The Gradient Flow for Entropy on Closed Planar Curves

Abstract

In this paper we consider the steepest descent $L^2$-gradient flow of the entropy functional. The flow expands convex curves, with the radius of an initial circle growing like the square root of time. Our main result is that, for any initial curve (either immersed locally strictly convex of class $C^2$ or embedded of class $W^{2,2}$ bounding a strictly convex body), the flow converges smoothly to a round expanding multiply-covered circle.

1 Introduction

Suppose $\gamma :{\mathbb {S}}^1\rightarrow {\mathbb {R}}^2$ is a locally strictly convex closed plane curve with turning number $\omega$ and consider the energy

$$\begin{aligned} {\mathcal {E}}(\gamma ) = \int _0^L k\log k\, \textrm{d}s = \int _0^{2\omega \pi } \log k\,\textrm{d}\theta . \end{aligned}$$

In the first equality $\gamma$ is parametrised by arc-length, and in the second parametrised by angle.

The $L^2(\textrm{d}\theta )$-gradient flow of ${\mathcal {E}}$ is the one-parameter family of smooth immersed curves $\gamma :{\mathbb {S}}^1\times (0,T)\rightarrow {\mathbb {R}}^2$ with normal velocity equal to $-{\text {grad}}_{L^2(\textrm{d}\theta )}({\mathcal {E}}(\gamma ))$ (see Section 2), that is,

$$\begin{aligned} \partial _t\gamma (\theta ,t) = -(k_{\theta \theta }(\theta ,t) + k(\theta ,t))N(\theta ). \end{aligned}$$

(EF)

Note that in the above evolution equation the $\theta$ and t variables are independent.

The entropy flow (EF) is a highly degenerate system of fourth order parabolic partial differential equations. The $\partial _\theta$ derivative involves division by the curvature scalar k. Expressing (EF) in the arbitrary parametrisation, the leading-order term $k_{\theta \theta }$ is

$$\begin{aligned} - \frac{|\gamma _u|}{\langle \gamma _{uu},N\rangle } \Bigg ( \frac{|\gamma _u|}{\langle \gamma _{uu},N\rangle } \bigg ( \frac{\langle \gamma _{uu},N\rangle }{|\gamma _u|^{2}} \bigg )_u \Bigg )_u, \end{aligned}$$

(1)

whose highest-order component is

$$\begin{aligned} - \frac{\langle \gamma _{uuuu},N\rangle }{\langle \gamma _{uu},N\rangle ^2}. \end{aligned}$$

Since $\langle \gamma _{uu},N\rangle = k|\gamma _u|^2$, and $|\gamma _u|\ne 0$ along any regular curve, the degeneracy in the highest-order term in the operator is proportional to $k^{-2}$. This is much more than the usual “powers of the gradient $|\gamma _u|$” degeneracy that one typically finds in curvature flow of curves. For local existence, we see this in the hypothesis of our first local existence theorem (Theorem A.1). This result says that the flow exists uniquely from locally strictly convex initial data of class $W^{3,\infty }(du)$. The flow instantly becomes smooth (that is, for $t>0$), and if $T<\infty$ then the $W^{3,\infty }(du)$ norm explodes as $t\nearrow T$. The proof of Theorem A.1 is via the method of maximal regularity and analytic semigroups.

The entropy flow expands any initial circle, with radius over time given by $r(t) = \sqrt{r_0^2 + 2t}$ ($r_0$ is the radius of the initial circle). A natural question is on the stability of this homothetic solution. This correlates well with other work on fourth-order flows, for instance stability of circles under curve diffusion [8, 16, 21], the elastic flow [9], and Chen’s flow [5, 7].

Here there are a few troubling aspects of the entropy flow from the outset.

First, the energy is not bounded from below and so we should be concerned about ${\mathcal {E}}(t)\searrow -\infty$ as $t\nearrow T < \infty$.

Second, the flow only makes sense in the class of locally strictly convex curves. It is not expected that any kind of positivity preserving principle holds for higher-order evolution equations [4, 11]. The failure of closed strictly convex plane curves to remain convex under a large class of fourth-order curvature flows is well-known (see Blatt [6], Giga-Ito [12, 13], Elliott-MaierPaape [10] for related results).

The paper of Andrews [3] stands in stark contrast to the literature above. There, the gradient flow for the affine length functional is considered. The main result is that any embedded convex curve flows for infinite time, becoming larger and closer to an homothetically expanding ellipse. A basic property of the flow is that convexity is preserved, despite the flow being of fourth-order. The idea is that the degeneracy in the operator is so powerful that the flow moves away from singular curves – those that are not strictly convex.

Our work here adds another example to the one of Andrews, of a higher-order flow with strong degeneracy that has a similar ‘singularity avoidance’ property. Despite the energy functional ${\mathcal {E}}$ having no lower bound, and the flow being fourth-order defined only on locally strictly convex curves, the resultant flow (EF) exhibits several beautiful characteristics. These include concavity of evolving length and convexity of the energy over time. The main consequence is a gradient estimate for the curvature and preservation of local strict convexity for all time (Proposition 3.17). Another remarkable aspect of (EF) then comes into play: monotonicity of the Dirichlet energy for the associated support function. This powerful diffusive effect allows us to obtain convergence of a rescaling of the flow in the $C^1$-topology.

None of these results require any kind of smallness condition or closeness to a circle. Furthermore, they work together to give a powerful smoothing effect, beyond that already attained by the $W^{3,\infty }(du)$ theorem mentioned above. In the class of embeddings, we push this to initial curves that have curvature function in $L^2(\textrm{d}s)$, and bound a convex domain. For immersions we are able to treat locally strictly convex curves with continuous curvature function.

Theorem 1.1

Suppose $\gamma _0:{\mathbb {S}}^1\rightarrow {\mathbb {R}}^2$ is either

• An immersed locally strictly convex closed curve of class $C^2(\textrm{d}s)$ with turning number $\omega$; or

• An embedded curve with $k\in L^2(\textrm{d}s)$ that bounds a strictly convex body.

The entropy flow $\gamma :{\mathbb {S}}^1\times (0,T)\rightarrow {\mathbb {R}}^2$ with $\gamma _0$ as initial data is a one-parameter family of regular smooth immersed locally strictly convex closed curves that exists for all time ($T=\infty$). The rescaling $\eta = \gamma /\sqrt{L_0^2 + 8\omega ^2\pi ^2 t}$ converges to a round $\omega$-circle in the sense that the rescaled support function converges to a constant in $C^\infty (\textrm{d}\theta )$.

Above we use the terminology from [19]: a strictly convex body is a non-empty, compact, strictly convex subset of ${\mathbb {R}}^2$.

Remark 1.2

Given that our flow is higher-order, the initial data being either immersed of class $C^2(\textrm{d}s)$ or embedded of class $H^2(\textrm{d}s)$ is quite weak. A similar phenomenon (but lower order) has been observed by Andrews [2, Theorem I2.1].

This paper is organised as follows. Section 2 is concerned with setting notation, giving some well-known properties of strictly convex plane curves, and the derivation of the flow as the steepest descent $L^2(\textrm{d}\theta )$-gradient flow for ${\mathcal {E}}$. The diligent reader may then continue with Appendix A, which gives an outline of our first local existence proof that requires $W^{3,\infty }(du)$ data. This is an important foundation for Section 3. The main goal of Section 3 is to derive powerful a-priori estimates that yield preservation of convexity and a smoothing effect. The latter gives, via a compactness argument, Theorem 3.24. This proof uses the $W^{3,\infty }(du)$ result from Appendix A. The global existence part of Theorem 1.1 is also established in Section 3 (see Corollary 3.18). Section 4 studies the global behaviour of the flow, and is concerned with giving estimates for a scale-invariant quantity that imply it is eventually monotone. In Section 5 we also prove a sharp lower bound for kL, which is an optimal convexity estimate. Finally, in Section 5, we study the rescaled flow $\eta$. We prove uniform estimates for rescaled length and curvature, then use these to show exponential decay of $||h^\eta _{\theta ^p}||_2^2(t)$ for all $p\in {\mathbb {N}}$, establishing the asymptotic roundness part of Theorem 1.1 in a standard way.

2 Preliminaries

To any regular closed curve $\gamma :{\mathbb {S}}^1\rightarrow {\mathbb {R}}^2$ with $\gamma \in C^1(du)$ we can associate an angle function $\theta :{\mathbb {S}}^1\rightarrow [0,2\omega \pi )$ (here $\omega$ is the turning number, and note that $\theta$ is not periodic), defined by

$$\begin{aligned} N(\theta (u)) = -(\cos \theta (u), \sin \theta (u)), \end{aligned}$$

where N is the inward unit normal vector.

If $\gamma \in C^2$ is strictly locally convex, the mapping $s\mapsto \theta (s)$ (here s is the Euclidean arc-length) satisfies

$$\begin{aligned} \theta _s = k > 0. \end{aligned}$$

This makes $\theta$ invertible on $[0,2\omega \pi )$, and so we may use $\theta$ as an independent variable. We define the support function $h:[0,2\omega \pi )\rightarrow (0,\infty )$ via

$$\begin{aligned} h(\theta ) = \langle \gamma (\theta ),-N(\theta )\rangle . \end{aligned}$$

Note that our energy (and the resultant flow) is invariant under translation, so we may without loss of generality assume that $\gamma ({\mathbb {S}}^1)$ encloses the origin so that the support function is positive.

If $\gamma \in W^{2,2}(\textrm{d}s)$ is an embedding, the Jordan curve theorem implies that ${\mathbb {R}}^n{\setminus }\gamma ({\mathbb {S}}^1)$ consists of two connected components: one unbounded and the other bounded, commonly referred to in this context as the ‘interior’ of $\gamma$. Here, we use the notation ${\mathcal {K}}(\gamma )$ for this ‘interior’ set together with its boundary. In words, we say that ${\mathcal {K}}(\gamma )$ is the strictly convex body associated to $\gamma$. As suggested by the terminology, we assume that ${\mathcal {K}}(\gamma )$ is a non-empty compact strictly convex set, so the support function $h:[0,2\pi )\rightarrow (0,\infty )$ again exists but with the more usual convex geometry definition that

$$\begin{aligned} h(\theta ) = h_{{\mathcal {K}}(\gamma )}(H_\theta ) = \sup _{x\in {\mathcal {K}}(\gamma )} \langle x,(\cos \theta ,\sin \theta )\rangle . \end{aligned}$$

Here we used $H_\theta$ to denote the supporting line to ${\mathcal {K}}(\gamma )$ that is normal to $(\cos \theta , \sin \theta )$. The strict convexity of ${\mathcal {K}}(\gamma )$ ensures that the support function h is of class $C^1$ and almost everywhere of class $C^2$. Note that the condition that $k\in L^2(\textrm{d}s)$ implies that the tangent vector to $\gamma$ is uniformly Hölder continuous of class $C^{0,1/2}$ with respect to arclength. We refer the reader to [19] for the essential theory of convex bodies and related facts on support functions.

We work for the remainder of this section with smooth curves. We have the fundamental relations (via the chain rule)

$$\begin{aligned} k\,\partial _\theta = \partial _s\quad {\text {and}}\quad \textrm{d}\theta = k\,\textrm{d}s. \end{aligned}$$

Note that then $\partial _\theta \partial _\theta = k^{-2}\partial _s\partial _s - k_sk^{-3}\partial _s$, so

$$\begin{aligned} h_{\theta \theta } + h&= \langle \gamma _{\theta \theta },-N\rangle + 2\langle \gamma _\theta ,-N_\theta \rangle - \langle \gamma ,N_{\theta \theta }+N\rangle \\&= \langle \gamma _{\theta \theta },-N\rangle + 2\langle \gamma _\theta ,-N_\theta \rangle \\&= \langle k^{-2}\kappa - k_sk^{-3}T,-N\rangle + 2\langle \frac{1}{k} T,T\rangle \\&= -\frac{1}{k} + 2\frac{1}{k} = \frac{1}{k}. \end{aligned}$$

In the second equality we used $N_{\theta \theta } = -T_\theta = -N$, which follows from the definition of $\theta$.

Summarising, the curvature of $\gamma$ satisfies

$$\begin{aligned} k(\theta ) = \frac{1}{h_{\theta \theta }(\theta ) + h(\theta )}. \end{aligned}$$

(2)

We are interested in the steepest descent $L^2$-gradient flow for the entropy of $\gamma$ in the $L^2(\textrm{d}\theta )$ metric. To this end, suppose $\gamma :{\mathbb {S}}^1\times (0,T_0)\rightarrow {\mathbb {R}}^2$ is now a one-parameter family of smooth locally convex curves differentiable in time with associated support functions $h:[0,2\omega \pi )\times (0,T_0)\rightarrow (0,\infty )$ satisfying

$$\begin{aligned} h_t = F(h). \end{aligned}$$

Note that here the t and $\theta$ derivatives are independent. We now have:

Lemma 2.1

Define ${\mathcal {E}}(t):= {\mathcal {E}}(\gamma (\cdot ,t))$. Then

$$\begin{aligned} {\mathcal {E}}'(t) = -\int _0^{2\omega \pi } F(k_{\theta \theta }+k)\,\textrm{d}\theta . \end{aligned}$$

Proof

We calculate using (2) and integration by parts:

$$\begin{aligned} {\mathcal {E}}'(t)&= \int _0^{2\omega \pi } \frac{k_t}{k}\,\textrm{d}\theta \\&= -\int _0^{2\omega \pi } \frac{1}{k}\frac{(h_{\theta \theta }+h)_t}{(h_{\theta \theta }+h)^2}\,\textrm{d}\theta \\&= -\int _0^{2\omega \pi } k(h_{\theta \theta }+h)_t\,\textrm{d}\theta \\&= -\int _0^{2\omega \pi } k(F_{\theta \theta }+F)\,\textrm{d}\theta \\&= -\int _0^{2\omega \pi } F(k_{\theta \theta }+k)\,\textrm{d}\theta . \end{aligned}$$

$\square$

Lemma 2.1 allows us to conclude that the steepest descent $L^2(\textrm{d}\theta )$-gradient flow for ${\mathcal {E}}$ is the evolution equation

$$\begin{aligned} h_t = k_{\theta \theta } + k \end{aligned}$$

where $h:[0,2\omega \pi )\times (0,T_0)\rightarrow (0,\infty )$ is the support function associated to a family of curves $\gamma :{\mathbb {S}}^1\times (0,T_0)\rightarrow (0,\infty )$. Note that we may sometimes write $h:{\mathbb {S}}^1\times (0,T_0)\rightarrow (0,\infty )$, where in this expression we understand ${\mathbb {S}}^1$ to denote $[0,2\omega \pi )$.

3 Fundamental Estimates and Rough Data

In this section, we establish bespoke estimates to consider general initial data. This is either locally convex and in $C^2(\textrm{d}s)$ or an embedding bounding a convex planar domain with curvature in $L^2(\textrm{d}s)$.

The fact that the entropy flow is the steepest descent $L^2(\textrm{d}\theta )$ flow for ${\mathcal {E}}$ immediately implies

$$\begin{aligned} \int _0^t \Vert F\Vert _2^2({{\hat{t}}})\,\textrm{d}{{\hat{t}}} = {\mathcal {E}}(0) - {\mathcal {E}}(t). \end{aligned}$$

However note that this is of limited use a-priori since ${\mathcal {E}}(t)$ is, for general convex curves, unbounded from below.

First, we must rule out that for $t\nearrow T_0<\infty$, ${\mathcal {E}}(t)\searrow -\infty$. We begin with the following generic estimate:

Lemma 3.1

Suppose $\gamma :{\mathbb {S}}^1\rightarrow {\mathbb {R}}^2$ is a locally convex curve. The entropy ${\mathcal {E}}(\gamma )$ is bounded from below by the logarithm of the reciprocal of length:

$$\begin{aligned} {\mathcal {E}}(\gamma ) \leqq 2\omega \pi \log \frac{2\omega \pi }{L(\gamma )}. \end{aligned}$$

Proof

Since $\log$ is concave, we have by Jensen’s inequality

$$\begin{aligned} \log \Bigg (\frac{1}{2\omega \pi }\int \frac{1}{k}\,\textrm{d}\theta \Bigg ) \leqq \frac{1}{2\omega \pi }\int \log \frac{1}{k}\,\textrm{d}\theta . \end{aligned}$$

Now just recall that the RHS of the above estimate is $-{\mathcal {E}}(\gamma )$ and

$$\begin{aligned} L(\gamma ) = \int \frac{1}{k}\,\textrm{d}\theta . \end{aligned}$$

This implies

$$\begin{aligned} \log \frac{L(\gamma )}{2\omega \pi } \leqq -\frac{1}{2\omega \pi }{\mathcal {E}}(\gamma ). \end{aligned}$$

Multiplying by $-2\omega \pi$ yields the claimed estimate. $\square$

The following lemma shows that our initial data hypotheses bound the initial energy:

Lemma 3.2

Suppose $\gamma :{\mathbb {S}}^1\rightarrow {\mathbb {R}}^2$ is either

• An immersed locally strictly convex closed curve of class $C^2(\textrm{d}s)$ with turning number $\omega$; or

• An embedded curve with $k\in L^2(\textrm{d}s)$ that bounds a strictly convex body.

Then

$$\begin{aligned} \bigg |\int \log k\,\textrm{d}\theta \bigg | = |{\mathcal {E}}(\gamma )| < \infty . \end{aligned}$$

(3)

Proof

In the first case, we can work pointwise. Let $\underline{k} = \inf k$, ${\overline{k}} = \sup k$. We know that $0< \underline{k} \leqq {\overline{k}} < \infty$ due to the fact that $\gamma$ is of class $C^2(\textrm{d}s)$. Then $\log \underline{k} \leqq \log k \leqq \log {\overline{k}}$ and (3) follows immediately by integration.

In the second case, we estimate from below using Lemma 3.1. The estimate from above is as follows:

$$\begin{aligned} {\mathcal {E}}(\gamma ) = \int _0^{L} k\log k\,\textrm{d}s \leqq \int _0^{L} k(k-1)\,\textrm{d}s \leqq ||k||_{L^2(\textrm{d}s)}^2. \end{aligned}$$

$\square$

Let us now show that the energy is convex in time.

Lemma 3.3

Suppose $\gamma :{\mathbb {S}}^1\times (0,T)\rightarrow {\mathbb {R}}^2$ is a smooth entropy flow. Then

$$\begin{aligned} {\mathcal {E}}' = -\int (k_{\theta \theta } + k)^2\,\textrm{d}\theta = - \int F^2\,\textrm{d}\theta \end{aligned}$$

(4)

(where we recall that $F = k_{\theta \theta } + k$) and

$$\begin{aligned} {\mathcal {E}}'' = 2\int \big (k(F_{\theta \theta } + F)\big )^2\,\textrm{d}\theta . \end{aligned}$$

(5)

Proof

The equality (4) follows immediately from Lemma 2.1 and the definition of the flow. Differentiating again, we find that

$$\begin{aligned} {\mathcal {E}}''&= -\frac{\textrm{d}}{\textrm{d}t}\int (k_{\theta \theta } + k)^2\,\textrm{d}\theta \nonumber \\&= -2\int (k_{\theta \theta } + k)(k_{t\theta \theta } + k_t)\,\textrm{d}\theta \nonumber \\&= -2\int (k_{\theta \theta } + k)_{\theta \theta }k_{t}\,\textrm{d}\theta + -2\int (k_{\theta \theta } + k)k_t\,\textrm{d}\theta \nonumber \\&= -2\int k_t\Big (F_{\theta \theta } + F\Big )\,\textrm{d}\theta . \end{aligned}$$

(6)

Note that

$$\begin{aligned} k_t = \Bigg (\frac{1}{h_{\theta \theta }+h}\Bigg )_t = -k^2 (F_{\theta \theta } + F) \end{aligned}$$

(7)

which, upon combining with (6), yields (5). $\square$

The convexity of ${\mathcal {E}}$ yields the following a priori estimate:

Corollary 3.4

Suppose $\gamma :{\mathbb {S}}^1\times (0,T)\rightarrow {\mathbb {R}}^2$ is a smooth entropy flow. Fix $\delta \in (0,T)$. Then for all $t\in (0,T)$ we have

$$\begin{aligned} \int k_{\theta \theta }^2 + k^2\,\textrm{d}\theta \bigg |_t \leqq 2\int k_\theta ^2\,\textrm{d}\theta \bigg |_t + c_0(\delta ), \end{aligned}$$

(8)

where

$$\begin{aligned} c_0(\delta ) = \Vert F\Vert _2^2(\delta ). \end{aligned}$$

Proof

Integrating (5) yields

$$\begin{aligned} {\mathcal {E}}'(t_0) + 2\int _\delta ^{t_0} \int \big (k(F_{\theta \theta } + F)\big )^2\,\textrm{d}\theta \,dt = {\mathcal {E}}'(\delta ) \end{aligned}$$

which implies

$$\begin{aligned} \Vert F\Vert _2^2(t_0) \leqq c_0(\delta ). \end{aligned}$$

(9)

Integration by parts on the cross term in $\Vert F\Vert _2^2(t)$ yields

$$\begin{aligned} \Vert F\Vert _2^2 = \int k_{\theta \theta }^2 + k^2\,\textrm{d}\theta + 2\int k_{\theta \theta }k\,\textrm{d}\theta = \int k_{\theta \theta }^2 + k^2\,\textrm{d}\theta - 2\int k_{\theta }^2\,\textrm{d}\theta . \end{aligned}$$

(10)

Combining (9) with (10) gives the claimed estimate (8). $\square$

Lemma 3.1 implies that any possible finite-time explosion of the energy to $-\infty$ would require a finite-time explosion to $+\infty$ of the evolving length. This leads us to investigate the dynamics of L.

Lemma 3.5

Suppose $\gamma :{\mathbb {S}}^1\times (0,T)\rightarrow {\mathbb {R}}^2$ is a smooth entropy flow. The length functional L is monotone increasing and concave; in particular,

$$\begin{aligned} L' = \frac{\textrm{d}}{\textrm{d}t} \int h\,\textrm{d}\theta = \int k\,\textrm{d}\theta \end{aligned}$$

(11)

and

$$\begin{aligned} L'' \leqq -\frac{1}{2}\int kk_{\theta \theta }^2\,\textrm{d}\theta - \frac{1}{3} \int k^3\,\textrm{d}\theta . \end{aligned}$$

Proof

Recalling (7) and $h_t = F = k_{\theta \theta } + k$, we calculate

$$\begin{aligned} L' = \frac{\textrm{d}}{\textrm{d}t}\int h\,\textrm{d}\theta = \int k_{\theta \theta } + k\,\textrm{d}\theta = \int k\,\textrm{d}\theta \,, \end{aligned}$$

which is positive.

Differentiating again, we find that

$$\begin{aligned} L''&= \frac{\textrm{d}}{\textrm{d}t}\int k\,\textrm{d}\theta \\&= \int k_t\,\textrm{d}\theta \\&= -\int k^2(F_{\theta \theta } + F)\,\textrm{d}\theta \\&= 2\int kk_\theta F_\theta \,\textrm{d}\theta - \int k^2F\,\textrm{d}\theta \\&= - 2\int (kk_\theta )_\theta k_{\theta \theta }\,\textrm{d}\theta + 2\int kk_\theta ^2\,\textrm{d}\theta + \int (k^2)_\theta k_\theta \,\textrm{d}\theta - \int k^3\,\textrm{d}\theta \\&= - 2\int kk_{\theta \theta }^2\,\textrm{d}\theta - 2\int k^2k_{\theta \theta }\,\textrm{d}\theta - \int k^3\,\textrm{d}\theta \\&\leqq - \frac{1}{2}\int kk_{\theta \theta }^2\,\textrm{d}\theta - \frac{1}{3}\int k^3\,\textrm{d}\theta . \end{aligned}$$

In the last step we used an instance of the Cauchy inequality: $ab \leqq (3/4)a^2 + (1/3)b^2$. $\square$

Similarly to Corollary 3.4, this yields a uniform estimate. Note that in terms of the initial regularity, the estimate requires only that the $L^1(\textrm{d}\theta )$-norm of k at initial time be bounded. (In the arclength parametrisation, this is the $L^2(\textrm{d}s)$-norm of k.)

Corollary 3.6

Suppose $\gamma :{\mathbb {S}}^1\times [0,T)\rightarrow {\mathbb {R}}^2$ is a smooth entropy flow with $\gamma (\cdot ,0)$ satisfying the hypothesis of Lemma 3.2. Then

$$\begin{aligned} \int k\,\textrm{d}\theta + \int _0^t\int \Bigg (\frac{1}{2}kk_{\theta \theta }^2 + \frac{1}{3} k^3\Bigg )\,\textrm{d}\theta \,\textrm{d}{{\hat{t}}} \leqq c_1, \end{aligned}$$

(12)

where $c_1 = \Vert k\Vert _1(0)$.

We now wish to control the long-time asymptotic growth of length from above and below along the flow (although we don’t have long-time existence yet, once we do, this will be decisive). An easy estimate from above of the form

$$\begin{aligned} L(t) \leqq L_0 + c_1t \end{aligned}$$

follows by combining Corollary 3.6 with (11). However this is far from sharp (the length of growing circles is like $\sqrt{t}$), and we can do significantly better.

Lemma 3.7

Suppose $\gamma :{\mathbb {S}}^1\times (0,T)\rightarrow {\mathbb {R}}^2$ is a smooth entropy flow. The rate of blowup for length is asymptotically $\sqrt{t}$; more precisely

$$\begin{aligned} \sqrt{L_0^2 + 8\omega ^2\pi ^2t} \leqq L(t) \leqq L_0 + 4\omega \pi c_1^{-1}\Bigg ( \sqrt{4\omega ^2\pi ^2 + tc_1^2} - 2\omega \pi \Bigg ). \end{aligned}$$

Proof

For the estimate from below, we use again Jensen’s inequality with length. Since $t\mapsto 1/t$ is convex, we find

$$\begin{aligned} \frac{1}{\frac{1}{2\omega \pi }\int k\,\textrm{d}\theta } \leqq \frac{1}{2\omega \pi } \int \frac{1}{k}\,\textrm{d}\theta . \end{aligned}$$

Therefore

$$\begin{aligned} \int k\,\textrm{d}\theta \leqq \frac{4\omega ^2\pi ^2}{L}. \end{aligned}$$

Combining this with (11) yields

$$\begin{aligned} (L^2)' \leqq 8\omega ^2\pi ^2 \end{aligned}$$

and the bound from below follows by integration.

First, note that we may estimate

$$\begin{aligned} - 2\int kk_{\theta \theta }^2\,\textrm{d}\theta - 2\int k^2k_{\theta \theta }\,\textrm{d}\theta - \int k^3\,\textrm{d}\theta \leqq -\frac{1}{2}\int k^3\,\textrm{d}\theta \end{aligned}$$

by using $2ab \leqq 2a^2 + \frac{1}{2}b^2$. (This does not yield the good higher order term from Lemma 3.5, but it is better for our present aim.)

Using Hölder we find

$$\begin{aligned} \frac{1}{2}\int k^3\,\textrm{d}\theta \leqq \frac{1}{2} \frac{1}{4\omega ^2\pi ^2} \bigg (\int k\,\textrm{d}\theta \bigg )^3 = \frac{1}{8\omega ^2\pi ^2}(L')^3. \end{aligned}$$

Using this estimate we find

$$\begin{aligned} L'' \leqq -\frac{1}{8\omega ^2\pi ^2} (L')^3(t) \end{aligned}$$

which implies

$$\begin{aligned} |L'| \leqq \frac{2\omega \pi }{\sqrt{4\omega ^2\pi ^2c_1^{-2} + t}}. \end{aligned}$$

(13)

(Note that $L'(t) > 0$ so the absolute value sign is not strictly needed.) This yields the estimate

$$\begin{aligned} L(t) \leqq L_0 + 4\omega \pi c_1^{-1}\bigg ( \sqrt{4\omega ^2\pi ^2 + tc_1^2} - 2\omega \pi \bigg ) \end{aligned}$$

which is the upper bound, and finishes the proof. $\square$

Remark 3.8

Note that the estimate (13) implies that $\int k\,\textrm{d}\theta$ asymptotically decays (assuming long time existence), with rate $\frac{1}{\sqrt{t}}$.

Remark 3.9

The length estimate presented here is the remarkably powerful statement that all convex curves have length evolving with the same asymptotic power of t (which is of course the same as the circle). This will be more interesting after we have shown that generic curves exist globally in time. An initial circle with radius $r_0>0$ has support function $h(\theta ,t) = \sqrt{r_0^2 + 2t}$, and length

$$\begin{aligned} L^{\textrm{circ}}(t) = \sqrt{8\omega ^2\pi ^2t + L_0^2}. \end{aligned}$$

This shows that the lower bound is sharp. The rate ($\sqrt{t}$) of the upper bound is sharp but the form it is in could be tighter. For the evolving circle, we have $c_1 = \frac{2\omega \pi }{r_0} = \frac{4\omega ^2\pi ^2}{L_0}$, so the corresponding upper bound of Lemma 3.7 reads as

$$\begin{aligned} L_0 + \frac{L_0}{\omega \pi } \bigg ( \sqrt{4\omega ^2\pi ^2 + 16t\,\pi ^4\omega ^4} - 2\omega \pi \bigg ) = L_0 + 2\bigg ( \sqrt{4\omega ^2\pi ^2 t + L_0^2} - L_0\bigg ); \end{aligned}$$

nevertheless, it is good enough for our purposes.

Remark 3.10

Given our study of length, it is natural to consider the signed enclosed area. We find A is increasing monotonically, blowing up as $t\nearrow \infty$. The rate is approximately linear in time; in particular,

$$\begin{aligned} A' = \frac{\textrm{d}}{\textrm{d}t} \frac{1}{2}\int \frac{h}{k}\,\textrm{d}\theta = 2\pi + \int ((\log k)_\theta )^2\,\textrm{d}\theta . \end{aligned}$$

This yields a sharp bound from below: $A(t) \leqq A_0 + 2\pi t$. The isoperimetric inequality and Lemma 3.7 gives a bound from above.

Combining Lemma 3.7 with Lemma 3.1 allows us to rule out the possibility of the energy exploding to $-\infty$ in finite time.

Corollary 3.11

Suppose $\gamma :{\mathbb {S}}^1\times [0,T)\rightarrow {\mathbb {R}}^2$ is a smooth entropy flow with $\gamma (\cdot ,0)$ satisfying the hypothesis of Lemma 3.2. Then

$$\begin{aligned} {\mathcal {E}}_0 \leqq {\mathcal {E}}(t) \leqq 2\omega \pi \log \frac{2\omega \pi }{L_0 + 4\omega ^2\pi ^2c_1^{-1}\big ( \sqrt{4\omega ^2\pi ^2 + tc_1^2} - 2\omega \pi \big )}. \end{aligned}$$

The length estimate also allows us to conclude a smoothing-type effect for the velocity in $L^2$.

