1 Introduction

We study the 2D Euler equation for \(u:{\mathbb {R}}^2\rightarrow {\mathbb {R}}^2\)

$$\begin{aligned} \partial _tu+u\cdot \nabla u+\nabla p=0, \nabla \cdot u=0 \end{aligned}$$

The equation can be rewritten in terms of the vorticity

$$\begin{aligned} \omega =\partial _1u_2-\partial _2u_1 \end{aligned}$$

as follows:

$$\begin{aligned} \partial _t\omega (x)&=u(x)\cdot \nabla \omega (x)\\ u&=K*\omega \end{aligned}$$

where, by rescaling time to avoid factors of \(2\pi \), we can take

$$\begin{aligned} K(x)=x^\perp /|x|^2. \end{aligned}$$

Thus the vorticity is transported by u, which is generated as a singular integral of the vorticity. In this paper, we construct solutions that consist of three vortex patches (not necessarily smooth), each growing slowly in time, with the distance between the patches growing as \(\sqrt{t}\). We will obtain that the trajectories of the centers of mass of these patches behave approximately like point vortices, so we will first discuss what is known about point vortex systems. A point vortex system consists of n point vortices, with masses \(\Omega _i\) and positions \({\zeta _i}\), whose motion is described by the ODE system

$$\begin{aligned} \frac{d}{dt}{\zeta _i}=\sum _{j\ne i}\Omega _j\frac{({\zeta _i}-{\zeta _j})^\perp }{|{\zeta _i}-{\zeta _j}|^2}. \end{aligned}$$

In the present paper, we will always assume \(\Omega _i\ne 0\). Here, as with the vorticity formulation of 2D Euler, we rescaled time to avoid factors of \(2\pi \). This ODE is meant to model a fluid in which vorticity is highly concentrated around a few points. Some information about the behavior of solutions to this ODE can be found in [1]. While there are specific solutions in which vortices collide, for generic initial data, this does not happen [7, 24]. Rigorous justification of the point vortex model is provided in [23]. They show that if one replaces point vortices by signed \(L^\infty \) localized vorticity, the solution to Euler will approximate the solution to the ODE over a fixed time interval. The assumptions on the \(L^\infty \) bound of the solution are then significantly weakened in independent and simultaneous works by Marchioro in [22] and by Serfati in [27], with [27] giving better approximation of point vortices. One can view these as being almost \(L^1\) results. Both the assumptions on the \(L^\infty \) bound on vorticity and the conclusion are further improved by Serfati in [28].

There are several observations about the long-term behavior of solutions to point vortex systems. First, if all masses are positive, then the solution will remain bounded. Second, it is easy to obtain solutions of two or more point vortices where a pair of point vortices with masses \(\Omega \) and \(-\Omega \) go off to infinity with their velocity approaching some nonzero limit. Third, there are point vortex systems that expand and spiral in a self-similar way so that the distance between the point vortices grows as \(\sqrt{t}\), specifically vortex i has trajectory

$$\begin{aligned} \sqrt{t}R_{\kappa \log t}{\eta _i} \end{aligned}$$

for some \(\kappa \) and \({\eta _1},\ldots ,y_n\). An analysis of such self-similarly evolving 3-vortex systems can be found in [1]. Some self-similarly evolving 4 and 5 vortex systems are constructed and analyzed in [25]. Some numerics for self-similarly evolving systems with more vortices may be found in [18].

The main theorem we prove can be stated roughly as follows:

Theorem 1

(Rough version, see Theorem 4 for a more precise statement). Given a self-similarly expanding solution

$$\begin{aligned} \sqrt{t}R_{\kappa \log t}{\eta _i}, \end{aligned}$$

and a small parameter \(\epsilon \), we can replace each of the point vortices with any vortex patch of constant sign and the same total vorticity, subject to certain bounds. Then at later times, the centers of mass of each of the vortex patches will be at

$$\begin{aligned} \gamma (t) R_{\beta (t)}{\xi _i}(t), \end{aligned}$$

where \(\gamma =\sqrt{t}(1+O(\epsilon ))\) and \(|{\xi _i}(t)-{\eta _i}|<\epsilon \), and each of the three patches will have diameter \(O(t^{1/4+\epsilon })\).

The centers of mass of the patches we construct will behave approximately like a self-similarly expanding 3-vortex system, see Figure 1 (for the precise statement, see Theorem 4 below). This is the first construction where the support of the vorticity is known to go off to infinity that does not rely on symmetry.

Fig. 1
figure 1

An example of a self-similarly expanding 3-vortex system and a corresponding solution involving three vortex patches

Full size image

Before understanding the behavior of the three vortex patches, we first need to understand a single vortex patch, so we will now discuss the previously known results regarding vortex patches. Yudovich [29] showed global well-posedness for solutions with \(\omega \in L^1\cap L^\infty \). Given global well-posedness, it is natural to study the long-term behavior of vorticity. It was shown by Kirchhoff that elliptical patches will rotate uniformly [17]. Other rotating solutions with m-fold symmetry bifurcating from the disk, called V-states, were found numerically by Deem–Zabusky [6] and proved to exist by Burbea [3]. For other results about rotating solutions, see [9] and results they cite. Aside from such special solutions, it is known that if the vorticity is the indicator function of a set with \(C^{k,\gamma }\) boundary, then this regularity of the boundary will continue for all time, as shown by Chemin [4], Bertozzi–Constantin [2], and Serfati [26]. For other results concerning regularity and long-term behavior of vortex patches, see [8] and results they cite.

However, very little is known if no additional regularity is assumed. In particular, one can ask what happens if vorticity is initially compactly supported and \(L^\infty \). We will go over some results bounding the expansion of the support, known as vorticity confinement results. It is easy to see that the radius of the support can grow at most linearly, since u is bounded. If vorticity is of a definite sign, then the radius of the support grows more slowly. In fact, it is not known whether it ever goes to infinity. Marchioro [20] showed an upper bound of \(t^{1/3}\) by using conservation of the second moment of vorticity. This was independently improved to \((t\log t)^{1/4}\) by Iftimie–Sideris–Gamblin [16] and to \(t^{1/4}\log \circ \cdots \circ \log t\) by Serfati [27], with the improvement coming largely from using conservation of the center of mass of the vorticity. Compare this with the present work, in which we get the slightly worse bound of \(t^{1/4+\epsilon }\) for each of the vortex patches. There are also several other vorticity confinement results, including getting similar confinement bounds, but depending on the \(L^{q}\) norm rather than the \(L^\infty \) norm of vorticity [19]. Compare this to the result in [27], which requires an \(L^\infty \) bound, but the constant in the confinement bound has very weak dependence on the \(L^\infty \) norm (it is linear in \(\log \circ \cdots \circ \log ||\omega ||_{L^\infty })\), depending mostly on the \(L^1\) norm. There are also various bounds on confinement of positive compactly supported vorticity in other domains. In particular, on the upper half-plane, the x coordinate of the center of mass of vorticity is at least ct and the y coordinate of points in the support is bounded by \(C(t\log t)^{1/3}\) [10], while the x coordinate of points in the support is at least \(-C(t\log t)^{1/2}\) [12]. The latter work also analyzes what possible weak limits a positive vorticity solution can have on a half-plane (under appropriate rescaling). In exterior domains, the radius of the support is bounded by \(Ct^{1/2}\), with further improvements in the exponent when the domain is the exterior of a disk [14, 21]. On \({\mathbb {T}}\times {\mathbb {R}}\), the y coordinate of points in the support is bounded by \(Ct^{1/3}\log ^2 t\) [5]. A survey of various related results can be found in [11]. However, things are very different if you allow mixed-sign vorticity. [16] contains a construction of a compactly supported positive vorticity vortex patch in the first quadrant of the plane, reflected into the other quadrants with changing sign, whose support grows linearly. However, the proof relies heavily on the symmetry, so it is very unstable and can only give this result for a system with total vorticity 0.

Returning to the plane, but now without a definite sign, there is a vorticity confinement result by Iftimie–Lopes Filho–Nussenzveig Lopes that addresses the question of weak limits under appropriate rescaling [13]. This result states that if we define

$$\begin{aligned} {{\tilde{\omega }}}_\alpha (x,t)=t^{2\alpha }\omega (t^\alpha x,t) \end{aligned}$$

for \(\alpha >1/2\), then

in the weak-* sense for measures where \(m=\int \omega dx\) and \(\delta _0\) is the Dirac delta. The authors interpret this as showing confinement of net vorticity to a radius of \(\sqrt{t}\), but this result still allows for strange examples like having both positive and negative vortex patches moving away from the origin in different directions and being at distance \(t^{2/3}\log t\). Our result shows that we cannot take \(\alpha =1/2\) in the statement of [13] and thus, in a sense, net vorticity is moving off to infinity. In fact, the solution given here should, modulo a rotation by a logarithmically growing angle, weakly converge to a sum of three delta masses under the rescaling with \(\alpha =1/2\).

The last previous result we discuss is a paper by Iftimie–Marchioro [15], which looked at a toy model of the construction given here and showed confinement of the vortex patches. The toy model consists of taking a self-similarly expanding point vortex system, replacing only one of the point vortices with a patch, assuming that the trajectory of the other point vortices is fixed, and seeing how the patch evolves. The purpose of looking at the toy model was to sidestep the issue of stability for self-similar point vortex systems and only worry about confinement of vorticity. For some configurations, they bound the radius of the support of the patch as growing no faster than \(t^{(1+\alpha )/3}\) for \(\alpha \) some constant that depends on the configuration, is always positive, and is less than 1/2. This means that the vortex patch size grows slower than the distance between patches.

The improvements of the present paper over [15] are replacing each of the three vortices with vortex patches, allowing a generic self-similarly expanding configuration of 3 vortices, taking an actual solution of 2D Euler, and obtaining that the radius of support grows at most as \(t^{1/4+\epsilon }\) for any given \(\epsilon >0\) instead of \(t^{(1+\alpha )/3}\). The improvement in the exponent from 1/3 to 1/4 comes from actually analyzing the stability and keeping track of the center of mass, in the same way that using conservation of center of mass gives the same improvement for a single vortex patch. To get rid of \(\alpha \), we obtain a better bound on moment growth by noticing that one of the expressions that shows up in the expression for moment derivatives is the approximate derivative of another expression and thus obtaining some cancellation in the most troublesome term (one can also think of this as a renormalization of the moments). Our result is limited to 3 vortex systems due to the stability analysis, and if one found stably growing systems of 4 or more vortices, the confinement result would most likely carry over with little modification. However, one needs sufficiently good stability results; orbital linear stability, which may not be hard to obtain for some systems, is not enough. Our assumptions on the patches, same as in [15], will be that the vorticity \(\omega \) is compactly supported and \(L^{q}\) for \(q>2\). Technically, we assume that \(\omega \in L^\infty \) to make use of the well-posedness theory, but all constants in the proof will only depend on the \(L^{q}\) bound. We need this bound in many places in the proofs in order to bound integrals of \(\frac{1}{|x-a|}\omega (x)\) using Holder’s inequality.

2 Preliminary Lemmas

In order to state the main result, we first need to analyze some properties of expanding systems of three point vortices. Take any three vortex system. It has four conserved quantities, each of which is easy to check.

  1. 1.

    \(X=\sum _i \Omega _i{\zeta _i}\) (this has two components)

  2. 2.

    \(I=\sum _i \Omega _i|{\zeta _i}|^2\)

  3. 3.

    \(E=\sum _{i< j} \Omega _i\Omega _j\log |{\zeta _i}-{\zeta _j}|\).

Now suppose we take a self-similarly expanding solution with three point vortices and nonzero total mass. By moving the origin, we can assume that \(X=0\). Then self-similarity ensures that \(I=Ct\) for some constant C. Then conservation of I and E ensure that

$$\begin{aligned} \Omega _1\Omega _2+\Omega _1\Omega _3+\Omega _2\Omega _3&=0 \end{aligned}$$
(1)
$$\begin{aligned} I&=0 \end{aligned}$$
(2)

We move the rest of the analysis to the following lemma, proved in the appendix:

Lemma 2

Suppose we have a point vortex system \(({\eta _i},\Omega _i)\) satisfying \(X=0\), (1), and (2). such that

$$\begin{aligned} (\sqrt{t}R_{\kappa \log t}{\eta _i}) \end{aligned}$$

is a point vortex solution. Then

  1. 1.

    The point vortices are not collinear and not the vertices of an equilateral triangle.

  2. 2.

    \(\sum \Omega _i\ne 0\).

  3. 3.

    Take the subspace V of vortex locations that satisfy \(X=0\). There exists some neighborhood \(U\subset V\) of \(({\eta _1},{\eta _2},{\eta _3})\) and some two-dimensional surface S through \(({\eta _i})\) so that I and E are coordinates on \(S\cap U\). Furthermore, S and U may be chosen so that for \(\zeta \in U\), we have unique \(\gamma \in {\mathbb {R}}^+\) and \(\beta \in (-\pi ,\pi )\) satisfying \(\xi (\zeta )=\frac{1}{\gamma }R_{-\beta }\zeta \in S\). Furthermore, S and U may be chosen so that \(\beta ,\gamma ,I(\xi (\zeta )),E(\xi (\zeta ))\) give coordinates on U.

