid for the full system (3).

3.1. Controlling the Fourier wavenumbers

We start with a technical lemma.

Lemma 1**.**

There is a $0<\delta<\frac{1}{4}$ such that if the initial condition $\psi$ of (3) satisfies:

[TABLE]

and

[TABLE]

for $t\in(0,\tau)$, then $\hat{k}\left(0,t\right)$ is non decreasing.

Proof.

From the hypothesis of the lemma,

[TABLE]

and the implicit constnat in the big $O$ notation does not depend on $\mathcal{Z}$. Hence, for $\delta>0$ small enough, the term $\dfrac{1}{p}\hat{k}\left(0,t\right)$ dominates the the term $U$ in differential equation for $\hat{k}\left(0,t\right)$. This implies that $\dfrac{d}{dt}\hat{k}\left(0,t\right)>0$, and the conclusion of the lemma follows.

∎

Now we show some control estimates for the Fourier modes,

Lemma 2**.**

There is a $0<\delta<\frac{1}{4}$ such that if the initial condition $\psi$ of (3) satisfies:

[TABLE]

holds, and for $t\in(0,\tau)$

[TABLE]

and

[TABLE]

Then

[TABLE]

Proof.

Let us consider the quantity $R_{n}=\left|\dfrac{\hat{k}\left(n,t\right)}{\hat{\psi}(0)}\right|,$ and we prove that is nonincreasing for $n$ fixed.

We compute:

[TABLE]

where the $B_{i}$ terms are given by:

[TABLE]

We bound $B_{1}$,

[TABLE]

Now if $\mathbf{q}\in\mathcal{C}_{n}\cap\mathcal{Z}^{p+2}$, then

[TABLE]

and hence

[TABLE]

with $C^{\prime}_{p}$ independent of $\mathcal{Z}$. Since $|\hat{k}\left(n,t\right)|=|\hat{k}\left(0,t\right)|/n^{2}$, we get

[TABLE]

Splitting the sums and using similar calculations as in (5), we obtain

[TABLE]

and again $C^{\prime\prime}_{p}$ is independent of $\mathcal{Z}$.

Since the sum of the absolute value of all these terms can be made smaller than

[TABLE]

for $n\geq 2$ by taking $\delta>0$ small enough, then the $R_{m}$ term is non increasing for $\delta>0$ small enough. From this the conclusion of the lemma follows. ∎

As we have been doing so far, in what follows the wave numbers are restricted to a fixed but arbitrary set $\mathcal{Z}$, so keep this in mind. And, as announced at be beginning of this section, since the estimates are independent of $\mathcal{Z}$, a limiting procedure will give the result for when we take as $\mathcal{Z}$ the whole set of integers.

Lemma 3**.**

Let $\psi$ be such that $Q\left(\psi\right)=0$. There is a $0<\delta^{\prime}<\frac{1}{4}$ such that if $0<\delta\leq\delta^{\prime}$ and $\delta\hat{\psi}\left(0\right)\geq\left|\hat{\psi}\left(\pm 1\right)\right|$ then whenever $\left|\hat{k}\left(n,t\right)\right|n^{2}\leq\delta\hat{k}\left(0,t\right)$ for all $\left|n\right|\geq 2$, for all $t\in\left[0,\tau\right]$, then we also have $\left|\hat{k}\left(q,t\right)\right|q^{2}\leq\delta\hat{k}\left(0,t\right)$ for $q=\pm 1$ on the same time interval.

Proof.

Since we have that $\delta\hat{\psi}\left(0\right)\geq\left|\hat{\psi}\left(\pm 1\right)\right|$, we can choose a $\tau^{\prime}\in\left[0,\tau\right]$ such that $\left|\hat{k}\left(\pm 1,t\right)\right|\leq 2\delta\hat{k}\left(0,t\right)$ on $\left[0,\tau^{\prime}\right]$ (remember we are working with an arbitrary but final dimensional approximation of the $p$-curve shortening flow). We have the following identity

[TABLE]

where $z=z(\theta,t)=\sum_{q\neq 0}\frac{\hat{k}(q,t)}{\hat{k}\left(0,t\right)}e^{iq\theta}$. It can be easily seen that for the Fourier modes of $z$ we have:

[TABLE]

Now taking the Fourier transform, this implies

[TABLE]

Since $k$ is the curvature of a convex curve, we have $Q\left(k\right)=0$, so

[TABLE]

then

[TABLE]

In order to estimate the sum in the right side, let us notice that

[TABLE]

Now we proceed to estimate $\displaystyle{S_{m}=\sum_{q_{1}+\cdots+q_{m}=-1}\hat{z}({q_{1}},t)\cdots\hat{z}({q_{m}},t)}$,

[TABLE]

where $C_{1}$ is a constant independent of $\mathcal{Z}$ and $\delta$.