Corollary 3.12

Suppose $\gamma :{\mathbb {S}}^1\times [0,T)\rightarrow {\mathbb {R}}^2$ is a smooth entropy flow with $\gamma (\cdot ,0)$ satisfying the hypothesis of Lemma 3.2. For each $t\in (0,T)$,

$$\begin{aligned} \Vert F\Vert _2^2(t) < \infty . \end{aligned}$$

(14)

Proof

We have

$$\begin{aligned} \int _0^t \Vert F\Vert _2^2({{\hat{t}}})\,\textrm{d}{{\hat{t}}} = {\mathcal {E}}_0 - {\mathcal {E}}(t). \end{aligned}$$

Note that here we used Lemma 3.2 to ensure that ${\mathcal {E}}_0$ on the RHS of the above is finite.

Using Corollary 3.11 we refine this to

$$\begin{aligned} \int _0^t \Vert F\Vert _2^2({{\hat{t}}})\,\textrm{d}{{\hat{t}}} \leqq {\mathcal {E}}_0 + 2\omega \pi \log \frac{L_0 + 4\omega ^2\pi ^2c_1^{-1}\big ( \sqrt{4\omega ^2\pi ^2 + tc_1^2} - 2\omega \pi \big )}{2\omega \pi }. \end{aligned}$$

(15)

This implies that $\int _0^t \Vert F\Vert _2^2({{\hat{t}}})\,\textrm{d}{{\hat{t}}}$, where $t\in (0,T)$, is uniformly bounded (by a constant depending only on ${\mathcal {E}}_0$, $L_0$, $c_1$, $\omega$ and T). Thus there exists a $t_0\in (0,t)$ such that $\Vert F\Vert _2^2(t_0) < \infty$. Then, (5) implies that $\Vert F\Vert _2^2(t') \leqq \Vert F\Vert _2^2(t_0) < \infty$ for all $t'\leqq t_0$. In particular, this holds for $t'=t$ which proves (14). $\square$

Corollary 3.12 gives that $||F||_2^2(t)$ is instantaneously bounded, even if $||F||_2^2(0)$ is undefined. The next result shows that $||F||_2^2(t)$ decays with rate $\frac{\log t}{t}$.

Corollary 3.13

Suppose $\gamma :{\mathbb {S}}^1\times [0,T)\rightarrow {\mathbb {R}}^2$ is a smooth entropy flow with $\gamma (\cdot ,0)$ satisfying the hypothesis of Lemma 3.2. Then for each $T_0\in (0,T)$,

$$\begin{aligned} \Vert F\Vert _2^2(T_0) \leqq \frac{1}{T_0}\left( {\mathcal {E}}_0 + 2\omega \pi \log \frac{L_0 + 4\omega ^2\pi ^2c_1^{-1}\big ( \sqrt{4\omega ^2\pi ^2 + T_0c_1^2} - 2\omega \pi \big )}{2\omega \pi }\right) . \end{aligned}$$

Proof

Calculate

$$\begin{aligned} (t\Vert F\Vert _2^2(t))'&= \Vert F\Vert _2^2(t) - 2t\int (k(F_{\theta \theta } + F))^2\,\textrm{d}\theta \\&\leqq \Vert F\Vert _2^2(t). \end{aligned}$$

Here we used Lemma 3.3. Integrating from zero to $T_0$ and using (15) we find

$$\begin{aligned} T_0\Vert F\Vert _2^2(T_0)&\leqq \int _0^{T_0} \Vert F\Vert _2^2({{\hat{t}}})\,\textrm{d}{{\hat{t}}} \\&\leqq {\mathcal {E}}_0 + 2\omega \pi \log \frac{L_0 + 4\omega ^2\pi ^2c_1^{-1}\big ( \sqrt{4\omega ^4\pi ^4 + T_0c_1^2} - 2\omega ^2\pi ^2 \big )}{2\omega \pi }. \end{aligned}$$

Dividing through by $T_0$ gives the result. $\square$

Control on $\Vert F\Vert _2^2$ allows us to obtain control on the $L^2(\textrm{d}\theta )$-norm of curvature.

Proposition 3.14

Suppose $\gamma :{\mathbb {S}}^1\times [0,T)\rightarrow {\mathbb {R}}^2$ is a smooth entropy flow with $\gamma (\cdot ,0)$ satisfying the hypothesis of Lemma 3.2. Let $[t_0,t_1]\subset \subset (0,T)$ be a compact time interval. For all $t\in [t_0,t_1]$ we have the estimate

$$\begin{aligned} \int k^2\,\textrm{d}\theta \leqq C(t_0,t_1,{\mathcal {E}}_0,\omega ,\Vert k\Vert _{L^1(\textrm{d}\theta )}(0)). \end{aligned}$$

In particular, we have the following estimate on the whole time interval: For $t\in (0,T)$

$$\begin{aligned} \int k^2\,\textrm{d}\theta \leqq C({\mathcal {E}}_0,L_0,\omega ,\Vert k\Vert _{L^1(\textrm{d}\theta )}(0))\Big (t + \frac{1}{t}\Big ). \end{aligned}$$

Proof

We calculate

$$\begin{aligned} \frac{\textrm{d}}{\textrm{d}t}\int k^2\,\textrm{d}\theta&= -2\int k^3(F_{\theta \theta } + F)\,\textrm{d}\theta \\&= -2\int (k_{\theta \theta }+k)((k^3)_{\theta \theta } + k^3)\,\textrm{d}\theta \\&= -2\int (k_{\theta \theta }+k)((3k^2k_\theta )_{\theta } + k^3)\,\textrm{d}\theta \\&= -2\int (k_{\theta \theta }+k)(3k^2k_{\theta \theta } + 6kk_\theta ^2 + k^3)\,\textrm{d}\theta \\&= -2\int 3k^2k_{\theta \theta }^2 + k_{\theta \theta }(6kk_\theta ^2 + k^3)\,\textrm{d}\theta -2\int - 3k^2k_\theta ^2 + k^4\,\textrm{d}\theta \\&= -6\int k^2k_{\theta \theta }^2\,\textrm{d}\theta -2\int k^4\,\textrm{d}\theta + 4\int k_\theta ^4 \,\textrm{d}\theta + 12\int k^2k_{\theta }^2\,\textrm{d}\theta \\&\leqq -6\int k^2k_{\theta \theta }^2\,\textrm{d}\theta -\int k^4\,\textrm{d}\theta + 40\int k_\theta ^4 \,\textrm{d}\theta . \end{aligned}$$

In the last line we used the estimate

$$\begin{aligned} 12\int k^2k_{\theta }^2\,\textrm{d}\theta \leqq \int k^4\,\textrm{d}\theta + 36\int k_\theta ^4\,\textrm{d}\theta . \end{aligned}$$

Set

$$\begin{aligned} C_0(t) = \frac{1}{t}\left( {\mathcal {E}}_0 + 2\omega \pi \log \frac{L_0 + 4\omega ^2\pi ^2c_1^{-1}\big ( \sqrt{4\omega ^2\pi ^2 + tc_1^2} - 2\omega \pi \big )}{2\omega \pi }\right) . \end{aligned}$$

(16)

The reverse Sobolev inequality, Corollary 3.4, combined with the decay estimate of Corollary 3.13 yields

$$\begin{aligned} \int k_{\theta \theta }^2 + k^2\,\textrm{d}\theta \leqq 2\int k_\theta ^2\,\textrm{d}\theta + C_0(t). \end{aligned}$$

(17)

Estimating the right hand side (with integration by parts and $ab \leqq \frac{1}{4}a^2 + b^2$), we find

$$\begin{aligned} \int k_{\theta \theta }^2\,\textrm{d}\theta&\leqq \frac{1}{2}\int k_{\theta \theta }^2\,\textrm{d}\theta + \int k^2\,\textrm{d}\theta + C_0(t) \end{aligned}$$

which implies

$$\begin{aligned} \int k_{\theta \theta }^2\,\textrm{d}\theta \leqq 2\int k^2\,\textrm{d}\theta + 2C_0(t). \end{aligned}$$

(18)

Above we have written explicitly that $C_0$ is multiplied by 2.

We use (18) in the following way. Hölder and Poincaré first imply

$$\begin{aligned} \int k_\theta ^4 \,\textrm{d}\theta \leqq \Vert k_\theta \Vert _\infty ^2\Vert k_\theta \Vert _2^2 \leqq c\Vert k_{\theta \theta }\Vert _2^4 \end{aligned}$$

which combines with (18) and then Hölder to yield

$$\begin{aligned} \int k_\theta ^4 \,\textrm{d}\theta \leqq c\Bigg (\int k^2\,\textrm{d}\theta \Bigg )^2 + c\,C_0^2(t) \leqq c\int k\,\textrm{d}\theta \int k^3\,\textrm{d}\theta + c\,C_0^2(t). \end{aligned}$$

Using this estimate in the evolution equation for $\int k^2\,\textrm{d}\theta$ we find

$$\begin{aligned} \frac{\textrm{d}}{\textrm{d}t}\int k^2\,\textrm{d}\theta&\leqq - 12\int k^2k_{\theta \theta }^2\,\textrm{d}\theta - \int k^4\,\textrm{d}\theta + c\,c_1\int k^3\,\textrm{d}\theta + c\,C_0^2(t) \,, \end{aligned}$$

where we used Corollary 3.6.

Now we compute

$$\begin{aligned}&\frac{\textrm{d}}{\textrm{d}t}\Bigg ( (t-t_0/2)_+\int k^2\,\textrm{d}\theta \Bigg ) \leqq - 12(t-t_0/2)_+\int k^2k_{\theta \theta }^2\,\textrm{d}\theta - (t-t_0/2)_+\int k^4\,\textrm{d}\theta \\ {}&\quad + c\,c_1(t-t_0/2)_+\int k^3\,\textrm{d}\theta + c\,C_0^2(t)(t-t_0/2)_+ \\ {}&\quad + \chi _{t\leqq t_0/2}\int k^2\,\textrm{d}\theta \\&\leqq - 12(t-t_0/2)_+\int k^2k_{\theta \theta }^2\,\textrm{d}\theta - \frac{1}{2}(t-t_0/2)_+\int k^4\,\textrm{d}\theta \\ {}&\quad + c(c_1^2+C_0^2(t))(t-t_0/2)_+ + c\frac{\chi _{t\leqq t_0/2}^2}{(t-t_0/2)_+}. \end{aligned}$$

Above we interpolated the terms $||k||_3^3$ and $||k||_2^2$. Integrating yields

$$\begin{aligned} \int k^2\,\textrm{d}\theta&\leqq c\,c_1^2(t-t_0/2)_+ + c(t-t_0/2)_+^{-1}\int _{t_0}^t\,C_0^2({{\tilde{t}}})({{\tilde{t}}}-t_0/2)_+\,d{{\tilde{t}}} \nonumber \\ {}&\quad + c\chi _{t\leqq t_0/2}^2(t-t_0/2)_+^{-1}\log \frac{(t-t_0/2)_+}{t_0/2}. \end{aligned}$$

(19)

Let $\alpha ^2(t) = t^2C_0^2(t)$. Then the integral on the right hand side above satisfies

$$\begin{aligned} \int _{t_0}^t\,C_0^2({{\tilde{t}}})({{\tilde{t}}}-t_0/2)_+\,d{{\tilde{t}}}&\leqq \alpha ^2(t)\Big (\log \frac{t}{t_0} - \frac{t_0}{2}\Big ( \frac{1}{t_0} - \frac{1}{t} \Big )\,\Big ) \\ {}&= \alpha ^2(t)\Big (\log \frac{t}{t_0} + \frac{t_0}{2t} - \frac{1}{2}\Big ). \end{aligned}$$

Note that with the factor $(t-t_0/2)_+^{-1}$ in front, this integral decays to zero for large t.

The right hand side of (19) is uniformly bounded for $t\in [t_0,t_1]$, which finishes the proof of the first claimed estimate. For the second estimate, we may apply the first on an interval $[t_0,t_1]$ where $t_0\in (0,t)$ and take $t_0\searrow 0$, noting that the asymptotics (see (19)) for small t are controlled by $\frac{1}{t}$ and for large t are controlled by t. Since the resultant estimate holds for all $t_1<T$ and in fact does not depend on $t_1$, it holds for all $t<T$. $\square$

This then allows us to conclude an estimate for $k_{\theta \theta }$ in $L^2(\textrm{d}\theta )$.

Corollary 3.15

Suppose $\gamma :{\mathbb {S}}^1\times [0,T)\rightarrow {\mathbb {R}}^2$ is a smooth entropy flow with $\gamma (\cdot ,0)$ satisfying the hypothesis of Lemma 3.2. For $t\in (0,T)$ we have the estimate

$$\begin{aligned} \int k_{\theta \theta }^2\,\textrm{d}\theta \leqq C({\mathcal {E}}_0,L_0,\omega ,\Vert k\Vert _{L^1(\textrm{d}\theta )}(0))\Big (t + \frac{1}{t}\Big ). \end{aligned}$$

Proof

Combine estimate (18) from the proof of Proposition 3.14 with the conclusion of Proposition 3.14. $\square$

Proposition 3.16

Suppose $\gamma :{\mathbb {S}}^1\times [0,T)\rightarrow {\mathbb {R}}^2$ is a smooth entropy flow with $\gamma (\cdot ,0)$ satisfying the hypothesis of Lemma 3.2. For $t\in (0,T)$ we have the estimate

$$\begin{aligned} \Vert k_{\theta }\Vert _{L^\infty (\textrm{d}\theta )}^2 \leqq C({\mathcal {E}}_0,L_0,\omega ,\Vert k\Vert _{L^1(\textrm{d}\theta )}(0))\Big (t + \frac{1}{t}\Big ):= c_2(t)^2. \end{aligned}$$

Proof

Use Corollary 3.15 with the Sobolev and then the Hölder inequalities to get that

$$\begin{aligned} k_\theta ^2&\leqq 2\omega \pi \int _0^{2\omega \pi } k_{\theta \theta }^2\,\textrm{d}\vartheta \leqq C({\mathcal {E}}_0,L_0,\omega ,\Vert k\Vert _{L^1(\textrm{d}\theta )}(0))\Big (t + \frac{1}{t}\Big ). \end{aligned}$$

Taking a supremum finishes the proof. $\square$

We now use the gradient estimate to obtain preservation of $k>0$.

Proposition 3.17

Suppose $\gamma :{\mathbb {S}}^1\times [0,T)\rightarrow {\mathbb {R}}^2$ is a smooth entropy flow with $\gamma (\cdot ,0)$ satisfying the hypothesis of Lemma 3.2. Then the flow remains strictly convex for all time, and if $T<\infty$, the infimum of the curvature as $t\nearrow T$ remains uniformly positive. More precisely, the infimum of the curvature at time t satisfies

$$\begin{aligned} k_0 \leqq C\frac{t}{t^2+1}\frac{1}{\exp \Bigg (C\Big (t+\frac{1}{t}\Big )\Big (\frac{L_0}{2}+2\omega \pi c_1^{-1}\big ( \sqrt{4\omega ^2\pi ^2 + tc_1^2} - 2\omega \pi \big )\Big )\Bigg )-1}, \end{aligned}$$

where $c_1 = \Vert k\Vert _{L^1(\textrm{d}\theta )}(0)$ and $C=C({\mathcal {E}}_0,L_0,\omega ,\Vert k\Vert _{L^1(\textrm{d}\theta )}(0))$.

Proof

Parameterise $\gamma$ on $[-\omega \pi ,\omega \pi )$ so that $k_0(t):= k(0,t) = \inf _{\theta \in [-\omega \pi ,\omega \pi )}k(\theta ,t)$. Observe that the gradient bound above implies $k(\theta ) \leqq \inf k + c_2(t)|\theta |$, so

$$\begin{aligned} L&= \int _{-\omega \pi }^{\omega \pi } \frac{1}{k}\,\textrm{d}\theta \\&\leqq \int _{-\omega \pi }^{\omega \pi } \frac{1}{k_0 + c_2(t)|\theta |}\,\textrm{d}\theta \\&= 2\int _{0}^{\omega \pi } \frac{1}{k_0 + c_2(t)\theta }\,\textrm{d}\theta \\&= \frac{2}{c_2(t)}\log \Bigg (\frac{k_0 + c_2(t)\omega \pi }{k_0}\Bigg ) = \frac{2}{c_2(t)}\log \Bigg (1 + \frac{c_2(t)\omega \pi }{k_0}\Bigg ). \end{aligned}$$

Rearranging yields

$$\begin{aligned} k_0 \leqq \frac{1}{c_2(t)\omega \pi }\frac{1}{e^{c_2(t)L/2}-1}. \end{aligned}$$

Combining this with Lemma 3.7 and using the expression for $c_2(t)$ from Proposition 3.16 yields the claimed estimate. $\square$

While the form of the estimate for k from below in Proposition 3.17 is somewhat messy, and degenerates as $t\searrow 0$ or as $t\nearrow \infty$, the important point is that it is strictly positive for any particular $t>0$. This is enough to preserve strict convexity, independent of final time (unless the final time is infinite). Furthermore, Corollary 3.15 implies $W^{3,\infty }(du)$ regularity up to and including final time. We may then apply Theorem A.1 in a standard way to obtain infinite maximal time of existence, so long as we have $W^{3,\infty }$ initial data.

Corollary 3.18

Suppose $\gamma :{\mathbb {S}}^1\times [0,T)\rightarrow {\mathbb {R}}^2$ is a smooth entropy flow with $\gamma (\cdot ,0)$ satisfying the hypothesis of Lemma 3.2. If $||\gamma (\cdot ,0)||_{W^{3,\infty }(du)} < \infty$, then the maximal time of existence is infinite.

We wish to push the initial regularity requirement as far as we can; in fact, we wish to remove the $W^{3,\infty }$ condition entirely. Our estimates are well-suited to this, as we have been careful to require as little initial regularity as possible. Up to now, we have a strictly (locally) convex initial curve that has curvature in $L^1(\textrm{d}\theta )$. Lemma 3.2 shows that this implies the initial entropy ${\mathcal {E}}_0$ is well-defined (and bounded). This is all we have used to ensure that the flow smoothes out and remains strictly convex.

Our strategy is to use a compactness argument with these estimates. The main missing component is uniform interior estimates for the flow, which we now establish.

We first give some special cases: the support function and its first two derivatives in $L^2(\textrm{d}\theta )$.

Proposition 3.19

Suppose $\gamma :{\mathbb {S}}^1\times [0,T)\rightarrow {\mathbb {R}}^2$ is a smooth entropy flow with $\gamma (\cdot ,0)$ satisfying the hypothesis of Lemma 3.2. The support function satisfies

$$\begin{aligned} \Vert h\Vert _2^2(t)= & {} \Vert h\Vert _2^2(0) + 4t\omega \pi , \\ (\Vert h_\theta \Vert _2^2)'= & {} -2\int k^{-2}k_\theta ^2\,\textrm{d}\theta , \end{aligned}$$

and

$$\begin{aligned} (\Vert h_{\theta \theta }\Vert _2^2)' = -2\int k^{-2}k_{\theta \theta }^2\textrm{d}\theta +4\int k^{-4}k_\theta ^4\,\textrm{d}\theta . \end{aligned}$$

Proof

Differentiating, we find

$$\begin{aligned} \frac{\textrm{d}}{\textrm{d}t}\int h^2\,\textrm{d}\theta&= 2\int h(k_{\theta \theta } + k)\,\textrm{d}\theta \\&= 2\int k(h_{\theta \theta } + h)\,\textrm{d}\theta = 4\omega \pi . \end{aligned}$$

For the derivative estimate, we calculate

$$\begin{aligned} \frac{\textrm{d}}{\textrm{d}t}\int h_\theta ^2\,\textrm{d}\theta&= 2\int h_\theta (k_{\theta \theta } + k)_\theta \,\textrm{d}\theta \\&= -2\int h_{\theta \theta }(k_{\theta \theta } + k)\,\textrm{d}\theta \\&= -2\int k^{-1}(k_{\theta \theta } + k)\,\textrm{d}\theta + 2\int h(k_{\theta \theta } + k)\,\textrm{d}\theta \\&= -2\int k^{-1}(k_{\theta \theta } + k)\,\textrm{d}\theta + 2\int k(h_{\theta \theta } + h)\,\textrm{d}\theta \\&= -2\int k^{-1}k_{\theta \theta }\,\textrm{d}\theta = -2\int k^{-2}k_{\theta }^2\,\textrm{d}\theta . \end{aligned}$$

For the second derivative we compute

$$\begin{aligned} \frac{\textrm{d}}{\textrm{d}t}\int h_{\theta \theta }^2\,\textrm{d}\theta&= 2\int h_{\theta \theta }(k_{\theta \theta } + k)_{\theta \theta }\,\textrm{d}\theta \\&= 2\int h_{\theta ^4}(k_{\theta \theta } + k)\,\textrm{d}\theta \\&= 2\int (h_{\theta ^6} + h_{\theta ^4})k\,\textrm{d}\theta \\&= 2\int (1/k)_{\theta ^4}k\,\textrm{d}\theta = 2\int (1/k)_{\theta \theta }k_{\theta \theta }\,\textrm{d}\theta \\&= 2\int (-k^{-2}k_{\theta \theta } + 2k^{-3}k_\theta ^2)k_{\theta \theta }\,\textrm{d}\theta \\&= -2\int k^{-2}k_{\theta \theta }^2\textrm{d}\theta +4\int k^{-4}k_\theta ^4\,\textrm{d}\theta . \end{aligned}$$

$\square$

We note that we also have the following pointwise estimates for the support function and curvature (from above):

Lemma 3.20

Suppose $\gamma :{\mathbb {S}}^1\times [0,T)\rightarrow {\mathbb {R}}^2$ is a smooth entropy flow with $\gamma (\cdot ,0)$ satisfying the hypothesis of Lemma 3.2. Then $||h_\theta ||_2^2(0) = \lim _{t\searrow 0} ||h_\theta ||_2^2(t)$ exists, and

$$\begin{aligned} ||h||_\infty \leqq C(1+\sqrt{t}), \end{aligned}$$

where $C=C(||h_\theta ||_2^2(0),||k||_1(0),\omega )$.

Proof

We find

$$\begin{aligned} h = h - {\overline{h}} + {\overline{h}} \leqq \int |h_\theta |\,\textrm{d}\theta + \frac{1}{2\omega \pi }L \leqq \sqrt{2\omega \pi }||h_\theta ||_2 + \frac{1}{2\omega \pi }L \end{aligned}$$

using the Hölder inequality, fundamental theorem of calculus and formula $L = \int h\,\textrm{d}\theta$. Since

$$\begin{aligned} \int h_\theta ^2\,\textrm{d}\theta = -\int h_{\theta \theta }h\,\textrm{d}\theta = -\int \frac{h}{k}\,\textrm{d}\theta + \int h^2\,\textrm{d}\theta = -2A+||h||_2^2 \end{aligned}$$

we know that $||h_\theta ||_2^2(0) = \lim _{t\searrow 0} ||h_\theta ||_2^2(t)$ exists.

Then Lemma 3.7 and Proposition 3.19 imply

$$\begin{aligned} ||h||_\infty \leqq \sqrt{2\omega \pi }||h_\theta ||_2(0) + \frac{1}{2\omega \pi }L_0 + 2c_1^{-1}\Bigg ( \sqrt{4\omega ^2\pi ^2 + tc_1^2} - 2\omega \pi \Bigg ). \end{aligned}$$

$\square$

Lemma 3.21

Suppose $\gamma :{\mathbb {S}}^1\times [0,T)\rightarrow {\mathbb {R}}^2$ is a smooth entropy flow with $\gamma (\cdot ,0)$ satisfying the hypothesis of Lemma 3.2. Then

$$\begin{aligned} k^2 \leqq C({\mathcal {E}}_0,L_0,\omega ,\Vert k\Vert _{L^1(\textrm{d}\theta )}(0))\Big (t + \frac{1}{t}\Big ). \end{aligned}$$

Proof

Using Corollary 3.6 and Proposition 3.16 we calculate

$$\begin{aligned} k = {\overline{k}} + \int |k_\theta |\,\textrm{d}\theta \leqq \frac{1}{2\omega \pi }c_1 2\omega \pi \,c_2(t). \end{aligned}$$

$\square$

While the norm $||h_{\theta \theta }||_2^2$ is not decaying or have a particularly simple evolution, due to Proposition 3.16 and Proposition 3.17 on a compact time interval $[t_1,t_2]\subset \subset (0,T)$ we are able to uniformly estimate the reaction term $\int k^{-4}k_\theta ^4\,\textrm{d}\theta$ by a constant. The next proposition uses this kind of crude technique to obtain uniform estimates on compact time intervals for $||h_{\theta ^p}||_2^2$ for any p.

Proposition 3.22

Suppose $\gamma :{\mathbb {S}}^1\times [0,T)\rightarrow {\mathbb {R}}^2$ is a smooth entropy flow with $\gamma (\cdot ,0)$ satisfying the hypothesis of Lemma 3.2. Let $[t_0,t_1]\subset \subset (0,T)$ be a compact time interval, and let $p\in {\mathbb {N}}_0$. For all $t\in [t_0,t_1]$ we have the estimate

$$\begin{aligned} \int h_{\theta ^p}^2\,\textrm{d}\theta \leqq C(p,t_0,t_1,{\mathcal {E}}_0,\omega ,\Vert k\Vert _1(0),||h_\theta ||_2(0)). \end{aligned}$$

(20)

Proof

First, we establish the required estimate for $p\in \{0,1,2,3,4,5,6\}$. Here and throughout the proof we take $t\in [t_1,t_2]$, the constant C depends on $p. t_0, t_1, {\mathcal {E}}_0, \omega$, and $\Vert k\Vert _{L^1(\textrm{d}\theta )}(0)$, and may vary from line to line.

Step 1. $p=0$ and $p=1$. The estimate (20) follows from Proposition 3.19.

Step 2. $p=2$ and $p=3$. Since

$$\begin{aligned} h_{\theta \theta } = \frac{1}{k} - h, {\text { and }}, h_{\theta \theta \theta } = -k^{-2}k_\theta - h_\theta , \end{aligned}$$

(21)

we find

$$\begin{aligned} ||h_{\theta \theta }||_2^2&\leqq 2\omega \pi ||1/k||^2_\infty \leqq C\,, {\text { and}} \\ ||h_{\theta \theta \theta }||_2^2&\leqq 2\int k^{-4}k_\theta ^2\,\textrm{d}\theta + 2\int h_\theta ^2\,\textrm{d}\theta \leqq 2\omega \pi ||k^{-1}||^4_\infty ||k_\theta ||^2_\infty + 2||h_\theta ||_2^2(0) \leqq C \end{aligned}$$

using Proposition 3.16, Proposition 3.17 and Proposition 3.19 (and recalling $t\in [t_1,t_2]$ plus what C depends on).

Step 3. $p=4$, $p=5$ and $p=6$. We calculate

$$\begin{aligned} \frac{\textrm{d}}{\textrm{d}t} \int h_{\theta ^{4}}^2\,\textrm{d}\theta&= 2\int h_{\theta ^{4}}(k_{\theta \theta } + k)_{\theta ^{4}}\,\textrm{d}\theta \nonumber \\&= 2\int (h_{\theta ^{6}} + h_{\theta ^{4}})k_{\theta ^{4}}\,\textrm{d}\theta \nonumber \\&= 2\int (k^{-1})_{\theta ^{4}}k_{\theta ^{4}}\,\textrm{d}\theta . \end{aligned}$$

(22)

Using

$$\begin{aligned} (k^{-1})_{\theta ^4}&= (-k^{-2}k_\theta )_{\theta ^3} \nonumber \\ {}&= ( - k^{-2}k_{\theta \theta } + 2k^{-3}k_\theta ^2 )_{\theta ^2} \nonumber \\ {}&= ( - k^{-2}k_{\theta ^3} + 6k^{-3}k_{\theta \theta }k_\theta - 6k^{-4}k_\theta ^3 )_{\theta } \nonumber \\ {}&= - k^{-2}k_{\theta ^4} + 8k^{-3}k_{\theta ^3}k_{\theta } + 6k^{-3}k_{\theta \theta }^2 - 36k^{-4}k_{\theta \theta }k_\theta ^2 + 24k^{-5}k_\theta ^4 \end{aligned}$$

(23)

in (22) and then estimating all occurrences of $k^{-1}$ and $k_\theta$ by constants, we find

$$\begin{aligned} \frac{\textrm{d}}{\textrm{d}t} \int h_{\theta ^{4}}^2\,\textrm{d}\theta&\leqq -2\int k^{-2}k_{\theta ^{4}}^2\,\textrm{d}\theta + C\int |k_{\theta ^{4}}| (|k_{\theta ^3}| + k_{\theta \theta }^2 + |k_{\theta \theta }| + 1) \,\textrm{d}\theta . \end{aligned}$$

Interpolating refines this to

$$\begin{aligned} \frac{\textrm{d}}{\textrm{d}t} \int h_{\theta ^{4}}^2\,\textrm{d}\theta&\leqq -\frac{3}{2}\int k^{-2}k_{\theta ^{4}}^2\,\textrm{d}\theta + C\int k_{\theta ^3}^2 + k_{\theta \theta }^4 \,\textrm{d}\theta + C \nonumber \\ {}&\leqq -c\int k_{\theta ^{4}}^2\,\textrm{d}\theta + C\int k_{\theta \theta }^4 \,\textrm{d}\theta + C. \end{aligned}$$

(24)

Note that we used integration by parts and interpolation on the $||k_{\theta ^3}||_2^2$ term, and we applied Lemma 3.21 (with the assumption $t\in [t_0,t_1]$) to estimate $-k^{-1}$ from above.

Let $A\in {\mathbb {R}}$ be a constant and $\delta >0$ be fixed. For the remaining term, we use the following estimate

$$\begin{aligned} A\int k_{\theta \theta }^4 \,\textrm{d}\theta \leqq \delta \int k_{\theta ^4}^2 \,\textrm{d}\theta + C(A) \end{aligned}$$

(25)

with $\delta = \frac{c}{2}$. To see this, use integration by parts, interpolation, and Proposition 3.16 to find

$$\begin{aligned} A\int k_{\theta \theta }^4 \,\textrm{d}\theta \leqq AC\int |k_{\theta ^3}|\,k_{\theta \theta }^2\,\textrm{d}\theta \leqq \frac{A}{2} \int k_{\theta \theta }^4 \,\textrm{d}\theta + AC \int k_{\theta ^3}^2 \,\textrm{d}\theta . \end{aligned}$$

Then, absorbing the first term on the left, we continue to estimate

$$\begin{aligned} A\int k_{\theta \theta }^4 \,\textrm{d}\theta \leqq 2AC \int |k_{\theta ^4}|\,|k_{\theta \theta }| \,\textrm{d}\theta \leqq \frac{\delta }{4} \int k_{\theta ^4}^2 \,\textrm{d}\theta + \frac{A}{2} \int k_{\theta \theta }^4 \,\textrm{d}\theta + C(A). \end{aligned}$$

Absorbing one more time on the left finishes the proof of (25). Note that above we used the inequality $ab \leqq \varepsilon _0 a^2 + \varepsilon _1 b^4 + C(\varepsilon _0,\varepsilon _1)$ which follows from two applications of Young’s inequality, and holds for all $\varepsilon _0, \varepsilon _1 > 0$.