Note that for the second part of the lemma statement, it is important that I is evaluated at the point \(\xi (\zeta )\in S\), not at the original point \(\zeta \).

The conditions in the lemma statement are generic for self-similarly expanding 3-vortex configurations; one system satisfying the hypotheses of the lemma is the following example, taken (up to sign reversal) from [24]. Let \(\Omega _1=-2\), \(\Omega _2=-2\), \(\Omega _3=1\) and \({\eta _1}=(-1,0)\), \({\eta _2}=(1,0)\), \({\eta _3}=(1,\sqrt{2})\). Then translate to achieve \(X=0\). This example is shown in Figure 1.

We also have the following result about Taylor expanding the kernel K, which we use multiple times.

Lemma 3

If \(z_1\in {\mathbb {R}}^2\backslash \{0\}\) and \(|z_2-z_1|<|z_1|/2\), then

$$\begin{aligned} K(z_2)=K(z_1)+A_{z_1}(z_2-z_1)+O\left( \frac{|z_2-z_1|^2}{|z_1|^3}\right) \end{aligned}$$

where

$$\begin{aligned} A_{z_1}(z_2-z_1)=-(z_2-z_1)\cdot z_1\frac{z_1^\perp }{|z_1|^4}-(z_2-z_1)\cdot z_1^\perp \frac{z_1}{|z_1|^4} \end{aligned}$$

is linear in \(z_2-z_1\).

Proof

If \(z_1=(1,0)\), then we apply Taylor’s formula. A direct computation gives us the linear term in the lemma statement and Taylor’s theorem gives us an error of

$$\begin{aligned} \frac{|z_2-z_1|^2}{2}|\nabla ^2 K(z)| \end{aligned}$$

for some z on the line segment between \(z_1\) and \(z_2\). Since z is constrained to be on a fixed disk away from the origin and \(\nabla ^2K\) is bounded on that disk, we obtain the lemma statement for \(z_1=(1,0)\).

For any other \(z_1\ne 0\), we have that \(z_1=|z_1|R_{\phi }(1,0)\) for some angle \(\phi \) and \(z_2=|z_1|R_{\phi }{{\tilde{z}}}_2\) for some \(\tilde{z}_2\). Then we apply the lemma statement to \({{\tilde{z}}}_2\) and \(\tilde{z}_1=(1,0)\) and take advantage of the scaling of K to obtain

$$\begin{aligned} K(z_2)&=|z_1|^{-1}R_{\phi }K({{\tilde{z}}}_2)=|z_1|^{-1}R_{\phi }K({{\tilde{z}}}_1)+|z_1|^{-1}R_{\phi }A_{{{\tilde{z}}}_1}({{\tilde{z}}}_2-{{\tilde{z}}}_1)+|z_1|^{-1}R_{\phi }O\left( \frac{|{{\tilde{z}}}_2-{{\tilde{z}}}_1|^2}{|{{\tilde{z}}}_1|^3}\right) \\&=K(z_1)+A_{z_1}( z_2-z_1)+O\left( \frac{|z_2-z_1|^2}{|z_1|^3}\right) . \end{aligned}$$

\(\square \)

3 Result Statement and Stability of Centers of Mass

We can now state the main result precisely.

Theorem 4

Take an arrangement of three points \(({\eta _i},\Omega _i)\) satisfying the conditions of Lemma 2. Let \(\epsilon >0\), be arbitrary and small, \(M>0\), \(\rho >0\) be fixed and sufficiently large. Then there exists some \(T>0\) so that for \(t_0>T\), if we take the solution and replace each point vortex \(\{(\sqrt{t_0}R_{\kappa \log t_0}{\eta _i},\Omega _i)\}\) with an \(L^\infty \) vorticity function \(\omega _i(t_0,\cdot )\) such that:

  1. 1.

    The center of mass of the whole system is still 0.

  2. 2.

    \({{\,\mathrm{supp}\,}}\omega _i(t_0,\cdot )\subseteq B(\sqrt{t_0}R_{\kappa \log t_0}{\eta _i},\rho )\).

  3. 3.

    \(||\omega _i(t_0,\cdot )||_{L^{q}}\le M\).

  4. 4.

    \(\omega _i(t_0,\cdot )\) has definite sign.

  5. 5.

    \(\int \omega _i(t_0,\cdot )=\Omega _i\).

then at each later time t, there exists some \(\{({\xi _i}(t))\}\in S\) with \(|{\xi _i}-{\eta _i}|<\epsilon \), some angle \(\beta (t)\), and some real factor \(\gamma (t)=(1+O(\epsilon ))\sqrt{t}\) such that, letting \({\zeta _i}=\gamma R_{\beta } {\xi _i}\), the solution at time t is \(\sum \omega _i\) with:

  1. 1.

    \({\zeta _i}=\frac{1}{\Omega _i}\int \omega _i(x)x dx\).

  2. 2.

    \({{\,\mathrm{supp}\,}}\omega _i\subseteq B({\zeta _i},\epsilon t^{1/4+\epsilon })\).

  3. 3.

    \(\int \omega _i=\Omega _i\).

  4. 4.

    \(\omega _i\) has definite sign.

There are a couple of comments about the statement. First, note that it is possible to have \(\lim _{t\rightarrow \infty }{\xi _i}(t)\ne {\eta _i}\). Since the \(L^\infty \) vorticity follows a rescaled copy of \({\xi _i}\), this is saying that a small amount of drift in the configuration is possible. Second, because of how we defined \({\zeta _i}\) and S, we have that \(\beta \) and \(\gamma \) are uniquely defined by the arrangement. Finally, condition 1 in the theorem statement is simply for convenience–since \(\sum \Omega _i\ne 0\), we could restate the theorem without this condition, but adding a translation to move the center of mass to 0.

Proof

\(k>2\) will denote an even integer and \(\delta \) will denote some sufficiently small constant that can depend on k and \(\epsilon \) and R will denote some large constant that depends on \(\delta \). At the end of the proof, we will choose k, then \(\delta \), then R depending on the initial configuration of point vortices, as well as \(q, \epsilon , \rho , M\). We will then choose T large enough depending on \(q, \epsilon , \rho , M,k,\delta \). All constants C in the statement and proof (including implicit constants hidden by O notation) can depend on the initial configuration of point vortices, as well as \(q, \epsilon , \rho , M,k\), but not on R or \(\delta \). The letters C may be used for different constants on different lines. We will use \({{\hat{O}}}\) if we’re allowing the implicit constant to depend on the initial configuration of point vortices and on nothing else. At any time t, let

$$\begin{aligned} {\zeta _i}(t)&=\frac{1}{\Omega _i}\int x\omega _i(x)dx\\ I_{k,i}(t)&=\int |x- {\zeta _i}|^k\omega _i(x)dx. \end{aligned}$$

We will have the following bootstrap assumptions:

  1. 1.

    \({\zeta _i}=\gamma R_{\beta } {\xi _i}\) with \(|{\xi _i}-{\eta _i}|<\epsilon ^2\) for some angle \(\beta (t)\), and some factor \(\gamma (t)\) satisfying \(|\frac{\gamma }{\sqrt{t}}-1|<\epsilon \)

  2. 2.

    \(I_{2,i}< t^{\epsilon /2}\)

  3. 3.

    \(I_{k,i}< t^{k(1+\epsilon )/4}\)

  4. 4.

    \(\omega _1, \omega _2, \omega _3\) are three \(L^{q}\) compactly supported functions of definite sign, \(\Vert \omega _i\Vert _{L^q}\le M\)

  5. 5.

    \({{\,\mathrm{supp}\,}}\omega _i\subseteq B({\zeta _i},\epsilon t^{1/4+\epsilon })\), that is \(|p-{\zeta _i}|<\epsilon t^{1/4+\epsilon }\) for any \(p\in {{\,\mathrm{supp}\,}}\omega _i\).

These assumptions hold at initial time \(t_0\) as long as T is big enough. If they always hold, we are done, so we can assume that the first time when one of them fails is \(T_*<\infty \).

First, we want to understand the ODE satisfied by the triple \(({\zeta _1},{\zeta _2},{\zeta _3})\) of centers of mass of the patches in order to verify bootstrap assumption 1. First note that from the conservation of the center of mass of the vorticity, we get that \(\sum \Omega _i{\zeta _i}=0\), so the center of mass of \({\xi _i}\) stays at 0. We will use the notation \(I_{\zeta }=\sum \Omega _i|{\zeta _i}|^2\) and \(I_{\xi }=\sum \Omega _i|{\xi _i}|^2\). We similarly define \(I_{\eta },E_{\zeta },E_{\xi },E_{\eta }\). For this calculation, we note that from the bootstrap assumptions, for \(x\in {{\,\mathrm{supp}\,}}\omega _i\), \(y\in {{\,\mathrm{supp}\,}}\omega _j\) with \(j\ne i\), we use Lemma 3 (essentially, Taylor expand) to obtain

$$\begin{aligned} \frac{(x-y)^\perp }{|x-y|^2}=\frac{({\zeta _i}-{\zeta _j})^\perp }{|{\zeta _i}-{\zeta _j}|^2}+\mathcal A_1(x-{\zeta _i})+\mathcal A_2(y-{\zeta _j})+O\left( \frac{|x-{\zeta _i}|^2+|y-{\zeta _j}|^2}{|{\zeta _i}-{\zeta _j}|^3}\right) \end{aligned}$$

where \({\mathcal {A}}_1,{\mathcal {A}}_2\) are some linear functions dependent on t. Then, in order to track the centers of mass, we use

$$\begin{aligned} {\zeta _i}=\frac{1}{\Omega _i}\int x\omega _i(x)dx. \end{aligned}$$

Since the vorticity of patch i is advected by the velocity arising from patch i and the velocity arising from other patches, we obtain

$$\begin{aligned} \frac{d}{dt}{\zeta _i}&=\frac{1}{\Omega _i}\left[ \iint \omega _i(x)\omega _i(y)\frac{(x-y)^\perp }{|x-y|^2} dxdy+\sum _{j\ne i} \iint \omega _i(x)\omega _j(y)\frac{(x-y)^\perp }{|x-y|^2} dxdy\right] \nonumber \\&=\frac{1}{\Omega _i}\Bigg [\frac{1}{2}\iint \omega _i(x)\omega _i(y)\left( \frac{(x-y)^\perp }{|x-y|^2}+\frac{(y-x)^\perp }{|y-x|^2}\right) dxdy\nonumber \\&\qquad +\sum _{j\ne i}\Omega _i\Omega _j\frac{({\zeta _i}-{\zeta _j})^\perp }{|{\zeta _i}-{\zeta _j}|^2}+O\left( \sum _{j=1}^3\frac{ I_{2,j}}{t^{3/2}}\right) \Bigg ]\nonumber \\&=\sum _{j\ne i}\Omega _j\frac{({\zeta _i}-{\zeta _j})^\perp }{|{\zeta _i}-{\zeta _j}|^2}+O(t^{\epsilon /2-3/2}). \end{aligned}$$
(3)

This means that the system \(({\zeta _i})\) evolves as point vortices, up to some error. We now look at the nearly-conserved quantities \(I_\zeta ,E_\zeta \). By bootstrap assumption 1, we have that \(|{\zeta _i}|=O(\sqrt{t})\). We use (3) and to calculate

$$\begin{aligned} \left| \frac{dI_{\zeta }}{dt}\right|&=\left| \frac{d}{dt}\sum _i \Omega _i|{\zeta _i}|^2\right| =\left| \sum _i 2\Omega _i{\zeta _i}\cdot \left( \sum _{j\ne i}\Omega _j\frac{({\zeta _i}-{\zeta _j})^\perp }{|{\zeta _i}-{\zeta _j}|^2}+O(t^{\epsilon /2-3/2})\right) \right| \\&=\left| \sum _{1\le i\le 3}\sum _{j\ne i} 2\Omega _i\Omega _j {\zeta _i}\cdot \left( \frac{({\zeta _i}-{\zeta _j})^\perp }{|{\zeta _i}-{\zeta _j}|^2}\right) \right| +O(t^{\epsilon /2-1})\\&=\left| \sum _{1\le i<j\le 3} 2\Omega _i\Omega _j \left( \frac{({\zeta _i}-{\zeta _j})\cdot ({\zeta _i}-{\zeta _j})^\perp }{|{\zeta _i}-{\zeta _j}|^2}\right) \right| +O(t^{\epsilon /2-1})\\&=O(t^{\epsilon /2-1}) \end{aligned}$$

where the cancellation is the same cancellation that gave us the conserved quantity I in the first place. Furthermore, at time \(t_0\), we have \(I_{\zeta }=O(t_0^{1/2})\). Integrating in t, we get \(I_{\zeta }=O(t_0^{1/2}+t^{\epsilon /2})\). This then gives us that

$$\begin{aligned} I_{\xi }=O(I_{\zeta }/t)=O\left( \frac{t_0^{1/2}}{t}+t^{\epsilon /2-1}\right) =O(t^{-1/2})=O(T^{-1/2}). \end{aligned}$$
(4)

