Mixing solutions for the Muskat problem with variable speed

Abstract

We provide a quick proof of the existence of mixing weak solutions for the Muskat problem with variable mixing speed. Our proof is considerably shorter and extends previous results in Castro et al. (Mixing solutions for the Muskat problem, 2016, arXiv:1605.04822) and Förster and Székelyhidi (Comm Math Phys 363(3):1051–1080, 2018).

1 Introduction

The mathematical model for the evolution of two incompressible fluids moving in a porous medium, such as oil and water in sand, was introduced by Morris Muskat in his treatise [22] and is based on Darcy’s law (see also [25, 30]). In this paper, we focus on the case of constant permeability under the action of gravity so that, after non-dimensionalizing, the equations describing the evolution of density $\rho$ and velocity u are given by (see [9, 24] and references therein)

$$\begin{aligned} \partial _t \rho + \text {div }(\rho u)&= 0, \end{aligned}$$

(1)

$$\begin{aligned} \text {div }u&= 0, \end{aligned}$$

(2)

$$\begin{aligned} u + \nabla p&= -(0,\rho ), \end{aligned}$$

(3)

$$\begin{aligned} \rho (x,0)&= \rho _0(x). \end{aligned}$$

(4)

We assume that at the initial time the two fluids, with densities $\rho ^+$ and $\rho ^-$, are separated by an interface which can be written as the graph of a function over the horizontal axis is,

$$\begin{aligned} \rho _0(x) = {\left\{ \begin{array}{ll} \rho ^+ &{} x_2 > z_0(x_1),\\ \rho ^- &{} x_2 < z_0(x_1). \end{array}\right. } \end{aligned}$$

(5)

Thus, the interface separating the two fluids at the initial time is given by $\Gamma _0 := \{(s,z_0(s))| s \in \mathbb {R}\}$. Assuming that $\rho (x,t)$ remains in the form (5) for positive times, the system reduces to a non-local evolution problem for the interface $\Gamma$. If the sheet can be presented as a graph as above, one can show (see for example [9]) that the equation for z(s, t) is given by

$$\begin{aligned} \partial _t z(s,t) = \frac{\rho ^- - \rho ^+}{2\pi } \int _{- \infty }^{\infty } \frac{(\partial _{s} z(s,t) - \partial _{s} z(\xi ,t))(s-\xi )}{(s-\xi )^2 + (z(s,t) - z(\xi ,t))^2} \mathrm{d}\xi . \end{aligned}$$

(6)

Linearising (6) around the flat interface $z=0$ reduces to $\partial _tf=\frac{\rho ^+-\rho ^-}{2}{\mathcal {H}}(\partial _sf)$, where ${\mathcal {H}}$ denotes the classical Hilbert transform. Thus one distinguishes the following cases: The case $\rho ^+ > \rho ^-$ is called the unstable regime and amounts to the situation where the heavier fluid is on top. The case $\rho ^+< \rho ^-$ is called the stable regime. In the stable case, this equation is locally well-posed in $H^3(\mathbb {R})$, see [7, 9], as well as [1, 2, 19, 23] for recent developments. In the unstable case, however, which is our focus in this article, we have an ill-posed problem (see [9, 25]), and there are no general existence results for (6) known. Thus, the description of (1)–(4) as a free boundary problem seems not suitable for the unstable regime. Indeed, as shown in experiments [30], in this regime the sharp interface seems to break down and the two fluids start to mix on a mesoscopic scale. In a number of applications [22, 30], it is precisely this mixing process in the unstable regime which turns out to be highly relevant, calling for an amenable mathematical framework.

1.1 Mixing solutions and admissible subsolutions

A notion of solution, which allows for a meaningful existence theory and at the same time able to represent the physical features of the problem such as mixing, was introduced in [28]; it is based on the concept of subsolution, which appears naturally when considering stability of the nonlinear system (1)–(4) under weak convergence [13]. This point of view was pioneered by L. Tartar in the 1970s-1980s in his study of compensated compactness [29], and experienced renewed interest in the past 10 years in connection with the theory of convex integration, applied to weak solutions in fluid mechanics [8, 11, 12, 26]. In order to state the definition, we recall that after applying a simple affine change of variables we may assume $|\rho ^\pm |=1$. In particular, for the rest of the paper we will be concerned with the unstable case, so that

$$\begin{aligned} \rho ^+=+1,\quad \rho ^-=-1. \end{aligned}$$

With this normalization subsolutions are defined as follows (c.f. [28, Definition 4.1]).

Definition 1.1

Let $T > 0$. We call a triple $(\rho ,u,m) \in L^{\infty }(\mathbb {R}^2 \times [0,T))$ an admissible subsolution of (1)–(4) if there exist open domains $\Omega ^{\pm }, \Omega _\mathrm{mix}$ with $\overline{\Omega ^+} \cup \overline{\Omega ^-} \cup \Omega _\mathrm{mix} = \mathbb {R}^2\times [0,T)$ such that

• (i) The system

  $$\begin{aligned} \begin{aligned} \partial _t \rho + \text {div }m&= 0\\ \text {div }u&= 0\\ \text {curl }u&= -\partial _{x_1} \rho \\ \rho \vert _{t=0}&= \rho _0 \end{aligned} \end{aligned}$$

  (7)

  holds in the sense of distributions in $\mathbb {R}^2 \times [0,T)$;

• (ii) The pointwise inequality

  $$\begin{aligned} \left| m-\rho u + \frac{1}{2} (0,1-\rho ^2) \right| \le \frac{1}{2} \left( 1-\rho ^2 \right) , \end{aligned}$$

  (8)

  holds almost everywhere;

• (iii) $|\rho (x,t)| = 1$ in $\Omega ^{+}\cup \Omega ^-$;

• (iv) In $\Omega _\mathrm{mix}$, the triple $(\rho ,u,m)$ is continuous and (8) holds with a strict inequality.

Observe that whenever $\rho =\pm 1$ for an admissible subsolution, then by (8) the system (7) reduces to (1)–(3). Conversely, in the set $\Omega _\mathrm{mix}$ the subsolution $\rho$ represents the coarse-grained density of a microscopically mixed state. More precisely, we have the following theorem from [5, 28]:

Theorem 1.2

Suppose there exists an admissible subsolution $({\bar{\rho }},{\bar{u}},{\bar{m}})$ to (1)–(4). Then there exist infinitely many admissible weak solutions $(\rho ,u)$ with the following additional mixing property: For any $r>0$, $0<t_0<T$ and $x_0\in \mathbb {R}^2$ such that $B:=B_r(x_0,t_0)\subset \Omega _\mathrm{mix}$, both sets $\{(x,t)\in B:\,\rho (x,t)=\pm 1\}$ have strictly positive Lebesgue measure.

Furthermore, there exists a sequence of such admissible weak solutions $(\rho _k,u_k)$ such that $\rho _k\overset{*}{\rightharpoonup }{\bar{\rho }}$ as $k\rightarrow \infty$.

In other words ${\bar{\rho }}$ represents a sort of “coarse-grained, average” density. Recently, this coarse-graining property of Theorem 1.2 was sharpened in [6] along the lines of [12] to the statement that, essentially, ${{\bar{\rho }}}$ denotes the average density not just on space-time balls B but also space-balls for every time $t>0$.

Theorem 1.2 is based on a general and very robust Baire-category type argument (c.f. [13] as well as [28, Appendix]) and basically highlights the key observation that a central object in the study of unstable hydrodynamic interfaces is a suitably defined subsolution. In recent years, this approach has been successfully applied in various contexts: for the incompressible Euler system in the presence of a Kelvin–Helmholtz instability [4, 21, 27], the density-driven Rayleigh–Taylor instability [16, 17], the Muskat problem with fluids of different mobilities [20], as well as in the context of the compressible Euler system [10]. This type of research is driven by the theoretical expectation, underpinned by recent numerical work [14, 18], that whereas the microscopic evolution in such problems is ill-posed, certain statistical quantities of interest obey a well-defined, “universal” macroscopic evolution, thus exhibiting universal features among the very large number of weak solutions.

1.2 Evolution of the mixing region: the pseudo-interface

An interesting phenomenon concerning the evolution of the coarse-grained interface was discovered by A. Castro, D. Cordoba and D. Faraco in [5]: for general (sufficiently smooth) initial curves $\Gamma _0$, the mixing (sub)solutions exhibit a two-scale dynamics. On a fast scale, the sharp interface diffuses to a mixing zone $\Omega _\mathrm{mix}$ at some speed $c>0$, which has a stabilizing effect on the overall dynamics. On a slower scale, the mixing zone itself begins to twist and evolve according to the now regularized evolution of the mid-section of $\Omega _\mathrm{mix}$, called a pseudo-interface.