With the estimate (25), we return to (24) and continue to find

$$\begin{aligned} \frac{\textrm{d}}{\textrm{d}t} \int h_{\theta ^{4}}^2\,\textrm{d}\theta&\leqq -c\int k_{\theta ^{4}}^2\,\textrm{d}\theta + C\int k_{\theta \theta }^4 \,\textrm{d}\theta + C \nonumber \\ {}&\leqq -\frac{c}{2}\int k_{\theta ^{4}}^2\,\textrm{d}\theta + C. \end{aligned}$$

(26)

Now we cut off in time. We use (26) to calculate

$$\begin{aligned} \frac{\textrm{d}}{\textrm{d}t}\Bigg ((t-t_0/2)_+^2\int h_{\theta ^{4}}^2\,\textrm{d}\theta \Bigg )\leqq & {} - \frac{c}{2}(t-t_0/2)_+^2\int k_{\theta ^4}^2\,\textrm{d}\theta + C(t-t_0/2)_+^2\nonumber \\{} & {} + 2(t-t_0/2)_+\int h_{\theta ^{4}}^2\,\textrm{d}\theta . \end{aligned}$$

(27)

In view of (21), (23) and Proposition 3.17, we see that

$$\begin{aligned} ||h_{\theta ^{5}}||_2^2&\leqq 2||h_{\theta ^{3}}||_2^2 + 2||(k^{-1})_{\theta ^3}||_2^2 \\&\leqq C(1 + ||k_{\theta ^3}||_2^2 + ||k_{\theta ^{2}}||_2^2). \end{aligned}$$

Applying this estimate after interpolation (and the $p=3$ step) we find

$$\begin{aligned} 2(t-t_0/2)_+\int h_{\theta ^{4}}^2\,\textrm{d}\theta&\leqq C\int h_{\theta ^{5}}^2\,\textrm{d}\theta + C\int h_{\theta ^{3}}^2\,\textrm{d}\theta \\&\leqq C\int k_{\theta ^{3}}^2+k_{\theta ^{2}}^2\,\textrm{d}\theta + C \\&\leqq \frac{c}{4} (t-t_0/2)_+^2\int k_{\theta ^4}^2\,\textrm{d}\theta + C. \end{aligned}$$

We use this to absorb the rightmost term in (27), giving

$$\begin{aligned} \frac{\textrm{d}}{\textrm{d}t}\Bigg ((t-t_0/2)_+^2\int h_{\theta ^{4}}^2\,\textrm{d}\theta \Bigg ) \leqq C. \end{aligned}$$

Integration gives the estimate (20), finishing the proof for $p=4$.

For $p=5$, we begin with

$$\begin{aligned} (k^{-1})_{\theta ^5} \nonumber&= (- k^{-2}k_{\theta ^4} + 8k^{-3}k_{\theta ^3}k_{\theta } + 6k^{-3}k_{\theta \theta }^2 - 36k^{-4}k_{\theta \theta }k_\theta ^2 + 24k^{-5}k_\theta ^4)_\theta \nonumber \\ {}&= - k^{-2}k_{\theta ^5} + k^{-3}(10k_{\theta ^4}k_\theta + 20k_{\theta ^3}k_{\theta \theta })\nonumber \\&\quad +k^{-4}(-60k_{\theta ^3}k_{\theta }^2 -90k_{\theta \theta }^2k_\theta ) +276k^{-5}k_{\theta \theta }k_\theta ^3 -120k^{-6}k_\theta ^5. \end{aligned}$$

(28)

Using (28) (and Proposition 3.16, Proposition 3.17, Lemma 3.21) we find

$$\begin{aligned} \frac{\textrm{d}}{\textrm{d}t} \int h_{\theta ^{5}}^2\,\textrm{d}\theta&= 2\int (k^{-1})_{\theta ^{5}}k_{\theta ^{5}}\,\textrm{d}\theta \\&\leqq -2c\int k_{\theta ^{5}}^2\,\textrm{d}\theta + C\int |k_{\theta ^{5}}| (|k_{\theta ^4}| + |k_{\theta ^3}|\,|k_{\theta \theta }|\\&\quad + |k_{\theta ^3}| + |k_{\theta \theta }|^2 + |k_{\theta \theta }| + 1) \,\textrm{d}\theta . \end{aligned}$$

Interpolating refines this to

$$\begin{aligned} \frac{\textrm{d}}{\textrm{d}t} \int h_{\theta ^{5}}^2\,\textrm{d}\theta&\leqq -\frac{3}{2}c\int k_{\theta ^{5}}^2\,\textrm{d}\theta + C\int k_{\theta ^4}^2 + k_{\theta ^3}^2 + k_{\theta \theta }^2 + k_{\theta ^3}^2 k_{\theta \theta }^2 \,\textrm{d}\theta + C \nonumber \\ {}&\leqq -c\int k_{\theta ^{5}}^2\,\textrm{d}\theta + \delta \int k_{\theta ^3}^4 \,\textrm{d}\theta + C\delta ^{-1}\int k_{\theta \theta }^4 \,\textrm{d}\theta + C. \end{aligned}$$

(29)

The $\delta ^{-1}||k_{\theta \theta }||_4^4$ term is dealt with by estimate (25) (and then interpolation). For the $||k_{\theta ^3}||_4^4$ term, we wish to use a similar estimate. We take this opportunity to derive a general version.

Let $A\in {\mathbb {R}}$ be a constant. Assume $||k_{\theta ^{m-2}}||_\infty \leqq C$. We claim

$$\begin{aligned} A \int k_{\theta ^{m+1}}^2\,k_{\theta ^{m-1}}^2 + k_{\theta ^{m+1}}^\frac{8}{3} + k_{\theta ^{m}}^4\,\textrm{d}\theta \leqq AC \int k_{\theta ^{m+2}}^2 \,\textrm{d}\theta \end{aligned}$$

(30)

First, estimate

$$\begin{aligned}{} & {} A\int k_{\theta ^m}^4 \,\textrm{d}\theta \leqq AC\int |k_{\theta ^{m+1}}|\,k_{\theta ^m}^2|k_{\theta ^{m-1}}|\,\textrm{d}\theta \leqq \frac{A}{2} \int k_{\theta ^m}^4 \,\textrm{d}\theta \\{} & {} \quad + AC \int k_{\theta ^{m+1}}^2\,k_{\theta ^{m-1}}^2\,\textrm{d}\theta \,\textrm{d}\theta . \end{aligned}$$

Absorbing the first term on the left, we have

$$\begin{aligned} A\int k_{\theta ^m}^4 \,\textrm{d}\theta&\leqq AC \int k_{\theta ^{m+1}}^2\,k_{\theta ^{m-1}}^2\,\textrm{d}\theta \,\textrm{d}\theta . \end{aligned}$$

Continuing, we find that

$$\begin{aligned} A\int k_{\theta ^m}^4 \,\textrm{d}\theta + AC\int k_{\theta ^{m+1}}^2\,k_{\theta ^{m-1}}^2\,\textrm{d}\theta&\leqq AC \int |k_{\theta ^{m-2}}|(|k_{\theta ^{m}}|\,k_{\theta ^{m+1}}^2 + |k_{\theta ^{m+2}}|\,|k_{\theta ^{m+1}}|\,|k_{\theta ^{m-1}}|) \,\textrm{d}\theta \\ {}&\leqq AC \int |k_{\theta ^{m}}|\,k_{\theta ^{m+1}}^2 + |k_{\theta ^{m+2}}|\,|k_{\theta ^{m+1}}|\,|k_{\theta ^{m-1}}| \,\textrm{d}\theta \\ {}&\leqq AC\int k_{\theta ^{m+2}}^2 \,\textrm{d}\theta + \frac{A}{2} \int k_{\theta ^{m}}^4 \,\textrm{d}\theta + A^2C \int k_{\theta ^{m+1}}^\frac{8}{3} \,\textrm{d}\theta \\ {}&\quad + \frac{AC}{2} \int k_{\theta ^{m+1}}^2\,k_{\theta ^{m-1}}^2 \,\textrm{d}\theta . \end{aligned}$$

Absorbing again, we find that

$$\begin{aligned} A \int k_{\theta ^{m+1}}^2\,k_{\theta ^{m-1}}^2\,\textrm{d}\theta + AC \int k_{\theta ^{m}}^4\,\textrm{d}\theta&\leqq AC\int k_{\theta ^{m+2}}^2 \,\textrm{d}\theta + AC \int k_{\theta ^{m+1}}^\frac{8}{3} \,\textrm{d}\theta \end{aligned}$$

(31)

Finally, we estimate

$$\begin{aligned}&AC \int k_{\theta ^{m+1}}^\frac{8}{3} \,\textrm{d}\theta \leqq AC\int k_{\theta ^{m+1}}^{\frac{2}{3}}|k_{\theta ^{m+2}}|\,|k_{\theta ^m}|\,\textrm{d}\theta \leqq \frac{AC}{2}\int k_{\theta ^{m+1}}^{\frac{8}{3}}\,\textrm{d}\theta \\ {}&\quad + AC\int k_{\theta ^{m+2}}^\frac{4}{3} k_{\theta ^m}^\frac{4}{3} \,\textrm{d}\theta \end{aligned}$$

which gives, absorbing once again,

$$\begin{aligned} AC \int k_{\theta ^{m+1}}^\frac{8}{3} \,\textrm{d}\theta \leqq AC\int k_{\theta ^{m+2}}^\frac{4}{3} k_{\theta ^m}^\frac{4}{3} \,\textrm{d}\theta . \end{aligned}$$

Using this now in (31) we have

$$\begin{aligned} A \int k_{\theta ^{m+1}}^2\,k_{\theta ^{m-1}}^2 + k_{\theta ^{m+1}}^\frac{8}{3} + k_{\theta ^{m}}^4\,\textrm{d}\theta&\leqq A \int k_{\theta ^{m+1}}^2\,k_{\theta ^{m-1}}^2\,\textrm{d}\theta + AC \int k_{\theta ^{m+1}}^\frac{8}{3} \,\textrm{d}\theta + AC \int k_{\theta ^{m}}^4\,\textrm{d}\theta \\ {}&\leqq AC\int k_{\theta ^{m+2}}^2 \,\textrm{d}\theta + AC\int k_{\theta ^{m+2}}^\frac{4}{3} k_{\theta ^m}^\frac{4}{3} \,\textrm{d}\theta \\ {}&\leqq AC\int k_{\theta ^{m+2}}^2 \,\textrm{d}\theta + \frac{A}{2}\int k_{\theta ^m}^4 \,\textrm{d}\theta . \end{aligned}$$

Absorbing one final time gives the claimed estimate (30).

With the estimate (30) and using (25) on the term $\delta ^{-1}||k_{\theta \theta }||_4^4$ followed by interpolation, we return to (29) to find that

$$\begin{aligned} \frac{\textrm{d}}{\textrm{d}t} \int h_{\theta ^{5}}^2\,\textrm{d}\theta&\leqq -\frac{c}{2}\int k_{\theta ^{5}}^2\,\textrm{d}\theta + C. \end{aligned}$$

(32)

Now we cut off in time, as before in the $p=4$ case. The details are similar, so we will be brief. We use (32) to calculate

$$\begin{aligned} \frac{\textrm{d}}{\textrm{d}t}\Bigg ((t-t_0/2)_+^2\int h_{\theta ^{5}}^2\,\textrm{d}\theta \Bigg )\leqq & {} - \frac{c}{2}(t-t_0/2)_+^2\int k_{\theta ^5}^2\,\textrm{d}\theta + C(t-t_0/2)_+^2 \nonumber \\{} & {} + 2(t-t_0/2)_+\int h_{\theta ^{5}}^2\,\textrm{d}\theta . \end{aligned}$$

(33)

Since

$$\begin{aligned} ||h_{\theta ^{6}}||_2^2&\leqq C(1 + ||k_{\theta ^4}||_2^2 + ||k_{\theta ^{3}}||_2^2 + ||k_{\theta ^{2}}||_2^2) \,, \end{aligned}$$

we find that

$$\begin{aligned} 2(t-t_0/2)_+\int h_{\theta ^{5}}^2\,\textrm{d}\theta&\leqq C\int h_{\theta ^{6}}^2\,\textrm{d}\theta + C\int h_{\theta ^{4}}^2\,\textrm{d}\theta \leqq C\int k_{\theta ^{4}}^2+k_{\theta ^{3}}^2+k_{\theta ^{2}}^2\,\textrm{d}\theta + C \\&\leqq \frac{c}{4} (t-t_0/2)_+^2\int k_{\theta ^5}^2\,\textrm{d}\theta + C. \end{aligned}$$

We use this to absorb the rightmost term in (33), giving

$$\begin{aligned} \frac{\textrm{d}}{\textrm{d}t}\Bigg ((t-t_0/2)_+^2\int h_{\theta ^{5}}^2\,\textrm{d}\theta \Bigg ) \leqq C. \end{aligned}$$

Integration gives the estimate (20), finishing the proof for $p=5$.

Finally $p=6$. Analogously to before, we begin with $(k^{-1})_{\theta ^6}$, but this time we write it more succinctly:

$$\begin{aligned} (k^{-1})_{\theta ^6}&= (- k^{-2}k_{\theta ^5} + k^{-3}(10k_{\theta ^4}k_\theta + 20k_{\theta ^3}k_{\theta \theta })\nonumber \\&\quad +k^{-4}(-60k_{\theta ^3}k_{\theta }^2 -90k_{\theta \theta }^2k_\theta ) +276k^{-5}k_{\theta \theta }k_\theta ^3 -120k^{-6}k_\theta ^5)_\theta \nonumber \\ {}&= -k^{-2}k_{\theta ^6} + \sum _{j=2}^6 k^{-1-j} \sum _{i_1+\cdots +i_j} c(i_1,\ldots ,i_j)k_{\theta ^{i_1}}\cdots k_{\theta ^{i_j}}. \end{aligned}$$

(34)

Above we implicitly assume each $i_j\in {\mathbb {N}}$, and the constants $c(i_1,\ldots ,i_j)$ are universal. The $p=5$ case just treated implies that $k_{\theta \theta }$ in addition to $k^{-1}$ and $k_\theta$ is uniformly bounded in $L^\infty$ on $[t_1,t_2]$, because

$$\begin{aligned} |k_{\theta \theta }|= & {} \bigg |\Bigg (\frac{1}{h_{\theta \theta }+h}\Bigg )_{\theta \theta }\bigg | = \bigg |\Bigg (-\frac{h_{\theta ^3}+h_\theta }{(h_{\theta \theta }+h)^2}\Bigg )_{\theta }\bigg |\nonumber \\= & {} \bigg | \frac{-(h_{\theta \theta }+h)(h_{\theta ^4}+h_{\theta ^2}) + 2(h_{\theta ^3}+h_\theta )^2}{(h_{\theta \theta }+h)^3} \bigg |, \end{aligned}$$

(35)

and $||h_{\theta ^m}||_\infty \leqq C$ for $m\in \{0,1,2,3,4\}$. As with earlier cases, we find that

$$\begin{aligned} \frac{\textrm{d}}{\textrm{d}t} \int h_{\theta ^{6}}^2\,\textrm{d}\theta&= 2\int (k^{-1})_{\theta ^{6}}k_{\theta ^{6}}\,\textrm{d}\theta \nonumber \\ {}&\leqq -2c\int k_{\theta ^{6}}^2\,\textrm{d}\theta + C\int |k_{\theta ^{6}}| (|k_{\theta ^5}| + |k_{\theta ^4}| + |k_{\theta ^3}|^2 + 1) \,\textrm{d}\theta \nonumber \\ {}&\leqq -\frac{3}{2}c\int k_{\theta ^{6}}^2\,\textrm{d}\theta + C\int k_{\theta ^5}^2 + k_{\theta ^4}^2 + k_{\theta ^3}^4 \,\textrm{d}\theta + C \nonumber \\ {}&\leqq -c\int k_{\theta ^{6}}^2\,\textrm{d}\theta + \int k_{\theta ^3}^4 \,\textrm{d}\theta + C. \end{aligned}$$

(36)

We now apply the estimate (30) followed by interpolation to refine (36) to

$$\begin{aligned} \frac{\textrm{d}}{\textrm{d}t} \int h_{\theta ^{6}}^2\,\textrm{d}\theta&\leqq -\frac{c}{2}\int k_{\theta ^{6}}^2\,\textrm{d}\theta + C. \end{aligned}$$

(37)

Now we cut off in time. The details are similar, so we will be brief. We use (32) to calculate

$$\begin{aligned} \frac{\textrm{d}}{\textrm{d}t}\Bigg ((t-t_0/2)_+^2\int h_{\theta ^{5}}^2\,\textrm{d}\theta \Bigg )\leqq & {} - \frac{c}{2}(t-t_0/2)_+^2\int k_{\theta ^5}^2\,\textrm{d}\theta + C(t-t_0/2)_+^2 \nonumber \\{} & {} + 2(t-t_0/2)_+\int h_{\theta ^{5}}^2\,\textrm{d}\theta \leqq C, \end{aligned}$$

(38)

where we interpolated the last term on the RHS using an argument completely analogous to the $p=5$ case, so we omit it. Integrating (38) gives the estimate (20), finishing the proof for $p=6$.

Step 4. Induction for large p. Suppose (20) holds for some $p\leqq 6$. We aim to show that it holds for $p+1$, thus finishing the proof by induction.

First, (20) implies $||h_{\theta ^{p-1}}||_\infty \leqq C$. It thus follows from $k = (h_{\theta \theta }+h)^{-1}$ and Proposition 3.17 (see also (35)) that

$$\begin{aligned} ||k_{\theta ^{p-3}}||_\infty \leqq C. \end{aligned}$$

(39)

Let us now note the following formula, which is the general version of (34):

$$\begin{aligned} (k^{-1})_{\theta ^m} = - k^{-2}k_{\theta ^m} + \sum _{q=2}^m k^{-(q+1)} \sum _{i_1+\cdots +i_q=m} c(i_1,\ldots ,i_q) k_{\theta ^{i_1}} \cdots k_{\theta ^{i_q}}. \end{aligned}$$

(40)

Above we implicitly assume each $i_j\in {\mathbb {N}}$. The last term in the sum is $-k^{-m-1}k_{\theta }^m$, has $q=m$, $i_1=i_2=\cdots =1$ and $c(1,\ldots ,1) = (-1)^m\,m!$, which is uniformly bounded. In general the pattern we saw earlier will continue: many terms will be bounded from work in previous steps, and the remaining terms can be estimated. Let us carry out the details.

For $p+1$ (which is at least 7) we compute

$$\begin{aligned} \frac{\textrm{d}}{\textrm{d}t} \int h_{\theta ^{p+1}}^2\,\textrm{d}\theta&= 2\int (k^{-1})_{\theta ^{p+1}}k_{\theta ^{p+1}}\,\textrm{d}\theta . \end{aligned}$$

Using (40) and then (39), we estimate

$$\begin{aligned} \frac{\textrm{d}}{\textrm{d}t} \int h_{\theta ^{(p+1)}}^2\,\textrm{d}\theta&\leqq -\int k^{-2}k_{\theta ^{(p+1)}}^2\,\textrm{d}\theta \\ {}&\quad +C\sum _{q=2}^{p+1} \sum _{i_1+\cdots +i_q={p+1}} c(i_1,\ldots ,i_q) \int k_{\theta ^{i_1}}^2 \cdots k_{\theta ^{i_q}}^2 \,\textrm{d}\theta \\&\leqq -\int k^{-2}k_{\theta ^{(p+1)}}^2\,\textrm{d}\theta + C + C\sum _{q=2}^{4} \sum _{i_1+\cdots +i_q={p+1}} c(i_1,\ldots ,i_q) \int k_{\theta ^{i_1}}^2 \cdots k_{\theta ^{i_q}}^2 \,\textrm{d}\theta \\&\leqq -\int k^{-2}k_{\theta ^{(p+1)}}^2\,\textrm{d}\theta + C \\&\quad + C\sum _{i_1+i_2={p+1}} \int k_{\theta ^{i_1}}^2 k_{\theta ^{i_2}}^2 \,\textrm{d}\theta + C\sum _{i_1+i_2+i_3={p+1}} \int k_{\theta ^{i_1}}^2 k_{\theta ^{i_2}}^2 k_{\theta ^{i_3}}^2 \,\textrm{d}\theta \\&\quad + C\sum _{i_1+i_2+i_3+i_4={p+1}} \int k_{\theta ^{i_1}}^2 k_{\theta ^{i_2}}^2 k_{\theta ^{i_3}}^2 k_{\theta ^{i_4}}^2 \,\textrm{d}\theta . \end{aligned}$$

In the summations above we use again (39) to estimate all factors of the form $k_{\theta ^m}$ for $m\leqq p-3$ (note that m is at least 3), and find

$$\begin{aligned} \frac{\textrm{d}}{\textrm{d}t} \int h_{\theta ^{(p+1)}}^2\,\textrm{d}\theta&\leqq -\int k^{-2}k_{\theta ^{(p+1)}}^2\,\textrm{d}\theta + C \\&\quad + C\int k_{\theta ^{p}}^2 k_{\theta }^2 + k_{\theta ^{p-1}}^2 k_{\theta ^2}^2 + k_{\theta ^{p-2}}^2 k_{\theta ^3}^2 \,\textrm{d}\theta \\&\quad + C\int k_{\theta ^{p-1}}^2 k_{\theta }^2 k_{\theta }^2 + k_{\theta ^{p-2}}^2 k_{\theta ^2}^2 k_{\theta }^2 \,\textrm{d}\theta \\&\quad + C\int k_{\theta ^{p-2}}^2 k_{\theta }^2 k_{\theta }^2 k_{\theta }^2 \,\textrm{d}\theta \\&\leqq -\int k^{-2}k_{\theta ^{(p+1)}}^2\,\textrm{d}\theta +C\int k_{\theta ^{p}}^2 \,\textrm{d}\theta \\&\quad +C\int k_{\theta ^{(p-1)}}^2 \,\textrm{d}\theta +C\int k_{\theta ^{(p-2)}}^2 \,\textrm{d}\theta +C. \end{aligned}$$

Lemma 3.21, integration by parts and interpolation yields

$$\begin{aligned} \frac{\textrm{d}}{\textrm{d}t} \int h_{\theta ^{(p+1)}}^2\,\textrm{d}\theta&\leqq -c\int k_{\theta ^{(p+1)}}^2\,\textrm{d}\theta +C\int k_{\theta ^{p}}^2 \,\textrm{d}\theta +C \leqq -c\int k_{\theta ^{p+1}}^2\,\textrm{d}\theta + C. \end{aligned}$$

From here we calculate

$$\begin{aligned} \frac{\textrm{d}}{\textrm{d}t}\bigg ((t-t_0/2)_+^2\int h_{\theta ^{p+1}}^2\,\textrm{d}\theta \bigg )\leqq & {} - c(t-t_0/2)_+^2\int k_{\theta ^p}^2\,\textrm{d}\theta + C(t-t_0/2)_+^2 \nonumber \\{} & {} + 2(t-t_0/2)_+\int h_{\theta ^{p+1}}^2\,\textrm{d}\theta . \end{aligned}$$

(41)

In view of (21), (40) and Proposition 3.17, we see that

$$\begin{aligned} ||h_{\theta ^{p+2}}||_2^2&\leqq 2||h_{\theta ^{p}}||_2^2 + 2||(k^{-1})_{\theta ^p}||_2^2 \\&\leqq C(1 + ||k_{\theta ^p}||_2^2 + ||k_{\theta ^{p-1}}||_2^2 + ||k_{\theta ^{p-2}}||_2^2). \end{aligned}$$

Then we estimate using interpolation (and choosing $\varepsilon$ in the last step)

$$\begin{aligned} 2(t-t_0/2)_+\int h_{\theta ^{p+1}}^2\,\textrm{d}\theta&\leqq \varepsilon (t-t_0/2)_+^2\int h_{\theta ^{p+2}}^2\,\textrm{d}\theta + C_\varepsilon \int h_{\theta ^{p}}^2\,\textrm{d}\theta \\&\leqq \varepsilon (t-t_0/2)_+^2\int k_{\theta ^{p}}^2+k_{\theta ^{p-1}}^2+k_{\theta ^{p-2}}^2\,\textrm{d}\theta + C_\varepsilon \int h_{\theta ^{p}}^2\,\textrm{d}\theta \\&\leqq \frac{c}{2} (t-t_0/2)_+^2\int k_{\theta ^p}^2\,\textrm{d}\theta + C \end{aligned}$$

We use this to absorb the rightmost term in (41), giving

$$\begin{aligned} \frac{\textrm{d}}{\textrm{d}t}\bigg ((t-t_0/2)_+^2\int h_{\theta ^{p+1}}^2\,\textrm{d}\theta \bigg ) \leqq C. \end{aligned}$$

Integration gives the estimate (20), finishing the proof. $\square$

We will need to also employ the following remarkably strong $L^2(\textrm{d}\theta )$-uniqueness property of the flow.

Proposition 3.23

Let $\{h_n\}$ be a sequence of support functions of smooth solutions to the entropy flow such that for every $t\in (0,t_1]$, $h_n(\cdot ,t)$ converges to some $h(\cdot ,t)$ in the $L^2(\textrm{d}\theta )$-norm. Suppose that $\{h_n(\cdot ,0)\}$ converges in $L^2(\textrm{d}\theta )$ to $h_0$, where $h_0$ is the support function of a curve satisfying the conditions of Theorem 1.1. Then $h(\cdot ,t)$ converges to $h_0$ in $L^2(\textrm{d}\theta )$ as $t\searrow 0$.

Proof

Let $h^1 = h_n$ and $h^2 = h_m$ (we also use superscripts 1 and 2 throughout to refer to quantities corresponding to the curves generated by $h^1$ and $h^2$ respectively), and calculate

$$\begin{aligned} \frac{\textrm{d}}{\textrm{d}t}\int |h^1 - h^2|^2\,\textrm{d}\theta&= 2\int (h^1-h^2)( (k^1)_{\theta \theta } + k^1 - (k^2)_{\theta \theta } - k^2 )\,\textrm{d}\theta \\&= 2\int (h^1-h^2)( (k^1-k^2)_{\theta \theta } + k^1 - k^2)\,\textrm{d}\theta \\&= 2\int ((h^1-h^2)_{\theta \theta } + h^1-h^2)(k^1 - k^2)\,\textrm{d}\theta \\&= 2\int (1/k^1 - 1/k^2)(k^1 - k^2)\,\textrm{d}\theta \\&= -2\int \frac{(k^2-k^1)^2}{k^1k^2} \,\textrm{d}\theta \leqq 0. \end{aligned}$$

We therefore have

$$\begin{aligned} \int |h^1(\theta ,t) - h^2(\theta ,t)|^2\,\textrm{d}\theta\leqq & {} \int |h^1(\theta ,0) - h^2(\theta ,0)|^2\,\textrm{d}\theta \\ {}\leqq & {} 2\int |h^1(\theta ,0) - h_0(\theta )|^2\,\textrm{d}\theta \\{} & {} + 2\int |h_0(\theta ) - h^2(\theta ,0)|^2\,\textrm{d}\theta . \end{aligned}$$

Observing that $h_n(\cdot ,0) \rightarrow h_0(\cdot )$ in $L^2(\textrm{d}\theta )$ by assumption finishes the proof. $\square$

Now we are able to conclude the existence of a global solution with weak data. This is the first half of Theorem 1.1, and state it as follows:

Theorem 3.24

Suppose $\gamma _0:{\mathbb {S}}^1\rightarrow {\mathbb {R}}^2$ is either

• (I) an immersed locally convex closed curve of class $C^2(\textrm{d}s)$ with turning number $\omega$; or

• (E) an embedded curve of with $k\in L^2(\textrm{d}s)$ bounding a convex planar domain (which has $\omega =1$).

The entropy flow $\gamma :{\mathbb {S}}^1\times (0,T)\rightarrow {\mathbb {R}}^2$ with $\gamma _0$ as initial data exists uniquely, is smooth, and global ($T=\infty$). The flow attains its initial data in $C^2(\textrm{d}s)$ for case (I) and in $H^2(\textrm{d}s)$ for case (E).

Proof

Consider the support function $h_0$ corresponding to $\gamma _0$ and take a sequence $\{h_n^0\}$ of smooth functions such that $h_n^0 \rightarrow h_0$ in $C^2(\textrm{d}s)$ for case (I), or $H^2(\textrm{d}s)$ for case (E). Let $h_n$ be the corresponding smooth entropy flow with $h_n^0$ as initial data, whose existence is guaranteed by Theorem A.1. Each flow $h_n$ exists globally by Corollary 3.18.

We have uniform estimates for all derivatives of $\{h_n\}$ over every compact subset of ${\mathbb {S}}^1\times (0,\infty )$ by Proposition 3.22. Note that, for any fixed compact subset, these estimates depend only on universal quantities: $\Vert \gamma _0\Vert _{H^2(\textrm{d}s)}$, $\Vert h_\theta \Vert _{L^2(\textrm{d}\theta )}$ ${\mathcal {E}}_0$, $\omega$. Since the convergence of $h_n^0 \rightarrow h_0$ is in (at least) $H^2(\textrm{d}s)$, these quantities are also uniformly bounded along the sequence $\{h_n^0\}$. By a diagonal subsequence argument, we find a sequence $\{h_{n_j}\}$ converging smoothly in every compact subset of ${\mathbb {S}}^1\times (0,\infty )$ to a smooth function $h:{\mathbb {S}}^1\times (0,\infty )\rightarrow {\mathbb {R}}$. The smooth convergence implies that h is the support function of an entropy flow (satisfying $h_t = k_{\theta \theta } + k$). By Proposition 3.23, $h(\cdot ,t)$ converges to $h_0$ in the $L^2(\textrm{d}\theta )$ topology as $t\searrow 0$. Interpolation and our uniform estimates upgrade this convergence to the regularity of the initial curve: either $C^2(\textrm{d}s)$ (locally convex immersion (I)) or $H^2(\textrm{d}s)$ (convex embedding (E)). $\square$

4 Global Analysis

It remains to establish the second half of Theorem 1.1. This will be completed by the end of Section 5. The entropy flow is expanding, and so to examine its asymptotic shape, one approach is to consider appropriate parabolic rescaling. Setting $h^\lambda (\theta ,t) = \lambda h(\theta , t/\lambda ^2)$, we see that $k^\lambda (\theta ,t) = \lambda ^{-1}k(\theta , t/\lambda ^2)$, and

$$\begin{aligned} h^\lambda _t(\theta ,t) = \lambda ^{-1} h_t(\theta , t/\lambda ^2) = \lambda ^{-1}(k_{\theta \theta }+k)(\theta , t/\lambda ^2) = (k^\lambda _{\theta \theta }+k^\lambda )(\theta , t/\lambda ^2) \end{aligned}$$

so $h^\lambda$ is again an entropy flow. Note that the $\theta$-derivative is scale-invariant.