Finally, from (1), we have that \(E_{\zeta }=E_{\xi }\), so

$$\begin{aligned} \left| \frac{dE_{\xi }}{dt}\right|&=\left| \frac{dE_{\zeta }}{dt}\right| =\left| \frac{d}{dt}\sum _{i<j} \Omega _i\Omega _j\log |{\zeta _i}-{\zeta _j}|\right| \\&=\sum _{i< j} \Omega _i\Omega _j\frac{{\zeta _i}-{\zeta _j}}{|{\zeta _i}-{\zeta _j}|^2}\cdot \sum _{\ell \ne i,j}\Omega _\ell \left( \frac{({\zeta _i}-{\zeta _\ell })^\perp }{|{\zeta _i}-{\zeta _\ell }|^2}-\frac{({\zeta _j}-{\zeta _\ell })^\perp }{|{\zeta _j}-{\zeta _\ell }|^2}+O(t^{\epsilon /2-3/2})\right) \\&=\Omega _1\Omega _2\Omega _3\bigg (\frac{({\zeta _1}-{\zeta _2})\cdot ({\zeta _1}-{\zeta _3})^\perp +({\zeta _1}-{\zeta _3})\cdot ({\zeta _1}-{\zeta _2})^\perp }{|{\zeta _1}-{\zeta _2}|^2|{\zeta _1}-{\zeta _3}|^2}\\&\qquad +\frac{({\zeta _2}-{\zeta _1})\cdot ({\zeta _2}-{\zeta _3})^\perp +({\zeta _2}-{\zeta _3})\cdot ({\zeta _2}-{\zeta _1})^\perp }{|{\zeta _2}-{\zeta _1}|^2|{\zeta _2}-{\zeta _3}|^2}\\&\qquad +\frac{({\zeta _3}-{\zeta _1})\cdot ({\zeta _3}-{\zeta _2})^\perp +({\zeta _3}-{\zeta _2})\cdot ({\zeta _3}-{\zeta _1})^\perp }{|{\zeta _1}-{\zeta _2}|^2|{\zeta _1}-{\zeta _3}|^2}\bigg )+O(t^{\epsilon /2-2})\\&=O(t^{\epsilon /2-2}) \end{aligned}$$

where once again the non-error terms canceled precisely. Integrating in t, and recalling that \(t_0>T\), we get that

$$\begin{aligned} |E_{\xi }(t)-E_{\xi }(t_0)|= \int _{t_0}^t \frac{dE_{\xi }}{dt}(s) ds=O(t_0^{\epsilon /2-1})=O(T^{\epsilon /2-1}). \end{aligned}$$
(5)

From (1), (2), and the initial conditions, we have \(|E_{\xi }(t_0)-E_{\eta }|=O(t_0^{-1/2})\) and \(I_{\eta }=0\), so by choosing T sufficiently large, we can guarantee that both \(I_{\xi }-I_{\eta }\le \epsilon ^3\) and \(E_{\xi }-E_{\eta }\le \epsilon ^3\). Since E and I are coordinates locally, we get that for \(\epsilon \) small enough, \(|{\xi _i}-{\eta _i}|<\epsilon ^2\), so that part of bootstrap assumption 1 is maintained.

Now we use (3) and the fact that \({\xi _i}=\gamma R_\beta {\xi _i}\) to obtain that

$$\begin{aligned} \frac{d}{dt}|{\zeta _1}|^2&=2{\zeta _1}\cdot \sum _{j\ne 1}\Omega _j\frac{({\zeta _1}-{\zeta _j})^\perp }{|{\zeta _1}-{\zeta _j}|^2}+O(t^{\epsilon /2-1})=2{\xi _1}\cdot \sum _{j\ne 1}\Omega _j\frac{({\xi _1}-{\xi _j})^\perp }{|{\xi _1}-{\xi _j}|^2}+O(t^{\epsilon /2-1})\\&=2{\eta _1}\cdot \sum _{j\ne 1}\Omega _j\frac{({\eta _1}-{\eta _j})^\perp }{|{\eta _1}-{\eta _j}|^2}+{{\hat{O}}}(\epsilon ^2)+O(t^{\epsilon /2-1}). \end{aligned}$$

The principal term here can also come from the self-similarly expanding solution \(\sqrt{t}R_{\kappa \log t}y\) at time \(t=1\), so

$$\begin{aligned} \frac{d}{dt}|{\zeta _1}|^2= & {} \frac{d}{dt}\left| \sqrt{t}R_{\kappa \log t}{\eta _1}\right| ^2+{{\hat{O}}}(\epsilon ^2)+O(t^{\epsilon /2-1})=\frac{d}{dt}(t|{\eta _1}|^2)+{{\hat{O}}}(\epsilon ^2)+O(t^{\epsilon /2-1})\\&=|{\eta _1}|^2+{{\hat{O}}}(\epsilon ^2)+O(t^{\epsilon /2-1}). \end{aligned}$$

From this, we get that

$$\begin{aligned} \left| \gamma (t)^2-t\right| =\left| \frac{|{\zeta _1}|^2}{|{\xi _1}|^2}-t\right| =\left| \frac{|{\zeta _1}|^2}{|{\eta _1}|^2}-t-{{\hat{O}}}(t\epsilon ^2)\right| ={{\hat{O}}}(\epsilon ^2t)+O(t^{\epsilon /2}). \end{aligned}$$

From this, and assuming that \(\epsilon \) is sufficiently small while T is sufficiently large, we get that

$$\begin{aligned} \left| \frac{\gamma }{\sqrt{t}}-1\right| <\epsilon \end{aligned}$$

which is the last remaining part of bootstrap assumption 1. Bootstrap assumption 4 is simply a consequence of the vorticity being transported by an incompressible flow (generated by divergence-free vector field u). For the other bootstrap assumptions, there are two cases: \(T_*<t_0+t_0^{9/10}\) and \(T_*\ge t_0+t_0^{9/10}\). We will handle the latter case first, since the former case uses weaker versions of estimates that we’ll need to derive along the way. \(\square \)

4 Long Time Behavior

In this section, we assume that \(T_*\ge t_0+t_0^{9/10}\).

We first prove that bootstrap assumptions 2 and 3 are maintained by bounding \(\frac{d}{dt}I_{k,i}\). We will mostly treat them together, as many of the calculations can be done for any k, and we will simply plug in 2 for k when we need to. For definiteness, we will take \(i=1\) below, and we assume that \(\omega _1\ge 0\). The proof for \(\omega _2\) and \(\omega _3\) and for negative vorticity is identical. We want to bound the growth of \(I_{k,1}\). Let \(v_j\) be the velocity field generated by \(\omega _j\). Then for \(x\in {{\,\mathrm{supp}\,}}(\omega _1)\), we apply Lemma 3 (that is, we Taylor expand the kernel) to get that for some linear function \(A_{{\zeta _1}-{\zeta _2}}\),

$$\begin{aligned} v_2(x)&=\int K(x-y) \omega _2(y)dy\nonumber \\&=\frac{({\zeta _1}-{\zeta _2})^\perp }{|{\zeta _1}-{\zeta _2}|^2}\int \omega _2(y)dy+\int -A_{{\zeta _1}-{\zeta _2}}(y-{\zeta _2}) \omega _2(y)dy\nonumber \\&\qquad +\left( -(x-{\zeta _1})\cdot ({\zeta _1}-{\zeta _2})\frac{({\zeta _1}-{\zeta _2})^\perp }{|{\zeta _1}-{\zeta _2}|^4}-(x-{\zeta _1})\cdot ({\zeta _1}-{\zeta _2})^\perp \frac{{\zeta _1}-{\zeta _2}}{|{\zeta _1}-{\zeta _2}|^4}\right) \int \omega _2(y)dy\nonumber \\&\qquad +O\left( \frac{|x-{\zeta _1}|^2}{|{\zeta _1}-{\zeta _2}|^3}\int \omega _2(y)dy\right) +O\left( \int \frac{|y-{\zeta _1}|^2}{|{\zeta _1}-{\zeta _2}|^3}\omega _2(y)dy\right) \nonumber \\&=\Omega _2\left( \frac{({\zeta _1}-{\zeta _2})^\perp }{|{\zeta _1}-{\zeta _2}|^2}-(x-{\zeta _1})\cdot ({\zeta _1}-{\zeta _2})\frac{({\zeta _1}-{\zeta _2})^\perp }{|{\zeta _1}-{\zeta _2}|^4}-(x-{\zeta _1})\cdot ({\zeta _1}-{\zeta _2})^\perp \frac{{\zeta _1}-{\zeta _2}}{|{\zeta _1}-{\zeta _2}|^4}\right) \nonumber \\&\qquad +O\left( \frac{|x-{\zeta _1}|^2}{|{\zeta _1}-{\zeta _2}|^3}\right) +O\left( \frac{I_{2,2}}{|{\zeta _1}-{\zeta _2}|^3}\right) \nonumber \\&=\Omega _2\left( \frac{({\zeta _1}-{\zeta _2})^\perp }{|{\zeta _1}-{\zeta _2}|^2}-(x-{\zeta _1})\cdot ({\zeta _1}-{\zeta _2})\frac{({\zeta _1}-{\zeta _2})^\perp }{|{\zeta _1}-{\zeta _2}|^4}-(x-{\zeta _1})\cdot ({\zeta _1}-{\zeta _2})^\perp \frac{{\zeta _1}-{\zeta _2}}{|{\zeta _1}-{\zeta _2}|^4}\right) \nonumber \\&\qquad +O\left( t^{-3/2}|x-{\zeta _1}|^2\right) +O\left( t^{\epsilon /2-3/2}\right) . \end{aligned}$$
(6)

where we used that \({\zeta _2}\) is the center of mass of \(\omega _2\) to eliminate one of the terms and then used the bootstrap assumptions to bound the error terms.

We have a similar expression for \(v_3(x)\). One consequence is that, plugging in \(x={\zeta _1}\) and combining with (3), we obtain

$$\begin{aligned} \frac{d{\zeta _1}}{dt}=v_2({\zeta _1})+v_3({\zeta _1})+O(t^{\epsilon /2-3/2}). \end{aligned}$$
(7)

From (7), we then have (using Holder’s inequality for the last step)

$$\begin{aligned} \frac{d}{dt}I_{k,1}&=\frac{d}{dt}\int |x-{\zeta _1}|^k\omega _1(x)dx\nonumber \\&=\iint k|x-{\zeta _1}|^{k-2}\frac{(x-{\zeta _1})\cdot (x-y)^{\perp }}{|x-y|^2}\omega _1(x)\omega _1(y)dxdy\nonumber \\&\qquad +\int k|x-{\zeta _1}|^{k-2}(x-{\zeta _1})\cdot (v_2(x)+v_3(x))\omega _1(x)dx\nonumber .\\&\qquad +\int -k|x-{\zeta _1}|^{k-2}(x-{\zeta _1})\cdot (v_2({\zeta _1})+v_3({\zeta _1})+O(t^{\epsilon /2-3/2}))\omega _1(x)dx\nonumber \\&=\iint k|x-{\zeta _1}|^{k-2}\frac{(x-{\zeta _1})\cdot (x-y)^{\perp }}{|x-y|^2}\omega _1(x)\omega _1(y)dxdy\nonumber \\&\qquad +\int k|x-{\zeta _1}|^{k-2}(x-{\zeta _1})\cdot (v_2(x)-v_2({\zeta _1})+v_3(x)-v_3({\zeta _1}))\omega _1(x)dx\nonumber \\&\qquad +O\left( t^{\epsilon /2-3/2}\int k|x-{\zeta _1}|^{k-1}\omega _1(x)dx\right) \nonumber \\&=\iint k|x-{\zeta _1}|^{k-2}\frac{(x-{\zeta _1})\cdot (x-y)^{\perp }}{|x-y|^2}\omega _1(x)\omega _1(y)dxdy\nonumber \\&\qquad +\int k|x-{\zeta _1}|^{k-2}(x-{\zeta _1})\cdot (v_2(x)-v_2({\zeta _1})+v_3(x)-v_3({\zeta _1}))\omega _1(x)dx\nonumber \\&\qquad +O\left( t^{\frac{\epsilon -3}{2}}\right) I_{k,1}^{\frac{k-1}{k}}. \end{aligned}$$
(8)

We first deal with the first term in the same way that it is done in [16]. First, we note that in the special case \(k=2\), we symmetrize in x and y, and that term vanishes. For even \(k>2\), we use the fact that \({\zeta _1}\) is the center of mass to subtract 0. Note that because \(|(x-{\zeta _1})\cdot ({\zeta _1}-y)^{\perp }|=|(x-y)\cdot ({\zeta _1}-y)^{\perp }|\le |x-y||{\zeta _1}-y|\), all terms in the expressions below are absolutely integrable, so the rearrangements and splitting are all valid.