The authors in [5] showed that appearance of a mixing zone with speed c is compatible with the requirements of Definition 1.1 provided $c<2$ (for the flat initial condition $c<2$ was also the upper bound reached in [28], in agreement with the relaxation approach in [24]) and by a suitable ansatz exhibited the regularized evolution of the pseudo-interface as a nonlinear and non-local evolution equation of the form (see (1.11)–(1.12) in [5])

$$\begin{aligned} \partial _tz={\mathcal {F}}(z). \end{aligned}$$

(9)

In a technical tour de force, they were able to show well-posedness of (9) for initial data $z_0\in H^5(\mathbb {R})$. Roughly speaking, the key point is that the linearization of (6), in Fourier space written as $\partial _t{\hat{f}}=|\xi |{\hat{f}}$, is modified by the appearance of the mixing zone to

$$\begin{aligned} \partial _t{\hat{f}}=\frac{|\xi |}{1+ct|\xi |}{{\hat{f}}}, \end{aligned}$$

(10)

which leads to ${\hat{f}}=(1+ct|\xi |)^{1/c}{{\hat{f}}}_0$. The analysis of this equation was performed for constant $c=1$ in [5], and recently extended to variable $c=c(s)$ in [3] (in which case (10) has to be interpreted as a pseudo-differential equation) under certain restrictive conditions. In particular, the analysis in [3] applies under the condition that the range of mixing speed $c_1\le c(s)\le c_2<2$ is small: $0<c_1\le c_2\le \frac{c_1}{1-c_1}$; furthermore, high regularity is required: $c\in W^{k,\infty }(\mathbb {R})$ with $k>c^{-1}$ (c.f. [3, Definition 4]).

It was observed in [15] that the ansatz of [5] is too restrictive since after a short initial time the macroscopic evolution of the pseudo-interface quite likely becomes non-universal. Thus, the authors in [15] replaced the equation (9) by a simple expansion in time up to second order,

$$\begin{aligned} z(s,t)=z_0(s)+tz_1(s)+\tfrac{1}{2}t^2z_2(s), \end{aligned}$$

(11)

and showed that a suitable choice of $z_1$ and $z_2$ leads to an evolution which is compatible with Definition 1.1 for any constant speed $c\in (0,2)$. More precisely, in the expansion (11) $z_1=\partial _tz|_{t=0}:=u_{\nu }|_{t=0}$ is chosen as the normal velocity induced on the interface by (6) at time $t=0$, whereas $z_2$ involves a non-local operator of the same type applied to $z_0$ plus a local curvature term:

$$\begin{aligned} z_2:=T[z_0]+c\frac{1-(\partial _sz_0)^2}{(1+(\partial _sz_0)^2)^{1/2}}\kappa _0, \end{aligned}$$

where $\kappa _0=\kappa _0(s)$ is the curvature of the initial interface $\Gamma _0$. This expansion reveals an important difference to the approach in [5]: the regularity of the pseudo-interface does not deteriorate with small $c\rightarrow 0$, in sharp contrast to the evolution in (10). From a physical point of view, this is natural to expect, if one takes into account the scale separation in the two dynamics: once a mixing zone appears, the pseudo-interface is a matter of arbitrary choice, the only relevant object for the system (7)–(8) being the set $\Omega _\mathrm{mix}$. Thus, a coupling between the two dynamics should appear at most in terms of higher order fluctuations; indeed, a closer look [3, 5] reveals that the deterioration of regularity observed in (10) in fact applies to $f=\partial _s^4z$.

Motivated by this heuristic, in this short note we extend and simplify the analysis of [15] by

• (1) allowing for variable mixing speed $c=c(s)$ within the whole range $0<\inf _{\mathbb {R}}c\le \sup _{\mathbb {R}}c<2$, with no degeneration of regularity;

• (2) allowing for asymptotically vanishing mixing speed $c(s)\rightarrow 0$ as $|s|\rightarrow \infty$ in case the initial interface $z_0$ is asymptotically horizontal.

Moreover, our analysis shows that the expansion (11) above, obtained in [15], remains valid up to second order even in this generality, thus giving further evidence towards universality, in the sense explained above, of the macroscopic evolution.

1.3 The main result

In this section, we state the precise form of our main result.

Our assumption on the initial datum is that the initial interface is asymptotically flat with some given slope $\beta \in \mathbb {R}$, i.e. $\rho _0$ is given by (5) with

$$\begin{aligned} z_0(s) = \beta s + {\tilde{z}}_0(s) \end{aligned}$$

(12)

for some ${\tilde{z}}_0$ with sufficiently fast decay at infinity, using—as in [15]—the following weighted Hölder norms: for any $0<\alpha <1$ set

$$\begin{aligned} \Vert f\Vert _0^*:=\sup _{s\in \mathbb {R}}(1+|s|^{1+\alpha })|f(s)|. \end{aligned}$$

Furthermore, we define the associated Hölder (semi-)norms as follows. We set

$$\begin{aligned}{}[f]^*_{\alpha }:=\sup _{|\xi |\le 1,s\in \mathbb {R}}(1+|s|^{1+\alpha })\frac{|f(s-\xi )-f(s)|}{|\xi |^{\alpha }}, \end{aligned}$$

and for any $k\in \mathbb {N}$

$$\begin{aligned} \Vert f\Vert _{k,\alpha }^*:=\sup _{s\in \mathbb {R},j\le k}(1+|s|^{1+\alpha })|\partial _s^jf(s)|+[\partial _s^kf]_{\alpha }^*. \end{aligned}$$

We denote by $C_*^{k,\alpha }(\mathbb {R}):=\{f\in C^{k,\alpha }(\mathbb {R}):\,\Vert f\Vert _{k,\alpha }^*<\infty \}$.

Next, we describe the geometry of the coarse-grained evolution. Given a pseudo-interface $z:\mathbb {R}\times [0,T] \rightarrow \mathbb {R}$ and mixing speed $c:\mathbb {R}\rightarrow (0,\infty )$ define $\Omega ^{\pm }(t)$ and $\Omega _\mathrm{mix}(t)$ as

$$\begin{aligned} \begin{aligned} \Omega ^+(t)&= \{x \in \mathbb {R}^2 | x_2 > z(x_1,t) + c(x_1)t\},\\ \Omega _\mathrm{mix}(t)&= \{x \in \mathbb {R}^2 |z(x_1,t) - c(x_1)t< x_2< z(x_1,t) + c(x_1)t\},\\ \Omega ^-(t)&= \{x \in \mathbb {R}^2 | x_2 < z(x_1,t) - c(x_1)t\},\\ \end{aligned} \end{aligned}$$

(13)

and set

$$\begin{aligned} \Omega ^{\pm }=\bigcup _{t>0}\Omega ^{\pm }(t),\quad \Omega _\mathrm{mix}=\bigcup _{t>0}\Omega _\mathrm{mix}(t). \end{aligned}$$

Theorem 1.3

Let $z_0(s) = \beta s + {\overline{z}}_0(s)$ with ${\overline{z}}_0 \in C^{3,\alpha }_*(\mathbb {R})$ for some $0<\alpha <1$ and $\beta \in \mathbb {R}$. Let $c=c(s)>0$ with $\sup _sc(s)<2$ and $\partial _sc\in C^{\alpha }_*(\mathbb {R})$. If $\inf _sc(s)=0$, assume in addition that $\beta =0$ and there exists $c_\mathrm{min}>0$ such that

$$\begin{aligned} c(s)\ge c_\mathrm{min}(1+|s|^{2\alpha /3})^{-1}. \end{aligned}$$

Then there exists $T>0$ such that there exists a pseudo-interface $z\in C^2([0,T];C^{1,\alpha }(\mathbb {R}))$ with $z|_{t=0}=z_0$ for which the mixing zone defined in (13) admits admissible subsolutions on [0, T]. In particular, there exist infinitely many admissible weak solutions to (1)–(4) on [0, T] with mixing zone given by (13).

Observe that under the conditions in the theorem the function c has limits at infinity $s\rightarrow \pm \infty$.

The paper is organised as follows. In Sect. 2, we show that an admissible subsolution exists provided certain smallness conditions are satisfied on the temporal expansion of the pseudo-interface—see Proposition 2.1. This section closely follows the construction in [15], in particular the construction of symmetric piecewise constant densities in [15, Section 5].

Then in Sect. 3, we obtain a regular expansion in time t for the normal component of the velocity across interfaces for arbitrary mixing speeds. Our key result in this section is Proposition 3.8, see also Remark 3.10 for a simplified statement. It is worth pointing out that validity of the expansion requires minimal smoothness assumptions on the pseudo-interface and, at variance with the approach in [3], does not degenerate as $c\rightarrow 0$ or $c\rightarrow 2$. Finally, in Sect. 4 we complete the proof of Theorem 1.3.