As we have that $\left|\hat{z}({\pm 1},t)\right|\leq 2\delta$, we get

[TABLE]

Therefore

[TABLE]

as long as $\delta\leq\frac{1}{2\left(4+C_{1}\right)}$. ∎

Lemma 4**.**

There is a $0<\delta<\frac{1}{4}$, independent of $\mathcal{Z}$, such that if the initial condition $\psi$ of (3) satisfies $Q(\psi)=0$ and

[TABLE]

then for all times $t$ (as long as the solution to (4) exists),

[TABLE]

Proof.

There exists a $\tau>0$ be such that the interval $[0,\tau]$ is maximal with respect to the following property: for all $t\in[0,\tau]$ we have

[TABLE]

and

[TABLE]

Now applying Lemmas 1 and 2, we obtain that

[TABLE]

whenever $\left|q\right|\geq 2$. Now Applying Lemma 3, we get that

[TABLE]

when $t\in[0,\tau]$. Hence we have that $\delta\psi(0)\geq q^{2}\left|\hat{k}(q,\tau)\right|$ and if we apply the same arguments as before we can show that there is a $\tau_{1}>\tau$ such that if $t\in\left[0,\tau_{1}\right]$ then $\delta\cdot\hat{\psi}(0)\geq q^{2}\left|\hat{k}(q,t)\right|$, contradicting the maximality of $\left[0,\tau\right]$. ∎

For $\delta>0$ small enough, assuming that for the initial condition we have $\delta\hat{\psi}\left(0\right)\geq q^{2}|\hat{\psi}(q)|$, we have now control over all the Fourier wavenumbers of the solution. The arguments in [5] now apply almost verbatim: see the upcoming sections.

3.2. Decay of the Fourier wavenumbers

Again, all the estimates proved in this section are valid for any choice of $\mathcal{Z}$, and are also independent of the choice. Our main purpose is to show that the Fourier wavenumbers $\hat{k}\left(n,t\right)$, $n\neq 0$, go to 0 as $t\rightarrow T$. To begin, we have, as in [5], the Trapping Lemma (Lemma 3.2 in [5]). Keep in mind that we are always under the assumption that $\psi>0$ is the curvature function of a simple convex closed curve (or equivalently, the identity $Q\left(\psi\right)=0$ holds).

Theorem 3** (Trapping Lemma).**

There exists a constant $c_{p}>0$ independent of the choice of $\mathcal{Z}$ such that if the initial datum $\psi$ satisfies the following inequality:

[TABLE]

then there exists a $\gamma>0$ that depends on $\psi$ such that the solution to (4) satisifies:

[TABLE]

Also, in the same way as Lemmas 3.3 and 3.4 are obtained in [5], we have a Blow-up Lemma.

Lemma 5** (Blow-up).**

There is a $c_{p}>0$ (the same as in the Trapping Lemma) such that if the initial condition $\psi$ of (3) satisfies

[TABLE]

then there are constants a number $c,c^{\prime}>0$ such that:

[TABLE]

From now on, we assume that $\psi$ satisfies

[TABLE]

where $c_{p}$ is such that the Trapping Lemma holds.

We also have a few important observations. First, integrating the ODE for $\hat{k}\left(n,t\right)$, we obtain

[TABLE]

where $h(t)$ is given by:

[TABLE]

Applying the Trapping Lemma, we get

[TABLE]

Also, from the Trapping Lemma, there exists $C,\mu>0,$ such that

[TABLE]

We shall use these observations in proving the following decay (in time) estimates for the Fourier wave numbers of $k$.

Proposition 6**.**

There exists $\epsilon_{0}>0$ which depends on $p$, and a $\mu>0$ that depends also on $p$ and on $\psi$, such that if $t>\frac{T}{2}>0$ then there is a constant $b>0$ such that for any $0<\epsilon<\epsilon_{0}$, for $n\neq 0,\pm 1$, the following estimate holds for the solution of (3),

[TABLE]

Proof.

(See also the proof of Lemma 3.6 in [5]) First we have

[TABLE]

where

[TABLE]

We are going to estimate the term inside the integral in the last inequality. As before we split $h(s)$ into sums of the form

[TABLE]

using this, the Trapping Lemma and the observations after its statement, we get

[TABLE]

and since $l\leq p$, we finally obtain

[TABLE]

Then we have

[TABLE]

Without lost of generality we can assume $\eta\leq\frac{1}{2}$. Now using again (10) and the fact that $\alpha(n,p)\geq\alpha(2,p)>0$, obtain

[TABLE]

then we have,

[TABLE]

for any $0<\epsilon<\min\left\{\frac{1}{2}\alpha\left(2,p\right),1-\frac{p}{p+1}\right\}=\epsilon_{0}$. ∎

Next we are going to improve on the decay estimates of the Fourier coefficients. To be able to do this, we will need the following lemma.

Lemma 7**.**

There is a $t_{0}$ such that if $t\in(t_{0},T)$, then we have the estimates

[TABLE]

and

[TABLE]

Proof.