$$\begin{aligned} \iint&|x-{\zeta _1}|^{k-2}\frac{(x-{\zeta _1})\cdot (x-y)^{\perp }}{|x-y|^2}\omega _1(x)\omega _1(y)dxdy=\nonumber \\&=\iint \left( |x-{\zeta _1}|^{k-2}\frac{(x-{\zeta _1})\cdot ({\zeta _1}-y)^{\perp }}{|x-y|^2}-|x-{\zeta _1}|^{k-2}\frac{(x-{\zeta _1})\cdot ({\zeta _1}-y)^{\perp }}{|x-{\zeta _1}|^2}\right) \omega _1(x)\omega _1(y)dxdy\nonumber \\&=\iint _{S_1} \left( |x-{\zeta _1}|^{k-2}\frac{(x-{\zeta _1})\cdot ({\zeta _1}-y)^{\perp }}{|x-y|^2}-|x-{\zeta _1}|^{k-2}\frac{(x-{\zeta _1})\cdot ({\zeta _1}-y)^{\perp }}{|x-{\zeta _1}|^2}\right) \omega _1(x)\omega _1(y))dxdy\nonumber \\&\qquad +\iint _{S_2}\left( |x-{\zeta _1}|^{k-2}\frac{(x-{\zeta _1})\cdot ({\zeta _1}-y)^{\perp }}{|x-y|^2}-|x-{\zeta _1}|^{k-2}\frac{(x-{\zeta _1})\cdot ({\zeta _1}-y)^{\perp }}{|x-{\zeta _1}|^2}\right) \omega _1(x)\omega _1(y)dxdy\nonumber \\&\qquad +\iint _{S_3}\left( |x-{\zeta _1}|^{k-2}\frac{(x-{\zeta _1})\cdot ({\zeta _1}-y)^{\perp }}{|x-y|^2}-|x-{\zeta _1}|^{k-2}\frac{(x-{\zeta _1})\cdot ({\zeta _1}-y)^{\perp }}{|x-{\zeta _1}|^2}\right) \omega _1(x)\omega _1(y)dxdy \end{aligned}$$
(9)

where

$$\begin{aligned} S_1&=\{|x-{\zeta _1}|<|y-{\zeta _1}|/2\}\\ S_2&=\{|y-{\zeta _1}|/2\le |x-{\zeta _1}|\le 2|y-{\zeta _1}|\}\\ S_3&=\{|x-{\zeta _1}|>2|y-{\zeta _1}|\} \end{aligned}$$

For the first term in (9), we use

$$\begin{aligned}&\left| \iint _{S_1} \left( |x-{\zeta _1}|^{k-2}\frac{(x-{\zeta _1})\cdot ({\zeta _1}-y)^{\perp }}{|x-y|^2}-|x-{\zeta _1}|^{k-2}\frac{(x-{\zeta _1})\cdot ({\zeta _1}-y)^{\perp }}{|x-{\zeta _1}|^2}\right) \omega _1(x)\omega _1(y)dxdy\right| \nonumber \\&\qquad \le \iint _{S_1} 2|x-{\zeta _1}|^{k-3}|y-{\zeta _1}|\omega _1(x)\omega _1(y)dxdy\nonumber \\&\qquad \le \iint _{S_1} |x-{\zeta _1}|^{k-4}|y-{\zeta _1}|^2\omega _1(x)\omega _1(y)dxdy\nonumber \\&\qquad \le CI_{k,1}^{\frac{k-4}{k}}I_{2,1}\le Ct^{\epsilon /2}I_{k,1}^{\frac{k-4}{k}}. \end{aligned}$$
(10)

For the second term in (9), we symmetrize in x and y and get

$$\begin{aligned}&\left| \iint _{S_2}\left( |x-{\zeta _1}|^{k-2}\frac{(x-{\zeta _1})\cdot ({\zeta _1}-y)^{\perp }}{|x-y|^2}-|x-{\zeta _1}|^{k-2}\frac{(x-{\zeta _1})\cdot ({\zeta _1}-y)^{\perp }}{|x-{\zeta _1}|^2}\right) \omega _1(x)\omega _1(y)\right| \nonumber \\&\qquad \le \left| \iint _{S_2}\frac{1}{2}(|x-{\zeta _1}|^{k-2}-|y-{\zeta _1}|^{k-2})\frac{(x-{\zeta _1})\cdot ({\zeta _1}-y)^{\perp }}{|x-y|^2}\omega _1(x)\omega _1(y)dxdy\right| \nonumber \\&\qquad \qquad +\left| \iint _{S_2}|x-{\zeta _1}|^{k-4}(x-{\zeta _1})\cdot ({\zeta _1}-y)^{\perp }\omega _1(x)\omega _1(y)dxdy\right| \nonumber \\&\qquad \le \iint _{S_2}\Bigg |(k/4-1/2)\big (|x-{\zeta _1}|^{k-4}+|y-{\zeta _1}|^{k-4}\big )\big (|x-{\zeta _1}|^2-|y-{\zeta _1}|^2\big )\nonumber \\&\qquad \qquad \times \frac{(x-{\zeta _1})\cdot (x-y)^{\perp }}{|x-y|^2}\Bigg |\omega _1(x)\omega _1(y)dxdy\nonumber \\&\qquad \qquad +2\left| \iint _{S_2}|x-{\zeta _1}|^{k-4}|y-{\zeta _1}|^2\omega _1(x)\omega _1(y)dxdy\right| \nonumber \\&\qquad \le C\left| \iint _{S_2}|x-{\zeta _1}|^{k-4}|y-{\zeta _1}|^2\omega _1(x)\omega _1(y)dxdy\right| \nonumber \\&\qquad \qquad +2\left| \iint _{S_2}|x-{\zeta _1}|^{k-4}|y-{\zeta _1}|^2\omega _1(x)\omega _1(y)dxdy\right| \nonumber \\&\qquad \le CI_{k,1}^{\frac{k-4}{k}}I_{2,1}\le Ct^{\epsilon /2}I_{k,1}^{\frac{k-4}{k}}. \end{aligned}$$
(11)

For the third term in (9), we take advantage of the term we subtracted off to get

$$\begin{aligned}&\left| \iint _{S_3} \left( |x-{\zeta _1}|^{k-2}\frac{(x-{\zeta _1})\cdot ({\zeta _1}-y)^{\perp }}{|x-y|^2}-|x-{\zeta _1}|^{k-2}\frac{(x-{\zeta _1})\cdot ({\zeta _1}-y)^{\perp }}{|x-{\zeta _1}|^2}\right) \omega _1(x)\omega _1(y)dxdy\right| \nonumber \\&\qquad \le C\iint _{S_3} |x-{\zeta _1}|^{k-4}|y-{\zeta _1}|^2\omega _1(x)\omega _1(y)dxdy\nonumber \\&\qquad \le CI_{k,1}^{\frac{k-4}{k}}I_{2,1}\le Ct^{\epsilon /2}I_{k,1}^{\frac{k-4}{k}}. \end{aligned}$$
(12)

Plugging (10), (11), (12) into (9) when \(k>2\) and remembering that the term goes away when \(k=2\), we get that for \(k\ge 2\) even

$$\begin{aligned} \left| \iint |x-{\zeta _1}|^{k-2}\frac{(x-{\zeta _1})\cdot (x-y)^{\perp }}{|x-y|^2}\omega _1(x)\omega _1(y)dxdy\right| \le C(k-2)t^{\epsilon /2}I_{k,1}^{\frac{k-4}{k}}. \end{aligned}$$
(13)

To deal with the second term in (8), we plug (6) into it to get

$$\begin{aligned} \int&|x-{\zeta _1}|^{k-2}(x-{\zeta _1})\cdot (v_2(x)-v_2({\zeta _1}))\omega _1(x)dx=\nonumber \\&=\int -2\Omega _2|x-{\zeta _1}|^{k-2}(x-{\zeta _1})\cdot ({\zeta _1}-{\zeta _2})^\perp \frac{(x-{\zeta _1})\cdot ({\zeta _1}-{\zeta _2})}{|{\zeta _1}-{\zeta _2}|^4}\omega _1(x)\\&\qquad +O\left( t^{-3/2}|x-{\zeta _1}|^{k+1}\right) \omega _1(x)+O\left( t^{\epsilon /2-3/2}|x-{\zeta _1}|^{k-1}\right) \omega _1(x)dx\\&=\int -\frac{\Omega _2\sin (2\theta (x))|x-{\zeta _1}|^{k}}{|{\zeta _1}-{\zeta _2}|^2}\omega _1(x)+\left( t^{-3/2}|x-{\zeta _1}|^{k+1}\right) \omega _1(x)\\&\qquad +O\left( t^{\epsilon /2-3/2}|x-{\zeta _1}|^{k-1}\right) \omega _1(x)dx \end{aligned}$$

where \(\theta (x)\) is the angle between \(x-{\zeta _1}\) and \({\zeta _1}-{\zeta _2}\). Now, using the bootstrap assumptions on \(\sup |x-{\zeta _1}|\), \(I_{2,2}\), and \(|{\zeta _1}-{\zeta _2}|\), as well as using Holder’s inequality on the last term, we get

$$\begin{aligned} \int&|x-{\zeta _1}|^{k-2}(x-{\zeta _1})\cdot (v_2(x)-v_2({\zeta _1}))\omega _1(x)dx=\nonumber \\&=\int -\frac{\Omega _2\sin (2\theta (x))|x-{\zeta _1}|^{k}}{|{\zeta _1}-{\zeta _2}|^2}\omega _1(x)dx+O\left( t^{\epsilon -5/4}\right) I_{k,1}+O\left( t^{\frac{\epsilon -3}{2}}\right) I_{k,1}^{\frac{k-1}{k}}. \end{aligned}$$
(14)

If we were to use crude bounds for the first term of (14), bounding the numerator by \(I_{k,1}\), we would achieve vorticity confinement that is worse by some factor of \(t^\alpha \), with \(\alpha \) depending on the configuration of point vortices \((\Omega _i,{\eta _i})\). In fact, for some configurations, our confinement result would be worse than \(t^{1/2}\), so it wouldn’t be enough to prevent the patches from interacting strongly, causing the whole proof to break down. For this reason, we want a better bound on this term. There is little hope of getting one for each time, but we note that we want to bound the expression in (14) because it appears in the derivative of \(I_{k,1}\), so it is enough to get a better bound on its time average (as long as the time interval we are averaging over isn’t too long). More precisely, define \(H:{\mathbb {R}}^2\times {\mathbb {R}}^2\rightarrow {\mathbb {R}}\) by

$$\begin{aligned} H(\lambda ,\mu ):=-\cos \big (2(\arg \lambda -\arg \mu )\big )|\lambda |^{k+2} \end{aligned}$$
(15)

where \(\arg \lambda \) is the angle of \(\lambda \) and we define

$$\begin{aligned} f_{k,2}(x):=H(x-{\zeta _1},{\zeta _1}-{\zeta _2})=-\cos (2\theta (x))|x-{\zeta _1}|^{k+2} \end{aligned}$$
(16)

where \(\theta (x)\) is the angle between \(x-{\zeta _1}\) and \({\zeta _1}-{\zeta _2}\).

Note that \(f_{k,2}(x)\) also has implicit dependence on time through its dependence on \({\zeta _1}\) and \({\zeta _2}\). We then use the following estimate, which will be proved in section 4.1:

$$\begin{aligned} \frac{d}{dt}\int f_{k,2}(x)\omega _1(x)dx=\Omega _1\int 2\sin (2\theta )|x-{\zeta _1}|^k\omega _1(x) dx+O(\delta I_{k,1}+R^k\delta ^{-k}).\nonumber \\ \end{aligned}$$
(17)

One way of thinking about this estimate is as a renormalization of the moments \(I_{k,i}\), where we define new quantities of the form

$$\begin{aligned} {{\hat{I}}}_{k,i}=I_{k,i}+C\frac{\int f_{k,2}(x)\omega _1(x)dx}{t} \end{aligned}$$

and bound their time derivatives. This is equivalent to the argument below, though the notation and organization are different.