We remark in passing that if $\beta \ne 0$, the statement of the theorem continues to hold provided the lower bound on c(s) is strengthened to

$$\begin{aligned} c(s)\ge c_\mathrm{min}(1+|s|^{\alpha /2})^{-1}. \end{aligned}$$

The proof of this requires minor modifications in Proposition 2.1, in particular replacing the term $c^{3/2}$ in (21) by $c^2$. As such modifications unnecessarily complicate the presentation without added value, we chose not to include the details here.

2 Subsolutions with variable mixing speed

We start by fixing $N\in \mathbb {N}$ and setting

$$\begin{aligned} c_i(s)=\frac{2i-1}{2N-1}c(s)\quad \text { for }i=1,\dots ,N. \end{aligned}$$

(14)

Define the density $\rho (x,t)$ to be the piecewise constant function

$$\begin{aligned} \rho (x,t) = {\left\{ \begin{array}{ll} 1 &{} x \in \Omega ^+(t),\\ \frac{i}{N} &{} c_i(x_1)t<x_2-z(x_1,t)<c_{i+1}(x_1)t\\ 0&{} -c_1(x_1)t<x_2-z(x_1,t)<c_1(x_1)t\\ -\frac{i}{N} &{} c_i(x_1)t<z(x_1,t)-x_2<c_{i+1}(x_1)t\\ -1 &{} x \in \Omega ^-(t), \end{array}\right. } \end{aligned}$$

(15)

with $i=1,\dots ,N-1$. This definition of $\rho$ already determines the velocity u by the kinematic part of (7), namely the Biot–Savart law (see Sect. 3 below)

$$\begin{aligned} \begin{aligned} \text {div }u&=0,\\ \text {curl }u&=-\partial _{x_1}\rho . \end{aligned} \end{aligned}$$

(16)

Observe that $\rho$ is piecewise constant, with jump discontinuities across 2N interfaces

$$\begin{aligned} \Gamma ^{(\pm i)}(t)=\left\{ (s,z^{(\pm i)}(s,t)):\,s\in \mathbb {R}\right\} ,\quad z^{(\pm i)}(s,t)=z(s,t)\pm c_i(s)t. \end{aligned}$$

(17)

It is well known [9] that, provided the interfaces are sufficiently regular, the solution u to (16) is then globally bounded, smooth in $\mathbb {R}^2\setminus \bigcup _{i}\Gamma ^{(i)}$ with well-defined traces on $\Gamma ^{\pm }$, and the normal component

$$\begin{aligned} u_{\nu }^{(i)}(s,t):=u(s,z^{(i)}(s,t),t)\cdot \begin{pmatrix}-\partial _{s}z^{(i)}(s,t)\\ 1\end{pmatrix} \end{aligned}$$

(18)

is continuous across the interfaces $\Gamma ^{(i)}$ for $i=\pm 1,\dots ,\pm N$.

Indeed, we will see in the next section that this is the case provided

$$\begin{aligned} z(s,t)=\beta s+{\tilde{z}}(s,t), \end{aligned}$$

(19)

with ${{\tilde{z}}}(\cdot ,t) \in C^{1,\alpha }_*(\mathbb {R})$, $\partial _sc\in C^{\alpha '}_*(\mathbb {R})$ and some $0<\alpha ,\alpha '<1$ and $\beta \in \mathbb {R}$.

Our main result in this section is as follows:

Proposition 2.1

Let z(s, t) and c(s) be as in (19) with ${{\tilde{z}}}\in C^1([0,T];C^{1,\alpha }_*(\mathbb {R}))$, $\partial _sc\in C^{\alpha '}_*(\mathbb {R})$ and $0<c(s)\le c_\mathrm{max}$ $\forall s$ for some $c_\mathrm{max}<2$. Let $N\in \mathbb {N}$ be the smallest integer such that $c_\mathrm{max}<\frac{2N-1}{N}$ and set $\rho$ to be defined by (15) and u the corresponding velocity field u according to (16), with normal traces as in (18). Assume that

$$\begin{aligned}&\lim _{t\rightarrow 0}\sup _{s}\frac{1}{c(s)}\left| \partial _tz(s,t)-u_{\nu }^{(\pm i)}(s,t)\right| = 0\,\text { for all }i, \end{aligned}$$

(20)

$$\begin{aligned}&\lim _{t\rightarrow 0}\sup _s\frac{1}{tc^{3/2}(s)}\left| \int _0^s\partial _tz(s',t)-\sum _{i=1}^N\frac{u_\nu ^{(+i)}(s',t)+u_{\nu }^{(-i)}(s',t)}{2N}\,\mathrm{d}s'\right| =0, \end{aligned}$$

(21)

and furthermore, there exists $M>0$ such that

$$\begin{aligned} |\partial _s z(s,t)|,|\partial _s c(s)|\le Mc^{1/2}(s)\quad \forall \,s\in \mathbb {R}, t\in [0,T]. \end{aligned}$$

(22)

Then there exists $T'\in (0,T]$, depending on $c_\mathrm{max}<2$, $\tilde{z},c$ as well as on the rate of convergences in (20)–(21), and a vectorfield $m:\mathbb {R}^2\times (0,T')\rightarrow \mathbb {R}^2$ such that $(\rho ,u,m)$ is an admissible subsolution on $[0,T')$ with mixing zone $\Omega _\mathrm{mix}$ given by (13).

Remark 2.2

Formally, in the proof below the time of existence $T'$ also depends on N, but this dependency is already taken into account with $c_\mathrm{max}$.

Proof

The proof follows closely the proof of Theorem 5.1 in [15], adapted here to the variable speed setting. Set

$$\begin{aligned} m = \rho u - (1-\rho ^2) (\gamma + \tfrac{1}{2}e_2) \end{aligned}$$

for some $\gamma =\gamma (x,t)$, with $\gamma \equiv 0$ in $\Omega ^{\pm }$. Then (8) amounts to the condition

$$\begin{aligned} |\gamma | < \frac{1}{2}\quad \text { in }\Omega _\mathrm{mix}, \end{aligned}$$

whereas (7) is equivalent to $\text {div }\gamma =0$ in $\Omega _\mathrm{mix}\setminus \bigcup _{i=1}^N(\Gamma ^{(i)}\cup \Gamma ^{(-i)})$ together with $2(N-1)$ jump conditions

$$\begin{aligned}{}[\rho ]_{\Gamma ^{(i)}}\partial _tz^{(i)}+[m]_{\Gamma ^{(i)}}\cdot \begin{pmatrix}\partial _{x_1}z^{(i)}\\ -1\end{pmatrix}=0\quad \text { on }\Gamma ^{(i)} \end{aligned}$$

(23)

for $i=\pm 1,\dots ,\pm N$, where $[\cdot ]_{\Gamma ^{(i)}}$ denotes the jump on $\Gamma ^{(i)}$. Observe that

$$\begin{aligned} \Omega _\mathrm{mix}\setminus \bigcup _{i=1}^N(\Gamma ^{(i)}\cup \Gamma ^{(-i)})=\bigcup _{i=-N+1}^{N-1}\Omega ^{(i)}, \end{aligned}$$

with connected open sets $\Omega ^{(i)}$ defined as

$$\begin{aligned} \begin{aligned} \Omega ^{(i)}&= \{z(x_1,t)+c_{i}(x_1)t< x_2< z(x_1,t)+c_{i+1}(x_1)t\},\\ \Omega ^{(-i)}&= \{z(x_1,t)-c_{i+1}(x_1)t< x_2 < z(x_1,t)-c_{i}(x_1)t\}, \end{aligned} \end{aligned}$$

(24)

for $i=1,\dots ,N-1$ and

$$\begin{aligned} \Omega ^{(0)}= \{z(x_1,t)-c_{1}(x_1)t< x_2 < z(x_1,t)+c_{1}(x_1)t\} \end{aligned}$$

The divergence-free condition is then taken care by setting

$$\begin{aligned} \gamma =\nabla ^\perp g^{(i)}\,\text { in } \Omega ^{(i)},\quad i=-N\dots N, \end{aligned}$$

where $g^{(N)}=g^{{(-N)}}=0$ and $g^{(i)}\in C^1(\overline{\Omega ^{(i)}})$ for $i=-(N-1)\dots (N-1)$ are to be determined. Then, (8) amounts to the conditions

$$\begin{aligned} |\nabla g^{(i)}|<\frac{1}{2}\quad \text { in }\Omega ^{(i)}\quad i=-(N-1)\dots (N-1), \end{aligned}$$

(25)

and (23) reduces to conditions on the tangential derivatives on each interface: for any $i=1,\dots ,N$ we require on $\Gamma ^{(\pm i)}$

$$\begin{aligned} \partial _\tau g^{(\pm (i-1))}&=\frac{1}{1-(\frac{i-1}{N})^2}\left\{ \frac{c_i}{N}-\frac{2i-1}{2N^2}+\left( 1-(\tfrac{i}{N})^2\right) \partial _{\tau }g^{(\pm i)}\pm \frac{\partial _tz-u_\nu ^{(\pm i)}}{N}\right\} \nonumber \\&=h^{(\pm i)}+\frac{1-(\tfrac{i}{N})^2}{1-(\frac{i-1}{N})^2}\partial _{\tau } g^{(\pm i)}\quad \text { on }\Gamma ^{\pm i}, \end{aligned}$$