Now $\Vert h^\lambda \Vert _2^2(t) = \lambda ^2\Vert h\Vert _2^2(t/\lambda ^2)$, and $\Vert h^\lambda _\theta \Vert _2^2(t) = \lambda ^2\Vert h_\theta \Vert _2^2(t/\lambda ^2)$. Take a sequence of times $\{t_j\}\rightarrow \infty$. Then $||h||_2^2(t_j) = ||h||_2^2(0) + 4\omega \pi \,t_j$. Set $\lambda _j = (||h||_2^2(0) + 4\omega \pi \,t_j)^{-1/2}$ and consider the sequence of rescalings $h^{\lambda _j}$. Then $||h^{\lambda _j}||_2^2(t_j)$ $= 1$ and

$$\begin{aligned} \Vert h^{\lambda _j}_\theta \Vert _2^2(t) = (||h||_2^2(0) + 4\omega \pi \,t_j)^{-1} \Vert h_\theta \Vert _2^2(t/\lambda ^2) \leqq (||h||_2^2(0) + 4\omega \pi \,t_j)^{-1} \Vert h_\theta \Vert _2^2(0) \rightarrow 0 \end{aligned}$$

for any $t\in [0,\infty )$. In particular, this holds for $t=t_j$ and suggests that $h^{\lambda _j}(\cdot ,t_j)$ converges to a circle (with support function equal to $\frac{1}{\sqrt{2\omega \pi }}$) as $j\rightarrow \infty$.

Another classical approach is to use a continuous rescaling (as for instance used by Huisken [14]). This is what we do in our treatment here of the entropy flow. Given a solution $\gamma$ to the entropy flow, we construct a rescaling $\eta$ by setting

$$\begin{aligned} \eta (\theta ,t) = \frac{\gamma (\theta ,t)}{\sqrt{L_0^2 + 8\omega ^2\pi ^2 t}} = \frac{\gamma (\theta ,t)}{\phi (t)}, \end{aligned}$$

where we have used $\phi (t) = \sqrt{L_0^2 + 8\omega ^2\pi ^2 t}$.

The rescaling $\eta$ is strictly convex with support function $h^\eta$ satisfying

$$\begin{aligned} h^\eta (\cdot ,t) = \frac{1}{\phi (t)}\, h^\gamma (\cdot , t), \end{aligned}$$

and curvature satisfying

$$\begin{aligned} k^\eta (\cdot ,t) = \phi (t)k^\gamma (\cdot ,t). \end{aligned}$$

We can calculate

$$\begin{aligned} \partial _th^\eta (\cdot ,t)&= -\frac{\phi '(t)}{\phi ^2(t)}\, h^\gamma (\cdot , t) + \frac{1}{\phi (t)}\, \partial _th^\gamma (\cdot , t) \\ {}&= -\frac{4\omega ^2\pi ^2}{(L_0^2 + 8\omega ^2\pi ^2 t)^{\frac{3}{2}}}\, h^\gamma (\cdot , t) + \frac{1}{\phi (t)}\, (k^\gamma _{\theta \theta }+k^\gamma )(\cdot , t) \\ {}&= -\frac{4\omega ^2\pi ^2}{L_0^2 + 8\omega ^2\pi ^2 t}\, h^\eta (\cdot , t) + \frac{1}{L_0^2 + 8\omega ^2\pi ^2 t}\, (k^\eta _{\theta \theta }+k^\eta )(\cdot , t) \\ {}&= \frac{1}{L_0^2 + 8\omega ^2\pi ^2 t} \bigg ( k^\eta _{\theta \theta }+k^\eta - 4\omega ^2\pi ^2\, h^\eta \bigg )(\cdot , t). \end{aligned}$$

Then, reparametrise time with a new variable $t^\eta$ defined by

$$\begin{aligned} \partial _{t^\eta } = \phi ^2 \partial _t , \end{aligned}$$

so that by the chain rule $h^\eta$ satisfies a new rescaled flow equation. In particular we find

$$\begin{aligned} h^\eta _{t^\eta } = k^\eta _{\theta \theta } + k^\eta - 4\omega ^2\pi ^2\, h^\eta . \end{aligned}$$

Our eventual goal will be to prove that $h^\eta$ converges smoothly exponentially fast to a round circle. In this section, we prove the remaining estimates needed on the un-scaled flow, whereas in Section 5 we study directly the rescaled flow.

First, let us use the global existence established above to show that eventually a certain scale-invariant quantity is small.

Lemma 4.1

Consider an entropy flow $\gamma :{\mathbb {S}}^1\times (0,\infty )\rightarrow {\mathbb {R}}^2$ with initial data $\gamma _0$ satisfying the conditions of Theorem 1.1. For any $\varepsilon >0$ there exists a $t^1_\varepsilon \in (0,\infty )$ such that

$$\begin{aligned} \int k^{-2}k_\theta ^2\,\textrm{d}\theta \bigg |_{t=t^1_\varepsilon } \leqq \varepsilon . \end{aligned}$$

Proof

Proposition 3.19 implies

$$\begin{aligned} \int _0^t\int k^{-2}k_\theta ^2\,\textrm{d}\theta \,\textrm{d}{{\hat{t}}} = \Vert h_\theta \Vert _2^2(0) - \Vert h_\theta \Vert _2^2(t^1_\varepsilon ) \leqq \Vert h_\theta \Vert _2^2(0). \end{aligned}$$

Note that $\Vert h_\theta \Vert _2^2(0)$ exists by Lemma 3.20. Taking $t\rightarrow \infty$ yields that $\int k^{-2}k_\theta ^2\,\textrm{d}\theta \in L^1((0,\infty ))$, which implies the result. $\square$

Now, we need to show that sufficient eventual pointwise in time smallness of $\Vert k^{-1}k_\theta \Vert _2^2$ is preserved.

Proposition 4.2

Consider an entropy flow $\gamma :{\mathbb {S}}^1\times (0,\infty )\rightarrow {\mathbb {R}}^2$ with initial data $\gamma _0$ satisfying the conditions of Theorem 1.1. For any $\varepsilon \in (0,1/108\omega \pi ]$, we have

$$\begin{aligned} \int k^{-2}k_\theta ^2\,\textrm{d}\theta \bigg |_{t} \leqq \varepsilon ,\quad {\text { for all }}\; t\leqq t^1_\varepsilon . \end{aligned}$$

(42)

Proof

First, let us calculate (recall (7))

$$\begin{aligned} \frac{\textrm{d}}{\textrm{d}t} \int k_\theta ^2k^{-2}\,\textrm{d}\theta&= 2\int k_\theta k^{-2}(-k^2(F_{\theta \theta }+F))_\theta \,\textrm{d}\theta -2\int k_\theta ^2k^{-3}(-k^2(F_{\theta \theta }+F))\,\textrm{d}\theta \\ {}&= -2\int k_\theta F_{\theta ^3}\,\textrm{d}\theta -2\int k_\theta F_\theta \,\textrm{d}\theta -4\int k_\theta ^2 k^{-1}(F_{\theta \theta }+F)\,\textrm{d}\theta \\ {}&\quad +2\int k_\theta ^2k^{-1}(F_{\theta \theta }+F)\,\textrm{d}\theta \\ {}&= -2\int F_{\theta }^2\,\textrm{d}\theta -2\int k_\theta ^2 k^{-1}(F_{\theta \theta }+F)\,\textrm{d}\theta . \end{aligned}$$

Now, integrating by parts and simplifying, we find that

$$\begin{aligned} \frac{\textrm{d}}{\textrm{d}t} \int k_\theta ^2k^{-2}\,\textrm{d}\theta&= -2\int F_{\theta }^2\,\textrm{d}\theta -2\int k_\theta ^2 k^{-1}(F_{\theta \theta }+F)\,\textrm{d}\theta \\&= -2\int F_{\theta }^2\,\textrm{d}\theta +2\int (k_\theta ^2 k^{-1})_\theta F_{\theta }\,\textrm{d}\theta -2\int k_\theta ^2 k^{-1}F\,\textrm{d}\theta \\&= -2\int F_{\theta }^2\,\textrm{d}\theta +2\int (2k_{\theta \theta } k_\theta k^{-1} - k_\theta ^3 k^{-2}) F_{\theta }\,\textrm{d}\theta \\ {}&\quad -2\int k_{\theta \theta }k_\theta ^2 k^{-1}\,\textrm{d}\theta -2\int k_\theta ^2\,\textrm{d}\theta \\&= -2\int F_{\theta }^2\,\textrm{d}\theta +4\int k_{\theta \theta } k_\theta k^{-1} F_{\theta }\,\textrm{d}\theta -2\int k_\theta ^3 k^{-2} k_{\theta ^3}\,\textrm{d}\theta \\ {}&\quad -\frac{8}{3}\int k_{\theta }^4k^{-2}\,\textrm{d}\theta -2\int k_\theta ^2\,\textrm{d}\theta \\&= -2\int F_{\theta }^2\,\textrm{d}\theta +4\int k_{\theta \theta } k_\theta k^{-1} F_{\theta }\,\textrm{d}\theta \\ {}&\quad +2\int (3k_{\theta \theta } k_\theta ^2 k^{-2} -2 k_\theta ^4 k^{-3}) k_{\theta \theta }\,\textrm{d}\theta \\ {}&\quad -\frac{8}{3}\int k_{\theta }^4k^{-2}\,\textrm{d}\theta -2\int k_\theta ^2\,\textrm{d}\theta \\&= -2\int k_{\theta ^3}^2\,\textrm{d}\theta -2\int k_\theta ^2\,\textrm{d}\theta +4\int k_{\theta \theta }^2\,\textrm{d}\theta \\ {}&\quad - 2\int k_{\theta \theta }^2 ( k_{\theta \theta } k^{-1} - k_\theta ^2 k^{-2} ) \,\textrm{d}\theta + \frac{4}{3}\int k_\theta ^4 k^{-2}\,\textrm{d}\theta \\ {}&\quad + 6\int k_{\theta \theta }^2 k_\theta ^2 k^{-2} \,\textrm{d}\theta - \frac{12}{5}\int k_\theta ^6 k^{-4} \,\textrm{d}\theta \\ {}&\quad -\frac{8}{3}\int k_{\theta }^4k^{-2}\,\textrm{d}\theta -2\int k_\theta ^2\,\textrm{d}\theta \\&= -2\int k_{\theta ^3}^2\,\textrm{d}\theta -4\int k_\theta ^2\,\textrm{d}\theta -4\int k_{\theta ^3}k_{\theta }\,\textrm{d}\theta - 2\int k_{\theta \theta }^3 k^{-1} \,\textrm{d}\theta \\ {}&\quad + 8\int k_{\theta \theta }^2 k_\theta ^2 k^{-2} \,\textrm{d}\theta - \frac{12}{5}\int k_\theta ^6 k^{-4} \,\textrm{d}\theta - \frac{4}{3}\int k_{\theta }^4k^{-2}\,\textrm{d}\theta . \end{aligned}$$

We need to estimate the third and fourth terms. Observe that integration by parts implies

$$\begin{aligned} - 2\int k_{\theta \theta }^3 k^{-1} \,\textrm{d}\theta&= - 2\int k_{\theta }^2 k_{\theta \theta }^2 k^{-2} \,\textrm{d}\theta + 4\int k_{\theta } k_{\theta \theta } k_{\theta ^3} k^{-1} \,\textrm{d}\theta \\&\leqq \delta \int k_{\theta ^3}^2\,\textrm{d}\theta + \Big (- 2 + \frac{4}{\delta }\Big ) \int k_{\theta }^2 k_{\theta \theta }^2 k^{-2} \,\textrm{d}\theta . \end{aligned}$$

This estimates the fourth term. For the third term, we simply use $ab \leqq \delta a^2 + \frac{1}{4\delta }b^2$. All together we have

$$\begin{aligned} \frac{\textrm{d}}{\textrm{d}t} \int k_\theta ^2k^{-2}\,\textrm{d}\theta&\leqq (5\delta -2)\int k_{\theta ^3}^2\,\textrm{d}\theta - (4-1/\delta )\int k_\theta ^2\,\textrm{d}\theta \\ {}&\quad + (6 + 4/\delta )\int k_{\theta \theta }^2 k_\theta ^2 k^{-2} \,\textrm{d}\theta - \frac{12}{5}\int k_\theta ^6 k^{-4} \,\textrm{d}\theta - \frac{4}{3}\int k_{\theta }^4k^{-2} \,\textrm{d}\theta . \end{aligned}$$

Take $\delta = 1/3$, then, using also the Poincaré and Hölder inequalities, we find that

$$\begin{aligned} \frac{\textrm{d}}{\textrm{d}t} \int k_\theta ^2k^{-2}\,\textrm{d}\theta&\leqq -\frac{1}{3}\int k_{\theta ^3}^2\,\textrm{d}\theta -\int k_\theta ^2\,\textrm{d}\theta + 18\int k_{\theta \theta }^2 k_\theta ^2 k^{-2} \,\textrm{d}\theta \\ {}&\quad - \frac{12}{5}\int k_\theta ^6 k^{-4} \,\textrm{d}\theta - \frac{4}{3}\int k_{\theta }^4k^{-2} \,\textrm{d}\theta \\ {}&\leqq -\bigg (\frac{1}{3} - 36\omega \pi \int k_\theta ^2 k^{-2}\,\textrm{d}\theta \bigg )\int k_{\theta ^3}^2\,\textrm{d}\theta -\int k_\theta ^2\,\textrm{d}\theta \\ {}&\quad - \frac{12}{5}\int k_\theta ^6 k^{-4} \,\textrm{d}\theta - \frac{4}{3}\int k_{\theta }^4k^{-2} \,\textrm{d}\theta . \end{aligned}$$

Above we used the estimate

$$\begin{aligned} \int k_{\theta \theta }^2 k_\theta ^2 k^{-2} \,\textrm{d}\theta\leqq & {} ||k_{\theta \theta }||_\infty ^2 \int k_\theta ^2 k^{-2} \,\textrm{d}\theta \leqq \bigg (\int |k_{\theta ^3}|\,\textrm{d}\theta \bigg )^2 \int k_\theta ^2 k^{-2} \,\textrm{d}\theta \\\leqq & {} 2\omega \pi \int k_{\theta ^3}^2\,\textrm{d}\theta \int k_\theta ^2 k^{-2} \,\textrm{d}\theta . \end{aligned}$$

Now, Lemma 4.1 implies that $\Vert k^{-1}k_\theta \Vert _2^2(t^1_\varepsilon ) \leqq \varepsilon$. Recall that $\varepsilon \leqq 1/108\omega \pi$, so that at $t=t^1_\varepsilon$ the coefficient of the highest order term is non-positive. Therefore

$$\begin{aligned} \frac{\textrm{d}}{\textrm{d}t} \int k_\theta ^2k^{-2}\,\textrm{d}\theta \bigg |_{t=t^1_\varepsilon } \leqq -\int k_\theta ^2\,\textrm{d}\theta - \frac{12}{5}\int k_\theta ^6 k^{-4} \,\textrm{d}\theta - \frac{4}{3}\int k_{\theta }^4k^{-2} \,\textrm{d}\theta \leqq 0. \end{aligned}$$

(43)

This preserves the smallness condition, yielding (43) for every $t\leqq t^1_\varepsilon$; therefore we have (42) and we are done. $\square$

We can now dramatically upgrade our preservation of convexity (Proposition 3.17) to an estimate that is as strong as possible (equality for circles). At the same time we note some long-time asymptotic information on the curvature.

Proposition 4.3

Consider an entropy flow $\gamma :{\mathbb {S}}^1\times (0,\infty )\rightarrow {\mathbb {R}}^2$ with initial data $\gamma _0$ satisfying the conditions of Theorem 1.1. Let $t_0>0$. There exists a $c_0>0$ depending only on $t_0$, ${\mathcal {E}}_0,L_0,\omega ,\Vert k\Vert _{L^1(\textrm{d}\theta )}(0))$ such that

$$\begin{aligned} k(\theta ,t)L(t) \leqq c_0,\quad {\text {for all }}\; t>t_0. \end{aligned}$$

Furthermore, $||k^{-1}k_\theta ||_2^2(t)\rightarrow 0$ and $\log k$ converges to its average in the topology $C^0({\mathbb {S}}^1)$ as $t\nearrow \infty$.

Proof

As we already know the flow is strictly convex, it remains to improve the estimate of Proposition 3.17 outisde a compact time interval. Applying Proposition 4.2 with $\varepsilon = \frac{1}{108\omega \pi }$ gives that

$$\begin{aligned} ||k^{-1}k_\theta ||_2^2(t) \leqq \frac{1}{108\omega \pi } \end{aligned}$$

(44)

for $t\leqq t^1_{1/108\omega \pi }$. Let us set $t^1 = t^1_{1/108\omega \pi }$. Then

$$\begin{aligned} \log k - \overline{\log k} \leqq -\int k^{-1}|k_\theta |\,\textrm{d}\theta \leqq -\sqrt{2\omega \pi } \bigg (\int k^{-2}k_\theta ^2\,\textrm{d}\theta \bigg )^{\frac{1}{2}}. \end{aligned}$$

Using (44) and the lower bound for the entropy (Corollary 3.11) yields, for $t\leqq t^1$,

$$\begin{aligned} \log k \leqq - \sqrt{\frac{\omega \pi }{108\omega \pi }} + \log \frac{2\omega \pi }{L_0 + 4\omega ^2\pi ^2c_1^{-1}\big ( \sqrt{4\omega ^2\pi ^2 + tc_1^2} - 2\omega \pi \big )} \end{aligned}$$

or

$$\begin{aligned} k \leqq \frac{2\omega \pi }{L_0 + 4\omega ^2\pi ^2c_1^{-1}\big ( \sqrt{4\omega ^2\pi ^2 + tc_1^2} - 2\omega \pi \big )}\exp \bigg (- \frac{1}{6\sqrt{3}}\bigg ) \quad {\text {for }}\; t\leqq t^1. \end{aligned}$$

Using Lemma 3.7 first and then the estimate above yields

$$\begin{aligned} kL&\leqq k\sqrt{L_0^2 + 8\omega ^2\pi ^2t} \nonumber \\&\leqq \frac{2\omega \pi \sqrt{L_0^2 + 8\omega ^2\pi ^2t}}{L_0 + \sqrt{2}\,\omega \pi \big ( \sqrt{32\omega ^4\pi ^4c_1^{-2} + 8\omega ^2\pi ^2t} - 2\omega \pi \big )}\exp \bigg (- \frac{1}{6\sqrt{3}}\bigg ) \end{aligned}$$

(45)

which is uniformly bounded from below for $t\in [t^1,\infty )$. Furthermore, Proposition 3.17 implies that kL is uniforomly bounded from below on $[t_0,t^1]$, by a constant depending only on $t_0$, $t^1$ and ${\mathcal {E}}_0,L_0,\omega ,\Vert k\Vert _{L^1(\textrm{d}\theta )}(0)$. This proves the first claim.

Then, using the bound from below on k in (43) we find that

$$\begin{aligned} \frac{\textrm{d}}{\textrm{d}t} \int k_\theta ^2k^{-2}\,\textrm{d}\theta \leqq -\int k_\theta ^2\,\textrm{d}\theta \leqq -c_0^{2}L^{-2}\int k^{-2}k_\theta ^2\,\textrm{d}\theta . \end{aligned}$$

Observe that

$$\begin{aligned} -\int _{t^1}^t L^{-2}\,dt&\leqq - \int _{t^1}^t \frac{1}{\bigg (L_0 + 4\omega \pi c_1^{-1}\bigg ( \sqrt{4\omega ^2\pi ^2 + tc_1^2} - 2\omega \pi \bigg )\bigg )^2}\,dt \\ {}&= -\frac{1}{2\omega \pi } \log \bigg (\frac{L_0-8\omega ^2\pi ^2c_1^{-1} + 4\omega \pi \sqrt{4\omega ^2\pi ^2c_1^{-2}+t}}{L_0-8\omega ^2\pi ^2c_1^{-1} + 4\omega \pi \sqrt{4\omega ^2\pi ^2c_1^{-2}+t^1}}\bigg ) \\ {}&\quad - \frac{L_0-8\omega ^2\pi ^2c_1^{-1}}{2\omega \pi } \bigg ( \frac{1}{L_0-8\omega ^2\pi ^2c_1^{-1} + 4\omega \pi \sqrt{4\omega ^2\pi ^2c_1^{-2}+t}} \\ {}&\quad - \frac{1}{L_0-8\omega ^2\pi ^2c_1^{-1} + 4\omega \pi \sqrt{4\omega ^2\pi ^2c_1^{-2}+t^1}} \bigg ) \\ {}&\leqq -\frac{1}{2\omega \pi } \log \Big (L_0-8\omega ^2\pi ^2c_1^{-1} + 4\omega \pi \sqrt{4\omega ^2\pi ^2c_1^{-2}+t}\Big ) + C \,, \end{aligned}$$

so

$$\begin{aligned} \int k^{-2}k_\theta ^2\,\textrm{d}\theta&\leqq C||k^{-1}k_\theta ||_2^2(0)\exp \bigg ( -\frac{c_0^2}{2\omega \pi } \log \Big (L_0-8\omega ^2\pi ^2c_1^{-1} + 4\omega \pi \sqrt{4\omega ^2\pi ^2c_1^{-2}+t}\Big ) \bigg ) \\ {}&\leqq C||k^{-1}k_\theta ||_2^2(0) \Big (L_0-8\omega ^2\pi ^2c_1^{-1} + 4\omega \pi \sqrt{4\omega ^2\pi ^2c_1^{-2}+t}\Big )^{-\frac{c_0^2}{2\omega \pi }}. \end{aligned}$$

This implies that $||k^{-1}k_\theta ||_2^2(t)\rightarrow 0$ (second claim), and so,

$$\begin{aligned} |\log k - \overline{\log k}|(\theta ,t) \leqq \sqrt{2\omega \pi }\,||k^{-1}k_\theta ||_2(t) \rightarrow 0, \end{aligned}$$

which is the third claim and finishes the proof. $\square$

Note that the curvature of growing circles decreases under the entropy flow with asymptotic rate $1/\sqrt{t}$, with kL constant along the flow. Proposition 4.3 is strong enough that along the continuous rescaling the estimate for curvature from below will be uniform.

5 The Rescaled Flow

We briefly recall the continuous rescaling $\eta$ from the introduction of Section 4. Given an entropy flow $\gamma$ satisfying the hypotheses of Theorem A.1, we rescale by setting $\eta (\theta ,t) = \gamma (\theta ,t)/\phi (t)$ where where $\phi (t) = \sqrt{L_0^2 + 8\omega ^2\pi ^2 t}$. Then, reparametrise time with a new variable $t^\eta$ defined by $\partial _{t^\eta } = \phi ^2 \partial _t$.

The rescaled flow equation is

$$\begin{aligned} h^\eta _{t^\eta } = k^\eta _{\theta \theta } + k^\eta - 4\omega ^2\pi ^2 h^\eta . \end{aligned}$$

From now until the end of this section, we drop the $\eta$ superscript. Let us record some immediate facts about the rescaled flow from our previous analysis.

Corollary 5.1

Consider the rescaling of an entropy flow generated by Theorem 3.24. The rescaled flow exists globally, and:

• Length is uniformly bounded, satisfying

  $$\begin{aligned} 1 \leqq L(t) \leqq c_L \end{aligned}$$

  (46)

  where $c_L = c_L(\omega ,L_0)$;

• Curvature is uniformly bounded from below, satisfying

  $$\begin{aligned} k(\theta ,t)\leqq c_0c_L^{-1} \end{aligned}$$

  where $c_0$ is defined in Proposition 4.3;

• We have $||k^{-1}k_\theta ||_2^2(t)\rightarrow 0$ and $\log k$ converges to its average in the topology $C^0({\mathbb {S}}^1)$ as $t\nearrow \infty$.

Proof

For length, the following estimate follows immediately from Lemma 3.7:

$$\begin{aligned} 1 \leqq L(t) \leqq \frac{L_0 - 8\omega ^2\pi ^2c_1^{-1}}{\sqrt{L_0^2 + 8\omega \pi t}} + 4\omega \pi \sqrt{\frac{4\omega ^2\pi ^2c_1^{-2} + t}{L_0^2 + 8\omega ^2\pi ^2 t}}. \end{aligned}$$

(47)

To see that this bound is uniform, note that the upper bound has limits both as $t\searrow 0$ and as $t\nearrow \infty$.

The estimate on curvature from below follows from Proposition 4.3. This is because kL is scale-invariant, so the main estimate from Proposition 4.3 yields $kL \leqq c_0L^{-1} \leqq c_0c_L^{-1}$. The quantity $||k^{-1}k_\theta ||_2^2$ is also scale-invariant, so its decay to zero also follows from Proposition 4.3. With this fact in hand, we can repeat the final steps of the proof of Proposition 4.3 but for the rescaled flow to conclude that $\log k$ converges to its average (which is uniformly bounded along the rescaled flow and thus convergent). $\square$

We have identification of the limit of the rescaling as a standard round $\omega$-circle. Our final task is to establish convergence in the smooth topology. For this we will focus on the rescaled support function.

Proposition 5.2

Consider the rescaling of an entropy flow generated by Theorem 3.24. Then $\Vert h\Vert _2^2(t) \rightarrow 2\omega \pi$, and

$$\begin{aligned} \Vert h_\theta \Vert _2^2(t) \leqq ||h_\theta ||_2^2(0) e^{-8\omega ^2\pi ^2\,t}. \end{aligned}$$

Proof

First, compute

$$\begin{aligned} \frac{\textrm{d}}{\textrm{d}t} \int h^2\,\textrm{d}\theta&= 2\int h(k_{\theta \theta } + k - 4\omega ^2\pi ^2 h)\,\textrm{d}\theta \\&= 2\int 1 -4\omega ^2\pi ^2 h^2\,\textrm{d}\theta = 4\omega \pi - 8\omega ^2\pi ^2 \int h^2\,\textrm{d}\theta \,, \end{aligned}$$

so

$$\begin{aligned} \frac{\textrm{d}}{\textrm{d}t} \Big (e^{8\omega ^2\pi ^2\,t} ||h||_2^2(t)\Big ) = 4\omega \pi e^{8\omega ^2\pi ^2\,t} \end{aligned}$$

and

$$\begin{aligned} e^{8\omega ^2\pi ^2\,t} ||h||_2^2(t) - ||h||_2^2(0) = \frac{e^{8\omega ^2\pi ^2\,t} - 1}{2\omega \pi } \end{aligned}$$

which implies

$$\begin{aligned} ||h||_2^2(t) = \frac{1}{2\omega \pi } + (||h||_2^2(0) - 1/2\omega \pi )e^{-8\omega ^2\pi ^2\,t}. \end{aligned}$$

This equation (note that $\Vert h\Vert _2^2$ is monotone for the original flow) implies that $\Vert h\Vert _2^2 \rightarrow 1/2\omega \pi$ as $t\rightarrow \infty$.

For $\Vert h_\theta \Vert _2^2$, we calculate

$$\begin{aligned} \frac{\textrm{d}}{\textrm{d}t} \int h_\theta ^2\,\textrm{d}\theta&= -2\int h_{\theta \theta }(k_{\theta \theta } + k - 4\omega ^2\pi ^2\,h)\,\textrm{d}\theta = -2\int (1/k)_{\theta \theta }k\,\textrm{d}\theta -8\omega ^2\pi ^2\int h_{\theta }^2\,\textrm{d}\theta \nonumber \\&= -2\int k_\theta ^2/k^2\,\textrm{d}\theta -8\omega ^2\pi ^2\int h_{\theta }^2\,\textrm{d}\theta \leqq -8\omega ^2\pi ^2\int h_{\theta }^2\,\textrm{d}\theta . \end{aligned}$$

(48)

Therefore

$$\begin{aligned} \Vert h_\theta \Vert _2^2(t) \leqq ||h_\theta ||_2^2(0) e^{-8\omega ^2\pi ^2\,t}, \end{aligned}$$

(49)

for $t\leqq 0$. $\square$

Now, as is standard, in order to obtain exponential decay for quantities of the form $||h_{\theta ^p}||_2^2$ it is enough to show that they are uniformly bounded and then apply interpolation with (49).

Proposition 5.3

Consider the rescaling of an entropy flow generated by Theorem 3.24. There exist $t_p\leqq 0$ such that for all $p\in {\mathbb {N}}$ we have

$$\begin{aligned} \Vert h_{\theta ^p}\Vert _2^2(t) \leqq C,\quad {\text {for}} \quad t\leqq t_p. \end{aligned}$$

Proof

Many of the estimates and calculations in this proof are in a sense rescaled analogies of the proof of Proposition 3.22. Proposition 5.2 covers the case $p=1$.

Case $p=2$. We calculate

$$\begin{aligned} \frac{\textrm{d}}{\textrm{d}t} \int h_{\theta \theta }^2\,\textrm{d}\theta&= 2\int h_{\theta ^4}(k_{\theta \theta } + k - 4\omega ^2\pi ^2\,h)\,\textrm{d}\theta \nonumber \\&= 2\int (1/k)_{\theta ^4}k\,\textrm{d}\theta -8\omega ^2\pi ^2\,\int h_{\theta \theta }^2\,\textrm{d}\theta \nonumber \\&= 2\int (1/k)_{\theta \theta }k_{\theta \theta }\,\textrm{d}\theta -8\omega ^2\pi ^2\,\int h_{\theta \theta }^2\,\textrm{d}\theta \nonumber \\&= -2\int (k_{\theta \theta }k^{-2} - 2k_\theta ^2k^{-3})k_{\theta \theta }\,\textrm{d}\theta -8\omega ^2\pi ^2\,\int h_{\theta \theta }^2\,\textrm{d}\theta \nonumber \\&= -2\int k_{\theta \theta }^2k^{-2}\,\textrm{d}\theta +4\int k_\theta ^4k^{-4}\,\textrm{d}\theta -8\omega ^2\pi ^2\,\int h_{\theta \theta }^2\,\textrm{d}\theta . \end{aligned}$$

(50)

Corollary 5.1 implies that $\log k$ is convergent to its average. The average of $\log k$ is $\frac{1}{2\omega \pi }{\mathcal {E}}(t)$. Applying the proof of Lemma 3.1 gives a uniform bound for ${\mathcal {E}}$ from below, due to the uniform length bound from Corollary 5.1. For the bound from above, note that at each t we have (using $k_0(t) = \inf k(\cdot ,t) \leqq c_0c_L^{-1}$)

$$\begin{aligned} |\log k - \log k_0| \leqq \sqrt{2\omega \pi }\sqrt{1/108\omega \pi } \end{aligned}$$

which implies a uniform bound from above for k and therefore also a uniform bound from above for ${\mathcal {E}}$. Thus the average of $\log k$ is bounded, and $\log k$ converges to it. Therefore k is convergent to a constant (also its average). Then

$$\begin{aligned} -\int k_{\theta \theta }^2k^{-2}\,\textrm{d}\theta +2\int k_\theta ^4k^{-4}\,\textrm{d}\theta \leqq - c||k_{\theta \theta }||_2^2 + C||k^{-1}k_\theta ||_2^2. \end{aligned}$$

Using (48) we have

$$\begin{aligned} \int _0^\infty \int k_\theta ^4k^{-4}\,\textrm{d}\theta \,dt \leqq C \end{aligned}$$

so, integrating (50), we find that

$$\begin{aligned} ||h_{\theta \theta }||_2^2(t) \leqq C. \end{aligned}$$

This holds for all $t\leqq 0$, so we can pick $t_2=t_1=0$.