To use (17), we plug it into into (14) to get

$$\begin{aligned} \int |x-{\zeta _1}|^{k-2}&(x-{\zeta _1})\cdot (v_2(x)-v_2({\zeta _1}))\omega _1(x)dx=\frac{-\Omega _2}{2\Omega _1|{\zeta _1}-{\zeta _2}|^2}\frac{d}{dt}\int f_{k,2}(x)\omega _1(x)dx\nonumber \\&+O\left( t^{\epsilon -5/4}I_{k,1}\right) +O\left( t^{\frac{\epsilon -3}{2}}I_{k,1}^{\frac{k-1}{k}}\right) +O\left( \frac{\delta I_{k,1}+R^k\delta ^{-k}}{t}\right) . \end{aligned}$$
(18)

If we introduce \(f_{k,3}\) as being entirely analogous to \(f_{k,2}\), but with \({\zeta _3}\) replacing \({\zeta _2}\), we can plug (13) and (18) and into (8) to get

$$\begin{aligned} \frac{d}{dt}I_{k,1}&=O\left( k(k-2)t^{\epsilon /2}I_{k,1}^{\frac{k-4}{k}}\right) -\frac{k\Omega _2}{2\Omega _1|{\zeta _1}-{\zeta _2}|^2}\frac{d}{dt}\int f_{k,2}(x)\omega _1(x)dx\nonumber \\&\qquad -\frac{k\Omega _3}{2\Omega _1|{\zeta _1}-{\zeta _3}|^2}\frac{d}{dt}\int f_{k,3}(x)\omega _1(x)dx+O\left( t^{\epsilon -5/4}\right) I_{k,1}\nonumber \\&\qquad +O\left( t^{\frac{\epsilon -3}{2}}\right) I_{k,1}^{\frac{k-1}{k}}+O\left( \frac{\delta I_{k,1}+R^k\delta ^{-k}}{t}\right) . \end{aligned}$$
(19)

We are now ready to confirm the bootstrap assumptions on \(I_{2,1}\) and \(I_{k,1}\). First, we note that

$$\begin{aligned} \left| \int f_{k,2}(x)\omega _1(x)dx\right| \le \left( \sup _{x\in {{\,\mathrm{supp}\,}}\omega _1}|x-{\zeta _1}|\right) ^2\int |x-{\zeta _1}|^{k}\omega _1(x)dx\le \epsilon ^2t^{1/2+2\epsilon }I_{k,1}.\nonumber \\ \end{aligned}$$
(20)

We now set \(k=2\) in order to prove that \(I_{2,1}(T_*)< T_*^{\epsilon /2}\). The term that has a factor of \(k-2\) in (19) goes away, and we get

$$\begin{aligned} \frac{d}{dt}I_{2,1}&=-\frac{2\Omega _2}{2\Omega _1|{\zeta _1}-{\zeta _2}|^2}\left( \frac{d}{dt}\int f_{2,2}(x)\omega _1(x)dx\right) -\frac{2\Omega _3}{2\Omega _1|{\zeta _1}-{\zeta _3}|^2}\left( \frac{d}{dt}\int f_{2,3}(x)\omega _1(x)dx\right) \\&\qquad +O\left( t^{\epsilon -5/4}I_{2,1}\right) +O\left( t^{\frac{\epsilon -3}{2}}I_{2,1}^{\frac{1}{2}}\right) +O\left( \frac{\delta I_{2,1}+R^k\delta ^{-k}}{t}\right) \end{aligned}$$

We integrate in t from \(t_1=T_*-T_*^{2/3}\ge t_0\) to \(T_*\) and plug in \(I_{2,1}(t)\le t^{\epsilon /2}\) to get

$$\begin{aligned} I_{2,1}(T_*)&\le -\int _{t_1}^{T_*}\frac{2\Omega _2}{2\Omega _1|{\zeta _1}-{\zeta _2}|^2}\left( \frac{d}{dt}\int f_{2,2}(x)\omega _1(x)dx\right) +\frac{2\Omega _3}{2\Omega _1|{\zeta _1}-{\zeta _3}|^2}\left( \frac{d}{dt}\int f_{2,3}(x)\omega _1(x)dx\right) dt\\&\qquad +C\int _{t_1}^{T_*}t^{3\epsilon /2-5/4}+t^{3\epsilon /4-3/2}+\delta t^{\epsilon /2-1}+R^k\delta ^{-k}t^{-1}dt+I_{2,1}(t_1). \end{aligned}$$

For the terms in the first line, we now integrate by parts in t. For the integral in the second line, we use that \(\epsilon \) is small and the fact that \(T_*\ge T\gg R,1/\delta \) to throw away all but the largest term at the cost of making the constant worse. We then obtain

$$\begin{aligned} I_{2,1}(T_*)&\le \left[ \frac{-2\Omega _2}{2\Omega _1|{\zeta _1}-{\zeta _2}|^2}\int f_{2,2}(x)\omega _1(x)dx\right] (T_*)+\left[ \frac{2\Omega _2}{2\Omega _1|{\zeta _1}-{\zeta _2}|^2}\int f_{2,2}(x)\omega _1(x)dx\right] (t_1)\nonumber \\&\qquad +\left[ \frac{-2\Omega _3}{2\Omega _1|{\zeta _1}-{\zeta _3}|^2}\int f_{2,3}(x)\omega _1(x)dx\right] (T_*)+\left[ \frac{2\Omega _3}{2\Omega _1|{\zeta _1}-{\zeta _3}|^2}\int f_{2,3}(x)\omega _1(x)dx\right] (t_1)\nonumber \\&\qquad +\int _{t_1}^{T_*}\left( \frac{d}{dt}\frac{2\Omega _2}{2\Omega _1|{\zeta _1}-{\zeta _2}|^2}\right) \int f_{2,2}(x)\omega _1(x)dxdt\nonumber \\&\qquad +\int _{t_1}^{T_*}\left( \frac{d}{dt}\frac{2\Omega _3}{2\Omega _1|{\zeta _1}-{\zeta _3}|^2}\right) \int f_{2,3}(x)\omega _1(x)dxdt\nonumber \\&\qquad +C\int _{t_1}^{T_*}\delta t^{\epsilon /2-1}dt+I_{2,1}(t_1). \end{aligned}$$
(21)

We now bound the boundary terms using the bootstrap assumptions on \(|{\zeta _1}-{\zeta _2}|\) as well as (20) (as well as the same bound for \(f_{2,3}\)). For the last line, we use the fact that \(T_*-t_1=T_*^{2/3}\) to bound the integral.

$$\begin{aligned} I_{2,1}(T_*)&\le CT_*^{-1/2+2\epsilon }(I_{2,1}(t_1)+I_{2,1}(T_*))+\int _{t_1}^{T_*}\left( \frac{d}{dt}\frac{2\Omega _2}{2\Omega _1|{\zeta _1}-{\zeta _2}|^2}\right) \int f_{2,2}(x)\omega _1(x)dxdt\nonumber \\&\qquad +\int _{t_1}^{T_*}\left( \frac{d}{dt}\frac{2\Omega _3}{2\Omega _1|{\zeta _1}-{\zeta _3}|^2}\right) \int f_{2,3}(x)\omega _1(x)dxdt+C\delta T_*^{\epsilon /2-1/3}+I_{2,1}(t_1). \end{aligned}$$
(22)

We now use use \(v_2=O(t^{-1/2})\) on \({{\,\mathrm{supp}\,}}\omega _1\) and (7) (as well as all the analogous statements where we permute the indices 1, 2, 3) to get

$$\begin{aligned} \frac{d}{dt}\frac{2\Omega _2}{2\Omega _1|{\zeta _1}-{\zeta _2}|^2}=O\left( t^{-3/2}\frac{d}{dt}|{\zeta _1}-{\zeta _2}|\right) \le O\left( t^{-2}\right) \end{aligned}$$
(23)

as well as all the analogous statements where we permute the indices 1, 2, 3. We plug this along with (20) into (22) to get

$$\begin{aligned} I_{2,1}(T_*)&\le CT_*^{-1/2+2\epsilon }(I_{2,1}(t_1)+I_{2,1}(T_*))+C\int _{t_1}^{T_*}\epsilon ^2t^{2\epsilon -3/2} I_{2,1}(t)dt+C\delta T_*^{\epsilon /2-1/3}+I_{2,1}(t_1)\nonumber \\&\le I_{2,1}(t_1)+CT_*^{-1/2+5\epsilon /2}+CT_*^{5\epsilon /2-3/2+2/3}+C\delta T_*^{\epsilon /2-1/3}\nonumber \\&\le I_{2,1}(t_1)+C\delta T_*^{\epsilon /2-1/3} \end{aligned}$$
(24)

where we used \(T_*>T\gg 1/\delta \). From (24) and \(I_{2,1}(t_1)<t_1^{\epsilon /2}\), and the fact that \(\delta \ll \epsilon \), we get

$$\begin{aligned} I_{2,1}(T_*)\le I_{2,1}(t_1)+C\delta T_*^{\epsilon /2-1/3}<t_1^{\epsilon /2}+\int _{t_1}^{T_*}\left( \frac{d}{dt} t^{\epsilon /2}\right) dt=T_* ^{\epsilon /2}. \end{aligned}$$

The same then applies to \(I_{2,2}\) and \(I_{2,3}\), so bootstrap assumption 2 is maintained.

We now take \(k>2\) even, which we treat similarly to the \(k=2\) case. We integrate (19) in time from \(t_1\) to \(T_*\) and use the bootstrap assumptions on \(I_{2,1}\) and \(I_{k,1}\) to get

$$\begin{aligned} I_{k,1}(T_*)&\le \int _{t_1}^{T_*}Ct^{\epsilon /2}I_{k,1}^{\frac{k-4}{k}}dt-\int _{t_1}^{T_*}\frac{k\Omega _2}{2\Omega _1|{\zeta _1}-{\zeta _2}|^2}\frac{d}{dt}\int f_{k,2}(x)\omega _1(x)dxdt\\&\qquad +\int _{t_1}^{T_*}-\frac{k\Omega _3}{2\Omega _1|{\zeta _1}-{\zeta _3}|^2}\frac{d}{dt}\int f_{k,3}(x)\omega _1(x)dx\\&\qquad +C\int _{t_1}^{T_*}t^{\frac{k(1+\epsilon )}{4}+\epsilon -\frac{5}{4}}+t^{\frac{(k-1)(1+\epsilon )}{4}+\frac{\epsilon -3}{2}}+\delta t^{\frac{k(1+\epsilon )}{4}-1}+\frac{R^k\delta ^{-k}}{t}dt+I_{k,1}(t_1). \end{aligned}$$

We integrate by parts in t and use (20), and the bootstrap assumptions to bound the boundary terms. In the last line, we are using \(T_*\ge T\gg R,1/\delta \) to get rid of all but the biggest term in the integral, at the price of making the constant bigger. We get

$$\begin{aligned} I_{k,1}(T_*)&\le C\int _{t_1}^{T_*}t^{\epsilon /2}I_{k,1}^{\frac{k-4}{k}}dt+CT_*^{-1/2+2\epsilon }(I_{k,1}(t_1)+I_{k,1}(T_*))\\&\qquad +\int _{t_1}^{T_*}\left( \frac{d}{dt}\frac{2\Omega _2}{2\Omega _1|{\zeta _1}-{\zeta _2}|^2}\right) \int f_{2,2}(x)\omega _1(x)dxdt\\&\qquad +\int _{t_1}^{T_*}\left( \frac{d}{dt}\frac{2\Omega _3}{2\Omega _1|{\zeta _1}-{\zeta _3}|^2}\right) \int f_{2,3}(x)\omega _1(x)dxdt+C\int _{t_1}^{T_*}\delta t^{\frac{k(1+\epsilon )}{4}-1}dt+I_{k,1}(t_1). \end{aligned}$$

We now use (23) and (20), as well as using the fact that \(T_*-t_1=T_*^{2/3}\), to get

$$\begin{aligned} I_{k,1}(T_*)&\le C\int _{t_1}^{T_*}t^{\epsilon /2}I_{k,1}^{\frac{k-4}{k}}dt+CT_*^{-1/2+2\epsilon }(I_{k,1}(t_1)+I_{k,1}(T_*))+C\int _{t_1}^{T_*}t^{2\epsilon -3/2}I_{k,1}(t)dt\\&\qquad +\int _{t_1}^{T_*}C\delta t^{\frac{k(1+\epsilon )}{4}-1}dt+I_{k,1}(t_1). \end{aligned}$$

We now use that \(T_*>T\gg 1/\delta \), as well as the bootstrap assumption 3, to get

$$\begin{aligned} I_{k,1}(T_*)\le t_1^{k(1+\epsilon )/4}+CT_*^{-1/2+2\epsilon }T_*^{k(1+\epsilon )/4}+C\int _{t_1}^{T_*} t^{\frac{(k-4)(1+\epsilon )}{4}}+\delta t^{\frac{k(1+\epsilon )}{4}-1}dt \end{aligned}$$

so, as long as \(\delta \) is sufficiently small, and \(T_*>T\) is sufficiently large in terms of \(\delta ,\epsilon ,k\), we get

$$\begin{aligned} I_{k,1}(T_*)< t_1^{k(1+\epsilon )/4}+\frac{1}{2}\left( T_*^{k(1+\epsilon )/4}-t_1^{k(1+\epsilon )/4}\right) +\frac{1}{2}\int _{t_1}^{T_*}\left( \frac{d}{dt} t^{k(1+\epsilon )/4}\right) dt =T_*^{k(1+\epsilon )/4}. \end{aligned}$$

The same then applies to \(I_{k,2}\) and \(I_{k,3}\), so bootstrap assumption 3 is maintained.