(26)

with

$$\begin{aligned} h^{(\pm i)}(s,t):=\frac{1}{1-(\frac{i-1}{N})^2}\left\{ \frac{c_i(s)}{N}-\frac{2i-1}{2N^2}\pm \frac{\partial _tz(s,t)-u_\nu ^{(\pm i)}(s,t)}{N}\right\} \end{aligned}$$

and

$$\begin{aligned} \partial _\tau g^{(i)}\big \vert _{\Gamma ^{(j)}}(s,t):=\partial _{s}\bigl [g^{(i)}(s,z^{(j)}(s,t),t)\bigr ]. \end{aligned}$$

Next, for $i\ne 0$ we make the choice that $g^{(i)}$ is a function of $x_1,t$ only. Then, using the fact that $g^{(\pm N)}=0$, we may use (26) to inductively define $g^{(\pm (N-1))}$, $g^{(\pm (N-2))},\dots ,g^{(\pm 1)}$ as

$$\begin{aligned} g^{(\pm i)}(x_1,t)&=\int _0^{x_1} h^{(\pm (i+1))}(s',t)+\frac{1-(\tfrac{i+1}{N})^2}{1-(\frac{i}{N})^2}\partial _{\tau } g^{(\pm (i+1))}(s',t)\,\mathrm{d}s'. \end{aligned}$$

(27)

In particular, we obtain

$$\begin{aligned} \begin{aligned} \partial _{x_1}g^{(\pm i)}&=\frac{1}{1-(\tfrac{i}{N})^2}\sum _{j=i+1}^N\left( \frac{c_j}{N}-\frac{2j-1}{2N^2}\pm \frac{\partial _t z-u_{\nu }^{(\pm j)}}{N}\right) \\&=\frac{N}{2N-1}\left( c-\frac{2N-1}{2N}\right) \pm \frac{N}{N^2-i^2}\sum _{j=i+1}^N[\partial _tz-u_{\nu }^{(j)}]\\&=-\frac{1}{2}+c(s)\left( \frac{N}{2N-1}+o(1)\right) , \end{aligned} \end{aligned}$$

where o(1) denotes terms going to zero uniformly in s as $t\rightarrow 0$ and we have used (20) in the last line. Since we also set $\partial _{x_2}g^{(i)}=0$, and $0<c(s)\le c_\mathrm{max}< \tfrac{2N-1}{N}$, we can deduce that (25) holds for sufficiently small $t>0$.

Next, we turn our attention to $g^{(0)}$ on $\Omega ^{(0)}$. For $s\in \mathbb {R}$, $t\in (0,T)$ and $\lambda \in [-1,1]$ define

$$\begin{aligned} {\hat{g}}(s,\lambda ,t):=g^{(0)}(s,z(s,t)+\lambda c_1(s)t,t). \end{aligned}$$

(28)

In order to satisfy (26), we set

$$\begin{aligned} {\hat{g}}(s,\pm 1,t):=\int _{0}^s h^{(\pm 1)}+\left( 1-\frac{1}{N^2}\right) \partial _{x_1}g^{(\pm 1)}\,\mathrm{d}s', \end{aligned}$$

and, more generally, for $\lambda \in [-1,1]$

$$\begin{aligned} {\hat{g}}(s,\lambda ,t):=\frac{1+\lambda }{2}{\hat{g}}(s,1,t)+\frac{1-\lambda }{2}{\hat{g}}(s,-1,t). \end{aligned}$$

Then

$$\begin{aligned} \partial _{\lambda }{\hat{g}}&=\frac{1}{N}\int _0^s\sum _{j=1}^N\left( \partial _tz-\frac{u_\nu ^{(+j)}+u_{\nu }^{(-j)}}{2}\right) \,\mathrm{d}s'\\&=tc^{3/2}(s)o(1),\\ \partial _s{\hat{g}}&=-\frac{1}{2}+\frac{1}{N}\sum _{j=1}^Nc_j+\frac{\lambda +1}{2N}\sum _{j=1}^N[\partial _t z - u_\nu ^{(+j)}] + \frac{\lambda -1}{2N} \sum _{j=1}^N[\partial _t z - u_\nu ^{(-j)}]\\&=-\frac{1}{2}+c(s)\left( \frac{N}{2N-1}+o(1)\right) \end{aligned}$$

where we have used (20)–(21). By differentiating (28) with respect to s and $\lambda$ and using (20)–(21), we deduce

$$\begin{aligned} \begin{aligned} \partial _{x_2}g^{(0)}&=c^{1/2}(s)o(1)\\ \partial _{x_1}g^{(0)}&=-\frac{1}{2}+c(s)(\tfrac{N}{2N-1}+o(1))+c^{1/2}(s)(|\partial _{s}z|+t|\partial _{s}c|)o(1)\\&=-\frac{1}{2}+c(s)(\tfrac{N}{2N-1}+o(1)) \end{aligned} \end{aligned}$$

(29)

where we have used (22) in the last line. Consequently

$$\begin{aligned} |\nabla g|^2&=\frac{1}{4}-c(s)(\tfrac{N}{2N-1}+o(1))+c^2(s)(\tfrac{N}{2N-1}+o(1))^2+c(s)o(1)\\&=\frac{1}{4}-c(s)(\tfrac{N}{2N-1}+o(1))+c^2(s)(1+o(1))\\&\le \frac{1}{4}-\tfrac{N}{2N-1}c(s)\bigl (1-\tfrac{N}{2N-1}c_\mathrm{max}+o(1)\bigr ). \end{aligned}$$

Since $\tfrac{N}{2N-1}c_\mathrm{max}<1$, we deduce that

$$\begin{aligned} |\nabla g^{(0)}| < \frac{1}{2}\quad \text { for sufficiently small }t>0. \end{aligned}$$

This concludes the proof. $\square$

3 The velocity u

In this section, we analyse more closely the normal component of the velocity, given in (18), where the velocity u is the solution of the system (16) with piecewise constant density $\rho$ given in (15). Following the computations in [9] and [15] we see that for any $t>0$

$$\begin{aligned} u_{\nu }^{(i)}(s,t)=\sum _{j}\frac{1}{2\pi N}PV\int _{\mathbb {R}}\frac{\partial _sz^{(j)}(s-\xi ,t)-\partial _sz^{(i)}(s,t)}{\xi }\Phi _{ij}(\xi ,s,t)\,\mathrm{d}\xi , \end{aligned}$$

(30)

where the sum is over $j=\pm 1,\dots ,\pm N$, the kernels $\Phi _{ij}(\xi ,s,t)$ are defined as

$$\begin{aligned} \Phi _{ij}(\xi ,s,t)=\frac{\xi ^2}{\xi ^2+(z^{(i)}(s,t)-z^{(j)}(s-\xi ,t))^2}, \end{aligned}$$

(31)

and

$$\begin{aligned} z^{(i)}(s,t)=z(s,t)+c_i(s)t. \end{aligned}$$

(32)

with the convention $c_{-i}(s)=-c_i(s)$, where $c_i(s)$ is defined in (14) for $i=1,\dots ,N$. The principal value integral here refers to $PV\int _{\mathbb {R}} = \underset{R\rightarrow \infty }{lim}\int _{-R}^R$. Next, we recall the operator $T_\Phi$ from [15], a weighted version of the Hilbert transform, defined for a weight function $\Phi =\Phi (\xi ,s)$ as

$$\begin{aligned} T_{\Phi }(g)(s):=\frac{1}{2\pi }PV\int _{\mathbb {R}}{\frac{\partial _sg(s-\xi )-\partial _sg(s)}{\xi }\Phi (\xi ,s)\mathrm{d}\xi }. \end{aligned}$$

(33)

Then (30) can be written as

$$\begin{aligned} u_{\nu }^{(i)}=\frac{1}{N}\sum _jT_{\Phi _{ij}}z^{(j)}+\frac{t}{N}\sum _{j\ne i}(\partial _sc_i-\partial _sc_j)I_{ij}, \end{aligned}$$

(34)

where we set

$$\begin{aligned} I_{ij}(s,t)=\frac{1}{2\pi }PV\int _{\mathbb {R}}\Phi _{ij}\frac{\mathrm{d}\xi }{\xi }. \end{aligned}$$

(35)

Observe that in $I_{ij}$ for $i\ne j$ it again suffices to consider the principal value integral as above, with regularization as $|\xi |\rightarrow \infty$. Nevertheless, for $i=j$ also a principal value regularization at $|\xi |\rightarrow 0$ is necessary—see below in Lemmas 3.6–3.7.