Case $p=3$. For the next case, we calculate

$$\begin{aligned} \frac{\textrm{d}}{\textrm{d}t} \int h_{\theta ^3}^2\,\textrm{d}\theta&= 2\int (1/k)_{\theta ^3}k_{\theta ^3}\,\textrm{d}\theta -8\omega ^2\pi ^2\int h_{\theta ^3}^2\,\textrm{d}\theta \nonumber \\&\leqq -c\int k_{\theta ^3}^2\,\textrm{d}\theta + C\int k_{\theta \theta }^2k_\theta ^2 + k_\theta ^6\,\textrm{d}\theta -8\omega ^2\pi ^2\int h_{\theta ^3}^2\,\textrm{d}\theta . \end{aligned}$$

(51)

We find

$$\begin{aligned} \int k_{\theta \theta }^2k_\theta ^2\,\textrm{d}\theta&= -\int k_{\theta ^3}k_\theta ^3 + 2k_{\theta \theta }^2k_\theta ^2\,\textrm{d}\theta \end{aligned}$$

which implies, after factorising and interpolating,

$$\begin{aligned} \int k_{\theta \theta }^2k_\theta ^2\,\textrm{d}\theta \leqq \delta \int k_{\theta ^3}^2\,\textrm{d}\theta + C(\delta )\int k_\theta ^6\,\textrm{d}\theta . \end{aligned}$$

(52)

Observe $k_\theta = (k-{\overline{k}})_\theta$ where ${\overline{k}}$ is the average of k. Then, integrating by parts, interpolating, and using the convergence of k to its average, we find that

$$\begin{aligned} \int k_\theta ^6\,\textrm{d}\theta&= -5\int k_{\theta \theta }k_{\theta }^4(k-{\overline{k}})\,\textrm{d}\theta \\ {}&\leqq \delta \int k_{\theta \theta }^2k_{\theta }^2\,\textrm{d}\theta + C(\delta ) \int k_{\theta }^6(k-{\overline{k}})^2\,\textrm{d}\theta \\ {}&\leqq \delta \int k_{\theta ^3}^2\,\textrm{d}\theta + \frac{1}{2}\int k_\theta ^6\,\textrm{d}\theta \end{aligned}$$

for $t>t_3$. This is the definition of $t_3$.

Thus

$$\begin{aligned} \int k_\theta ^6\,\textrm{d}\theta \leqq \delta \int k_{\theta ^3}^2\,\textrm{d}\theta \end{aligned}$$

(53)

for $t>t_3$. Combining (52), (53) and (51) yields

$$\begin{aligned} \frac{\textrm{d}}{\textrm{d}t} \int h_{\theta ^3}^2\,\textrm{d}\theta&\leqq 0 \end{aligned}$$

for $t>t_3$, which implies $||h_{\theta ^3}||_2^2(t) \leqq ||h_{\theta ^3}||_2^2(t_3)$.

Case $p=4$. For $||h_{\theta ^4}||_2^2$ we calculate

$$\begin{aligned} \frac{\textrm{d}}{\textrm{d}t} \int h_{\theta ^4}^2\,\textrm{d}\theta&= 2\int (1/k)_{\theta ^4}k_{\theta ^4}\,\textrm{d}\theta -8\omega ^2\pi ^2\int h_{\theta ^4}^2\,\textrm{d}\theta \nonumber \\&\leqq -c\int k_{\theta ^4}^2\,\textrm{d}\theta + C\int k_{\theta ^3}^2k_\theta ^2 + k_{\theta \theta }^2k_\theta ^4 + k_{\theta \theta }^4 + k_{\theta }^8 \,\textrm{d}\theta -8\omega ^2\pi ^2\int h_{\theta ^4}^2\,\textrm{d}\theta \nonumber \\&\leqq -c\int k_{\theta ^4}^2\,\textrm{d}\theta + C\int k_{\theta ^3}^2k_\theta ^2 + k_{\theta \theta }^4 + k_{\theta }^8 \,\textrm{d}\theta -8\omega ^2\pi ^2\int h_{\theta ^4}^2\,\textrm{d}\theta . \end{aligned}$$

(54)

First, use

$$\begin{aligned} \int k_{\theta ^3}^2k_\theta ^2 \,\textrm{d}\theta \leqq \delta \int k_{\theta ^4}^2 + k_{\theta \theta }^4\,\textrm{d}\theta + C(\delta )\int k_\theta ^8\,\textrm{d}\theta , \end{aligned}$$

(55)

where $\delta >0$ will be chosen.

To prove (55), first calculate

$$\begin{aligned} \int k_{\theta ^3}^2k_\theta ^2 \,\textrm{d}\theta&\leqq C\int |k_{\theta ^4}|\,|k_{\theta \theta }|\,k_\theta ^2 + |k_{\theta ^3}|\,k_{\theta \theta }^2\,|k_\theta | \,\textrm{d}\theta \\&\leqq \delta \int k_{\theta ^4}^2\,\textrm{d}\theta + C(\delta )\int k_{\theta \theta }^2k_\theta ^4\,\textrm{d}\theta + \frac{1}{2}\int k_{\theta ^3}^2k_\theta ^2\,\textrm{d}\theta + C\int k_{\theta \theta }^4 \,\textrm{d}\theta . \end{aligned}$$

Then absorb to find that

$$\begin{aligned} \int k_{\theta ^3}^2k_\theta ^2 \,\textrm{d}\theta&\leqq \delta \int k_{\theta ^4}^2\,\textrm{d}\theta + C(\delta )\int k_{\theta \theta }^2k_\theta ^4\,\textrm{d}\theta + C\int k_{\theta \theta }^4 \,\textrm{d}\theta \nonumber \\&\leqq \delta \int k_{\theta ^4}^2 + k_{\theta \theta }^4\,\textrm{d}\theta + C(\delta )\int k_\theta ^8\,\textrm{d}\theta \,, \end{aligned}$$

(56)

as required.

For the remaining terms, let us note two general estimates. The first is

$$\begin{aligned} \int k_{\theta }^{2q} \,\textrm{d}\theta&\leqq C(q)\int |k_{\theta \theta }|\,|k_\theta |^{2q-2}\,|k-{\overline{k}}|\,\textrm{d}\theta \nonumber \\&\leqq \delta \int k_{\theta \theta }^2\,k_\theta ^{2q-4}\,\textrm{d}\theta + C(\delta )||k-{\overline{k}}||_\infty \int k_\theta ^{2q}\,\,\textrm{d}\theta . \end{aligned}$$

(57)

The second requires two steps. Start by estimating

$$\begin{aligned} \int k_{\theta ^m}^{2q} \,\textrm{d}\theta&\leqq C\int |k_{\theta ^{m+1}}|\,k_{\theta ^m}^{2q-2}\,|k_{\theta ^{m-1}}|\,\textrm{d}\theta \\&\leqq \frac{1}{2}\int k_{\theta ^m}^{2q} \,\textrm{d}\theta + C\int k_{\theta ^{m+1}}^2k_{\theta ^m}^{2q-4}k_{\theta ^{m-1}}^2\,\textrm{d}\theta \end{aligned}$$

then absorb to find that

$$\begin{aligned} \int k_{\theta ^m}^{2q} \,\textrm{d}\theta&\leqq C\int k_{\theta ^{m+1}}^2k_{\theta ^m}^{2q-4}k_{\theta ^{m-1}}^2\,\textrm{d}\theta \end{aligned}$$

(58)

Now applying (57) with $q=4$ and then interpolating once more yields

$$\begin{aligned} \int k_{\theta }^{8} \,\textrm{d}\theta&\leqq \delta \int k_{\theta \theta }^2\,k_\theta ^{4}\,\textrm{d}\theta + C(\delta )||k-{\overline{k}}||_\infty \int k_\theta ^{8}\,\,\textrm{d}\theta \\&\leqq \frac{\delta ^2}{2}\int k_{\theta \theta }^4\,\textrm{d}\theta + (\frac{1}{2}+C(\delta )||k-{\overline{k}}||_\infty ) \int k_\theta ^{8}\,\,\textrm{d}\theta \end{aligned}$$

which is

$$\begin{aligned} \int k_{\theta }^{8} \,\textrm{d}\theta \leqq \delta ^2\int k_{\theta \theta }^4\,\textrm{d}\theta + C(\delta )||k-{\overline{k}}||_\infty \int k_\theta ^{8}\,\,\textrm{d}\theta \end{aligned}$$

(59)

after absorption.

Applying (58) with $q=2$ and $m=2$, then (56), then (59) (using ${\hat{\delta }}$ there for clarity) gives

$$\begin{aligned} \int k_{\theta \theta }^{4} \,\textrm{d}\theta&\leqq C\int k_{\theta ^3}^2k_\theta ^2\,\textrm{d}\theta \nonumber \\&\leqq \delta \int k_{\theta ^4}^2 + k_{\theta \theta }^4\,\textrm{d}\theta + C(\delta )\int k_\theta ^8\,\textrm{d}\theta \nonumber \\&\leqq (\delta +C(\delta ){\hat{\delta }}^2)\int k_{\theta ^4}^2 + k_{\theta \theta }^4\,\textrm{d}\theta + C(\delta )C({\hat{\delta }})||k-{\overline{k}}||_\infty \int k_\theta ^{8}\,\,\textrm{d}\theta . \end{aligned}$$

(60)

Add the estimates (56), (59), (60) together to find (also use (59) once more)

$$\begin{aligned} C\int k_{\theta ^3}^2k_\theta ^2 + k_{\theta \theta }^4 + k_{\theta }^8 \,\textrm{d}\theta&\leqq (2\delta +\delta ^2+C(\delta ){\hat{\delta }}^2)\int k_{\theta ^4}^2 + k_{\theta \theta }^4\,\textrm{d}\theta \\&\quad + C(\delta )C({\hat{\delta }})||k-{\overline{k}}||_\infty \int k_\theta ^{8}\,\,\textrm{d}\theta . \end{aligned}$$

Choose $\delta$, ${\hat{\delta }}$ small enough and let $t_4>0$ be large enough (recall k is converging pointwise to its average) such that

$$\begin{aligned} C\int k_{\theta ^3}^2k_\theta ^2 + k_{\theta \theta }^4 + k_{\theta }^8 \,\textrm{d}\theta \leqq \frac{c}{2}\delta \int k_{\theta ^4}^2 \,\textrm{d}\theta + \frac{1}{2}\int k_{\theta ^3}^2k_\theta ^2 + k_{\theta \theta }^4 + k_{\theta }^8 \,\textrm{d}\theta . \end{aligned}$$

Absorbing then combining with (54) gives

$$\begin{aligned} \frac{\textrm{d}}{\textrm{d}t} \int h_{\theta ^4}^2\,\textrm{d}\theta&\leqq -\frac{c}{2}\int k_{\theta ^4}^2\,\textrm{d}\theta -8\omega ^2\pi ^2\int h_{\theta ^4}^2\,\textrm{d}\theta \,, \end{aligned}$$

for $t\leqq t_4$, which clearly implies

$$\begin{aligned} ||h_{\theta ^4}||_2^2(t) \leqq ||h_{\theta ^4}||_2^2(t_4), \end{aligned}$$

as required.

Case $p=5$. For the next case we calculate

$$\begin{aligned} \frac{\textrm{d}}{\textrm{d}t} \int h_{\theta ^5}^2\,\textrm{d}\theta&= 2\int (1/k)_{\theta ^5}k_{\theta ^5}\,\textrm{d}\theta -8\omega ^2\pi ^2\int h_{\theta ^5}^2\,\textrm{d}\theta \nonumber \\&\leqq -c\int k_{\theta ^5}^2\,\textrm{d}\theta + C\int k_{\theta ^4}^2k_\theta ^2 + k_{\theta ^3}^2k_{\theta \theta }^2 + k_{\theta ^3}^2k_\theta ^4 + k_{\theta \theta }^4k_\theta ^2 \nonumber \\&\quad + k_{\theta \theta }^2k_\theta ^6 + k_{\theta }^{10} \,\textrm{d}\theta -8\omega ^2\pi ^2\int h_{\theta ^5}^2\,\textrm{d}\theta \nonumber \\&\leqq -c\int k_{\theta ^5}^2\,\textrm{d}\theta + C\int k_{\theta ^4}^2 \nonumber \\&\quad + k_{\theta ^3}^2k_{\theta \theta }^2 + k_{\theta ^3}^2 + k_{\theta \theta }^4 \nonumber \\&\quad + k_{\theta \theta }^2 \,\textrm{d}\theta + C -8\omega ^2\pi ^2\int h_{\theta ^5}^2\,\textrm{d}\theta . \end{aligned}$$

(61)

In the last step we used the fact that $||h_{\theta ^4}||_2^2(t)$ bounded implies $||k_{\theta }||_\infty$ bounded (see the proof of Proposition 3.22).

Interpolation and the estimate (58) gives

$$\begin{aligned}&C\int k_{\theta ^4}^2 + k_{\theta ^3}^2k_{\theta \theta }^2 + k_{\theta ^3}^2 + k_{\theta \theta }^4 + k_{\theta \theta }^2 \,\textrm{d}\theta \leqq -\frac{c}{4}\int k_{\theta ^5}^2\,\textrm{d}\theta \\ {}&\quad + C\int k_{\theta ^3}^2k_{\theta \theta }^2 + k_{\theta \theta }^4 \,\textrm{d}\theta + C. \end{aligned}$$

Now using (58) with $m=2$ and $p=2$ gives $||k_{\theta \theta }||_4^4(t) \leqq C||k_{\theta ^3}||_2^2(t)$ for $t>t_4$, which can be interpolated. For the remaining term, we estimate

$$\begin{aligned} \int k_{\theta ^3}^2k_{\theta \theta }^2\,\textrm{d}\theta \leqq \int |k_{\theta ^4}|\,|k_{\theta \theta }^3|\,\textrm{d}\theta - 2\int k_{\theta ^3}^2k_{\theta \theta }^2\,\textrm{d}\theta \end{aligned}$$

which implies

$$\begin{aligned} \int k_{\theta ^3}^2k_{\theta \theta }^2\,\textrm{d}\theta&\leqq \frac{1}{3}\int |k_{\theta ^4}|\,|k_{\theta \theta }^3|\,\textrm{d}\theta \\ {}&\leqq \delta \int k_{\theta \theta }^6\,\textrm{d}\theta + C(\delta )\int k_{\theta ^4}^2\,\textrm{d}\theta \\ {}&\leqq \delta \int k_{\theta ^3}^2k_{\theta \theta }^2k_{\theta }^2\,\textrm{d}\theta + C(\delta )\int k_{\theta ^4}^2\,\textrm{d}\theta \\ {}&\leqq C\delta \int k_{\theta ^3}^2k_{\theta \theta }^2\,\textrm{d}\theta + C(\delta )\int k_{\theta ^4}^2\,\textrm{d}\theta \, \end{aligned}$$

for $t>t_4$. Above we used (58) with $m=2$, $q=3$ and then the boundedness of $||k_\theta ||_\infty$. Absorbing, we find that

$$\begin{aligned} \int k_{\theta ^3}^2k_{\theta \theta }^2\,\textrm{d}\theta \leqq C(\delta )\int k_{\theta ^4}^2\,\textrm{d}\theta \leqq \frac{c}{4}\int k_{\theta ^5}^2\,\textrm{d}\theta + C. \end{aligned}$$

These estimates allow us to conclude, with (61), that $||h_{\theta ^5}||_2^2(t) \leqq C$ for $t>t_5$, where we simply choose $t_5=t_4$.¹

Case $p=6$. For the next case we calculate

$$\begin{aligned} \frac{\textrm{d}}{\textrm{d}t} \int h_{\theta ^6}^2\,\textrm{d}\theta&= 2\int (1/k)_{\theta ^6}k_{\theta ^6}\,\textrm{d}\theta -8\omega ^2\pi ^2\int h_{\theta ^6}^2\,\textrm{d}\theta \nonumber \\&\leqq -c\int k_{\theta ^6}^2\,\textrm{d}\theta -8\omega ^2\pi ^2\int h_{\theta ^6}^2\,\textrm{d}\theta \nonumber \\ {}&\qquad + C\int k_{\theta ^5}^2k_\theta ^2 + k_{\theta ^4}^2k_{\theta \theta }^2 + k_{\theta ^4}^2k_\theta ^4 + k_{\theta ^3}^4 \nonumber \\&\quad + k_{\theta ^3}^2k_{\theta \theta }^2k_\theta ^2 + k_{\theta ^3}^2k_\theta ^6 + k_{\theta ^2}^6 + k_{\theta ^2}^4k_\theta ^4 + k_{\theta ^2}^2k_\theta ^8 + k_{\theta }^{12} \,\textrm{d}\theta \nonumber \\&\leqq -c\int k_{\theta ^6}^2\,\textrm{d}\theta + C\int k_{\theta ^5}^2 + k_{\theta ^4}^2 + k_{\theta ^3}^4 + k_{\theta ^3}^2 \,\textrm{d}\theta \nonumber \\ {}&\quad + C -8\omega ^2\pi ^2\int h_{\theta ^6}^2\,\textrm{d}\theta . \end{aligned}$$

(62)

In the last step we used the fact that $||h_{\theta ^5}||_2^2(t)$ bounded implies $||k_{\theta \theta }||_\infty$ bounded (see the proof of Proposition 3.22).

Interpolation and the estimate (58) (for $q=2$, $m=3$) gives

$$\begin{aligned}&C\int k_{\theta ^5}^2 + k_{\theta ^4}^2 + k_{\theta ^3}^4 + k_{\theta ^3}^2 \,\textrm{d}\theta \leqq -\frac{c}{4}\int k_{\theta ^5}^2\,\textrm{d}\theta + C\int k_{\theta ^4}^2k_{\theta \theta }^2 \,\textrm{d}\theta \\ {}&\quad + C \leqq -\frac{c}{2}\int k_{\theta ^5}^2\,\textrm{d}\theta + C. \end{aligned}$$

These estimates allow us to conclude, with (62), that $||h_{\theta ^6}||_2^2(t) \leqq C$ for $t\leqq t_6$, where we simply choose $t_6=t_5$.

Final case: Induction for large p. Suppose there exist times $\{t_1,\ldots ,t_p\}$, $t_p>t_i$ for all $i\ne p$, such that

$$\begin{aligned} \Vert h_{\theta ^p}\Vert _2^2(t) \leqq \Vert h_{\theta ^p}\Vert _2^2(t_p) \end{aligned}$$

for all $t>t_p$. We wish to show that this implies

$$\begin{aligned} \Vert h_{\theta ^{p+1} }\Vert _2^2(t) \leqq \Vert h_{\theta ^{p+1} }\Vert _2^2(t_{p}) \end{aligned}$$

for all $t>t_{p}$.

As before, previous steps imply

$$\begin{aligned} ||k_{\theta ^{p-3}}||_\infty \leqq C. \end{aligned}$$

(63)

Note that as $p \leqq 7$, we know at least that the first four derivatives of k are bounded in $L^\infty$.

Recall (40), which we copy here for the convenience of the reader:

$$\begin{aligned} (k^{-1})_{\theta ^m} = - k^{-2}k_{\theta ^m} + \sum _{q=2}^m k^{-(q+1)} \sum _{i_1+\cdots +i_q=m} c(i_1,\ldots ,i_q) k_{\theta ^{i_1}} \cdots k_{\theta ^{i_q}}. \end{aligned}$$

We compute

$$\begin{aligned} \frac{\textrm{d}}{\textrm{d}t} \int h_{\theta ^{p+1}}^2\,\textrm{d}\theta&= 2\int (k^{-1})_{\theta ^{p+1}}k_{\theta ^{p+1}}\,\textrm{d}\theta - 8\omega ^2\pi ^2\int h_{\theta ^{p+1}}^2\,\textrm{d}\theta . \end{aligned}$$

Using (40) and then (63), we estimate

$$\begin{aligned}&-2\int (k^{-1})_{\theta ^{p+1}}k_{\theta ^{p+1}}\,\textrm{d}\theta \leqq -\int k^{-2}k_{\theta ^{(p+1)}}^2\,\textrm{d}\theta \\&\qquad +C\sum _{q=2}^{p+1} \sum _{i_1+\cdots +i_q={p+1}} c(i_1,\ldots ,i_q) \int k_{\theta ^{i_1}}^2 \cdots k_{\theta ^{i_q}}^2 \,\textrm{d}\theta \\&\quad \leqq -\int k^{-2}k_{\theta ^{(p+1)}}^2\,\textrm{d}\theta + C + C\sum _{q=2}^{4} \sum _{i_1+\cdots +i_q={p+1}} c(i_1,\ldots ,i_q) \int k_{\theta ^{i_1}}^2 \cdots k_{\theta ^{i_q}}^2 \,\textrm{d}\theta \\&\quad \leqq -\int k^{-2}k_{\theta ^{(p+1)}}^2\,\textrm{d}\theta + C \\&\qquad + C\sum _{i_1+i_2={p+1}} \int k_{\theta ^{i_1}}^2 k_{\theta ^{i_2}}^2 \,\textrm{d}\theta + C\sum _{i_1+i_2+i_3={p+1}} \int k_{\theta ^{i_1}}^2 k_{\theta ^{i_2}}^2 k_{\theta ^{i_3}}^2 \,\textrm{d}\theta \\&\qquad + C\sum _{i_1+i_2+i_3+i_4={p+1}} \int k_{\theta ^{i_1}}^2 k_{\theta ^{i_2}}^2 k_{\theta ^{i_3}}^2 k_{\theta ^{i_4}}^2 \,\textrm{d}\theta . \end{aligned}$$

In the summations above we use again (63) to estimate all factors of the form $k_{\theta ^m}$ for $m\leqq p-3$ (note that m is at least 3), and find that

$$\begin{aligned} \frac{\textrm{d}}{\textrm{d}t} \int h_{\theta ^{(p+1)}}^2\,\textrm{d}\theta&\leqq -c\int k^{-2}k_{\theta ^{(p+1)}}^2\,\textrm{d}\theta + C - 8\omega ^2\pi ^2\int h_{\theta ^{p+1}}^2\,\textrm{d}\theta \\&\quad + C\int k_{\theta ^{p}}^2 k_{\theta }^2 + k_{\theta ^{p-1}}^2 k_{\theta ^2}^2 + k_{\theta ^{p-2}}^2 k_{\theta ^3}^2 \,\textrm{d}\theta \\&\quad + C\int k_{\theta ^{p-1}}^2 k_{\theta }^2 k_{\theta }^2 + k_{\theta ^{p-2}}^2 k_{\theta ^2}^2 k_{\theta }^2 \,\textrm{d}\theta \\&\quad + C\int k_{\theta ^{p-2}}^2 k_{\theta }^2 k_{\theta }^2 k_{\theta }^2 \,\textrm{d}\theta \\&\leqq -\int k^{-2}k_{\theta ^{(p+1)}}^2\,\textrm{d}\theta +C\int k_{\theta ^{p}}^2 \,\textrm{d}\theta +C\int k_{\theta ^{(p-1)}}^2 \,\textrm{d}\theta \\&\quad +C\int k_{\theta ^{(p-2)}}^2 \,\textrm{d}\theta +C - 8\omega ^2\pi ^2\int h_{\theta ^{p+1}}^2\,\textrm{d}\theta . \end{aligned}$$

Corollary 5.1, integration by parts and interpolation yields

$$\begin{aligned} \frac{\textrm{d}}{\textrm{d}t} \int h_{\theta ^{(p+1)}}^2\,\textrm{d}\theta&\leqq -c\int k_{\theta ^{(p+1)}}^2\,\textrm{d}\theta +C\int k_{\theta ^{p}}^2 \,\textrm{d}\theta +C - 8\omega ^2\pi ^2\int h_{\theta ^{p+1}}^2\,\textrm{d}\theta \\ {}&\leqq -c\int k_{\theta ^{p+1}}^2\,\textrm{d}\theta + C - 8\omega ^2\pi ^2\int h_{\theta ^{p+1}}^2\,\textrm{d}\theta . \end{aligned}$$

This implies $||h_{\theta ^{(p+1)}}||_2^2(t) \leqq C$ for $t\leqq t_{p}$, as required. $\square$

Finally, we give uniform exponential decay estimates for all derivatives of the rescaled support function.

Proposition 5.4

Consider the rescaling of an entropy flow generated by Theorem 3.24. There exist $t_0>0$ such that for all $p\in {\mathbb {N}}$ we have

$$\begin{aligned} \Vert h_{\theta ^p}\Vert _2^2(t) \leqq C\,e^{-4\omega ^2\pi ^2 t},\quad {\text {for}} \quad t\leqq t_0. \end{aligned}$$

Proof

First, note that for Proposition 5.3 the times $t_p$ satisfy $t_p = t_{p-1}$ for $p\leqq 5$, so we may take $t_0 = \max \{t_1,t_2,t_3,t_4\}$.

For $p\leqq 2$ we interpolate to find that

$$\begin{aligned} \Vert h_{\theta ^p}\Vert _2^2(t) \leqq C\Vert h_{\theta ^{2p-2}}\Vert _2\Vert h_\theta \Vert _2 \leqq C\,e^{-4\omega ^2\pi ^2 t}, \end{aligned}$$

for $t\leqq t_0$. This uses Proposition 5.2 and Proposition 5.3. $\square$

Proposition 5.4 implies convergence as $t\nearrow \infty$ of the rescaled support function to that of a round circle in $C^\infty (\textrm{d}\theta )$ by a standard argument (c.f. [2]).

Appendix A: Local Well-Posedness

In this section, we establish the following local well-posedness theorem. This will be used in the compactness and smoothing argument in Section 3.

Theorem A.1

Suppose $\gamma _0:{\mathbb {S}}^1\rightarrow {\mathbb {R}}^2$ is a locally convex curve of class $W^{3,\infty }(du)$. Then there exists a $T>0$ and unique corresponding entropy flow $\gamma :{\mathbb {S}}^1\times (0,T)\rightarrow {\mathbb {R}}^2$ such that

• $\lim _{t\searrow 0}\gamma = \gamma _0$ in $W^{3,\infty }(du)$

• $\gamma (\cdot ,t)$ is of class $C^\infty$ and locally convex

• $T<\infty$ implies $\lim _{t\nearrow \infty } \Vert \gamma (\cdot ,t)\Vert _{W^{3,\infty }(du)} = +\infty$.

We will in fact establish existence for initial strictly convex immersed data of class $W^{4-4/p,p}$ with $p>5/2$. Of course the $W^{3,\infty }$ hypothesis implies this with for instance $p=4$. Formally Theorem A.1 is a corollary of Theorem A.11.