We now need to recover bootstrap assumption 5. For this, we let \(t_1=T_*-T_*^{2/3}>t_0\) and take some point \(p(t_1)\) that is in the support of \(\omega _1\), so we have \(|p(t_1)-{\zeta _1}(t_1)|<\epsilon t_1^{1/4+\epsilon }\). We then have p(t) solve \(p'(t)=u(t,p(t))\). We want to show that \(|p(T_*)-{\zeta _1}(T_*)|<\epsilon T_*^{1/4+\epsilon }\), which would show that bootstrap assumption 5 is maintained since the support of \(\omega _1\) is transported by u. Suppose this is false, that is \(|p(T_*)-{\zeta _1}(T_*)|\ge \epsilon T_*^{1/4+\epsilon }\). Then let

$$\begin{aligned} t_2=\sup \left\{ s\in [t_1,T_*]\mid s=t_1\text { or }|p(s)-{\zeta _1}(s)|<\frac{\epsilon }{2} T_*^{1/4+\epsilon }\right\} . \end{aligned}$$

Note for later use that whether or not \(t_2=t_1\), we have

$$\begin{aligned} |p(t_2)-\zeta _1(t_2)|\le \epsilon t_1^{1/4+\epsilon }. \end{aligned}$$
(25)

We will work on the interval \([t_2,T_*]\), where we are guaranteed that

$$\begin{aligned} \frac{\epsilon }{2} T_*^{1/4+\epsilon }\le |p-{\zeta _1}|\le \epsilon T_*^{1/4+\epsilon }. \end{aligned}$$
(26)

We calculate (using (7) to get to the second equality)

$$\begin{aligned} \frac{d}{dt}(p-{\zeta _1})&=v_2(p)+v_3(p)+\frac{(p-{\zeta _1})^\perp }{|p-{\zeta _1}|^2}\Omega _1+\nonumber \\&\qquad \qquad +\int \left( \frac{(p-x)^\perp }{|p-x|^2}-\frac{(p-{\zeta _1})^\perp }{|p-{\zeta _1}|^2}\right) \omega _1(x)dx-\frac{d{\zeta _1}}{dt}\nonumber \\&=v_2(p)-v_2({\zeta _1})+v_3(p)-v_3({\zeta _1})+\frac{(p-{\zeta _1})^\perp }{|p-{\zeta _1}|^2}\Omega _1+\nonumber \\&\qquad \qquad +\int \left( \frac{(p-x)^\perp }{|p-x|^2}-\frac{(p-{\zeta _1})^\perp }{|p-{\zeta _1}|^2}\right) \omega _1(x)dx+O(t^{\epsilon /2-3/2}). \end{aligned}$$
(27)

We now use (6) to get that

$$\begin{aligned} v_2(p)-v_2({\zeta _1})&=-\Omega _2(p-{\zeta _1})\cdot ({\zeta _1}-{\zeta _2})\frac{({\zeta _1}-{\zeta _2})^\perp }{|{\zeta _1}-{\zeta _2}|^4}\nonumber \\&\qquad -\Omega _2(p-{\zeta _1})\cdot ({\zeta _1}-{\zeta _2})^\perp \frac{{\zeta _1}-{\zeta _2}}{|{\zeta _1}-{\zeta _2}|^4}+O(\epsilon ^2 t^{2\epsilon -1}). \end{aligned}$$
(28)

We will also need a coarser form of this estimate, namely

$$\begin{aligned} v_2(p)-v_2({\zeta _1})=O( t^{\epsilon -3/4}). \end{aligned}$$
(29)

We now note that by Lemma 3 (essentially, Taylor expansion), there is some time-dependent matrix \({\mathcal {A}}\) with \(|{\mathcal {A}}|\le C|p-{\zeta _1}|^{-2}\) such that when \(|x-{\zeta _1}|<\frac{|p-{\zeta _1}|}{2}\), we have that

$$\begin{aligned} \frac{(p-x)^\perp }{|p-x|^2}-\frac{(p-{\zeta _1})^\perp }{|p-{\zeta _1}|^2}={\mathcal {A}}(x-{\zeta _1})+O\left( \frac{|x-{\zeta _1}|^2}{|p-{\zeta _1}|^3}\right) . \end{aligned}$$
(30)

Let

$$\begin{aligned} S_7&=\left\{ |x-{\zeta _1}|<\frac{\epsilon }{4} T_*^{1/4+\epsilon }\right\} \\ S_8&=\left\{ \epsilon T_*^{1/4+\epsilon }\ge |x-{\zeta _1}|\ge \frac{\epsilon }{4} T_*^{1/4+\epsilon }\right\} . \end{aligned}$$

Then, since \({\zeta _1}\) is the center of mass, we use (30) and the bound on \(|{\mathcal {A}}|\) in region \(S_7\) to obtain

$$\begin{aligned}&\left| \int \left( \frac{(p-x)^\perp }{|p-x|^2}-\frac{(p-{\zeta _1})^\perp }{|p-{\zeta _1}|^2}\right) \omega _1(x)dx\right| \\&\qquad =\left| \int \frac{(p-x)^\perp }{|p-x|^2}-\frac{(p-{\zeta _1})^\perp }{|p-{\zeta _1}|^2}-{\mathcal {A}}(x-{\zeta _1})\omega _1(x)dx\right| \\&\qquad =\int _{S_7} \left| \frac{(p-x)^\perp }{|p-x|^2}-\frac{(p-{\zeta _1})^\perp }{|p-{\zeta _1}|^2}-{\mathcal {A}}(x-{\zeta _1})\omega _1(x)\right| dx\\&\qquad \qquad +\int _{S_8} \left| \frac{(p-x)^\perp }{|p-x|^2}-\frac{(p-{\zeta _1})^\perp }{|p-{\zeta _1}|^2}-{\mathcal {A}}(x-{\zeta _1})\omega _1(x)\right| dx\\&\qquad \le \frac{C}{|p-{\zeta _1}|^3}\int _{S_7} |x-{\zeta _1}|^2 \omega _1(x)dx+C\int _{S_8}\left( \frac{1}{|p-x|}+\frac{1}{|p-{\zeta _1}|}+\frac{|x-{\zeta _1}|}{|p-{\zeta _1}|^{2}}\right) \omega _1(x)dx \end{aligned}$$

We now use (26), Cauchy-Schwarz, and bootstrap assumption 2 to obtain

$$\begin{aligned}&\left| \int \left( \frac{(p-x)^\perp }{|p-x|^2}-\frac{(p-{\zeta _1})^\perp }{|p-{\zeta _1}|^2}\right) \omega _1(x)dx\right| \nonumber \\&\quad \le \frac{CI_{2,1}}{|p-{\zeta _1}|^3}+C\left( \frac{\epsilon }{2} T_*^{1/4+\epsilon }\right) ^{-1}\int _{S_8}\omega _1(x)dx+C\int _{S_8}\frac{1}{|p-x|}\omega _1(x)dx+C\left( \frac{\epsilon }{2}T_*^{1/4+\epsilon }\right) ^{-2}I_{2,1}^{\frac{1}{2}}\left( \int _{S_8}\omega _1(x)dx\right) ^{\frac{1}{2}}\nonumber \\&\quad \le CT_*^{-3/4-5\epsilon /2}+CT_*^{-1/4-\epsilon }\int _{S_8}\omega _1(x)dx+C\int _{S_8}\frac{1}{|p-x|}\omega _1(x)dx+T_*^{-1/2-7\epsilon /4}\left( \int _{S_8}\omega _1(x)dx\right) ^{\frac{1}{2}} \end{aligned}$$
(31)

Now, from \(I_{k,1}(t)<T_*^{k(1+\epsilon )/4}\), we get that

$$\begin{aligned} \int _{S_8}\omega _1(x)dx\le C\frac{T_*^{k(1+\epsilon )/4}}{T_*^{k(1/4+\epsilon )}}=CT_*^{-3k\epsilon /4}. \end{aligned}$$
(32)

From this, we get by Holder’s inequality

$$\begin{aligned} \int _{S_8}\frac{1}{|p-x|}\omega _1(x)dx\le C\left\| \frac{1}{|p-\cdot |}\right\| _{L^{\bar{q}}(S_8)}\left\| \omega _1\right\| _{L^q(S_8)}^{\sigma }\left\| \omega _1\right\| _{L^{1}(S_8)}^{1-\sigma }\le CT_*^{\epsilon -3k\epsilon (1-\sigma )/4}\nonumber \\ \end{aligned}$$
(33)

for some \(\sigma \in (0,1),{{\bar{q}}}\in (2-\epsilon ,2)\) that depend only on q. We now choose k sufficiently large that \(3k\epsilon (1-\sigma )/4-\epsilon >2\). Then plugging (32) and (33) into (31), we get that

$$\begin{aligned} \int \left( \frac{(p-x)^\perp }{|p-x|^2}-\frac{(p-{\zeta _1})^\perp }{|p-{\zeta _1}|^2}\right) \omega _1(x)dx=O\left( T_*^{-3/4-5\epsilon /2}\right) . \end{aligned}$$
(34)

We now plug (34) and (29) along with the analogous estimate for \(v_3\) into (27) to get

$$\begin{aligned} \frac{d}{dt}(p-{\zeta _1})=\frac{(p-{\zeta _1})^\perp }{|p-{\zeta _1}|^2}\Omega _1+O(T_*^{\epsilon -3/4}). \end{aligned}$$
(35)

Now, for any \({{\tilde{t}}}\) with

$$\begin{aligned}{}[{{\tilde{t}}}, {{\hat{t}}}]:=\left[ {{\tilde{t}}},{{\tilde{t}}}+2\pi |p(\tilde{t})-{\zeta _1}({{\tilde{t}}})|^{2}/\Omega _1\right] \subset [t_2,T_*] \end{aligned}$$

we have that on the time interval \([{{\tilde{t}}}, {{\hat{t}}}]\), the total variation of \(|p-{\zeta _1}|\) is at most

$$\begin{aligned} Var_{[{{\tilde{t}}}, {{\hat{t}}}]}(|p-{\zeta _1}|)=O(T_*^{\epsilon -3/4}|p({{\tilde{t}}})-{\zeta _1}(\tilde{t})|^{2})=O(T_*^{3\epsilon -1/4}). \end{aligned}$$
(36)

We now define the angles \(\varphi =\arg (p-{\zeta _1})\) and \(\theta _2=\arg ({\zeta _1}-{\zeta _2})\). Combining (35) with (36) gives that the angular velocity for \(t\in [{{\tilde{t}}},{{\hat{t}}}]\) is

$$\begin{aligned} \frac{d}{dt}\varphi&=\frac{\Omega _1}{|p-\zeta _1|^2}+O\left( \frac{T_*^{\epsilon -3/4}}{|p-\zeta _1|}\right) =\frac{\Omega _1}{|p({{\tilde{t}}})-\zeta _1({{\tilde{t}}})|^2}+O\left( \frac{T_*^{3\epsilon -1/4}}{\left( T_*^{1/4+\epsilon }\right) ^3}\right) +O(T_*^{-1})\\&=\frac{\Omega _1}{|p({{\tilde{t}}})-\zeta _1(\tilde{t})|^2}+O\left( T_*^{-1}\right) \end{aligned}$$

so

$$\begin{aligned} \varphi (t)=\varphi ({{\tilde{t}}})+\frac{\Omega _1(t-{{\tilde{t}}})}{|p(\tilde{t})-x({{\tilde{t}}})|^2}+O\left( T_*^{2\epsilon -1/2}\right) . \end{aligned}$$
(37)

Also, we have \(\left| \frac{d}{dt}{\zeta _i}\right| =O(T_*^{-1/2})\) so

$$\begin{aligned} \left| \frac{d}{dt}\theta _2\right| =\left| \frac{d}{dt}\frac{{\zeta _1}-{\zeta _2}}{|{\zeta _1}-{\zeta _2}|}\right| =O(T_*^{-1}) \end{aligned}$$

so

$$\begin{aligned} \theta _2(t)=\theta _2({{\tilde{t}}})+O(T_*^{2\epsilon -1/2}) \end{aligned}$$
(38)

We now use (27) and (28) along with the analogous estimate for \({\zeta _3}\) and (34) to compute

$$\begin{aligned} \frac{d}{dt}|p-{\zeta _1}|^2&=-4\Omega _2(p-{\zeta _1})\cdot ({\zeta _1}-{\zeta _2})\frac{(p-{\zeta _1})\cdot ({\zeta _1}-{\zeta _2})^\perp }{|{\zeta _1}-{\zeta _2}|^4}\nonumber \\&\qquad -4\Omega _3(p-{\zeta _1})\cdot ({\zeta _1}-{\zeta _3})\frac{(p-{\zeta _1})\cdot ({\zeta _1}-{\zeta _3})^\perp }{|{\zeta _1}-{\zeta _3}|^4}\nonumber \\&\qquad +O(T_*^{(2\epsilon -1)+(1/4+\epsilon )})+O\left( T_*^{(-3/4-5\epsilon /2)+(1/4+\epsilon )}\right) \nonumber \\&=\frac{-2\Omega _2\sin (-2\theta _2+2\varphi )|p-{\zeta _1}|^2}{|{\zeta _1}-{\zeta _2}|^2}-\frac{2\Omega _3\sin (-2\theta _3+2\varphi )|p-{\zeta _1}|^2}{|{\zeta _1}-{\zeta _3}|^2}+O\left( T_*^{-1/2-3\epsilon /2}\right) . \end{aligned}$$
(39)