We next recall the following bound on $T_{\Phi }$ on Hölder-spaces from [15], where for the weight we use the following norms: first of all we assume that $\Phi ^{\infty }(s):=\lim _{|\xi |\rightarrow \infty }\Phi (\xi ,s)$ exists, $\Phi (\cdot ,s)\in C^1(\mathbb {R}\setminus \{0\})$, and set

$$\begin{aligned} \begin{aligned} {\bar{\Phi }}&=\xi \left( \Phi -\Phi ^{\infty }\right) ,\quad \Phi ^{\infty }=\lim _{|\xi |\rightarrow \infty }\Phi (\xi ,s),\\ {\tilde{\Phi }}&=\xi ^2\partial _{\xi }\left( \frac{1}{\xi }\Phi \right) =\xi \partial _{\xi }\Phi -\Phi . \end{aligned} \end{aligned}$$

(36)

We introduce the norms

$$\begin{aligned} \!|\!|\!|\Phi \!|\!|\!|_0&:=\sup _{s\in \mathbb {R},|\xi |\le 1}|\Phi (\xi ,s)|+\sup _{s\in \mathbb {R},|\xi |>1}(|{\bar{\Phi }}(\xi ,s)|+|{\tilde{\Phi }}(\xi ,s)|),\\ \!|\!|\!|\Phi \!|\!|\!|_{k,\alpha }&:=\max _{j\le k}\!|\!|\!|\partial _s^j\Phi \!|\!|\!|_0+[\partial _s^k\Phi ]_\alpha +\sup _{|\xi |>1}([\partial _s^k{\bar{\Phi }}(\xi ,\cdot )]_\alpha +[\partial _s^k{\tilde{\Phi }}(\xi ,\cdot )]_\alpha ), \end{aligned}$$

where we use the convention that $\Vert \Phi (\xi ,\cdot )\Vert$ denotes a norm in the second argument only and $\Vert \Phi \Vert$ denotes a norm joint in both variables. In particular, the Hölder-continuity of $\partial _s^k\Phi$ in both variables $\xi ,s$ is required in the norm $|\!|\!|\Phi \!|\!|\!|_{k,\alpha }$. Accordingly, we define the spaces

$$\begin{aligned} {\mathcal {W}}^0&=\{\Phi \in L^{\infty }(\mathbb {R}^2):\,\Phi ^\infty \text { and }\partial _\xi \Phi \text { exist, with }\!|\!|\!|\Phi \!|\!|\!|_0<\infty \},\\ {\mathcal {W}}^{k,\alpha }&=\{\Phi \in {\mathcal {W}}^0:\,\!|\!|\!|\Phi \!|\!|\!|_{k,\alpha }<\infty \} \end{aligned}$$

Then, the following version of the classical estimate on the Hilbert transform $T_1={\mathcal {H}}\nabla$ on Hölder-spaces holds [15, Theorem 3.1]:

Theorem 3.1

For any $\alpha >0$, $f\in C^{1,\alpha }_*(\mathbb {R})$ and $\Phi \in {\mathcal {W}}^0$ we have

$$\begin{aligned} \Vert T_{\Phi }(f)\Vert _0^*\le C \!|\!|\!|\Phi \!|\!|\!|_0\Vert f\Vert _{1,\alpha }^*. \end{aligned}$$

(37)

Moreover, for any $k\in \mathbb {N}$, $f\in C_*^{k+1,\alpha }(\mathbb {R})$ and $\Phi \in {\mathcal {W}}^{k,\alpha }$

$$\begin{aligned} \Vert T_{\Phi }(f)\Vert _{k,\alpha }^*\le C\!|\!|\!|\Phi \!|\!|\!|_{k,\alpha }\Vert f\Vert _{k+1,\alpha }^*. \end{aligned}$$

(38)

where the constant depends only on k and $\alpha$.

In the following, we analyse boundedness and continuity properties of the type of operators (33) arising in (34). This will ultimately enable us to derive an expansion in time for $t\rightarrow 0$ of the normal velocity components $u^{(i)}_\nu$ in (30).

Lemma 3.2

Let $z(s,t)=\beta s+{{\tilde{z}}}(s,t)$ with ${{\tilde{z}}}\in C^{l}([0,T];C^{k+1,\alpha }(\mathbb {R}))$ for some $l,k\in \mathbb {N}$, $0<\alpha <1$, $\beta \in \mathbb {R}$ and $T<\infty$, and let

$$\begin{aligned} \Phi (\xi ,s,t)=\frac{\xi ^2}{\xi ^2+(z(s,t)-z(s-\xi ,t))^2}. \end{aligned}$$

Then $\Phi \in C^{l}([0,T];{\mathcal {W}}^{k,\alpha })$.

Proof

We start by introducing the following notation: for $z=z(s,t)$ define

$$\begin{aligned} Z=Z(\xi ,s,t)=\frac{z(s,t)-z(s-\xi ,t)}{\xi }=\int _0^1\partial _sz(s-\tau \xi ,t)\,\mathrm{d}\tau , \end{aligned}$$

(39)

and furthermore, let

$$\begin{aligned} K(Z)=\frac{1}{1+Z^2}. \end{aligned}$$

(40)

Since $K\in C^{\infty }(\mathbb {R})$ with derivatives of any order uniformly bounded on $\mathbb {R}$, and since $\Phi =K\circ Z$, it follows easily from the chain rule that, for any $j\le l$, $\partial _t^j\Phi \in C^{k,\alpha }(\mathbb {R}^2)$ with

$$\begin{aligned} \sup _t\Vert \partial _t^j\Phi (\cdot ,\cdot ,t)\Vert _{C^{k,\alpha }(\mathbb {R}^2)}<\infty . \end{aligned}$$

Concerning the far-field terms, note that $\Phi ^{\infty }=\frac{1}{1+\beta ^2}$, hence

$$\begin{aligned} {\bar{\Phi }}=\frac{\beta +Z}{(1+\beta ^2)(1+Z^2)}({\tilde{z}}(s-\xi ,t)-{\tilde{z}}(s,t))=K_{\beta }(Z)({\tilde{z}}(s-\xi ,t)-{\tilde{z}}(s,t)), \end{aligned}$$

where $K_{\beta }(x)=\frac{\beta +x}{(1+\beta ^2)(1+x^2)}$ is again a function with derivatives of all order uniformly bounded on $\mathbb {R}$. Therefore the chain rule as above, together with the product rule, easily imply that, for any $j\le l$, $\partial _t^j{{\bar{\Phi }}}\in C^{k,\alpha }(\mathbb {R}^2)$ with

$$\begin{aligned} \sup _t\Vert \partial _t^j{{\bar{\Phi }}}(\cdot ,\cdot ,t)\Vert _{C^{k,\alpha }(\mathbb {R}^2)}<\infty . \end{aligned}$$

Similarly, $\xi \partial _\xi \Phi =K'(Z)\xi \partial _\xi Z$, where

$$\begin{aligned} \xi \partial _\xi Z=\partial _s{\tilde{z}}(s-\xi ,t)-\int _0^1\partial _s{\tilde{z}}(s-\tau \xi ,t)\,\mathrm{d}\tau , \end{aligned}$$

so that, once again for any $j\le l$, $\xi \partial _\xi (\partial _t^j\Phi )=\partial _t^j(\xi \partial _\xi \Phi )\in C^{k,\alpha }(\mathbb {R}^2)$ with

$$\begin{aligned} \sup _t\Vert \xi \partial _\xi \partial _t^j{{\bar{\Phi }}}(\cdot ,\cdot ,t)\Vert _{C^{k,\alpha }(\mathbb {R}^2)}<\infty . \end{aligned}$$

We deduce that $\partial _t^j\Phi \in {\mathcal {W}}^{k,\alpha }$ with

$$\begin{aligned} \sup _{t\in [0,T]}\!|\!|\!|\partial _t^j\Phi \!|\!|\!|_{k,\alpha }<\infty \end{aligned}$$

as required. $\square$

Using the notation introduced above in (39)–(40), we can write

$$\begin{aligned} \Phi _{ij}=K\left( Z^{(j)}+\frac{c_{ij}t}{\xi }\right) , \end{aligned}$$

(41)

where

$$\begin{aligned} Z^{(j)}(\xi ,s,t)=\frac{z^{(j)}(s,t)-z^{(j)}(s-\xi ,t)}{\xi },\quad c_{ij}(s)=c_i(s)-c_j(s). \end{aligned}$$

Observe that $c_{ii}=0$ so that Lemma 3.2 applies to $\Phi _{ii}$, but the second term requires more care. In the next lemmata, we address boundedness and continuity with respect to the functions z, c.