Suppose that $\gamma : {\mathbb {S}}^1 \times [0, T) \rightarrow {\mathbb {R}}^2$ lies in the maximal regularity space

$$\begin{aligned} {\mathbb {E}}^{1,p}((0,T); L^p({\mathbb {S}}^1; {\mathbb {R}}^2)) = W^{1,p}((0,T); L^p({\mathbb {S}}^1; {\mathbb {R}}^2)) \cap L^p((0,T); W^{4,p}({\mathbb {S}}^1; {\mathbb {R}}^2)) \end{aligned}$$

for some $1< p < \infty$ and it satisfies (EF). ${\mathbb {E}}^{1,p}((0,T); L^p({\mathbb {S}}^1; {\mathbb {R}}^2))$ is a Banach space when equipped with the norm

$$\begin{aligned} \Vert \gamma \Vert _{{\mathbb {E}}^{1,p}((0,T); L^p({\mathbb {S}}^1; {\mathbb {R}}^2))}:= \max \left\{ \Vert \gamma \Vert _{W^{1,p}((0,T); L^p({\mathbb {S}}^1; {\mathbb {R}}^2))}, \Vert \gamma \Vert _{L^p((0,T); W^{4,p}({\mathbb {S}}^1; {\mathbb {R}}^2))} \right\} . \end{aligned}$$

The optimal space of initial data in the class of maximal $L^p$-regularity is precisely immersed curves in

$$\begin{aligned} (L^p({\mathbb {S}}^1; {\mathbb {R}}^2), W^{4,p}({\mathbb {S}}^1; {\mathbb {R}}^2))_{1 - \frac{1}{p},p} = W^{4(1 - \frac{1}{p}),p}({\mathbb {S}}^1; {\mathbb {R}}^2) \end{aligned}$$

for which we shall also make the additional assumption that $\gamma _0$ is strictly locally convex so that the entropy is initially well defined. By expanding out (1) we see that the entropy flow in a general parameter u is equivalent to solving

$$\begin{aligned} {\left\{ \begin{array}{ll} \quad \partial _t \gamma &{}= - \frac{1}{\langle \gamma _{uu}, \nu \rangle ^2} \gamma _{u^4}^{\perp } - \frac{\langle \gamma _{u^3}, \nu \rangle }{|\gamma _u| \langle \gamma _{uu}, \nu \rangle }\gamma _{u^3}^{\perp } - \frac{1}{|\gamma _u|^2} \gamma _{uu}^{\perp } - \frac{4\langle \gamma _{u^3}, \gamma _u \rangle }{|\gamma _u|^2 \langle \gamma _{uu}, \nu \rangle }\nu + \frac{6 \langle \gamma _{uu}, \gamma _u \rangle ^2}{|\gamma _u|^4 \langle \gamma _{uu}, \nu \rangle } \nu , \quad t > 0 \\ \gamma (\cdot ,0) &{}= \gamma _0(\cdot ). \end{array}\right. } \end{aligned}$$

(64)

We shall linearise this operator by freezing the highest order term at the initial datum $\gamma _0$ such that we instead will consider

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t \gamma + \frac{1}{|(\gamma _0)_{uu}^{\perp }|^2} \gamma _{u^4}^{\perp } &{}= \left( \frac{1}{|(\gamma _0)_{uu}^{\perp }|^2} - \frac{1}{|\gamma _{uu}^{\perp }|^2} \right) \gamma _{u^4}^{\perp } + F(\partial ^3_u \gamma , \partial ^2_u \gamma , \partial _u \gamma ), \quad t > 0 \\ \qquad \qquad \ \ \gamma (\cdot ,0) &{}= \gamma _0(\cdot ) \end{array}\right. } \end{aligned}$$

(65)

where

$$\begin{aligned} F(\partial ^3_u \gamma , \partial ^2_u \gamma , \partial _u \gamma ) = - \frac{\langle \gamma _{u^3}, \nu \rangle }{|\gamma _u| \langle \gamma _{uu}, \nu \rangle }\gamma _{u^3}^{\perp } - \frac{1}{|\gamma _u|^2} \gamma _{uu}^{\perp } - \frac{4\langle \gamma _{u^3}, \gamma _u \rangle }{|\gamma _u|^2 \langle \gamma _{uu}, \nu \rangle }\nu + \frac{6 \langle \gamma _{uu}, \gamma _u \rangle ^2}{|\gamma _u|^4 \langle \gamma _{uu}, \nu \rangle } \nu . \end{aligned}$$

To establish local well-posedness we follow [18] by using contraction estimates on an appropriate subset of ${\mathbb {E}}^{1,p}((0,T); L^p({\mathbb {S}}^1; {\mathbb {R}}^2))$ and extending to all of ${\mathbb {E}}^{1,p}((0,T); L^p({\mathbb {S}}^1; {\mathbb {R}}^2))$ with an appropriate contradiction argument. To that end suppose for a given $R > 0$ we define

$$\begin{aligned} {\overline{B}}_R^{{\mathbb {E}}^{1,p}}({\bar{\gamma }}):= & {} \{ \gamma \in {\mathbb {E}}^{1,p}((0,T); L^p({\mathbb {S}}^1; {\mathbb {R}}^2)): \Vert \gamma - {\bar{\gamma }} \Vert _{{\mathbb {E}}^{1,p}((0,T); L^p({\mathbb {S}}^1; {\mathbb {R}}^2))}\\\leqq & {} R, \gamma (\cdot ,0) = {\bar{\gamma }}(\cdot ,0)\} \end{aligned}$$

where ${\bar{\gamma }}$ is a specified reference flow. We shall assume that R, T are bounded apriori by $R_0, T_0$ respectively and assume that ${\bar{\gamma }}$ is the solution to the linear evolution equation

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t {\bar{\gamma }} + \frac{1}{|(\gamma _0)_{uu}^{\perp }|^2} \gamma _{u^4}^{\perp } &{}= 0, \quad t > 0 \\ \qquad \qquad \ \ {\bar{\gamma }}(\cdot ,0) &{}= \gamma _0(\cdot ). \end{array}\right. } \end{aligned}$$

(66)

Such a reference flow is locally well-posed from the following theorem:

Theorem A.2

Let $1< p< \infty , 0 < T \leqq T_0$ and suppose that $a \in C^0({\mathbb {S}}^1; {\mathbb {R}}^2)$ with

$$\begin{aligned} (f,\gamma _0) \in L^p((0,T); L^p({\mathbb {S}}^1; {\mathbb {R}}^2)) \times W^{4(1 - \frac{1}{p}),p}({\mathbb {S}}^1; {\mathbb {R}}^2). \end{aligned}$$

Then there exists a unique $\sigma \in {\mathbb {E}}^{1,p}((0,T); L^p({\mathbb {S}}^1; {\mathbb {R}}^2))$ satisfying

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t \sigma + a \sigma _{u^4} &{}= f, \quad t > 0 \\ \quad \ \ \ \sigma (\cdot ,0) &{}= \gamma _0(\cdot ) \end{array}\right. } \end{aligned}$$

(67)

and there exists $C = C(T,p,a) > 0$ such that

$$\begin{aligned} \Vert \sigma \Vert _{{\mathbb {E}}^{1,p}((0,T); L^p({\mathbb {S}}^1; {\mathbb {R}}^2))} \leqq C\left( \Vert f \Vert _{L^p((0,T); L^p({\mathbb {S}}^1; {\mathbb {R}}^2))} + \Vert \gamma _0 \Vert _{W^{4(1 - \frac{1}{p}),p}({\mathbb {S}}^1; {\mathbb {R}}^2)}\right) . \end{aligned}$$

(68)

Proof

This follows from the maximal regularity of the operator $a\partial ^4_u$ as seen in [17] Theorem 6.3.2 with zero boundary data. $\square$

Our present goal then is to control the non-linearities present in F but also in controlling $|(\gamma _0)_{uu}^{\perp }|^{-2} - |\gamma _{uu}^{\perp }|^{-2}$. To do that, we shall employ the following lemmata and proposition:

Lemma A.3

([18] Proposition B.3) Let $k \in \{1,2,3,4\}, 1< p < \infty$ and $\rho _1,\rho _2 \in [1, \infty )$. Suppose there exists some $\theta \in [0,1]$ such that, for $k \in \{1,2,3,4\}$,

$$\begin{aligned} \frac{4 - k}{4}\theta - \frac{1}{p} \leqq - \frac{1}{\rho _1}, \quad (4-k)(1- \theta ) - \frac{1}{p} \leqq - \frac{1}{\rho _2}. \end{aligned}$$

Then, for any $0 < T \leqq T_0$,

$$\begin{aligned} \partial ^k_u : {\mathbb {E}}^{1,p}((0,T); L^p({\mathbb {S}}^1; {\mathbb {R}}^2)) \hookrightarrow L^{\rho _1}((0,T); L^{\rho _2}({\mathbb {S}}^1; {\mathbb {R}}^2)), \end{aligned}$$

with the estimate

$$\begin{aligned} \Vert \partial ^k_u \gamma \Vert _{L^{\rho _1}((0,T); L^{\rho _2}({\mathbb {S}}^1; {\mathbb {R}}^2))} \leqq C(T_0,k,\theta ,p,\rho _1,\rho _2) \Vert \gamma \Vert _{{\mathbb {E}}^{1,p}((0,T); L^p({\mathbb {S}}^1; {\mathbb {R}}^2))}. \end{aligned}$$

Lemma A.4

Let X be a Banach space of class $\mathcal{H}\mathcal{T}$ that is a Banach space for which the Hilbert Transform is a bounded linear operator on $L^2({\mathbb {R}}; X)$ and $I \subset {\mathbb {R}}$. Then

• (i) For $s \in (0,\infty ), 1 \leqq p \leqq \infty$

  $$\begin{aligned} W^{s,p}(I; X) \hookrightarrow C^{[\alpha ], \alpha - [\alpha ]}({{\overline{I}}}; X), \quad {\mathbb {N}} \not \ni \alpha = s - \frac{1}{p}. \end{aligned}$$

• (ii) If $k + \frac{1}{p}< s < k + 1 + \frac{1}{p}$ for some $k \in {\mathbb {N}}_0$

  $$\begin{aligned} W^{s,p}(I; X) \hookrightarrow BUC^k({{\bar{I}}}; X). \end{aligned}$$

  The same assertion holds if we replace $W^{s,p}(I; X)$ by $H^{s,p}(I; X)$.

• (iii) For $l \in \{0,1,2,3\}$, $1 \leqq p \leqq \infty$ such that

  $$\begin{aligned} {\mathbb {E}}^{1,p}((0,T); L^p({\mathbb {S}}^1; {\mathbb {R}}^2)) \hookrightarrow C^{0,\alpha }([0,T]; C^{l,\alpha }({\mathbb {S}}^1; {\mathbb {R}}^2)) \end{aligned}$$

  for $p > \frac{5}{4 - l}$. Moreover there exists some $C = C(T_0,p,l) > 0$ such that

  $$\begin{aligned} \Vert \gamma \Vert _{C^{0,\alpha }([0,T]; C^{l,\alpha }({\mathbb {S}}^1; {\mathbb {R}}^2))} \leqq C(T_0,p,l) \Vert \gamma \Vert _{{\mathbb {E}}^{1,p}((0,T); L^p({\mathbb {S}}^1; {\mathbb {R}}^2))}. \end{aligned}$$

Proof

• (i) By characterisation of $W^{s,p}(I;X)$ as an interpolation space, the monotonicity and embeddings of Besov spaces ([1] 3.2, 3.3 and [20] Theorem 1.5.1 (ii)) it holds that

  $$\begin{aligned} W^{s,p}(I;X) = B^s_{p,p}(I;X) \hookrightarrow B^s_{p,\infty }(I; X) \hookrightarrow B^{\alpha }_{\infty , \infty }({{\overline{I}}};X) = {\mathcal {C}}^{\alpha }({{\overline{I}}}; X) \end{aligned}$$

  where ${\mathcal {C}}^{\alpha }$ is the Hölder-Zygmund space of order $\alpha$. The claim then follows since by [20] Theorem 1.2.2 ${\mathcal {C}}^{\alpha }$ coincides with $C^{[\alpha ], \alpha - [\alpha ]}$ for ${\mathbb {N}} \not \ni \alpha > 0$.

• (ii) Follows directly from [15] Proposition 2.10.

• (iii) By properties of interpolation spaces for $\theta \in (0,1)$ such that $2m(1 - \theta ) \notin {\mathbb {N}}_0$

  $$\begin{aligned} {\mathbb {E}}^{1,p}((0,T); L^p({\mathbb {S}}^1; {\mathbb {R}}^2))&\hookrightarrow (W^{1,p}((0,T); L^p({\mathbb {S}}^1; {\mathbb {R}}^2)), L^p((0,T); W^{4,p}({\mathbb {S}}^1; {\mathbb {R}}^2)))_{\theta ,p} \\&= W^{\theta ,p}((0,T); W^{4(1 - \theta ),p}({\mathbb {S}}^1; {\mathbb {R}}^2)) \end{aligned}$$

  where we have identified the two spaces using [1] Theorem 3.1. By (i) for $\theta > \frac{1}{p}$ there exists some $(0,1) \ni \alpha _1 = \theta - \frac{1}{p}$ such that

  $$\begin{aligned} {\mathbb {E}}^{1,p}((0,T); L^p({\mathbb {S}}^1; {\mathbb {R}}^2)) \hookrightarrow C^{0,\alpha _1}([0,T]; W^{2m(1 - \theta ),p}({\mathbb {S}}^1; {\mathbb {R}}^2)). \end{aligned}$$

  Any $\gamma \in {\mathbb {E}}^{1,p}((0,T); L^p({\mathbb {S}}^1; {\mathbb {R}}^2))$ has a weak l-th order spatial derivative with $\partial ^l_u \gamma \in C^{0,\alpha _1}([0,T]; W^{4 - l - 4\theta , p}({\mathbb {S}}^1; {\mathbb {R}}^2))$ which embeds into

  $$\begin{aligned} C^{0, \alpha _1}([0,T]; C^{[\alpha _2],\alpha _2 - [\alpha _2]}({\mathbb {S}}^1; {\mathbb {R}}^2)) \end{aligned}$$

  for some $\alpha _2 = 4 - l - 4\theta - \frac{1}{p} > 0$ which requires $\theta < 1 - \left( \frac{l}{4} + \frac{1}{4p}\right)$. Thus for the embeddings to hold we will need to choose $\theta$ such that $4(1 - \theta ) \notin {\mathbb {N}}_0$ and

  $$\begin{aligned} \theta \in \left( \frac{1}{p}, 1 - \left( \frac{l}{4} + \frac{1}{4p}\right) \right) \end{aligned}$$

  for this interval to be well defined we must have $p > \frac{5}{4 - l}$. The choice of $\theta = \frac{1}{2} - \frac{l}{8} - \frac{5}{8p}$ is well defined for every $p > \frac{5}{4 - l}$ with the exception of $p = 5$ when $l = 1$, else the choice of $\theta = \frac{23}{80} - \frac{l}{16}$ in the case that $p = 5$ works. By identifying $B^{\alpha }_{\infty , \infty }$ with the Hölder-Zygmund spaces ${\mathcal {C}}^{\alpha }$ and the fact that $B^{s_0}_{p,q} \hookrightarrow B^{s_1}_{p,q}$ for any $s_0 \leqq s_1, 1 \leqq p,q \leqq \infty$ we have that for $\alpha := \min \{\alpha _1, \alpha _2\}$

  $$\begin{aligned} C^{[\alpha _2], \alpha _2 - [\alpha _2]}({\mathbb {S}}^1; {\mathbb {R}}^2)&= {\mathcal {C}}^{\alpha _2}({\mathbb {S}}^1; {\mathbb {R}}^2) = B^{\alpha _2}_{\infty ,\infty }({\mathbb {S}}^1; {\mathbb {R}}^2) \\&\hookrightarrow B^{\alpha }_{\infty , \infty }({\mathbb {S}}^1; {\mathbb {R}}^2) = {\mathcal {C}}^{\alpha }({\mathbb {S}}^1; {\mathbb {R}}^2) = C^{0, \alpha }({\mathbb {S}}^1; {\mathbb {R}}^2) \end{aligned}$$

  and thus there exists some $\alpha \in (0,1)$ such that

  $$\begin{aligned} {\mathbb {E}}^{1,p}((0,T); L^p({\mathbb {S}}^1; {\mathbb {R}}^2)) \hookrightarrow C^{0, \alpha }([0,T]; C^{l,\alpha }({\mathbb {S}}^1; {\mathbb {R}}^2)). \end{aligned}$$

$\square$

Lemma A.5

Let $l \in \{0,1,2,3\}$. For every $p > \frac{5}{4 - l}$, there exists some $T = T(T_0,R_0,p,l,{\bar{\gamma }}) > 0$ such that for some $R \in (0, R_0]$ and any $\gamma \in {\overline{B}}_R^{{\mathbb {E}}^{1,p}}({\bar{\gamma }})$

$$\begin{aligned} |\partial ^l_u \gamma (u,t)| \leqq \frac{1}{2} \inf _{v \in {\mathbb {S}}^1} |\partial ^l_u \gamma _0(v)| \end{aligned}$$

for every $(u,t) \in {\mathbb {S}}^1 \times [0, T)$. In particular if $|\partial ^l_u \gamma _0| > 0$ then $|\partial ^l_u \gamma (\cdot ,t)| > 0$ for every $t \in [0, T)$.

Proof

By A.4 for every $p > \frac{5}{4 - l}$ we have that

$$\begin{aligned} {\mathbb {E}}^{1,p}((0,T); L^p({\mathbb {S}}^1; {\mathbb {R}}^2)) \hookrightarrow C^{0,\alpha }([0,T]; C^{l,\alpha }({\mathbb {S}}^1; {\mathbb {R}}^2)) \end{aligned}$$

which implies that $\gamma \in C^{0,\alpha }([0,T]; C^l({\mathbb {S}}^1; {\mathbb {R}}^2))$ and there exists some $C = C(T_0,p,l) > 0$ such that for any $\gamma \in {\overline{B}}_R^{{\mathbb {E}}^{1,p}}({\bar{\gamma }})$

$$\begin{aligned} \Vert \gamma - \gamma _0 \Vert _{C^l({\mathbb {S}}^1; {\mathbb {R}}^2)}&\leqq \Vert \gamma \Vert _{C^{0,\alpha }([0,T]; C^l({\mathbb {S}}^1; {\mathbb {R}}^2))} T^{\alpha } \\&\leqq \left( \Vert \gamma - {\bar{\gamma }} \Vert _{C^{0,\alpha }([0,T]; C^l({\mathbb {S}}^1; {\mathbb {R}}^2))} + \Vert {\bar{\gamma }} \Vert _{C^{0,\alpha }([0,T]; C^l({\mathbb {S}}^1; {\mathbb {R}}^2))} \right) T^{\alpha } \\&\leqq \left( C(T_0,p,l) \Vert \gamma - {\bar{\gamma }} \Vert _{{\mathbb {E}}^{1,p}((0,T); L^p({\mathbb {S}}^1; {\mathbb {R}}^2))}\right. \\&\left. \quad + C(T_0,p,l) \Vert {\bar{\gamma }} \Vert _{{\mathbb {E}}^{1,p}((0,T); L^p({\mathbb {S}}^1; {\mathbb {R}}^2))} \right) T^{\alpha } \\&\leqq C(T_0,R_0,p,l, {\bar{\gamma }})T^{\alpha } \end{aligned}$$

where we have used the fact that $\gamma - {\bar{\gamma }}$ has zero temporal trace. Thus if we choose $T = T(T_0,R_0,p,l, {\bar{\gamma }}) > 0$ small enough

$$\begin{aligned} |\partial ^l_u \gamma (u,t)| \leqq \inf _{{\mathbb {S}}^1} |\partial ^l_u \gamma _0(u)| - |\partial ^l_u \gamma (u,t) - \partial ^l_u \gamma _0(u)| \leqq \frac{1}{2} \inf _{{\mathbb {S}}^1} |\partial ^l_u \gamma _0|. \end{aligned}$$

$\square$

Lemma A.6

Let $\gamma \in B_R^{{\mathbb {E}}^{1,p}}({\bar{\gamma }})$ for some $p > \frac{5}{2}$. Then there exists some $T = T(T_0,R_0,p,{\bar{\gamma }}) > 0$ such that

$$\begin{aligned} |\gamma _{uu}^{\perp }(u,t)| \leqq \frac{1}{2} \inf _{v \in {\mathbb {S}}^1}|\gamma _{uu}^{\perp }(v,0)| \end{aligned}$$

for every $(u,t) \in {\mathbb {S}}^1 \times [0, T)$. In particular if $|\gamma _{uu}^{\perp }(\cdot ,0)| > 0$ then $|\gamma _{uu}^{\perp }(\cdot ,t)| > 0$ for every $t \in [0,T)$.

Proof

Let $\gamma \in B_R^{{\mathbb {E}}^{1,p}}({\bar{\gamma }})$ then the following holds

$$\begin{aligned} |\langle \gamma _{uu}, \nu \rangle \nu - \langle (\gamma _0)_{uu}, \nu _0 \rangle \nu _0|&\leqq | \gamma _{uu} - (\gamma _0)_{uu}| + |\langle \gamma _{uu}, \tau \rangle \tau - \langle (\gamma _0)_{uu}, \tau _0 \rangle \tau _0| \nonumber \\&\leqq |\gamma _{uu} - (\gamma _0)_{uu}| + |\langle \gamma _{uu}, \tau \rangle \tau - \langle (\gamma _0)_{uu}, \tau \rangle \tau | \nonumber \\&\quad \ + |\langle (\gamma _0)_{uu}, \tau \rangle \tau - \langle (\gamma _0)_{uu}, \tau _0 \rangle \tau _0| \nonumber \\&\leqq 2|\gamma _{uu} - (\gamma _0)_{uu}| + |\langle (\gamma _0)_{uu}, \tau \rangle \tau - \langle (\gamma _0)_{uu}, \tau _0 \rangle \tau | \nonumber \\&\quad \ + |\langle (\gamma _0)_{uu}, \tau _0 \rangle \tau - \langle (\gamma _0)_{uu}, \tau _0 \rangle \tau _0| \nonumber \\&\leqq 2|\gamma _{uu} - (\gamma _0)_{uu}| + 2|(\gamma _0)_{uu}| |\tau - \tau _0| . \end{aligned}$$

(69)

To control $\tau - \tau _0$ we shall use the Mean Value Theorem to find

$$\begin{aligned} |\tau - \tau _0|&= \left| \frac{\gamma _u}{|\gamma _u|} - \frac{(\gamma _0)_u}{|(\gamma _0)_u|}\right| \nonumber \\&\leqq \left| \frac{1}{|(\gamma _0)_u|} \right| |\gamma _u - (\gamma _0)_u| + \left| \left( \frac{1}{|(\gamma _0)_u|} - \frac{1}{|\gamma _u|}\right) \gamma _u \right| \nonumber \\&\leqq \left( \inf _{{\mathbb {S}}^1} |(\gamma _0)_u|\right) ^{-1} |\gamma _u - (\gamma _0)_u| + \left( \inf _{{\mathbb {S}}^1} |(\gamma _0)_u|\right) ^{-2} |\gamma _u - (\gamma _0)_u| |\gamma _u| \nonumber \\&\leqq C(\gamma _0) |\gamma _u - (\gamma _0)_u| + C(\gamma _0) |\gamma _u - (\gamma _0)_u|^2 + C(\gamma _0) |\gamma _u - (\gamma _0)_u| | (\gamma _0)_u|. \end{aligned}$$

(70)

If we let $\alpha _1, \alpha _2$ be chosen such that the embeddings

$$\begin{aligned}{} & {} {\mathbb {E}}^{1,p}((0,T); L^p({\mathbb {S}}^1; {\mathbb {R}}^2)) \hookrightarrow C^{0,\alpha _1}([0,T]; C^{1,\alpha _1}({\mathbb {S}}^1; {\mathbb {R}}^2)) \\{} & {} {\mathbb {E}}^{1,p}((0,T); L^p({\mathbb {S}}^1; {\mathbb {R}}^2)) \hookrightarrow C^{0,\alpha _2}([0,T]; C^{2,\alpha _2}({\mathbb {S}}^1; {\mathbb {R}}^2)) \end{aligned}$$

hold and let $\alpha = \min \{\alpha _1,\alpha _2\}$ it then follows by A.4 (iii)

$$\begin{aligned}&\Vert \langle \gamma _{uu}, \nu \rangle \nu - \langle (\gamma _0)_{uu}, \nu _0 \rangle \nu _0 \Vert _{C^0({\mathbb {S}}^1; {\mathbb {R}}^2)} \nonumber \\ {}&\leqq 2 \Vert \gamma _{uu} - (\gamma _0)_{uu} \Vert _{C^0({\mathbb {S}}^1; {\mathbb {R}}^2)} \nonumber \\&\qquad + C(\gamma _0) \Vert (\gamma _0)_{uu} \Vert _{C^0({\mathbb {S}}^1; {\mathbb {R}}^2)} \Vert \gamma _u - (\gamma _0)_u \Vert _{C^0({\mathbb {S}}^1; {\mathbb {R}}^2)}\nonumber \\&\qquad + C(\gamma _0) \Vert \gamma _u - (\gamma _0)_u \Vert _{C^0({\mathbb {S}}^1; {\mathbb {R}}^2)}^2 \nonumber \\&\qquad + C(\gamma _0) \Vert \gamma _u - (\gamma _0)_u \Vert _{C^0({\mathbb {S}}^1; {\mathbb {R}}^2)} \Vert (\gamma _0)_u \Vert _{C^0({\mathbb {S}}^1; {\mathbb {R}}^2)} \nonumber \\&\quad \leqq 2 \Vert \gamma \Vert _{C^{0,\alpha }([0,T]; C^2({\mathbb {S}}^1; {\mathbb {R}}^2))} T^{\alpha } \nonumber \\&\qquad + C(\gamma _0) \Vert \gamma \Vert _{C^{0,\alpha }([0,T]; C^1({\mathbb {S}}^1; {\mathbb {R}}^2))}T^{\alpha } \nonumber \\&\qquad + C(\gamma _0) \Vert \gamma \Vert _{C^{0,\alpha }([0,T]; C^1({\mathbb {S}}^1; {\mathbb {R}}^2))}^2T^{2\alpha } \nonumber \\&\qquad + C(\gamma _0) \Vert \gamma \Vert _{C^{0,\alpha }([0,T]; C^1({\mathbb {S}}^1; {\mathbb {R}}^2))}T^{\alpha } \nonumber \\&\quad \leqq C(T_0,p,\gamma _0) \Vert \gamma \Vert _{{\mathbb {E}}^{1,p}((0,T); L^p({\mathbb {S}}^1; {\mathbb {R}}^2))} T^{\alpha } \nonumber \\&\qquad + C(T_0,p,\gamma _0) \Vert \gamma \Vert _{{\mathbb {E}}^{1,p}((0,T); L^p({\mathbb {S}}^1; {\mathbb {R}}^2))}^2 T_0^{\alpha }T^{\alpha } \nonumber \\&\quad \leqq C(T_0,p,\gamma _0) \Vert \gamma - {\bar{\gamma }} \Vert _{{\mathbb {E}}^{1,p}((0,T); L^p({\mathbb {S}}^1; {\mathbb {R}}^2))} T^{\alpha }\nonumber \\&\qquad + C(T_0,p,\gamma _0) \Vert {\bar{\gamma }} \Vert _{{\mathbb {E}}^{1,p}((0,T); L^p({\mathbb {S}}^1; {\mathbb {R}}^2))} T^{\alpha } \nonumber \\&\qquad + C(T_0,p,\gamma _0) \Vert \gamma - {\bar{\gamma }} \Vert _{{\mathbb {E}}^{1,p}((0,T); L^p({\mathbb {S}}^1; {\mathbb {R}}^2))}^2 T^{\alpha }\nonumber \\&\qquad + C(T_0,p,\gamma _0) \Vert \gamma - {\bar{\gamma }} \Vert _{{\mathbb {E}}^{1,p}((0,T); L^p({\mathbb {S}}^1; {\mathbb {R}}^2))} \Vert {\bar{\gamma }} \Vert _{{\mathbb {E}}^{1,p}((0,T); L^p({\mathbb {S}}^1; {\mathbb {R}}^2))}T^{\alpha } \nonumber \\&\qquad + C(T_0,p,\gamma _0) \Vert {\bar{\gamma }} \Vert _{{\mathbb {E}}^{1,p}((0,T); L^p({\mathbb {S}}^1; {\mathbb {R}}^2))}^2 T^{\alpha } \nonumber \\&\quad \leqq C(T_0,R_0,p,{\bar{\gamma }})T^{\alpha } . \end{aligned}$$

(71)

In particular for $T = T(T_0,R_0,p, {\bar{\gamma }}) > 0$ sufficiently small

$$\begin{aligned} |\langle \gamma _{uu}, \nu \rangle |&= |\gamma _{uu}^{\perp }| \\&\leqq \inf _{{\mathbb {S}}^1} |\gamma _{uu}^{\perp }(\cdot ,0)| - |\gamma _{uu}^{\perp }(\cdot ,t) - \gamma _{uu}^{\perp }(\cdot ,0)| \\&\leqq \frac{1}{2} \inf _{{\mathbb {S}}^1} |\gamma _{uu}^{\perp }(\cdot ,0)| = \frac{1}{2} \inf _{{\mathbb {S}}^1} |\langle (\gamma _0)_{uu}, \nu _0 \rangle |. \end{aligned}$$

$\square$

We first shall tackle the highest order term as it will require the least machinery to control.

Proposition A.7

Suppose that $\gamma \in B_R^{{\mathbb {E}}^{1,p}}({\bar{\gamma }})$. Then for sufficiently small $T,R > 0$ the map

$$\begin{aligned} \gamma \mapsto \left( \frac{1}{|(\gamma _0)_{uu}^{\perp }|^2} - \frac{1}{|\gamma _{uu}^{\perp }|^2} \right) \gamma _{u^4}: B_R^{{\mathbb {E}}^{1,p}}({\bar{\gamma }}) \rightarrow L^p((0,T); L^p({\mathbb {S}}^1; {\mathbb {R}}^2)) \end{aligned}$$

is a well defined $\varepsilon$-contraction that is a Lipschitz map with Lipschitz constant $\varepsilon$.

Proof

Let $\gamma , {\tilde{\gamma }} \in B_R^{{\mathbb {E}}^{1,p}}({\bar{\gamma }})$. Observe by the triangle inequality that

$$\begin{aligned}&\left| \left( \frac{1}{|(\gamma _0)_{uu}^{\perp }|^2} - \frac{1}{|\gamma _{uu}^{\perp }|^2} \right) \gamma _{u^4} - \left( \frac{1}{|(\gamma _0)_{uu}^{\perp }|^2} - \frac{1}{|{\tilde{\gamma }}_{uu}^{\perp }|^2}\right) {\tilde{\gamma }}_{u^4} \right| \\ {}&\quad \leqq \left| \frac{1}{|(\gamma _0)_{uu}^{\perp }|^2} - \frac{1}{|\gamma _{uu}^{\perp }|^2} \right| |\gamma _{u^4} - {\tilde{\gamma }}_{u^4}| \ + \left| \frac{1}{|\gamma _{uu}^{\perp }|^2} - \frac{1}{|{\tilde{\gamma }}_{uu}^{\perp }|^2}\right| |{\tilde{\gamma }}_{u^4}| \end{aligned}$$

from which it holds

$$\begin{aligned}&\left\| \left( \frac{1}{|(\gamma _0)_{uu}^{\perp }|^2} - \frac{1}{|\gamma _{uu}^{\perp }|^2} \right) \gamma _{u^4} - \left( \frac{1}{|(\gamma _0)_{uu}^{\perp }|^2} - \frac{1}{|{\tilde{\gamma }}_{uu}^{\perp }|^2}\right) {\tilde{\gamma }}_{u^4} \right\| _{L^p((0,T); L^p({\mathbb {S}}^1;{\mathbb {R}}^2))} \\&\quad \leqq \left\| \frac{1}{|(\gamma _0)_{uu}^{\perp }|^2} - \frac{1}{|\gamma _{uu}^{\perp }|^2} \right\| _{L^{\infty }((0,T); L^p({\mathbb {S}}^1))} \Vert \gamma _{u^4} - {\tilde{\gamma }}_{u^4} \Vert _{L^p((0,T); L^p({\mathbb {S}}^1; {\mathbb {R}}^2))} \\&\qquad \ + \left\| \frac{1}{|\gamma _{uu}^{\perp }|^2} - \frac{1}{|{\tilde{\gamma }}_{uu}^{\perp }|^2} \right\| _{L^{\infty }((0,T); L^p({\mathbb {S}}^1))} \Vert {\tilde{\gamma }}_{u^4} \Vert _{L^p((0,T); L^p({\mathbb {S}}^1; {\mathbb {R}}^2))}. \end{aligned}$$

We may improve this further by utilising the Mean Value Theorem and A.6

$$\begin{aligned} \left| \frac{1}{|(\gamma _0)_{uu}^{\perp }|^2} - \frac{1}{|\gamma _{uu}^{\perp }|^2}\right| \leqq 2\left( \frac{1}{2} \inf _{{\mathbb {S}}^1}|(\gamma _0)_{uu}^{\perp }|\right) ^{-3} |(\gamma _0)_{uu}^{\perp } - \gamma _{uu}^{\perp }| \leqq C(\gamma _0)|\gamma _{uu}^{\perp } - (\gamma _0)_{uu}^{\perp }| \end{aligned}$$

(72)

to find that

$$\begin{aligned} \left\| \frac{1}{|(\gamma _0)_{uu}^{\perp }|^2} - \frac{1}{|\gamma _{uu}^{\perp }|^2} \right\| _{L^{\infty }((0,T); L^p({\mathbb {S}}^1))}&\leqq C(\gamma _0) \sup _{t \in [0, T]} \Vert \gamma _{uu}^{\perp }(\cdot ,t) - \gamma _{uu}^{\perp }(\cdot ,0)\Vert _{C^0({\mathbb {S}}^1; {\mathbb {R}}^2)} \\&\leqq C(T_0,R_0,p,{\bar{\gamma }})T^{\alpha } \end{aligned}$$

where $\alpha$ is the same as in A.6. Combining this with the simple estimate

$$\begin{aligned} \Vert \partial ^4_u \gamma - \partial ^4_u {\tilde{\gamma }} \Vert _{L^p((0,T); L^p({\mathbb {S}}^1; {\mathbb {R}}^2))} \leqq \Vert \gamma - {\tilde{\gamma }} \Vert _{{\mathbb {E}}^{1,p}((0,T); L^p({\mathbb {S}}^1; {\mathbb {R}}^2))} \end{aligned}$$

shows that for sufficiently small $T = T(T_0,R_0,p,\varepsilon ,{\bar{\gamma }}) > 0$

$$\begin{aligned} \left\| \frac{1}{|(\gamma _0)_{uu}^{\perp }|^2} - \frac{1}{|\gamma _{uu}^{\perp }|^2} \right\| _{L^{\infty }((0,T); L^p({\mathbb {S}}^1))} \Vert \gamma _{u^4} - {\tilde{\gamma }}_{u^4} \Vert _{L^p({\mathbb {S}}^1; {\mathbb {R}}^2)} \leqq \frac{\varepsilon }{2} \Vert \gamma - {\tilde{\gamma }} \Vert _{{\mathbb {E}}^{1,p}((0,T); L^p({\mathbb {S}}^1; {\mathbb {R}}^2))}. \end{aligned}$$