We have that the following differential inequality

[TABLE]

holds for a constant $A>0$ independent of $t$. This is equivalent to

[TABLE]

Using Lemma 5, from the previous differential inequality we obtain

[TABLE]

The result follows by integration. For the other inequality, notice that there exists a constant $A^{\prime}$ so we also have the following differential inequality

[TABLE]

∎

In order to proceed we use previous proposition to estimate the integral

[TABLE]

from below, from previous lemma, since $\delta$ is small, using Taylor’s Theorem, this shows

[TABLE]

$c>0,$ and using this and (9), we get

[TABLE]

here

[TABLE]

Using the estimate of the proposition and the fact that if $\mathbf{q}\in\mathcal{A}_{n}$ then $\mathbf{q}$ has at least two entries different from [math],we get

[TABLE]

If we introduce the bound from Proposition 6 in (3.2) we get,

[TABLE]

for a well chosen $0<\mu^{\prime}<\mu$, and from which we obtain the estimate (we use again the fact $\alpha(n,p)\geq\alpha(2,p)$)

[TABLE]

If we assume that $\alpha(2,p)>1-\frac{p}{p+1}+2\epsilon,$ using this new bound an pluggin it into (3.2), we improve again our estimate on $\hat{k}\left(n,t\right)$:

[TABLE]

,

Finally , if we repeat this procedure a finite number of times we arrive at

[TABLE]

where $\xi$ is a constant independent of $n$, and $\alpha\left(2,p\right)$ is defined by (12) (so is value its $\left(3p-2\right)/\left(p+1\right)$).

Now, we must show now that the wavenumbers $\hat{k}\left(\pm 1,t\right)$ satisfy the same estimate. In this case we write

[TABLE]

And we have an identity which follows from $Q\left(k\right)=0$ (here we use that $\hat{z}\left(-1,t\right)$ is the conjugate of $\hat{z}\left(1,t\right)$, as $z$ is real valued)

[TABLE]

where the prime (′) in the inner sum of the righthand side indicates that at least one of the $q_{j}\neq\pm 1$. Using similar computations as in the proof of Lemma 3, together with (13), we can conclude that

[TABLE]

hence, if $\delta>0$ is small enough, we can deduce that

[TABLE]

which is just that $\left|\hat{k}\left(\pm 1,t\right)\right|\leq C\left(T-t\right)^{\frac{3p-2}{p+1}}$. So we have proved

Proposition 8**.**

Let $\psi>0$ is a smooth $2\pi$-periodic function which satisfies that $Q\left(\psi\right)=0$. There exists a positive constant $c_{p}$ such that if

[TABLE]

then a solution to (3) satisfies

[TABLE]

where $T$ is the blow-up time and $M_{p,k}$ is a constant that depends only on $p,k$ and $\psi$.

We normalize the solution of (3) by means of the following transformation:

[TABLE]

Applying chain rule, we obtain the following normalized version of (3)

[TABLE]

Using this normalisation, Proposition 8 translates into:

Corollary 9**.**

Let $\psi>0$ is a smooth $2\pi$-periodic function which satisfies that $Q\left(\psi\right)=0$. There exists a positive constant $c_{p}$ such that if

[TABLE]

then the normalization $\tilde{k}$ of $k$ satisfies:

[TABLE]

where $M_{p,k}$ is a positive constant that depends only on $p,k$ and $\psi$.

4. Exponential convergence of the normalised curvature towards 1: Proof of the main result

We will need the following improvement over Lemma 7.

Lemma 10**.**

The following estimates hold

[TABLE]

and

[TABLE]

Proof.

Notice that using (13) and the equation satisfied by $\hat{k}\left(0,t\right)$, we have the differential inequality, which is valid for a constant $A>0$

[TABLE]

Integrating, from $t$ to $T$, we obtain the second inequality. Analogously for a constant $A^{\prime}$, we have the differential inequality

[TABLE]

which by integration gives the first inequality. ∎

Finally we have our main result.

Theorem 4**.**

Let $\psi>0$ be the initial condition of (3) (so it is the curvature function of a convex simple curve). Then there exists a constant $c_{p}>0$ such that if

[TABLE]

then the solution to (14) satisfies

[TABLE]

where $C_{p,l}$ is a constant that only depends on the initial condition $\psi$ and $p$ and $l$.

Proof.

Let

[TABLE]

now we compute

[TABLE]

Analogously,

[TABLE]

In this case, $\displaystyle{e^{-\tau}=\left(\frac{T-t}{T}\right)^{\frac{1}{p+1}}}$, then

[TABLE]

Applying the triangular inequality and Corollary 9 we can conclude that

[TABLE]

for some constant $C_{p,l}$ that depends on $p$ and $l$.

∎

4.1. Final Remarks

The rate of convergence obatined in Theorem, seems to be the best possible in general. We have not been able to produce an example where the rate given in the theorem is met; however, to justify our claim, we refer to the comments after the statement of Theorem 2.2 in [5]: The first positive eigenvalue of the elliptic part of (14), i.e., the left hand side of the equation, when linearised around the steady solution $\tilde{k}\equiv 1$ is precisely $3p-1$.