Lemma 3.3

There exists a constant $C>1$ such that the following holds. Let $z(s)=\beta s+{{\tilde{z}}}(s)$ with ${{\tilde{z}}}\in C^1(\mathbb {R})$, $c\in C(\mathbb {R})$, and let

$$\begin{aligned} \Phi (\xi ,s)=K\left( Z(\xi ,s)+\frac{c(s)}{\xi }\right) , \end{aligned}$$

where $K(Z)=(1+Z^2)^{-1}$ as in (40) and $Z(\xi ,s)=\frac{z(s)-z(s-\xi )}{\xi }$. Then $\Phi \in {\mathcal {W}}^0$, with

$$\begin{aligned} \!|\!|\!|\Phi \!|\!|\!|_0\le C(1+\Vert {{\tilde{z}}}\Vert _1+\Vert c\Vert _0). \end{aligned}$$

(42)

Furthermore, if $\Phi _1,\Phi _2$ are defined as above with $\tilde{z}_1,{{\tilde{z}}}_2\in C^1(\mathbb {R})$, then

$$\begin{aligned} \!|\!|\!|\Phi _1-\Phi _2\!|\!|\!|_0\le C\Vert {{\tilde{z}}}_1-{{\tilde{z}}}_2\Vert _1. \end{aligned}$$

(43)

Proof

Using that $|K|\le 1$, we deduce $\sup _{\xi ,s}|\Phi |\le 1$. Moreover, since also $|K'|\le 1$, we have

$$\begin{aligned} \begin{aligned} |{\bar{\Phi }}|&=|\xi |\left| K(Z+\tfrac{c}{\xi })-K(\beta )\right| \le |\xi ||Z-\beta |+|c|\\&\le 2\Vert {{\tilde{z}}}\Vert _0+\Vert c\Vert _0. \end{aligned} \end{aligned}$$

Similarly, for $|\xi |\ge 1$ we also have

$$\begin{aligned} |\xi \partial _\xi \Phi |=\left| K'(Z+\tfrac{c}{\xi })\right| \left| \xi \partial _\xi Z-\tfrac{c}{\xi }\right| \le 2\Vert \partial _s{{\tilde{z}}}\Vert _0+\Vert c\Vert _0. \end{aligned}$$

The estimate (42) follows.

For the Lipschitz bound (43), we proceed entirely analogously, using the representation

$$\begin{aligned} \Phi _1-\Phi _2=\int _0^1K'\left( \tau Z_1+(1-\tau )Z_2+\frac{c}{\xi }\right) \mathrm{d}\tau \,(Z_1-Z_2) \end{aligned}$$

and the bound $|Z_1-Z_2|\le \Vert \partial _s{{\tilde{z}}}_1-\partial _s\tilde{z}_2\Vert _0$. $\square$

Next, we address continuity of the mapping $c\mapsto K(Z+\frac{c}{\xi })$ at the singularity $c=0$.

Lemma 3.4

There exists a constant $C>1$ such that the following holds. Let $z(s)=\beta s+{{\tilde{z}}}(s)$ with ${{\tilde{z}}}\in C(\mathbb {R})$, $c\in C(\mathbb {R})$, and let

$$\begin{aligned} \Phi (\xi ,s)=K\left( Z(\xi ,s)+\frac{c(s)}{\xi }\right) -K\left( Z(\xi ,s)\right) , \end{aligned}$$

where $K(Z)=(1+Z^2)^{-1}$ as in (40) and $Z(\xi ,s)=\frac{z(s)-z(s-\xi )}{\xi }$. Then, for any f with $\partial _sf\in C^{\alpha }_*(\mathbb {R})$ we have

$$\begin{aligned} \Vert T_{\Phi }f\Vert _0^*\le C[\partial _sf]_{\alpha }^*\Vert c\Vert _0^{\alpha }. \end{aligned}$$

(44)

Proof

Using the fact that $|K|,|K'|\le 1$, we have

$$\begin{aligned} \left| \Phi (\xi ,s)\right| =\left| K\left( Z+\tfrac{c}{\xi }\right) -K(Z)\right| \le C\min \left( 1,\frac{c(s)}{|\xi |}\right) . \end{aligned}$$

Next, recalling the definition of $T_{\Phi }f$ from (33) we have

$$\begin{aligned} \begin{aligned} |T_{\Phi }f(s)|&\le C \int _\mathbb {R}\left| \frac{\partial _s f(s-\xi )-\partial _s f(s)}{\xi }\right| \min \left( 1,\frac{c(s)}{|\xi |}\right) \,\mathrm{d}\xi \\&\le C\frac{[\partial _sf]_{\alpha }^*}{1+|s|^{1+\alpha }}\int _\mathbb {R}|\xi |^{\alpha -1}\min \left( 1,\frac{c(s)}{|\xi |}\right) \,\mathrm{d}\xi \\&= C\frac{[\partial _sf]_{\alpha }^*}{1+|s|^{1+\alpha }}c^{\alpha }(s). \end{aligned} \end{aligned}$$

$\square$

Lemma 3.5

There exists a constant $C>1$ such that the following holds. Let $z(s)=\beta s+{{\tilde{z}}}(s)$ with ${{\tilde{z}}}\in C^{1,\alpha }(\mathbb {R})$, $c\in C(\mathbb {R})$, and let

$$\begin{aligned} \begin{aligned} \Phi (\xi ,s)=K\left( Z(\xi ,s)+\frac{c(s)}{\xi }\right) +K\left( Z(\xi ,s)-\frac{c(s)}{\xi }\right) -2K(Z(\xi ,s)), \end{aligned} \end{aligned}$$

where $K(Z)=(1+Z^2)^{-1}$ as in (40) and $Z(\xi ,s)=\frac{z(s)-z(s-\xi )}{\xi }$. Then, for any $f\in C^{2,\alpha }_*(\mathbb {R})$ and any $s\in \mathbb {R}$ we have

$$\begin{aligned} (1+|s|^{1+\alpha })|T_{\Phi }f-|c|\sigma (\partial _sz)\partial _s^2f|\le C[f]_{2,\alpha }^*(1+[\partial _sz]_\alpha )|c|^{1+\alpha }, \end{aligned}$$

(45)

where

$$\begin{aligned} \sigma (a)=\frac{1-a^2}{(1+a^2)^2}. \end{aligned}$$

(46)

Proof

Let us fix $s\in \mathbb {R}$. Observe that if $c(s)=0$, (45) is obvious; therefore, we may assume in the following that $c(s)\ne 0$, without loss of generality $c(s)>0$. We may then change variables in the integral defining $T_{\Phi }f$ to obtain

$$\begin{aligned} T_{\Phi }f(s)=\frac{c(s)}{2\pi }\int _\mathbb {R}\frac{\partial _s f(s-c(s)\xi )-\partial _s f(s)}{c(s)\xi }\Phi _c(\xi ,s)\,\mathrm{d}\xi , \end{aligned}$$

(47)

where $\Phi _c(\xi ,s):=\Phi (c(s)\xi ,s)$. Note that

$$\begin{aligned} \begin{aligned} \Psi&=K\left( Z_c+\frac{1}{\xi }\right) +K\left( Z_c-\frac{1}{\xi }\right) -2K\left( Z_c\right) \\&=\frac{1}{\xi ^2}\int _0^1\left[ K''(Z_c+\tfrac{\tau }{\xi })+K''(Z_c-\tfrac{\tau }{\xi })\right] (1-\tau )\mathrm{d}\tau , \end{aligned} \end{aligned}$$

where we denoted $Z_c(\xi ,s)=Z(c(s)\xi ,s)$. Using that both K and $K''$ are uniformly bounded, it follows that

$$\begin{aligned} |\Phi _c|\le C\min \left( 1,\xi ^{-2}\right) . \end{aligned}$$

(48)

Furthermore, letting

$$\begin{aligned} \Phi _0(\xi ,s)=K\left( \partial _sz(s)+\frac{1}{\xi }\right) +K\left( \partial _sz(s)-\frac{1}{\xi }\right) -2K\left( \partial _sz(s)\right) , \end{aligned}$$

we also obtain, using that both K and $K''$ are uniformly Lipschitz, that

$$\begin{aligned} |\Phi _c-\Phi _0|\le C\min \left( 1,\xi ^{-2}\right) [\partial _s z]_{\alpha }(c\xi )^{\alpha }. \end{aligned}$$

(49)

Since $\xi \mapsto \Phi _0(\xi ,s)$ is a rational function of $\xi$, its precise integral in $\xi$ may be calculated by elementary methods (as done in [15]), leading to

$$\begin{aligned} \int _{\mathbb {R}}\left[ K(a+\tfrac{1}{\xi })+K(a-\tfrac{1}{\xi })-2K(a)\right] \,\mathrm{d}\xi =-2\pi \frac{1-a^2}{(1+a^2)^2}=-2\pi \sigma (a). \end{aligned}$$

Thus $\frac{1}{2\pi }\int _\mathbb {R}\Phi _0(\xi ,s)\,\mathrm{d}\xi =-\sigma (\partial _sz(s))$. We then write (47) as

$$\begin{aligned} \begin{aligned} T_{\Phi }f(s)=&c(s)\sigma (\partial _sz(s))\partial _s^2f(s)+\frac{c(s)}{2\pi }\partial _s^2f(s)\int _{\mathbb {R}}\Phi _0(\xi ,s)-\Phi _c(\xi ,s)\,\mathrm{d}\xi \\&+\frac{c(s)}{2\pi }\int _{\mathbb {R}}\left[ \frac{\partial _s f(s-c(s)\xi )-\partial _s f(s)}{c(s)\xi }+\partial _s^2f(s)\right] \Phi _c(\xi ,s)\,\mathrm{d}\xi . \end{aligned} \end{aligned}$$

Using the bounds (48)–(49) on the two integrals, we then deduce (45). $\square$

Finally, we turn our attention to $I_{ij}$ defined in (35).