For the remaining term we use (72) in conjunction with (69), (70) replacing $\gamma _0$ with ${\tilde{\gamma }}$ where appropriate to find that

$$\begin{aligned}&\left\| \frac{1}{|\gamma _{uu}^{\perp }|^2} - \frac{1}{|{\tilde{\gamma }}_{uu}^{\perp }|^2} \right\| _{L^{\infty }((0,T); L^p({\mathbb {S}}^1))} \\&\quad \leqq 2\left( \inf _{{\mathbb {S}}^1} |(\gamma _0)_{uu}^{\perp }|\right) ^{-3} \Vert \gamma _{uu}^{\perp } - {\tilde{\gamma }}_{uu}^{\perp } \Vert _{L^{\infty }((0,T); L^p({\mathbb {S}}^1; {\mathbb {R}}^2))} \\&\quad \leqq C(\gamma _0) \Vert \gamma _{uu} - {\tilde{\gamma }}_{uu} \Vert _{L^{\infty }((0,T); L^p({\mathbb {S}}^1; {\mathbb {R}}^2))} \\&\qquad \ + C(\gamma _0) \Vert {\tilde{\gamma }}_{uu}\Vert _{L^{\infty }((0,T); L^p({\mathbb {S}}^1; {\mathbb {R}}^2))} \Vert \tau - {\tilde{\tau }} \Vert _{L^{\infty }((0,T); L^p({\mathbb {S}}^1; {\mathbb {R}}^2))} \\&\quad \leqq C(p,\gamma _0) \Vert \gamma - {\tilde{\gamma }} \Vert _{C^{0,\alpha }((0,T); C^{2,\alpha }({\mathbb {S}}^1; {\mathbb {R}}^2))} \\&\qquad \ + C(p,\gamma _0) \Vert {\tilde{\gamma }} \Vert _{C^{0,\alpha }((0,T); C^{2,\alpha }({\mathbb {S}}^1; {\mathbb {R}}^2))} \Vert \gamma - {\tilde{\gamma }} \Vert _{C^{0,\alpha }((0,T); C^{1,\alpha }({\mathbb {S}}^1; {\mathbb {R}}^2))} \\&\qquad \ + C(p,\gamma _0) \Vert {\tilde{\gamma }} \Vert _{C^{0,\alpha }((0,T); C^{2,\alpha }({\mathbb {S}}^1; {\mathbb {R}}^2))} \Vert \gamma \\&\qquad - {\tilde{\gamma }} \Vert _{C^{0,\alpha }((0,T); C^{1,\alpha }({\mathbb {S}}^1; {\mathbb {R}}^2))} \Vert {\tilde{\gamma }} \Vert _{C^{0,\alpha }((0,T); C^{1,\alpha }({\mathbb {S}}^1; {\mathbb {R}}^2))} \\&\quad \leqq C(T_0,p,\gamma _0) \Vert \gamma - {\tilde{\gamma }} \Vert _{{\mathbb {E}}^{1,p}((0,T); L^p({\mathbb {S}}^1; {\mathbb {R}}^2))}T^{\alpha } \\ \end{aligned}$$

$$\begin{aligned}&\qquad \ + C(T_0,p,\gamma _0) \Vert {\tilde{\gamma }} - {\bar{\gamma }} \Vert _{C^{0,\alpha }([0,T]; C^{2,\alpha }({\mathbb {S}}^1; {\mathbb {R}}^2))} \Vert \gamma - {\tilde{\gamma }} \Vert _{{\mathbb {E}}^{1,p}((0,T); L^p({\mathbb {S}}^1; {\mathbb {R}}^2))} T^{\alpha } \\&\qquad \ + C(T_0,p,\gamma _0) \Vert {\bar{\gamma }} \Vert _{C^{0,\alpha }([0,T]; C^{2,\alpha }({\mathbb {S}}^1; {\mathbb {R}}^2))} \Vert \gamma - {\tilde{\gamma }} \Vert _{{\mathbb {E}}^{1,p}((0,T); L^p({\mathbb {S}}^1; {\mathbb {R}}^2))} T^{\alpha } \\&\qquad \ + C(T_0,p, \gamma _0) \Vert {\tilde{\gamma }} - {\bar{\gamma }} \Vert _{C^{0,\alpha }([0,T]; C^{2,\alpha }({\mathbb {S}}^1; {\mathbb {R}}^2))} \Vert \gamma \\&\qquad - {\tilde{\gamma }} \Vert _{{\mathbb {E}}^{1,p}((0,T); L^p({\mathbb {S}}^1; {\mathbb {R}}^2))} \Vert {\tilde{\gamma }} \Vert _{C^{0,\alpha }([0,T]; C^{1,\alpha }({\mathbb {S}}^1; {\mathbb {R}}^2))}T^{\alpha } \\&\qquad \ + C(T_0,p,\gamma _0) \Vert {\bar{\gamma }} \Vert _{C^{0,\alpha }([0,T]; C^{2,\alpha }({\mathbb {S}}^1; {\mathbb {R}}^2))} \Vert \gamma \\&\qquad - {\tilde{\gamma }} \Vert _{{\mathbb {E}}^{1,p}((0,T); L^p({\mathbb {S}}^1; {\mathbb {R}}^2))} \Vert {\tilde{\gamma }} \Vert _{C^{0,\alpha }([0,T]; C^{2,\alpha }({\mathbb {S}}^1; {\mathbb {R}}^2))}T^{\alpha } \\&\quad \leqq C(T_0,p,\gamma _0) \Vert \gamma - {\tilde{\gamma }} \Vert _{{\mathbb {E}}^{1,p}((0,T); L^p({\mathbb {S}}^1; {\mathbb {R}}^2))}T^{\alpha } \\&\qquad \ + C(T_0,p,\gamma _0) \Vert {\tilde{\gamma }} - {\bar{\gamma }} \Vert _{{\mathbb {E}}^{1,p}((0,T); L^p({\mathbb {S}}^1; {\mathbb {R}}^2))} \Vert \gamma - {\tilde{\gamma }} \Vert _{{\mathbb {E}}^{1,p}((0,T); L^p({\mathbb {S}}^1; {\mathbb {R}}^2))} T^{2\alpha } \\&\qquad \ + C(T_0,p,{\bar{\gamma }}) \Vert \gamma - {\tilde{\gamma }} \Vert _{{\mathbb {E}}^{1,p}((0,T); L^p({\mathbb {S}}^1; {\mathbb {R}}^2))} T^{\alpha } \\&\qquad \ + C(T_0,p, \gamma _0) \Vert {\tilde{\gamma }} - {\bar{\gamma }} \Vert _{{\mathbb {E}}^{1,p}((0,T); L^p({\mathbb {S}}^1; {\mathbb {R}}^2))} \Vert \gamma \\&\qquad - {\tilde{\gamma }} \Vert _{{\mathbb {E}}^{1,p}((0,T); L^p({\mathbb {S}}^1; {\mathbb {R}}^2))} \Vert {\tilde{\gamma }} \Vert _{C^{0,\alpha }([0,T]; C^{1,\alpha }({\mathbb {S}}^1; {\mathbb {R}}^2))}T^{2\alpha } \\&\qquad \ + C(T_0,p,{\bar{\gamma }}) \Vert \gamma - {\tilde{\gamma }} \Vert _{{\mathbb {E}}^{1,p}((0,T); L^p({\mathbb {S}}^1; {\mathbb {R}}^2))} \Vert {\tilde{\gamma }} \Vert _{C^{0,\alpha }([0,T]; C^{1,\alpha }({\mathbb {S}}^1; {\mathbb {R}}^2))}T^{\alpha } \\ \end{aligned}$$

$$\begin{aligned}&\quad \leqq C(T_0,p,{\bar{\gamma }}) \Vert \gamma - {\tilde{\gamma }} \Vert _{{\mathbb {E}}^{1,p}((0,T); L^p({\mathbb {S}}^1; {\mathbb {R}}^2))}T^{\alpha } \\&\qquad \ + C(T_0,R_0,p,\gamma _0) \Vert \gamma - {\tilde{\gamma }} \Vert _{{\mathbb {E}}^{1,p}((0,T); L^p({\mathbb {S}}^1; {\mathbb {R}}^2))} T_0^{\alpha }T^{\alpha } \\&\qquad \ + C(T_0,p, \gamma _0) \Vert {\tilde{\gamma }} - {\bar{\gamma }} \Vert _{{\mathbb {E}}^{1,p}((0,T); L^p({\mathbb {S}}^1; {\mathbb {R}}^2))} \\ {}&\quad \Vert \gamma - {\tilde{\gamma }} \Vert _{{\mathbb {E}}^{1,p}((0,T); L^p({\mathbb {S}}^1; {\mathbb {R}}^2))} \Vert {\tilde{\gamma }} \Vert _{C^{0,\alpha }([0,T]; C^{1,\alpha }({\mathbb {S}}^1; {\mathbb {R}}^2))}T_0^{\alpha }T^{\alpha } \\&\qquad \ + C(T_0,p,{\bar{\gamma }}) \Vert \gamma - {\tilde{\gamma }} \Vert _{{\mathbb {E}}^{1,p}((0,T); L^p({\mathbb {S}}^1; {\mathbb {R}}^2))} \Vert {\tilde{\gamma }} - {\bar{\gamma }} \Vert _{C^{0,\alpha }([0,T]; C^{1,\alpha }({\mathbb {S}}^1; {\mathbb {R}}^2))}T^{\alpha } \\&\qquad \ + C(T_0,p,{\bar{\gamma }}) \Vert \gamma - {\tilde{\gamma }} \Vert _{{\mathbb {E}}^{1,p}((0,T); L^p({\mathbb {S}}^1; {\mathbb {R}}^2))} \Vert {\bar{\gamma }} \Vert _{C^{0,\alpha }([0,T]; C^{1,\alpha }({\mathbb {S}}^1; {\mathbb {R}}^2))}T^{\alpha } \\&\quad \leqq C(T_0,R_0,p,{\bar{\gamma }}) \Vert \gamma - {\tilde{\gamma }} \Vert _{{\mathbb {E}}^{1,p}((0,T); L^p({\mathbb {S}}^1; {\mathbb {R}}^2))} T^{\alpha } \\&\qquad \ + C(T_0,R_0,p,\gamma _0) \Vert \gamma - {\tilde{\gamma }} \Vert _{{\mathbb {E}}^{1,p}((0,T); L^p({\mathbb {S}}^1; {\mathbb {R}}^2))} \Vert {\tilde{\gamma }} - {\bar{\gamma }} \Vert _{C^{0,\alpha }([0,T]; C^{1,\alpha }({\mathbb {S}}^1; {\mathbb {R}}^2))} T^{\alpha } \\&\qquad \ + C(T_0,R_0,p,\gamma _0) \Vert \gamma - {\tilde{\gamma }} \Vert _{{\mathbb {E}}^{1,p}((0,T); L^p({\mathbb {S}}^1; {\mathbb {R}}^2))} \Vert {\bar{\gamma }} \Vert _{C^{0,\alpha }([0,T]; C^{1,\alpha }({\mathbb {S}}^1; {\mathbb {R}}^2))} T^{\alpha } \\&\qquad \ + C(T_0,p,{\bar{\gamma }}) \Vert \gamma - {\tilde{\gamma }} \Vert _{{\mathbb {E}}^{1,p}((0,T); L^p({\mathbb {S}}^1; {\mathbb {R}}^2))} \Vert {\tilde{\gamma }} - {\bar{\gamma }} \Vert _{{\mathbb {E}}^{1,p}((0,T); L^p({\mathbb {S}}^1; {\mathbb {R}}^2))} T^{2\alpha } \\&\quad \leqq C(T_0,R_0,p,{\bar{\gamma }}) \Vert \gamma - {\tilde{\gamma }} \Vert _{{\mathbb {E}}^{1,p}((0,T); L^p({\mathbb {S}}^1; {\mathbb {R}}^2))} T^{\alpha } \\&\qquad \ + C(T_0,R_0,p,\gamma _0) \Vert \gamma - {\tilde{\gamma }} \Vert _{{\mathbb {E}}^{1,p}((0,T); L^p({\mathbb {S}}^1; {\mathbb {R}}^2))} \Vert {\tilde{\gamma }} - {\bar{\gamma }} \Vert _{{\mathbb {E}}^{1,p}((0,T); L^p({\mathbb {S}}^1; {\mathbb {R}}^2))} T^{2\alpha } \\&\quad \leqq C(T_0,R_0,p,{\bar{\gamma }}) \Vert \gamma - {\tilde{\gamma }} \Vert _{{\mathbb {E}}^{1,p}((0,T); L^p({\mathbb {S}}^1; {\mathbb {R}}^2))} T^{\alpha } \end{aligned}$$

where we have used the fact that $\gamma (\cdot ,0) = {\tilde{\gamma }}(\cdot ,0) = {\bar{\gamma }}(\cdot ,0) = \gamma _0(\cdot )$. As before we have that

$$\begin{aligned} \Vert {\tilde{\gamma }}_{u^4} \Vert _{L^p((0,T); L^p({\mathbb {S}}^1; {\mathbb {R}}^2))}\leqq & {} \Vert {\tilde{\gamma }} - {\bar{\gamma }} \Vert _{{\mathbb {E}}^{1,p}((0,T); L^p({\mathbb {S}}^1; {\mathbb {R}}^2))}\\{} & {} + \Vert {\bar{\gamma }} \Vert _{{\mathbb {E}}^{1,p}((0,T); L^p({\mathbb {S}}^1; {\mathbb {R}}^2))} \leqq R_0 + C({\bar{\gamma }}) \leqq C(R_0,{\bar{\gamma }}) \end{aligned}$$

so that it follows for sufficiently small $T = T(T_0,R_0,p,\varepsilon , {\bar{\gamma }}) > 0$

$$\begin{aligned}&\left\| \frac{1}{|\gamma _{uu}^{\perp }|^2} - \frac{1}{|{\tilde{\gamma }}_{uu}^{\perp }|^2} \right\| _{L^{\infty }((0,T); L^p({\mathbb {S}}^1))} \Vert {\tilde{\gamma }}_{u^4} \Vert _{L^p((0,T); L^p({\mathbb {S}}^1; {\mathbb {R}}^2))} \\ {}&\quad \leqq \frac{\varepsilon }{2} \Vert \gamma - {\tilde{\gamma }} \Vert _{{\mathbb {E}}^{1,p}((0,T); L^p({\mathbb {S}}^1; {\mathbb {R}}^2))}. \end{aligned}$$

$\square$

Having shown the $\varepsilon$-contractivity of the highest order term of the operator on the right hand side of (65) we now turn our attention to controlling the non-linearities present in F. The main obstacle to this is the presence of $|\gamma _u|^{-l}\langle \gamma _{uu}, \nu \rangle ^{-1}$ for some $l \in {\mathbb {N}}$ in most of the terms. The following lemma shows that we are able to control such quantities.

Lemma A.8

Let $l \in {\mathbb {N}}, 1 \leqq q \leqq \infty$. Suppose that $\varphi (\gamma _1, \ldots , \gamma _m)$ is a multilinear map such that for every $\gamma _1,\ldots ,\gamma _l \in {\mathbb {E}}^{1,p}((0,T); L^p({\mathbb {S}}^1; {\mathbb {R}}^2))$,

$$\begin{aligned} \Vert \varphi (\gamma _1, \ldots , \gamma _l) \Vert _{L^q((0,T); Z)} \leqq C \prod _{i = 1}^l \Vert \partial _u^{\beta _i} \gamma _i \Vert _{X_i} \end{aligned}$$

(73)

for some $\beta _1,\ldots ,\beta _l \in \{0,1,2,3\}$ and $Z, X_1,\ldots ,X_m$ are Banach spaces satisfying the following criteria:

• (i) $\partial _u^{\beta _i}: {\mathbb {E}}^{1,p}((0,T); L^p({\mathbb {S}}^1; {\mathbb {R}}^2)) \rightarrow X_i$ and for $\gamma \in {}_{0}{{\mathbb {E}}^{1,p}}((0,T); L^p({\mathbb {S}}^1; {\mathbb {R}}^2))$ there exists some constant $C = C(T_0) > 0$ such that

  $$\begin{aligned} \Vert \partial _u^{\beta _i} \gamma \Vert _{X_i} \leqq C(T_0,p) \Vert \gamma \Vert _{{\mathbb {E}}^{1,p}((0,T); L^p({\mathbb {S}}^1; {\mathbb {R}}^2))} \end{aligned}$$

  for each $i = 1,\ldots ,l$.

• (ii) One of the following holds:

  • (a) There exists some $\alpha > 0$ and a constant $C = C(T_0,p) > 0$ such that for every $\gamma \in {}_{0}{{\mathbb {E}}^{1,p}}((0,T); L^p({\mathbb {S}}^1; {\mathbb {R}}^2))$

    $$\begin{aligned} \Vert \partial _u^{\beta _i} \gamma \Vert _{X_i} \leqq C(T_0,p)T^{\alpha } \Vert \gamma \Vert _{{\mathbb {E}}^{1,p}((0,T); L^p({\mathbb {S}}^1; {\mathbb {R}}^2))}. \end{aligned}$$

  • (b) There is some $j \ne l$ such that for every $\gamma \in {}_{0}{{\mathbb {E}}^{1,p}}((0,T); L^p({\mathbb {S}}^1; {\mathbb {R}}^2))$

    $$\begin{aligned} \lim _{T \searrow 0} \Vert \partial _u^{\beta _j}\gamma \Vert _{X_j} = 0. \end{aligned}$$

Then if $\varphi (\gamma ):= \varphi (\gamma ,\ldots ,\gamma )$ we have that $\varphi (\gamma ) \in L^q((0,T); Z)$ for all $\gamma \in {\mathbb {E}}^{1,p}((0,T); L^p({\mathbb {S}}^1; {\mathbb {R}}^2))$ and for any $\varepsilon \in (0,1)$ there exists $R = R(T_0,R_0,p,q,l,\varepsilon , {\bar{\gamma }}), T = T(T_0,R_0,p,q,l,\varepsilon , {\bar{\gamma }}) > 0$ small enough such that for every $\gamma , {\tilde{\gamma }} \in B_R^{{\mathbb {E}}^{1,p}}({\bar{\gamma }})$,

$$\begin{aligned} \Vert \varphi (\gamma ) - \varphi ({\tilde{\gamma }}) \Vert _{L^q((0,T); Z)} \leqq \varepsilon \Vert \gamma - {\tilde{\gamma }} \Vert _{{\mathbb {E}}^{1,p}((0,T); L^p({\mathbb {S}}^1; {\mathbb {R}}^2))}. \end{aligned}$$

Moreover for any $l \in {\mathbb {N}}$ and for $Z = L^r({\mathbb {S}}^1; {\mathbb {R}}^2)$ for some $1 \leqq r \leqq \infty$ there exists $R = R(T_0,R_0,p,q,r,l,\varepsilon , {\bar{\gamma }})$, $T = T(T_0,R_0,p,q,r,l, \varepsilon , {\bar{\gamma }}) > 0$ small enough such that

$$\begin{aligned}{} & {} \Vert |\gamma _u|^{-l}\langle \gamma _{uu}, \nu \rangle ^{-1} \varphi (\gamma ) - |{\tilde{\gamma }}_u|^{-l}\langle {\tilde{\gamma }}_{uu}, {\tilde{\nu }} \rangle ^{-1} \varphi ({\tilde{\gamma }}) \Vert _{L^q((0,T); L^r({\mathbb {S}}^1; {\mathbb {R}}^2))} \\ {}{} & {} \quad \leqq \varepsilon \Vert \gamma - {\tilde{\gamma }} \Vert _{{\mathbb {E}}^{1,p}((0,T); L^p({\mathbb {S}}^1; {\mathbb {R}}^2))}. \end{aligned}$$

Proof

The first part of the proof follows as in [18]. Let $l \in {\mathbb {N}}$ be arbitrary. The simple estimate

$$\begin{aligned} |\gamma _u|^{-l} \langle \gamma _{uu}, \nu \rangle ^{-1}|\varphi (\gamma )| \leqq \left( \frac{1}{2} \inf _{{\mathbb {S}}^1} |(\gamma _0)_u|\right) ^{-l} \left( \frac{1}{2}\inf _{{\mathbb {S}}^1} \langle (\gamma _0)_{uu}, \nu _0 \rangle \right) ^{-1} |\varphi (\gamma )| \end{aligned}$$

shows that $\gamma \mapsto |\gamma _u|^{-l}(\langle \gamma _{uu}, \nu \rangle )^{-1}\varphi (\gamma )$ is well defined in the sense that it maps into the same space as $\varphi (\gamma )$. Let $\varepsilon \in (0,1)$ and $\gamma , {\tilde{\gamma }} \in B_R^{{\mathbb {E}}^{1,p}}({\bar{\gamma }})$. It then holds

$$\begin{aligned}&\Vert |\gamma _u|^{-l}\langle \gamma _{uu}, \nu \rangle ^{-1} \varphi (\gamma ) - |{\tilde{\gamma }}_u|^{-l}\langle {\tilde{\gamma }}_{uu}, {\tilde{\nu }} \rangle ^{-1} \varphi ({\tilde{\gamma }}) \Vert _{L^q((0,T); L^r({\mathbb {S}}^1; {\mathbb {R}}^2))} \\&\quad \leqq \Vert (|\gamma _u|^{-l} - |{\tilde{\gamma }}_u|^{-l})\langle \gamma _{uu}, \nu \rangle ^{-1} \varphi (\gamma ) \Vert _{L^q((0,T); L^r({\mathbb {S}}^1; {\mathbb {R}}^2))} \\&\qquad \ + \Vert |{\tilde{\gamma }}_u|^{-l}(\langle \gamma _{uu}, \nu \rangle ^{-1}\varphi (\gamma ) - \langle {\tilde{\gamma }}_{uu}, {\tilde{\nu }} \rangle ^{-1} \varphi ({\tilde{\gamma }})) \Vert _{L^q((0,T); L^r({\mathbb {S}}^1; {\mathbb {R}}^2))} \\&\quad \leqq \Vert |\gamma _u|^{-l} - |{\tilde{\gamma }}_u|^{-l} \Vert _{L^{\infty }((0,T); L^r({\mathbb {S}}^1))} \Vert \langle \gamma _{uu}, \nu \rangle ^{-1} \Vert _{L^{\infty }((0,T); L^r({\mathbb {S}}^1))} \Vert \varphi (\gamma ) \Vert _{L^q((0,T); L^r({\mathbb {S}}^1; {\mathbb {R}}^2))} \\&\qquad \ + \Vert |{\tilde{\gamma }}_u|^{-l} \Vert _{L^{\infty }((0,T); L^r({\mathbb {S}}^1))} \Vert \langle \gamma _{uu}, \nu \rangle ^{-1}\\&\qquad - \langle {\tilde{\gamma }}_{uu}, {\tilde{\nu }} \rangle ^{-1} \Vert _{L^{\infty }((0,T); L^r({\mathbb {S}}^1))} \Vert \varphi (\gamma ) \Vert _{L^q((0,T); L^r({\mathbb {S}}^1; {\mathbb {R}}^2))} \\&\qquad \ + \Vert |{\tilde{\gamma }}_u|^{-l} \Vert _{L^{\infty }((0,T); L^r({\mathbb {S}}^1))} \Vert \langle {\tilde{\gamma }}_{uu}, {\tilde{\nu }} \rangle ^{-1} \Vert _{L^{\infty }((0,T); L^r({\mathbb {S}}^1))} \Vert \varphi (\gamma ) - \varphi ({\tilde{\gamma }}) \Vert _{L^q((0,T); L^r({\mathbb {S}}^1; {\mathbb {R}}^2))}. \end{aligned}$$

We see from the Mean Value Theorem that

$$\begin{aligned} |\gamma _u|^{-l} - |{\tilde{\gamma }}_u|^{-l} \leqq l\left( \frac{1}{2}\inf _{{\mathbb {S}}^1} |(\gamma _0)_u|\right) ^{-(l+1)}|\gamma _u - {\tilde{\gamma }}_u| \end{aligned}$$

from which it follows there exists some $\alpha \in (0,1)$ such that

$$\begin{aligned} \Vert |\gamma _u|^{-l} - |{\tilde{\gamma }}_u|^{-l} \Vert _{L^{\infty }((0,T); L^r({\mathbb {S}}^1))}&\leqq C(l,\gamma _0) \Vert \gamma _u - {\tilde{\gamma }}_u \Vert _{L^{\infty }((0,T); L^r({\mathbb {S}}^1; {\mathbb {R}}^2))} \\&\leqq C(r,l,\gamma _0) \Vert \gamma - {\tilde{\gamma }} \Vert _{C^{0,\alpha }([0,T]; C^{1,\alpha }({\mathbb {S}}^1; {\mathbb {R}}^2))} \\&\leqq C(T_0,p,r,l,\gamma _0) \Vert \gamma - {\tilde{\gamma }} \Vert _{{\mathbb {E}}^{1,p}((0,T); L^p({\mathbb {S}}^1; {\mathbb {R}}^2))}T^{\alpha }. \end{aligned}$$

Similarly

$$\begin{aligned} \langle \gamma _{uu}, \nu \rangle ^{-1} - \langle {\tilde{\gamma }}_{uu}, {\tilde{\nu }} \rangle ^{-1}&\leqq \left( \inf _{{\mathbb {S}}^1} \langle (\gamma _0)_{uu}, \nu _0 \rangle \right) ^{-2}|\langle \gamma _{uu}, \nu \rangle - \langle {\tilde{\gamma }}_{uu}, {\tilde{\nu }} \rangle | \\&\leqq C(\gamma _0)(|\langle \gamma _{uu}, \nu \rangle - \langle {\tilde{\gamma }}_{uu}, \nu \rangle | + |\langle {\tilde{\gamma }}_{uu}, \nu \rangle - \langle {\tilde{\gamma }}_{uu}, {\tilde{\nu }} \rangle |) \\&\leqq C(\gamma _0)|\gamma _{uu} - {\tilde{\gamma }}_{uu}| + C(\gamma _0)|{\tilde{\gamma }}_{uu}||\nu - {\tilde{\nu }}| \\&= C(\gamma _0)|\gamma _{uu} - {\tilde{\gamma }}_{uu}| + C(\gamma _0)|{\tilde{\gamma }}_{uu}|| R_{\frac{\pi }{2}}(\tau - {\tilde{\tau }})| \\&= C(\gamma _0)|\gamma _{uu} - {\tilde{\gamma }}_{uu}| + C(\gamma _0)|{\tilde{\gamma }}_{uu}| |\tau - {\tilde{\tau }}| \end{aligned}$$

where we have exploited the fact that the rotation $R_{\frac{\pi }{2}}$ is a linear isometry of ${\mathbb {R}}^2$. Using (70) replacing $\tau _0$ with ${\tilde{\tau }}$ where appropriate we find using the Mean Value Theorem again,

$$\begin{aligned} |\tau - {\tilde{\tau }}|&\leqq \left( \frac{1}{2}\inf _{{\mathbb {S}}^1} |(\gamma _0)_u| \right) ^{-1}|\gamma _u - {\tilde{\gamma }}_u| + \left| \frac{1}{|\gamma _u|} - \frac{1}{|{\tilde{\gamma }}_u|} \right| |\gamma _u| \\&\leqq \left( \frac{1}{2}\inf _{{\mathbb {S}}^1} |(\gamma _0)_u| \right) ^{-1}|\gamma _u - {\tilde{\gamma }}_u| + \left( \frac{1}{2} \inf _{{\mathbb {S}}^1} |(\gamma _0)_u| \right) ^{-2}|\gamma _u - {\tilde{\gamma }}_u| |\gamma _u|. \end{aligned}$$

Thus

$$\begin{aligned}&\Vert \langle \gamma _{uu}, \nu \rangle ^{-1} - \langle {\tilde{\gamma }}_{uu}, {\tilde{\nu }} \rangle ^{-1} \Vert _{L^{\infty }((0,T); L^r({\mathbb {S}}^1))} \\&\quad \leqq C(\gamma _0) \Vert \gamma _{uu} - {\tilde{\gamma }}_{uu} \Vert _{L^{\infty }((0,T); L^r({\mathbb {S}}^1; {\mathbb {R}}^2))} \\&\qquad \ + C(\gamma _0) \Vert {\tilde{\gamma }}_{uu} \Vert _{L^{\infty }((0,T); L^r({\mathbb {S}}^1; {\mathbb {R}}^2))} \Vert \gamma _u - {\tilde{\gamma }}_u \Vert _{L^{\infty }((0,T); L^r({\mathbb {S}}^1; {\mathbb {R}}^2))} \\&\qquad \ + C(\gamma _0) \Vert {\tilde{\gamma }}_{uu} \Vert _{L^{\infty }((0,T); L^r({\mathbb {S}}^1; {\mathbb {R}}^2))} \Vert \gamma _u - {\tilde{\gamma }}_u \Vert _{L^{\infty }((0,T); L^r({\mathbb {S}}^1; {\mathbb {R}}^2))} \Vert \gamma _u \Vert _{L^{\infty }((0,T); L^r({\mathbb {S}}^1; {\mathbb {R}}^2))} \\&\quad \leqq C(T_0,p,r,\gamma _0) \Vert \gamma - {\tilde{\gamma }} \Vert _{{\mathbb {E}}^{1,p}((0,T); L^p({\mathbb {S}}^1; {\mathbb {R}}^2))}T^{\alpha } \\&\qquad \ + C(T_0,p,r,\gamma _0) \Vert {\tilde{\gamma }} - {\bar{\gamma }} \Vert _{{\mathbb {E}}^{1,p}((0,T); L^p({\mathbb {S}}^1; {\mathbb {R}}^2))} \Vert \gamma - {\tilde{\gamma }} \Vert _{{\mathbb {E}}^{1,p}((0,T); L^p({\mathbb {S}}^1; {\mathbb {R}}^2))}T^{2\alpha } \\&\qquad \ + C(T_0,p,r,\gamma _0) \Vert {\bar{\gamma }} \Vert _{C^{0,\alpha }([0,T]; C^{2,\alpha }({\mathbb {S}}^1; {\mathbb {R}}^2))} \Vert \gamma - {\tilde{\gamma }} \Vert _{{\mathbb {E}}^{1,p}((0,T); L^p({\mathbb {S}}^1; {\mathbb {R}}^2))} T^{\alpha } \\&\qquad \ + C(T_0,p,r,\gamma _0) \Vert {\tilde{\gamma }} - {\bar{\gamma }} \Vert _{{\mathbb {E}}^{1,p}((0,T); L^p({\mathbb {S}}^1; {\mathbb {R}}^2))}^2 \Vert \gamma - {\tilde{\gamma }} \Vert _{{\mathbb {E}}^{1,p}((0,T); L^p({\mathbb {S}}^1; {\mathbb {R}}^2))} T^{3\alpha } \\&\qquad \ + C(T_0,p,r,\gamma _0) \Vert {\tilde{\gamma }} - {\bar{\gamma }} \Vert _{{\mathbb {E}}^{1,p}((0,T); L^p({\mathbb {S}}^1; {\mathbb {R}}^2))} \Vert {\bar{\gamma }} \Vert _{C^{0,\alpha }([0,T]; C^{2,\alpha }({\mathbb {S}}^1; {\mathbb {R}}^2))} \Vert \gamma \\&\qquad - {\tilde{\gamma }} \Vert _{{\mathbb {E}}^{1,p}((0,T); L^p({\mathbb {S}}^1; {\mathbb {R}}^2))} T^{2\alpha } \\&\qquad \ + C(T_0,p,r,\gamma _0) \Vert {\tilde{\gamma }} - {\bar{\gamma }} \Vert _{{\mathbb {E}}^{1,p}((0,T); L^p({\mathbb {S}}^1; {\mathbb {R}}^2))} \Vert {\bar{\gamma }} \Vert _{C^{0,\alpha }([0,T]; C^{1,\alpha }({\mathbb {S}}^1; {\mathbb {R}}^2))} \Vert \gamma \\&\qquad - {\tilde{\gamma }} \Vert _{{\mathbb {E}}^{1,p}((0,T); L^p({\mathbb {S}}^1; {\mathbb {R}}^2))} T^{2\alpha } \\&\quad \leqq C(T_0,R_0,p,r,{\bar{\gamma }}) T^{\alpha } \Vert \gamma - {\tilde{\gamma }} \Vert _{{\mathbb {E}}^{1,p}((0,T); L^p({\mathbb {S}}^1; {\mathbb {R}}^2))}. \end{aligned}$$