By combining (38), (37), (36), and (23), we get that for \(t\in [{{\tilde{t}}},{{\hat{t}}}]\), we have

$$\begin{aligned}&\frac{-2\Omega _2\sin (-2\theta _2+2\varphi )|p-{\zeta _1}|^2}{|{\zeta _1}-{\zeta _2}|^2}=\nonumber \\&\qquad \qquad =-2\Omega _2\sin \left( -2\theta _2(\tilde{t})+2\varphi ({{\tilde{t}}})+\frac{2\Omega _1(t-{{\tilde{t}}})}{|p(\tilde{t})-x({{\tilde{t}}})|^2}\right) \frac{|p({{\tilde{t}}})-{\zeta _1}(\tilde{t})|^2}{|({\zeta _1}({{\tilde{t}}})-{\zeta _2}(\tilde{t}))|^2}+O\left( T_*^{4\epsilon -1}\right) . \end{aligned}$$
(40)

Then substituting (40) into (39) (along with the analogous estimate for \({\zeta _3}\)) and integrating from \({{\tilde{t}}}\) to \({{\hat{t}}}\), we note that the principal term of (40) cancels and we are left with

$$\begin{aligned} |p({{\hat{t}}})-{\zeta _1}({{\hat{t}}})|^2-|p({{\tilde{t}}})-{\zeta _1}(\tilde{t})|^2=O(({{\hat{t}}}-{{\tilde{t}}})T_*^{-3\epsilon /2-1/2}). \end{aligned}$$

We now take a new interval starting at \({{\hat{t}}}\). Tiling most of \([t_2,T_*]\) with such intervals, we get that

$$\begin{aligned} |p(T_*)-{\zeta _1}(T_*)|^2\le |p(t_2)-{\zeta _1}(t_2)|^2+O\left( (T_*-t_2)T_*^{-3\epsilon /2-1/2}\right) +\int _{t_3}^{T_*}\frac{d}{dt}|p-{\zeta _1}|^2dt\nonumber \\ \end{aligned}$$
(41)

where \(t_3\in [t_2,T_*]\) satisfies \(T_*-t_3=O(t^{1/2+2\epsilon })\). Using (35) to bound the last term of (41) and applying (25), we then get

$$\begin{aligned} |p(T_*)-{\zeta _1}(T_*)|^2\le |p(t_2)-{\zeta _2}(t_1)|^2+O(T_*^{1/6-3\epsilon /2})+O(T_*^{4\epsilon })\le (\epsilon t_1^{1/4+\epsilon })^2+O(T_*^{1/6-3\epsilon /2})<\left( \epsilon T_*^{1/4+\epsilon }\right) ^2, \end{aligned}$$

which verifies bootstrap assumption 5. Thus we have shown (modulo the proof of (17) in section 4.1) that we cannot have \(t_0+t_0^{9/10}\le T_*<\infty \).

4.1 Moment renormalization

In this section, we prove estimate (17). This estimate would follow from a short computation directly if all of the mass of the vortex patch were located precisely at \({\zeta _i}\) and all of the kth moment came from parts of the vortex patch at a large distance from \({\zeta _i}\). This cannot hold precisely, but we will obtain an approximation to this by proving that each \(\omega _i\) concentrates, as shown in Figure 2 (see (42) below for the precise statement).

Fig. 2
figure 2

The red mass is the vortex patch given by \(\omega _i\). Note that most of its mass is in a ball centered at some \(\tilde{\zeta _i}\) that is distinct from the center of mass \({\zeta _i}\)

Full size image

To prove the concentration result, we note that for any solution of 2D Euler with compactly supported \(L^\infty \) vorticity, the following is a conserved quantity:

$$\begin{aligned} L=\iint \log |x-y| \omega (x)\omega (y)dxdy. \end{aligned}$$

This quantity corresponds to physical energy of the fluid, and one can directly check that it is conserved with a simple computation. In our solution, for any two points \(x\in {{\,\mathrm{supp}\,}}\omega _i,y\in {{\,\mathrm{supp}\,}}\omega _j\) with \(i\ne j\), we have \(\log |x-y|=(\log t)/2+O(1)\). Thus

$$\begin{aligned} L&= \sum _{i\ne j}\Omega _i\Omega _j(\log t)/2+O(1)+\sum _{i=1}^3\iint \omega _i(x)\omega _i(y)\log |x-y| dxdy\\&=\sum _{i=1}^3\iint \omega _i(x)\omega _i(y)\log |x-y| dxdy+O(1) \end{aligned}$$

where we used (1). Now note that L is conserved, and using a combination of Holder’s and Young’s inequality, we get

$$\begin{aligned} \left| \iint \omega _i(x)\omega _i(y)(\log |x-y|)_- dxdy\right| \le \Vert \omega _i\Vert _{L^q}^2\Vert (\log |\cdot |)_-\Vert _{L^{1/(2-2/q)}}=O(1). \end{aligned}$$

Thus

$$\begin{aligned} \sum _{i=1}^3\iint \omega _i(x)\omega _i(y)(\log |x-y|)_{+} dxdy=O(1). \end{aligned}$$

Since \(\omega _i(x)\omega _i(y)\ge 0\), we then have that there is some constant C so that for \(1\le i\le 3\),

$$\begin{aligned} \iint \omega _i(x)\omega _i(y)(\log |x-y|)_{+} dxdy\le C. \end{aligned}$$

Thus there is some \(R=R(\delta )>0\) such that

$$\begin{aligned} \iint \omega _i(x)\omega _i(y)\mathbb {1}_{|x-y|>R} dxdy<\Omega _i\delta ^4, \end{aligned}$$

from which it follows that for some \(\tilde{\zeta _i}\in {{\,\mathrm{supp}\,}}\omega _i\), we have that the vorticity mass of \(\omega _i\) concentrates around \(\tilde{\zeta _i}\), meaning that

$$\begin{aligned} \int \omega _i(x)\mathbb {1}_{|x-\tilde{\zeta _i}|>R} dx<\delta ^4. \end{aligned}$$
(42)

We now recall that in (15) and (16), we defined \(f_{k,2}(x)\) and \(H:{\mathbb {R}}^2\times {\mathbb {R}}^2\rightarrow {\mathbb {R}}\). In the calculation below, we will use the following bounds on derivatives of H, where \(\partial _1H\) means a derivative with respect to \(\lambda \) and \(\partial _2H\) means a derivative with respect to \(\mu \). The bounds on derivatives come from H being homogeneous of order \(k+2\) with respect to \(\lambda \) and homogeneous of order 0 with respect to \(\mu \), so derivatives of H are also homogeneous of appropriate orders.

$$\begin{aligned} \begin{aligned} |H(\lambda ,\mu )|&\le C|\lambda |^{k+2}\\ |\partial _1 H(\lambda ,\mu )|&\le C|\lambda |^{k+1}\\ |\partial _1^2 H(\lambda ,\mu )|&\le C|\lambda |^{k}\\ |\partial _2 H(\lambda ,\mu )|&\le C\frac{|\lambda |^{k+2}}{|\mu |}. \end{aligned} \end{aligned}$$
(43)

We calculate

$$\begin{aligned} \frac{d}{dt}&\int f_{k,2}(x)\omega _1(x)dx=\frac{d}{dt}\int H(x-{\zeta _1},{\zeta _1}-{\zeta _2})\omega _1(x)dx=\nonumber \\&=\iint \partial _1H(x-{\zeta _1},{\zeta _1}-{\zeta _2})\cdot \frac{(x-y)^\perp }{|x-y|^2}\omega _1(x)\omega _1(y)dxdy\nonumber \\&\qquad +\int \partial _1H(x-{\zeta _1},{\zeta _1}-{\zeta _2})\cdot \left( v_2(x)+v_3(x)-\frac{d}{dt}\zeta _1\right) \omega _1(x)dx\nonumber \\&\qquad +\int \partial _2H(x-{\zeta _1},{\zeta _1}-{\zeta _2})\cdot \left( \frac{d}{dt}\zeta _1-\frac{d}{dt}\zeta _2\right) \omega _1(x)dx. \end{aligned}$$
(44)

In the calculation below, we will use

$$\begin{aligned} S_4&=\{|x-{\zeta _1}|\le \delta |y-{\zeta _1}|\}\\ S_5&=\{\delta |y-{\zeta _1}|<|x-{\zeta _1}|< |y-{\zeta _1}|/\delta \}\\ S_6&=\{|x-{\zeta _1}|\ge |y-{\zeta _1}|/\delta \}. \end{aligned}$$

Using \(v_2,v_3,\frac{d}{dt}\zeta _j=O(t^{-1/2})\), and symmetrizing in x and y for the integral over \(S_5\) we get

$$\begin{aligned} \frac{d}{dt}&\int f_{k,2}(x)\omega _1(x)dx\nonumber \\&=\iint _{S_4\cup S_5\cup S_6} \partial _1H(x-{\zeta _1},{\zeta _1}-{\zeta _2})\cdot \frac{(x-y)^\perp }{|x-y|^2}\omega _1(x)\omega _1(y)dxdy+ O\left( t^{-1/2}\right) \int |x-{\zeta _1}|^{k+1}\omega _1(x)dx\nonumber \\&\qquad + O\left( t^{-1}\right) \int |x-{\zeta _1}|^{k+2}\omega _1(x)dx\nonumber \\&=\iint _{S_4}\partial _1H(x-{\zeta _1},{\zeta _1}-{\zeta _2})\cdot \frac{(x-y)^\perp }{|x-y|^2}\omega _1(x)\omega _1(y)dxdy\nonumber \\&\qquad +\frac{1}{2}\iint _{S_5}(\partial _1H(x-{\zeta _1},{\zeta _1}-{\zeta _2})-\partial _1H(y-{\zeta _1},{\zeta _1}-{\zeta _2}))\cdot \frac{(x-y)^\perp }{|x-y|^2}\omega _1(x)\omega _1(y)dxdy\nonumber \\&\qquad +\iint _{S_6}\left( \partial _1H(x-{\zeta _1},{\zeta _1}-{\zeta _2})\cdot \frac{(x-y)^\perp }{|x-y|^2}-\partial _1H(x-{\zeta _1},{\zeta _1}-{\zeta _2})\cdot \frac{(x-{\zeta _1})^\perp }{|x-{\zeta _1}|^2}\right) \omega _1(x)\omega _1(y)dxdy\nonumber \\&\qquad +\iint _{S_4\cup S_5}- \partial _1H(x-{\zeta _1},{\zeta _1}-{\zeta _2})\cdot \frac{(x-{\zeta _1})^\perp }{|x-{\zeta _1}|^2}\omega _1(x)\omega _1(y)dxdy\nonumber \\&\qquad +\iint _{{\mathbb {R}}^3\times {\mathbb {R}}^3}\partial _1H(x-{\zeta _1},{\zeta _1}-{\zeta _2})\cdot \frac{(x-{\zeta _1})^\perp }{|x-{\zeta _1}|^2}\omega _1(x)\omega _1(y)dxdy\nonumber \\&\qquad + O\left( t^{-1/2}\sup _{x\in {{\,\mathrm{supp}\,}}\omega _1}|x-{\zeta _1}|\int |x-{\zeta _1}|^{k}\omega _1(x)dx\right) \nonumber \\&\qquad + O\left( t^{-1}\sup _{x\in {{\,\mathrm{supp}\,}}\omega _1}|x-{\zeta _1}|^2\int |x-{\zeta _1}|^{k}\omega _1(x)dx\right) . \end{aligned}$$
(45)

We now address each of these terms. First, we want to bound

$$\begin{aligned} \iint _{{\mathbb {R}}^3\times {\mathbb {R}}^3} \partial _1H(x-{\zeta _1},{\zeta _1}-{\zeta _2})\cdot \frac{(x-{\zeta _1})^\perp }{|x-{\zeta _1}|^2}\omega _1(x)\omega _1(y) dxdy. \end{aligned}$$

This is the principal term. The integral with respect to y factors out to give a factor of \(\Omega _1\). The integrand is just \(1/|x-{\zeta _1}|^2\) multiplied by the derivative of H with respect to \(\arg \lambda \). Since \(\theta =\arg (x-{\zeta _1})-\arg ({\zeta _1}-{\zeta _2})\), we have

$$\begin{aligned} \iint _{{\mathbb {R}}^3\times {\mathbb {R}}^3} \partial _1H(x-{\zeta _1},{\zeta _1}-{\zeta _2})\cdot \frac{(x-{\zeta _1})^\perp }{|x-{\zeta _1}|^2}\omega _1(x)\omega _1(y) dxdy=\Omega _1\int 2\sin (2\theta )|x-{\zeta _1}|^k\omega _1(x) dx.\nonumber \\ \end{aligned}$$
(46)

The remaining terms of (45) we need to bound with something small. The first term we bound by

$$\begin{aligned}&\left| \iint _{S_4}\partial _1H(x-{\zeta _1},{\zeta _1}-{\zeta _2})\cdot \frac{(x-y)^\perp }{|x-y|^2}\omega _1(x)\omega _1(y)dxdy\right| \nonumber \\&\qquad \le C\iint _{S_4}\frac{|x-{\zeta _1}|^{k+1}}{|y-{\zeta _1}|}\omega _1(x)\omega _1(y)dxdy\nonumber \\&\qquad \le C \delta \iint _{{\mathbb {R}}^3\times {\mathbb {R}}^3} |x-{\zeta _1}|^{k}\omega _1(x)\omega _1(y)dxdy\nonumber \\&\qquad \le \delta C I_{k,1}. \end{aligned}$$
(47)