Lemma 3.6

There exists a constant $C>1$ such that the following holds. Let $z(s,t)=\beta s+{{\tilde{z}}}(s,t)$ with ${{\tilde{z}}}\in C([0,T];C^{1,\alpha }(\mathbb {R}))$, and let

$$\begin{aligned} I(s,t)=PV\int _{\mathbb {R}}\frac{\xi }{\xi ^2+(z(s,t)-z(s-\xi ,t))^2}\mathrm{d}\xi . \end{aligned}$$

Then $I\in C([0,T];C(\mathbb {R}))$ with

$$\begin{aligned} |I(s,t)|\le C\Vert \partial _sz\Vert _{\alpha }. \end{aligned}$$

Proof

Using that $PV\int _{|\xi |<1}\frac{\xi }{\xi }=PV\int _{|\xi |>1}\frac{\mathrm{d}\xi }{\xi }=0$ and using the notation introduced in (39)–(40), we have

$$\begin{aligned} \begin{aligned} I(s,t)&=PV\int _{\mathbb {R}}K(Z)\frac{\mathrm{d}\xi }{\xi }\\&=\int _{|\xi |<1}[K(Z)-K(\partial _sz)]\frac{\mathrm{d}\xi }{\xi }+\int _{|\xi |>1}[K(Z)-K(\beta )]\frac{\mathrm{d}\xi }{\xi }. \end{aligned} \end{aligned}$$

Observe that the integrands in these two integrals are uniformly integrable. Indeed, $K:\mathbb {R}\rightarrow \mathbb {R}$ is uniformly Lipschitz continuous, so that for the first integral we may use $|Z-\partial _sz|\le [\partial _sz]_{\alpha }|\xi |^\alpha$ and for the second integral $|Z-\beta |\le 2\Vert {{\tilde{z}}}\Vert _{0}|\xi |^{-1}$. Thus

$$\begin{aligned} \left| \frac{K(Z)-K(\partial _sz)}{\xi }\right| \le [\partial _sz]_{\alpha }|\xi |^{\alpha -1},\quad \left| \frac{K(Z)-K(\beta )}{\xi }\right| \le 2\Vert \tilde{z}\Vert _0|\xi |^{-2}. \end{aligned}$$

The conclusion follows from Lebesgue’s dominated convergence theorem. $\square$

Lemma 3.7

There exists a constant $C>1$ such that the following holds. Let $z(s)=\beta s+{{\tilde{z}}}(s)$ with ${{\tilde{z}}}\in C^{1,\alpha }(\mathbb {R})$, $c\in C(\mathbb {R})$ with $c(s)\ne 0$, and let

$$\begin{aligned} I_c(s)=PV\int _{\mathbb {R}}K\left( Z(\xi ,s)+\tfrac{c(s)}{\xi }\right) \frac{\mathrm{d}\xi }{\xi }. \end{aligned}$$

Then

$$\begin{aligned} |I_c(s)|\le C(1+\Vert {{\tilde{z}}}\Vert _{1,\alpha }+\Vert \partial _sz\Vert _0^2), \end{aligned}$$

and moreover

$$\begin{aligned} \left| I_c(s)-PV\int _{\mathbb {R}} \left[ K(\partial _sz+\tfrac{1}{\xi })+K(Z)\right] \frac{\mathrm{d}\xi }{\xi }\right| \le C[\partial _sz]_{\alpha }c(s)^{\alpha }. \end{aligned}$$

(50)

Proof

Let us fix $s\in \mathbb {R}$ and assume without loss of generality that $c(s)>0$. We perform the change of variables $\xi \mapsto \frac{\xi }{c(s)}$ to obtain

$$\begin{aligned} \begin{aligned} I_c&=PV\int _{\mathbb {R}}K\left( Z+\tfrac{c}{\xi }\right) \frac{\mathrm{d}\xi }{\xi }=PV\int _{\mathbb {R}}K\left( Z_c+\tfrac{1}{\xi }\right) \frac{\mathrm{d}\xi }{\xi }\\&=\int _{|\xi |>1}\left[ K\left( Z_c+\tfrac{1}{\xi }\right) -K\left( Z_c\right) \right] \frac{\mathrm{d}\xi }{\xi }\\&\quad +PV\int _{|\xi |>1}K\left( Z_c\right) \frac{\mathrm{d}\xi }{\xi }+\int _{|\xi |<1}K\left( Z_c+\tfrac{1}{\xi }\right) \frac{\mathrm{d}\xi }{\xi }, \end{aligned} \end{aligned}$$

where we denoted, as in the proof of Lemma 3.5, $Z_c(\xi ,s)=Z(c(s)\xi ,s)$. Now we observe the following: using the uniform Lipschitz bound on K,

$$\begin{aligned} \left| K\left( Z_c+\tfrac{1}{\xi }\right) -K\left( Z_c\right) \right| \le \frac{1}{|\xi |}, \end{aligned}$$

so that the first integrand is dominated by $|\xi |^{-2}$ on $|\xi |>1$; furthermore, we have the lower bound

$$\begin{aligned} \xi ^2+(\xi Z_c+1)^2=(1+Z_c^2)\left( \xi +\tfrac{Z_c}{1+Z_c^2}\right) ^2+\frac{1}{1+Z_c^2}\ge \frac{1}{1+\Vert \partial _sz\Vert _0^2}, \end{aligned}$$

implying that the third integrand is uniformly bounded. Finally, for the second integral we proceed as in the proof of Lemma 3.6:

$$\begin{aligned} \begin{aligned} PV&\int _{|\xi |>1}K\left( Z_c\right) \frac{\mathrm{d}\xi }{\xi }=PV\int _{|\xi |>c}K\left( Z\right) \frac{\mathrm{d}\xi }{\xi }\\&=PV\int _{1>|\xi |>c}K\left( Z\right) \frac{\mathrm{d}\xi }{\xi }+PV\int _{|\xi |>1}K\left( Z\right) \frac{\mathrm{d}\xi }{\xi }\\&=\int _{1>|\xi |>c}[K(Z)-K(\partial _sz)]\frac{\mathrm{d}\xi }{\xi }+\int _{|\xi |>1}[K(Z)-K(\beta )]\frac{\mathrm{d}\xi }{\xi }. \end{aligned} \end{aligned}$$

Collecting the estimates above, we obtain the uniform bound on $I_c$.

In order to prove (50), we write

$$\begin{aligned} \begin{aligned}&PV\int _{\mathbb {R}} \left[ K(Z+\tfrac{c}{\xi })-K(Z)-K(\partial _sz+\tfrac{1}{\xi })\right] \frac{\mathrm{d}\xi }{\xi }=\\&\quad =PV\int _{\mathbb {R}} \left[ K(Z_c+\tfrac{1}{\xi })-K(Z_c)\right] \frac{\mathrm{d}\xi }{\xi }-PV\int _{\mathbb {R}}K(\partial _sz+\tfrac{1}{\xi }) \frac{\mathrm{d}\xi }{\xi }\\&\quad =PV\int _{|\xi |>1} \left[ K(Z_c+\tfrac{1}{\xi })-K(Z_c)-K(\partial _sz+\tfrac{1}{\xi })+K(\partial _sz)\right] \frac{\mathrm{d}\xi }{\xi }\\&\qquad +PV\int _{|\xi |<1}[K(\partial _sz)-K(Z_c)]\frac{\mathrm{d}\xi }{\xi }+PV\int _{|\xi |<1}K(Z_c+\tfrac{1}{\xi })-K(\partial _sz+\tfrac{1}{\xi })\frac{\mathrm{d}\xi }{\xi }. \end{aligned} \end{aligned}$$

For the first term, we use uniform boundedness of $K''$ to obtain the bound

$$\begin{aligned} \begin{aligned}&\left| K(Z_c+\tfrac{1}{\xi })-K(\partial _sz+\tfrac{1}{\xi })-K(Z_c)+K(\partial _sz)\right| \\&\quad =\frac{1}{|\xi |}\left| \int _0^1K'(Z_c+\tfrac{\tau }{\xi })-K'(\partial _sz+\tfrac{\tau }{\xi })\mathrm{d}\tau \right| \\&\quad \le C|\xi |^{-1}|Z_c-\partial _sz|\le C[\partial _sz]_{\alpha }c^{\alpha }|\xi |^{\alpha -1}, \end{aligned} \end{aligned}$$

which suffices to bound the integral. For the second and third terms, we obtain analogously

$$\begin{aligned} \left| K(Z_c)-K(\partial _sz)\right| +\left| K(Z_c+\tfrac{1}{\xi })-K(\partial _sz+\tfrac{1}{\xi })\right| \le 2|Z_c-\partial _sz|\le 2[\partial _sz]_{\alpha }(c|\xi |)^{\alpha }, \end{aligned}$$

which again allows to bound the integrals. The estimate (50) follows. $\square$

With Lemmas 3.2–3.7, we are now in a position to obtain an expansion in time as $t\rightarrow 0$ for the normal velocities $u^{(i)}_\nu$ for sufficiently regular z, c.