Similarly by the Mean Value Theorem and similar estimates as above

$$\begin{aligned} \Vert \langle \gamma _{uu}, \nu \rangle ^{-1} \Vert _{L^{\infty }((0,T); L^r({\mathbb {S}}^1))}&\leqq C(\gamma _0) \Vert \langle \gamma _{uu}, \nu \rangle \Vert _{L^{\infty }((0,T); L^r({\mathbb {S}}^1))} \\&\leqq C(\gamma _0) \Vert \gamma _{uu} \Vert _{L^{\infty }((0,T); L^r({\mathbb {S}}^1; {\mathbb {R}}^2))} \\&\leqq C(r,\gamma _0) \Vert \gamma - {\bar{\gamma }} \Vert _{C^{0,\alpha }([0,T]; C^{2,\alpha }({\mathbb {S}}^1; {\mathbb {R}}^2))} \\&\quad \ + C(r,\gamma _0) \Vert {\bar{\gamma }} \Vert _{C^{0,\alpha }([0,T]; C^{2,\alpha }({\mathbb {S}}^1; {\mathbb {R}}^2))} \\&\leqq C(T_0,p,r,\gamma _0) \Vert \gamma - {\bar{\gamma }} \Vert _{{\mathbb {E}}^{1,p}((0,T); L^p({\mathbb {S}}^1; {\mathbb {R}}^2))} T^{\alpha } + C(r,{\bar{\gamma }}) \\&\leqq C(T_0,R_0,p,r,{\bar{\gamma }}) \end{aligned}$$

with the same estimate replacing $\langle \gamma _{uu}, \nu \rangle ^{-1}$ with $\langle {\tilde{\gamma }}_{uu}, {\tilde{\nu }} \rangle ^{-1}$ and,

$$\begin{aligned} \Vert |{\tilde{\gamma }}_u|^{-l} \Vert _{L^{\infty }((0,T); L^r({\mathbb {S}}^1))}&\leqq C(l, \gamma _0) \Vert {\tilde{\gamma }}_u \Vert _{L^{\infty }((0,T); L^r({\mathbb {S}}^1; {\mathbb {R}}^2))} \\&\leqq C(l,r, \gamma _0) \Vert {\tilde{\gamma }} - {\bar{\gamma }} \Vert _{C^{0,\alpha }([0,T]; C^{1,\alpha }({\mathbb {S}}^1; {\mathbb {R}}^2))} \\&\quad \ + C(l,\gamma _0) \Vert {\bar{\gamma }} \Vert _{L^{\infty }((0,T); L^r({\mathbb {S}}^1; {\mathbb {R}}^2))} \\&\leqq C(T_0,p,r,l,\gamma _0) \Vert {\tilde{\gamma }} - {\bar{\gamma }} \Vert _{{\mathbb {E}}^{1,p}((0,T); L^p({\mathbb {S}}^1; {\mathbb {R}}^2))} T^{\alpha } + C(l,{\bar{\gamma }}) \\&\leqq C(T_0,R_0,p,r,l,{\bar{\gamma }}). \end{aligned}$$

From the simple estimate

$$\begin{aligned} \Vert \varphi (\gamma ) \Vert _{L^q((0,T); L^r({\mathbb {S}}^1; {\mathbb {R}}^2))}&\leqq \Vert \varphi (\gamma ) - \varphi ({\bar{\gamma }}) \Vert _{L^q((0,T); L^r({\mathbb {S}}^1; {\mathbb {R}}^2))} + \Vert \varphi ({\bar{\gamma }}) \Vert _{L^q((0,T); L^r({\mathbb {S}}^1; {\mathbb {R}}^2))} \\&\leqq {\bar{\varepsilon }} \Vert \gamma - {\bar{\gamma }} \Vert _{{\mathbb {E}}^{1,p}((0,T); L^p({\mathbb {S}}^1; {\mathbb {R}}^2))} + C({\bar{\gamma }}) \\&\leqq R_0 {\bar{\varepsilon }} + C({\bar{\gamma }}) \end{aligned}$$

for some ${\bar{\varepsilon }} > 0$ to be chosen, we thus find that

$$\begin{aligned} \Vert |\gamma _u|^{-l}\langle \gamma _{uu}, \nu \rangle ^{-1} \varphi (\gamma )&- |{\tilde{\gamma }}_u|^{-l}\langle {\tilde{\gamma }}_{uu}, {\tilde{\nu }} \rangle ^{-1} \varphi ({\tilde{\gamma }}) \Vert _{L^q((0,T); L^r({\mathbb {S}}^1; {\mathbb {R}}^2))} \\&\leqq C(T_0,R_0,p,q,r,l,{\bar{\gamma }}){\bar{\varepsilon }} T^{\alpha } \Vert \gamma - {\tilde{\gamma }} \Vert _{{\mathbb {E}}^{1,p}((0,T); L^p({\mathbb {S}}^1; {\mathbb {R}}^2))} \\&\quad \ + C(T_0,R_0,p,q,r,l,{\bar{\gamma }}) T^{\alpha } \Vert \gamma - {\tilde{\gamma }} \Vert _{{\mathbb {E}}^{1,p}((0,T); L^p({\mathbb {S}}^1; {\mathbb {R}}^2))} \\&\quad \ + C(T_0,R_0,p,q,r,l,{\bar{\gamma }}) {\tilde{\varepsilon }} T^{\alpha } \Vert \gamma - {\tilde{\gamma }} \Vert _{{\mathbb {E}}^{1,p}((0,T); L^p({\mathbb {S}}^1; {\mathbb {R}}^2))} \\&\leqq \varepsilon \Vert \gamma - {\tilde{\gamma }} \Vert _{{\mathbb {E}}^{1,p}((0,T); L^p({\mathbb {S}}^1; {\mathbb {R}}^2))} \end{aligned}$$

for sufficiently small ${\bar{\varepsilon }} = \bar{\varepsilon }(T_0,R_0,p,q,r,l,\varepsilon , {\bar{\gamma }})$, ${\tilde{\varepsilon }} = {\tilde{\varepsilon }}(T_0,R_0,p,q,r,l,\varepsilon , {\bar{\gamma }}) \in (0,1)$, adjusting $T = T(T_0,R_0,p,q,r,l,\varepsilon , {\bar{\gamma }})$, $R = R(T_0,R_0,p,q,r,l,\varepsilon , {\bar{\gamma }})$ if necessary. $\square$

This leads into contraction estimates on the operator F.

Proposition A.9

Let $\varepsilon \in (0,1)$, $p > \frac{5}{2}$. For sufficiently small $T, R > 0$ the maps

• (i) ${\overline{B}}_R^{{\mathbb {E}}^{1,p}}({\bar{\gamma }}) \rightarrow L^p((0,T); L^p({\mathbb {S}}^1; {\mathbb {R}}^2)), \quad \gamma \mapsto - \frac{\langle \gamma _{u^3}, \nu \rangle }{|\gamma _u|\langle \gamma _{uu}, \nu \rangle } \gamma _{u^3}^{\perp }$

• (ii) ${\overline{B}}_R^{{\mathbb {E}}^{1,p}}({\bar{\gamma }}) \rightarrow L^p((0,T); L^p({\mathbb {S}}^1; {\mathbb {R}}^2)), \quad \gamma \mapsto - \frac{1}{|\gamma _u|^2} \gamma _{uu}^{\perp }$

• (iii) ${\overline{B}}_R^{{\mathbb {E}}^{1,p}}({\bar{\gamma }}) \rightarrow L^p((0,T); L^p({\mathbb {S}}^1; {\mathbb {R}}^2)), \quad \gamma \mapsto - \frac{4\langle \gamma _{u^3}, \gamma _u \rangle }{|\gamma _u|^2 \langle \gamma _{uu}, \nu \rangle }\nu$

• (iv) ${\overline{B}}_R^{{\mathbb {E}}^{1,p}}({\bar{\gamma }}) \rightarrow L^p((0,T); L^p({\mathbb {S}}^1; {\mathbb {R}}^2)), \quad \gamma \mapsto \frac{6 \langle \gamma _{uu}, \gamma _u \rangle ^2}{|\gamma _u|^4\langle \gamma _{uu}, \nu \rangle }\nu$

are well defined $\varepsilon$-contractions.

Proof

By A.8 it is sufficient to show that the maps

• (i) ${\overline{B}}_R^{{\mathbb {E}}^{1,p}}({\bar{\gamma }}) \rightarrow L^p((0,T); L^p({\mathbb {S}}^1; {\mathbb {R}}^2)), \quad \gamma \mapsto \langle \gamma _{u^3}, \nu \rangle \gamma _{u^3}^{\perp }$

• (ii) ${\overline{B}}_R^{{\mathbb {E}}^{1,p}}({\bar{\gamma }}) \rightarrow L^p((0,T); L^p({\mathbb {S}}^1; {\mathbb {R}}^2)), \quad \gamma \mapsto \gamma _{uu}^{\perp }$

• (iii) ${\overline{B}}_R^{{\mathbb {E}}^{1,p}}({\bar{\gamma }}) \rightarrow L^p((0,T); L^p({\mathbb {S}}^1; {\mathbb {R}}^2)), \quad \gamma \mapsto \langle \gamma _{u^3}, \gamma _u \rangle \nu$

• (iv) ${\overline{B}}_R^{{\mathbb {E}}^{1,p}}({\bar{\gamma }}) \rightarrow L^p((0,T); L^p({\mathbb {S}}^1; {\mathbb {R}}^2)), \quad \gamma \mapsto \langle \gamma _{uu}, \gamma _u \rangle ^2 \nu$

satisfy the assumption of A.8 for which we shall be using A.3 and Hölder’s Inequality in space and time to verify this. (i) Observe that using Hölder’s Inequality in time along with the fact that $\nu$ is of unit length

$$\begin{aligned} \Vert \langle \gamma _{u^3}, \nu \rangle \gamma _{u^3}^{\perp } \Vert _{L^p((0,T); L^p({\mathbb {S}}^1; {\mathbb {R}}^2))} \leqq \Vert \gamma _{u^3} \Vert _{L^{2p}((0,T); L^p({\mathbb {S}}^1; {\mathbb {R}}^2))}^2 \end{aligned}$$

so that we are looking for $\theta _1 \in (0,1), \rho _1 \in [1, \infty )$ such that

$$\begin{aligned} \frac{\theta _1}{4} - \frac{1}{p} \leqq - \frac{1}{2p}, \quad 1 - \theta _1 - \frac{1}{p} \leqq - \frac{1}{\rho _1}. \end{aligned}$$

from which we find that $\theta _1 \in [\frac{2}{p}, 1)$ which is well defined for $p > \frac{5}{2}$. Thus for a choice of $\theta _1 = \frac{2}{p}$ we find that

$$\begin{aligned} 1 - \frac{2}{p} - \frac{1}{p} = 1 - \frac{3}{p} \leqq - \frac{1}{\rho _1}. \end{aligned}$$

For $p \leqq 3$, $1 - \frac{3}{p} \leqq 0$ so any choice of $\rho _1 > p$ say $\rho _1 = 2p$ will satisfy this inequality. Else for $\frac{5}{2}< p < 3$ we instead choose $\rho _1 = \frac{p}{3 - p}$ which is larger than p for every $p > 2$ which of course holds by assumption. Thus applying Hölder’s Inequality in the space variable we find that for $p \leqq 3$

$$\begin{aligned} \Vert \gamma _{u^3} \Vert _{L^{2p}((0,T); L^p({\mathbb {S}}^1; {\mathbb {R}}^2))}^2&\leqq C(p) \Vert \gamma _{u^3} \Vert _{L^{2p}((0,T); L^{2p}({\mathbb {S}}^1; {\mathbb {R}}^2))}^2 \\&\leqq C(T_0,p) \Vert \gamma \Vert _{{\mathbb {E}}^{1,p}((0,T); L^p({\mathbb {S}}^1; {\mathbb {R}}^2))}^2 \end{aligned}$$

else for $\frac{5}{2}< p < 3$

$$\begin{aligned} \Vert \gamma _{u^3} \Vert _{L^{2p}((0,T); L^p({\mathbb {S}}^1; {\mathbb {R}}^2))}^2&\leqq C(p) \Vert \gamma _{u^3} \Vert _{L^{2p}((0,T); L^{\frac{p}{3 - p}}({\mathbb {S}}^1; {\mathbb {R}}^2))}^2 \\&\leqq C(T_0,p) \Vert \gamma \Vert _{{\mathbb {E}}^{1,p}((0,T); L^p({\mathbb {S}}^1; {\mathbb {R}}^2))}^2 \end{aligned}$$

(ii) The estimate from applying Hölder’s Inequality in the time variable

$$\begin{aligned}{} & {} \Vert \gamma _{uu}^{\perp } \Vert _{L^p((0,T); L^p({\mathbb {S}}^1; {\mathbb {R}}^2))} \leqq \Vert \gamma _{uu} \Vert _{L^p((0,T); L^p({\mathbb {S}}^1; {\mathbb {R}}^2))} \\{} & {} \leqq C(T_0,p) \Vert \gamma _{uu} \Vert _{L^{2p}((0,T); L^p({\mathbb {S}}^1; {\mathbb {R}}^2))} \end{aligned}$$

shows that we must find $\theta _2 \in (0,1), \rho _2 \in [1, \infty )$ such that

$$\begin{aligned} \frac{\theta _2}{4} - \frac{1}{p} \leqq - \frac{1}{2p}, \quad 2(1 - \theta _2) - \frac{1}{p} \leqq - \frac{1}{\rho _2}. \end{aligned}$$

For a choice of $\theta _2 = \frac{2}{p}$ we need $\rho _2$ such that

$$\begin{aligned} 2 - \frac{4}{p} - \frac{1}{p} = 2 - \frac{5}{p} = \frac{2p - 5}{p} \leqq - \frac{1}{\rho _2} \end{aligned}$$

but since we have assumed $p> \frac{5}{2}, 2p - 5 > 0$ and thus any $\rho _2 > p$ can be chosen for which we again shall choose $\rho _2 = 2p$. Thus it holds

$$\begin{aligned} \Vert \gamma _{uu}^{\perp } \Vert _{L^p((0,T); L^p({\mathbb {S}}^1; {\mathbb {R}}^2))}\leqq & {} C(T_0,p) \Vert \gamma _{uu} \Vert _{L^{2p}((0,T); L^{2p}({\mathbb {S}}^1; {\mathbb {R}}^2))} \\\leqq & {} C(T_0,p) \Vert \gamma \Vert _{{\mathbb {E}}^{1,p}((0,T); L^p({\mathbb {S}}^1; {\mathbb {R}}^2))}. \end{aligned}$$

(iii) Observe that by using Hölder’s Inequality in time

$$\begin{aligned} \Vert \langle \gamma _{u^3}, \gamma _u \rangle \nu \Vert _{L^p((0,T); L^p({\mathbb {S}}^1; {\mathbb {R}}^2))}&\leqq \Vert \gamma _u \Vert _{L^{\infty }((0,T); L^p({\mathbb {S}}^1; {\mathbb {R}}^2))} \Vert \gamma _{u^3} \Vert _{L^p((0,T); L^p({\mathbb {S}}^1; {\mathbb {R}}^2))} \\&\leqq C(T_0,p) \Vert \gamma _u \Vert _{L^{\infty }((0,T); L^p({\mathbb {S}}^1; {\mathbb {R}}^2))} \Vert \gamma _{u^3} \Vert _{L^{2p}((0,T); L^p({\mathbb {S}}^1; {\mathbb {R}}^2))} \end{aligned}$$

with the estimate

$$\begin{aligned} \Vert \gamma _u \Vert _{L^{\infty }((0,T); L^p({\mathbb {S}}^1; {\mathbb {R}}^2))}\leqq & {} C(T_0,p) \Vert \gamma \Vert _{C^{0,\alpha }([0,T]; C^{1,\alpha }({\mathbb {S}}^1; {\mathbb {R}}^2))} \\\leqq & {} C(T_0,p)T^{\alpha } \Vert \gamma \Vert _{{\mathbb {E}}^{1,p}((0,T); L^p({\mathbb {S}}^1; {\mathbb {R}}^2))} \end{aligned}$$

for some $\alpha \in (0,1)$ it is then immediate from (i) that this term satisfies the hypotheses of A.8. (iv) Applying Hölder’s Inequality in the time variable again we find for the final term

$$\begin{aligned} \Vert \langle \gamma _{uu}, \gamma _u \rangle ^2 \nu \Vert _{L^p((0,T); L^p({\mathbb {S}}^1; {\mathbb {R}}^2))} \leqq \Vert \gamma _u \Vert _{L^{\infty }((0,T); L^p({\mathbb {S}}^1; {\mathbb {R}}^2))}^2 \Vert \gamma _{uu} \Vert _{L^{2p}((0,T); L^p({\mathbb {S}}^1; {\mathbb {R}}^2))}^2 \end{aligned}$$

from which applying Hölder’s Inequality in the space variable, (ii) and the same $L^{\infty }$ estimate on $\gamma _u$ as above we find that

$$\begin{aligned} \Vert \langle \gamma _{uu}, \gamma _u \rangle ^2 \nu \Vert _{L^p((0,T); L^p({\mathbb {S}}^1; {\mathbb {R}}^2))} \leqq C(T_0,p)T^{2\alpha } \Vert \gamma \Vert _{{\mathbb {E}}^{1,p}((0,T); L^p({\mathbb {S}}^1; {\mathbb {R}}^2))}^4. \end{aligned}$$

$\square$

We are now ready to establish the local well-posedness to (65) by use of a fixed point argument. We must first define an appropriate map between a linear problem and our abstract quasilinear problem which is the purpose of the following proposition.

Proposition A.10

Let $\sigma \in {\mathbb {E}}^{1,p}((0,T); L^p({\mathbb {S}}^1; {\mathbb {R}}^2))$ be the unique solution to

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t \sigma + \frac{\sigma _{u^4}}{|(\gamma _0)_u|^4} &{}= {{\textbf {F}}}(\gamma ) := \left( \frac{1}{|(\gamma _0)_u|^4} - \frac{1}{|\gamma _u|^4} \right) \gamma _{u^4} + F(\gamma _{u^3}, \gamma _{uu}, \gamma _u), \quad t > 0 \\ \qquad \ \ \sigma (\cdot ,0) &{}= \gamma _0(\cdot ) \end{array}\right. } \end{aligned}$$

which is guaranteed by A.2 and suppose we define $\Phi : {\overline{B}}_R^{{\mathbb {E}}^{1,p}}({\bar{\gamma }}) \rightarrow {\mathbb {E}}^{1,p}((0,T); L^p({\mathbb {S}}^1; {\mathbb {R}}^2))$ by $\Phi (\gamma ) = \sigma$. Then there exists $T = T(T_0,R_0,p,\varepsilon , {\bar{\gamma }}), R = R(T_0,R_0,p,\varepsilon , {\bar{\gamma }}) > 0$ such that $\Phi : {\overline{B}}_R^{{\mathbb {E}}^{1,p}}({\bar{\gamma }}) \rightarrow {\overline{B}}_R^{{\mathbb {E}}^{1,p}}({\bar{\gamma }})$ is a well defined $\varepsilon$-contraction.

Proof

Let $\varepsilon \in (0,1)$ and $\gamma , {\tilde{\gamma }} \in {\overline{B}}_R^{{\mathbb {E}}^{1,p}}({\bar{\gamma }})$ with $\sigma = \Phi (\gamma ), {\tilde{\sigma }} = \Phi ({\tilde{\gamma }})$. As ${{\textbf {F}}}$ is written as a linear combination of $\varepsilon$-contractions it then holds that ${{\textbf {F}}}$ is also a well-defined $\varepsilon$-contraction by reducing $T = T(T_0,R_0,p,\varepsilon , {\bar{\gamma }}), R = R(T_0,R_0,p,\varepsilon , {\bar{\gamma }}) > 0$ as needed. By (68) there exists some $C = C(T_0,p,\gamma _0) > 0$ such that

$$\begin{aligned} \Vert \sigma - {\tilde{\sigma }} \Vert _{{\mathbb {E}}^{1,p}((0,T); L^p({\mathbb {S}}^1; {\mathbb {R}}^2))}&\leqq C \Vert {{\textbf {F}}}(\gamma ) - {{\textbf {F}}}({\tilde{\gamma }}) \Vert _{L^p((0,T); L^p({\mathbb {S}}^1; {\mathbb {R}}^2))} \nonumber \\&\leqq C \frac{\varepsilon }{2C} \Vert \gamma - {\tilde{\gamma }} \Vert _{{\mathbb {E}}^{1,p}((0,T); L^p({\mathbb {S}}^1; {\mathbb {R}}^2))} \end{aligned}$$

(74)

reducing $T = T(T_0,R_0,p,\varepsilon , {\bar{\gamma }}), R = R(T_0,R_0, p, \varepsilon , {\bar{\gamma }})$ as necessary. To see that $\Phi$ is well defined we apply (74)

$$\begin{aligned} \Vert \sigma - {\bar{\gamma }} \Vert _{{\mathbb {E}}^{1,p}((0,T); L^p({\mathbb {S}}^1; {\mathbb {R}}^2))}&= \Vert \Phi (\gamma ) - \Phi (0) \Vert _{{\mathbb {E}}^{1,p}((0,T); L^p({\mathbb {S}}^1; {\mathbb {R}}^2))} \\&\leqq C \Vert {{\textbf {F}}}(\gamma ) - 0 \Vert _{L^p((0,T); L^p({\mathbb {S}}^1; {\mathbb {R}}^2))} \\&\leqq C (\Vert {{\textbf {F}}}(\gamma ) - {{\textbf {F}}}({\bar{\gamma }}) \Vert _{L^p((0,T); L^p({\mathbb {S}}^1; {\mathbb {R}}^2))}\\&\quad + \Vert {{\textbf {F}}}({\bar{\gamma }}) \Vert _{L^p((0,T); L^p({\mathbb {S}}^1; {\mathbb {R}}^2))}) \\&\leqq \frac{\varepsilon }{2} \Vert \gamma - {\bar{\gamma }} \Vert _{{\mathbb {E}}^{1,p}((0,T); L^p({\mathbb {S}}^1; {\mathbb {R}}^2))} \\&\quad + C \Vert {{\textbf {F}}}({\bar{\gamma }}) \Vert _{L^p((0,T); L^p({\mathbb {S}}^1; {\mathbb {R}}^2))} \\&\leqq \frac{\varepsilon }{2}R + C \Vert {{\textbf {F}}}({\bar{\gamma }}) \Vert _{L^p((0,T); L^p({\mathbb {S}}^1; {\mathbb {R}}^2))}. \end{aligned}$$

We can repeat similar estimates as in A.7, A.9 to find that there exists some $C = C(T_0,p,{\bar{\gamma }}) > 0$ such that

$$\begin{aligned} \Vert {{\textbf {F}}}({\bar{\gamma }}) \Vert _{L^p((0,T); L^p({\mathbb {S}}^1; {\mathbb {R}}^2))} \leqq C(T_0,p,{\bar{\gamma }})T^{\alpha } \leqq \frac{\varepsilon }{2C}R \end{aligned}$$

by restricting $T = T(T_0,R_0,p,\varepsilon , {\bar{\gamma }}) > 0$ as necessary. Thus

$$\begin{aligned} \Vert \Phi (\gamma ) - {\bar{\gamma }} \Vert _{{\mathbb {E}}^{1,p}((0,T); L^p({\mathbb {S}}^1; {\mathbb {R}}^2))} \leqq R \end{aligned}$$

for $R, T > 0$ small enough. $\square$

This naturally leads into the main existence theorem.

Theorem A.11

Let $\gamma _0 \in W_{{{\,\textrm{Imm}\,}}}^{4(1 - \frac{1}{p}),p}({\mathbb {S}}^1; {\mathbb {R}}^2)$ be a convex curve, for some $p > \frac{5}{2}$. Then there exists $T, R > 0$ such that (65) has a unique solution $\gamma \in {\mathbb {E}}^{1,p}((0,T); L^p({\mathbb {S}}^1; {\mathbb {R}}^2))$.

Proof

Choosing $T, R > 0$ as in A.10 with $\varepsilon = \frac{1}{2}$ the map $\Phi : {\overline{B}}_R^{{\mathbb {E}}^{1,p}}({\bar{\gamma }}) \rightarrow {\overline{B}}_R^{{\mathbb {E}}^{1,p}}({\bar{\gamma }})$ defines a contraction on the complete metric space ${\overline{B}}_R^{{\mathbb {E}}^{1,p}}({\bar{\gamma }})$ and hence by the Banach fixed point Theorem $\Phi$ has a unique fixed point $\gamma \in {\overline{B}}_R^{{\mathbb {E}}^{1,p}}({\bar{\gamma }})$. Since any fixed point of $\Phi$ is a solution to (65) in ${\overline{B}}_R^{{\mathbb {E}}^{1,p}}({\bar{\gamma }})$ and conversely a solution to (65) in ${\overline{B}}_R^{{\mathbb {E}}^{1,p}}({\bar{\gamma }})$ is a fixed point of $\Phi$ this establishes local well-posedness for (65) in ${\overline{B}}_R^{{\mathbb {E}}^{1,p}}({\bar{\gamma }})$. We note that any restriction of $\gamma \in {\overline{B}}_R^{{\mathbb {E}}^{1,p}}({\bar{\gamma }})$ to a smaller time interval $[0,{{\tilde{T}}}]$ is also a unique solution to (65) in ${\overline{B}}_R^{{\mathbb {E}}^{1,p}}({\bar{\gamma }})$. Suppose that $T_1, T_2 > 0$ and $\gamma _i \in {\mathbb {E}}^{1,p}((0,T_i); L^p({\mathbb {S}}^1; {\mathbb {R}}^2)), i = 1,2$ are two families of immersions satisfying (65) with $\gamma _0 \in W_{{{\,\textrm{Imm}\,}}}^{4(1 - \frac{1}{p}),p}({\mathbb {S}}^1; {\mathbb {R}}^2)$ which is convex. Without loss of generality we may assume that $T_1 \leqq T_2$. We claim that $\gamma _2 \vert _{[0,T_1]} = \gamma _1$. To verify this claim suppose we define ${{\hat{t}}} = \sup \{t \in [0, T_1): \gamma _1(\cdot ,{{\tilde{t}}}) = \gamma _2(\cdot ,{{\tilde{t}}}), \forall \ 0 \leqq {{\tilde{t}}} < t\}$ to be the maximal time where $\gamma _1, \gamma _2$ agree. ${{\hat{t}}}$ is well defined by use of the embedding

$$\begin{aligned} {\mathbb {E}}^{1,p}((0,T); L^p({\mathbb {S}}^1; {\mathbb {R}}^2)) \hookrightarrow BUC([0,T]; W^{4(1 - \frac{1}{p}),p}({\mathbb {S}}^1; {\mathbb {R}}^2)) \end{aligned}$$

which holds by [15] Theorem 4.2. Moreover since $\lim _{t \searrow 0} |\gamma _i - {\bar{\gamma }}| = 0$ for $i = 1,2$ it follows by the dominated convergence theorem that

$$\begin{aligned} \lim _{T \searrow 0} \Vert \gamma _i - {\bar{\gamma }} \Vert _{{\mathbb {E}}^{1,p}((0,T); L^p({\mathbb {S}}^1; {\mathbb {R}}^2))} = 0 \end{aligned}$$

and hence by restricting $T,R > 0$ as necessary we can guarantee that $\gamma _i \in {\overline{B}}_R^{{\mathbb {E}}^{1,p}}({\bar{\gamma }})$ and hence must agree for at least as long as both $\gamma _i$ remain in ${\overline{B}}_R^{{\mathbb {E}}^{1,p}}({\bar{\gamma }})$. We claim that ${{\hat{t}}} = T_1$, we have already shown above that ${{\hat{t}}} \leqq T > 0$. Suppose for the sake of contradiction that ${{\hat{t}}} < T_1$, since $\gamma _i \in {\mathbb {E}}^{1,p}((0,T_1); L^p({\mathbb {S}}^1; {\mathbb {R}}^2)) \hookrightarrow BUC([0,T_1]; W^{4(1 - \frac{1}{p}),p}({\mathbb {S}}^1; {\mathbb {R}}^2))$ and remain immersed for all times along with A.6 shows that while they might not necessarily remain convex it still remains that $|\gamma _{uu}^{\perp }| > 0$ for all times which is the only necessary assumption needed to control the inverse curvature terms present in the operator. Thus the curve $\gamma _0(\cdot ):= \gamma _1(\cdot ,{{\hat{t}}}) \in W^{4(1 - \frac{1}{p}),p}_{{{\,\textrm{Imm}\,}}}({\mathbb {S}}^1; {\mathbb {R}}^2)$ is well defined and thus there exists some $T,R > 0$ such that (65) with initial data $\gamma _0$ has a unique solution $\gamma \in {\overline{B}}_R^{{\mathbb {E}}^{1,p}}({\bar{\gamma }})$. Observing that $\gamma _i(\cdot , {{\hat{t}}} + t) \vert _{0 \leqq t \leqq T_1 - {{\hat{t}}}}$ are both solutions to (65) with initial data $\gamma _0$ we can follow a similar argument as above to show that $\gamma _1(\cdot , {{\hat{t}}} + \cdot ) = \gamma _2(\cdot , {{\hat{t}}} + \cdot )$ in [0, T) contradicting the definition of ${{\hat{t}}}$. $\square$