We bound the third term in (45) by

$$\begin{aligned}&\left| \iint _{S_6}\partial _1H(x-{\zeta _1},{\zeta _1}-{\zeta _2})\cdot \frac{(x-y)^\perp }{|x-y|^2}-\partial _1H(x-{\zeta _1},{\zeta _1}-{\zeta _2})\cdot \frac{(x-{\zeta _1})^\perp }{|x-{\zeta _1}|^2}\omega _1(x)\omega _1(y)dxd\right| \nonumber \\&\qquad \le C\iint _{S_5}|x-{\zeta _1}|^{k+1}\frac{|y-{\zeta _1}|}{|x-{\zeta _1}|^2}\omega _1(x)\omega _1(y)dxdy\nonumber \\&\qquad \le \delta C\iint _{S_4\cup S_5}|x-{\zeta _1}|^{k}\omega _1(x)\omega _1(y)dxdy\nonumber \\&\qquad \le \delta CI_{k,1}. \end{aligned}$$
(48)

We bound the second term in (45) by using the mean value theorem on \(\partial _1H(\cdot ,{\zeta _1}-{\zeta _2})\) to get that for some \(s\in (0,1)\), we have

$$\begin{aligned}&\left| \iint _{S_5}(\partial _1H(x-{\zeta _1},{\zeta _1}-{\zeta _2})-\partial _1H(y-{\zeta _1},{\zeta _1}-{\zeta _2}))\cdot \frac{(x-y)^\perp }{|x-y|^2}\omega _1(x)\omega _1(y)dxdy\right| \\&=\left| \iint _{S_5}(\partial _1^2H(sx+(1-s)y-{\zeta _1},{\zeta _1}-{\zeta _2})\cdot \frac{(x-y)^\perp }{|x-y|^2}\omega _1(x)\omega _1(y)dxdy\right| . \end{aligned}$$

We now use the bound on \(\partial _1^2H\) in (43) and the convexity of \(z\mapsto |z|^k\) to obtain

$$\begin{aligned}&\left| \iint _{S_5}(\partial _1H(x-{\zeta _1},{\zeta _1}-{\zeta _2})-\partial _1H(y-{\zeta _1},{\zeta _1}-{\zeta _2}))\cdot \frac{(x-y)^\perp }{|x-y|^2}\omega _1(x)\omega _1(y)dxdy\right| \nonumber \\&\qquad \le C\iint _{S_5}(|x-{\zeta _1}|^{k}+|y-{\zeta _1}|^{k})\frac{|x-y|^2}{|x-y|^2}\omega _1(x)\omega _1(y)dxdy\nonumber \\&\qquad \le C\iint _{S_5}|x-{\zeta _1}|^{k}\omega _1(x)\omega _1(y)dxdy\nonumber \\&\qquad \le C\iint _{S_4\cup S_5}|x-{\zeta _1}|^{k}\omega _1(x)\omega _1(y)dxdy. \end{aligned}$$
(49)

We bound the fourth term in (45) by

$$\begin{aligned}&\left| \iint _{S_4\cup S_5}- \partial _1H(x-{\zeta _1},{\zeta _1}-{\zeta _2})\cdot \frac{(x-{\zeta _1})^\perp }{|x-{\zeta _1}|^2}\omega _1(x)\omega _1(y)dxdy\right| \nonumber \\&\qquad \le C\iint _{S_4\cup S_5}|x-{\zeta _1}|^{k}\omega _1(x)\omega _1(y)dxdy. \end{aligned}$$
(50)

To bound this expression, we use the concentration result (42) and split into two cases. First, if \(|\tilde{\zeta _1}-{\zeta _1}|<2R\), then

$$\begin{aligned} \iint _{S_4\cup S_5}&|x-{\zeta _1}|^k\omega _1(x)\omega _1(y)dxdy\nonumber \\&\le \iint _{\{|y-{\zeta _1}|\ge 3R\}}|x-{\zeta _1}|^k\omega _1(x)\omega _1(y)dxdy+\iint _{\{|x-{\zeta _1}|\le 3R/\delta \}}|x-{\zeta _1}|^k\omega _1(x)\omega _1(y)dxdy\nonumber \\&\le \delta ^4 I_{k,1}+\Omega _1^2(3R/\delta )^k. \end{aligned}$$
(51)

The second case is \(|\tilde{\zeta _1}-{\zeta _1}|\ge 2R\). In this case, for any \(x\in B(\tilde{\zeta _1},R)\), we have

$$\begin{aligned} (x-{\zeta _1})\cdot \frac{\tilde{\zeta _1}-{\zeta _1}}{|\tilde{\zeta _1}-{\zeta _1}|}\ge \frac{|\tilde{\zeta _1}-{\zeta _1}|}{2}. \end{aligned}$$

Also, since \(\delta \) is sufficiently small, the total vorticity contained in \(B(\tilde{\zeta _1},R)\) is at least \(\Omega _1/2\). Thus

$$\begin{aligned} \left| \int _{B(\tilde{\zeta _1},R)}(x-{\zeta _1})\omega _1(x)dx\right| \ge \frac{\Omega _1}{2}\left( \frac{|\tilde{\zeta _1}-{\zeta _1}|}{2}\right) \ge \frac{|\tilde{\zeta _1}-{\zeta _1}|\Omega _1}{4}. \end{aligned}$$

Also, since \({\zeta _1}\) is the center of mass,

$$\begin{aligned} \int (x-{\zeta _1})\omega _1(x) dx=0 \end{aligned}$$

and by (42), we have

$$\begin{aligned} \int _{B({\zeta _1},|\tilde{\zeta _1}-{\zeta _1}|\Omega _1/(8\delta ^4))\backslash B(\tilde{\zeta _1},R)}|x-{\zeta _1}|\omega _1(x)dx\le \frac{|\tilde{\zeta _1}-{\zeta _1}|\Omega _1}{8}. \end{aligned}$$

Combining the last three inequalities, we get

$$\begin{aligned} \int _{\{|x-{\zeta _1}|>|\tilde{\zeta _1}-{\zeta _1}|\Omega _1/(8\delta ^4)\}}|x-{\zeta _1}|\omega _1(x)dx\ge \frac{|\tilde{\zeta _1}-{\zeta _1}|\Omega _1}{8} \end{aligned}$$

from which it follows that

$$\begin{aligned} I_{k,1}\ge \int _{\{|x-{\zeta _1}|>|\tilde{\zeta _1}-{\zeta _1}|\Omega _1/(8\delta ^4)\}}|x-{\zeta _1}|^k\omega _1(x)dx\ge \frac{|\tilde{\zeta _1}-{\zeta _1}|^k\Omega _1^k}{8^k\delta ^{4(k-1)}}. \end{aligned}$$

Thus

$$\begin{aligned} \int _{\{|x-{\zeta _1}|\le 2|\tilde{\zeta _1}-{\zeta _1}|/\delta \}}|x-{\zeta _1}|^k\omega _1(x)dx\le 2^k\Omega _1|\tilde{\zeta _1}-{\zeta _1}|^k/\delta ^k<\delta I_{k,1}. \end{aligned}$$

as long as \(\delta \) is sufficiently small. Thus (recalling that we are in the case where \(|{{\tilde{\zeta }}}_1-\zeta _1|\ge 2R\))

$$\begin{aligned} \iint _{S_4\cup S_5}&|x-{\zeta _1}|^{k}\omega _1(x)\omega _1(y)dxdy\nonumber \\&\le \iint _{\{|x-{\zeta _1}|<2|\tilde{\zeta _1}-{\zeta _1}|/\delta \}}|x-{\zeta _1}|^k\omega _1(x)\omega _1(y)dxdy\nonumber \\&+\iint _{\{|y-{\zeta _1}|\ge 2|\tilde{\zeta _1}-{\zeta _1}|\}}|x-{\zeta _1}|^k\omega _1(x)\omega _1(y)dxdy\nonumber \\&\le \delta I_{k,1}+\delta ^4 I_{k,1}\le 2\delta I_{k,1}. \end{aligned}$$
(52)

Thus, combining (51) and (52), we get that in either case, we have

$$\begin{aligned} \iint _{S_4\cup S_5}|x-{\zeta _1}|^k\omega _1(x)\omega _1(y)dxdy\le \delta CI_{k,1}+C\delta ^{-k}R^k. \end{aligned}$$
(53)

Finally, we bound the last two terms of (45) by using the bootstrap assumption 5 and the fact that \(t\ge T\gg 1/\delta \) to get

$$\begin{aligned} t^{-1/2}\sup _{x\in {{\,\mathrm{supp}\,}}\omega _1}&|x-{\zeta _1}|\int |x-{\zeta _1}|^{k}\omega _1(x)dx+ t^{-1}\sup _{x\in {{\,\mathrm{supp}\,}}\omega _1}|x-{\zeta _1}|^2\int |x-{\zeta _1}|^{k}\omega _1(x)dx\nonumber \\&\le t^{-1/4+\epsilon }I_{k,1}+t^{-1/2+2\epsilon }I_{k,1}\le C\delta I_{k,1}. \end{aligned}$$
(54)

Combining (45), with (46), and getting error bounds from (47), (48), (49), (50), (53), and (54), we get

$$\begin{aligned} \frac{d}{dt}\int f_{k,2}(x)\omega _1(x)dx=\Omega _1\int 2\sin (2\theta )|x-{\zeta _1}|^k\omega _1(x) dx+O(\delta I_{k,1}+R^k\delta ^{-k}). \end{aligned}$$

which completes the proof of (17).

5 Short Time Behavior

In this section, we assume that \(T_*<t_0+t_0^{9/10}\). We will use rougher versions of estimates from Section 4. In particular, we use the boundedness of \(\sin \) to turn (14) into

$$\begin{aligned} \int |x-{\zeta _1}|^{k-2}(x-{\zeta _1})\cdot (v_2(x)-v_2({\zeta _1}))\omega _1(x)dx=O\left( t^{-1}I_{k,1}\right) +O\left( t^{\epsilon -5/4}I_{k,1}\right) +O\left( t^{\frac{\epsilon -3}{2}}I_{k,1}^{\frac{k-1}{k}}\right) . \end{aligned}$$

We then plug this, the analogous bound for \(v_3\), and (13) into (8) to get that whenever \(I_{k,1}\ge 1\), we have

$$\begin{aligned} \frac{d}{dt}I_{k,1}\le C\frac{I_{k,1}}{t}+ C(k-2)t^{\epsilon /2}I_{k,1}^{\frac{k-4}{k}}. \end{aligned}$$
(55)

Plugging in \(k=2\), and using the fact that \(I_{2,1}(t_0)=O(1)\), we get that for all \(t\in [t_0,T_*]\), we have

$$\begin{aligned} I_{2,1}(t)=O(1)\exp \left( \int _{t_0}^{t_0+t_0^{9/10}}\frac{C}{t}dt\right) =O(1)<t^{\epsilon /2} \end{aligned}$$

so bootstrap assumption 2 is maintained. Now we apply (55) for more general k. Using bootstrap assumption 3, we get that

$$\begin{aligned} I_{k,1}(T_*)&\le C+C\int _{t_0}^{T_*}\frac{I_{k,1}}{t}+t^{\epsilon /2}I_{k,1}^{\frac{k-4}{k}}dt\\&\le C+Ct_0^{9/10}\left( t_0^{k(1+\epsilon )/4-1}+t_0^{\epsilon /2}t_0^{k(1+\epsilon )/4-(1+\epsilon )}\right) \\&<T_*^{k(1+\epsilon )/4} \end{aligned}$$

so bootstrap assumption 3 is maintained. Now, to verify bootstrap assumption 5, we do things similarly to section 4. We suppose that there is some \(p(t)\in {{\,\mathrm{supp}\,}}\omega _1\) transported by u such that at time \(T_*\), we have \(|p(T_*)-{\zeta _1}(T_*)|\ge \epsilon T_*^{1/4+\epsilon }\) and we define

$$\begin{aligned} t_2=\sup \left\{ s\in [t_0,T_*]\, \big \vert \, s=t_0\text { or }|p(s)-{\zeta _1}(s)|<\frac{\epsilon }{2} T_*^{1/4+\epsilon }\right\} . \end{aligned}$$

Since \(|p(t_0)-{\zeta _1}(t_0)|\le \rho \), we in fact have that \(t_2>t_0\) and that

$$\begin{aligned} |p(t_2)-x(t_2)|=\frac{\epsilon }{2} T_*^{1/4+\epsilon }. \end{aligned}$$

Then on the interval \([t_2,T_*]\), we have that (35) holds, so

$$\begin{aligned} |p(T_*)-x(T_*)|\le |p(t_2)-x(t_2)|+O\left( T_*^{9/10}T_*^{\epsilon -3/4}\right) < \epsilon T_*^{1/4+\epsilon }, \end{aligned}$$

which gives a contradiction, so bootstrap assumption 5 is maintained. Thus we have shown that we cannot have \(T_*<t_0+t_0^{9/10}\), so \(T_*=\infty \), completing the proof of Theorem 4.