Proposition 3.8

Let $z(s,t)=\beta s+{{\tilde{z}}}(s,t)$ with ${{\tilde{z}}}\in C^1([0,T];C^{1,\alpha }_*(\mathbb {R}))$, $c>0$ with $\partial _sc\in C^{\alpha }_*(\mathbb {R})$, and define $c_i, \Phi _{ij}, z^{(i)}$ as in (14), (31) and (32). Then there exists a constant $C_{z,c}$ depending on $\Vert \tilde{z}\Vert _{1,\alpha }^*$, $\Vert \partial _t{{\tilde{z}}}\Vert _{1,\alpha }^*$, $\Vert \partial _sc\Vert _{\alpha }^*$, and N, such that

$$\begin{aligned} \Vert u^{(i)}_\nu -T_{\Phi _0}z_0\Vert ^*_0\le C_{z,c}t^{\alpha }, \end{aligned}$$

(51)

where $z_0(s)=z(s,0)$, $\Phi _0(\xi ,s)=\Phi (\xi ,s,0)$ and

$$\begin{aligned} \Phi (\xi ,s,t)=\frac{2\xi ^2}{\xi ^2+(z(s,t)-z(s-\xi ,t))^2}. \end{aligned}$$

(52)

Furthermore

$$\begin{aligned} \left\| \frac{1}{2N}\sum _{i}u^{(i)}_\nu -\left( T_{\Phi _0}z_0+tT_{\Psi _0}z_0+tT_{\Phi _0}z_1+t{\bar{c}}\sigma (\partial _sz_0)\partial _s^2z_0\right) \right\| _0^*\le C_{z,c}t^{1+\alpha },\nonumber \\ \end{aligned}$$

(53)

where $z_1(s)=\partial _tz(s,0)$, $\Psi _0(\xi ,s)=\partial _t\Phi (\xi ,s,0)$ and

$$\begin{aligned} {\bar{c}}(s)=\frac{1}{4N^2}\sum _{i,j}|c_i(s)-c_j(s)|. \end{aligned}$$

(54)

Proof

Using the representation (34) and the identity $\sum _jc_j=0$ (recall that here and below $\sum _j$ means summing over $1\le |j|\le N$, and that $c_{-j}=-c_j$), we write

$$\begin{aligned} \begin{aligned} u^{(i)}_\nu&=\frac{1}{N}\sum _{j}T_{\Phi _{ij}}z^{(j)}+t(\partial _sc_i-\partial _sc_j)I_{ij}\\&= T_{\Phi _0}z_0+\frac{1}{N}\sum _{j}\left[ T_{(\Phi _{ij}-\tfrac{1}{2}\Phi _0)}(z_0+tc_j)+T_{\Phi _{ij}}(z-z_0)+t(\partial _sc_i-\partial _sc_j)I_{ij}\right] . \end{aligned} \end{aligned}$$

For the first term in the summation, we can argue as in the proof of Lemma 3.4: indeed, $\Phi _{ij}-\tfrac{1}{2}\Phi _0=K(Z^{(j)}+\tfrac{tc_{ij}}{\xi })-K(Z)$ and $|Z^{(j)}+\tfrac{tc_{ij}}{\xi }-Z|\le 2t\tfrac{\Vert c\Vert _0}{|\xi |}$. For the second term in the summation, we apply Lemma 3.3 and Theorem 3.1, and for the third term Lemma 3.7. This way we arrive at estimate (51).

Next, we calculate

$$\begin{aligned} \begin{aligned} \sum _{i}u^{(i)}_\nu&=\frac{1}{N}\sum _{i,j}T_{\Phi _{ij}}z^{(j)}+t(\partial _sc_i-\partial _sc_j)I_{ij}\\&=\frac{1}{2N}\sum _{i,j}(T_{\Phi _{ij}}z^{(j)}+T_{\Phi _{ji}}z^{(i)})+t(\partial _sc_i-\partial _sc_j)(I_{ij}-I_{ji})\\&=2N\,T_{\Phi _0}z_0+\frac{1}{2N}\sum _{i,j}T_{(\Phi _{ij}+\Phi _{ji}-\Phi _0)}z_0+\frac{1}{N}\sum _{i,j}T_{\Phi _{ij}}(z^{(j)}-z_0)\\&\quad +\frac{1}{2N}\sum _{i,j}t(\partial _sc_i-\partial _sc_j)(I_{ij}-I_{ji}) \end{aligned} \end{aligned}$$

Now, let us write

$$\begin{aligned} \begin{aligned}&\Phi _{ij}+\Phi _{ji}-\Phi _0=\left[ K(Z^{(j)}+\tfrac{c_{ij}t}{\xi })-K(Z_0+\tfrac{c_{ij}t}{\xi })-tK'(Z_0)Z^{(j)}_1\right] \\&\quad +\left[ K(Z^{(i)}-\tfrac{c_{ij}t}{\xi })-K(Z_0-\tfrac{c_{ij}t}{\xi })-tK'(Z_0)Z^{(i)}_1\right] \\&\quad +tK'(Z_0)(Z^{(i)}_1+Z^{(j)}_1)+\left[ K(Z_0+\tfrac{c_{ij}t}{\xi })+K(Z_0-\tfrac{c_{ij}t}{\xi })-2K(Z_0)\right] , \end{aligned} \end{aligned}$$

where $Z_0(\xi ,s)=\frac{z_0(s)-z_0(s-\xi )}{\xi }$, $Z_1^{(i)}(\xi ,s)=\partial _t|_{t=0}Z^{(i)}(\xi ,s,t)$ and $c_{ij}=c_i-c_j$. Recalling the convention that $c_{-i}=-c_i$, we see that

$$\begin{aligned} \sum _{i,j}K'(Z_0)(Z^{(i)}_1+Z^{(j)}_1)=8N^2K'(Z_0)Z_1=4N^2\Psi _0. \end{aligned}$$

On the first two terms, we can argue as in Lemmas 3.3 and 3.4, whereas Lemma 3.5 applies to the last term.

Furthermore, we have

$$\begin{aligned} \frac{1}{N}\sum _{i,j}T_{\Phi _{ij}}(z^{(j)}-z_0)&=2NT_{\Phi _0}(z-z_0)+\frac{1}{N}\sum _{i,j}T_{\Phi _{ij}-\tfrac{1}{2}\Phi _0}(z-z_0+tc_j), \end{aligned}$$

and for the second term on the RHS, we argue again as above, using the strategy of Lemma 3.4. Finally, Lemma 3.7 implies that

$$\begin{aligned} |I_{ij}-I_{ji}|\le Ct^{\alpha }. \end{aligned}$$

Collecting all the terms, we then conclude the proof of (53). $\square$

Remark 3.9

In the statement of Proposition 3.8, we left the expression for ${\bar{c}}$ in the general form (54), which is valid for any choice of $0<c_1(s)<c_2(s)<\dots <c_N(s)$ with $c_{-i}(s)=-c_i(s)$ and $\partial _sc_i\in C^\alpha _*(\mathbb {R})$. The explicit choice of $c_i(s)$ in (14) leads to the following simplified expression:

$$\begin{aligned} \begin{aligned} {\bar{c}}(s)&=\tfrac{1}{2N^2}\sum _{i,j=1}^N|c_i(s)-c_j(s)|+|c_i(s)+c_j(s)|\\&=\tfrac{1}{N^2}\sum _{i,j=1}^N\max (c_i(s),c_j(s))=\tfrac{c(s)}{N^2(2N-1)}\sum _{i,j=1}^N(2\max (i,j)-1)\\&=\frac{2N+1}{6N}c(s). \end{aligned} \end{aligned}$$

Remark 3.10

It is instructive to compare the expansions in Proposition 3.8 with the expansion of the sharp interface evolution in (6). In particular, recall that the right hand side of (6) is the normal component of the velocity at the interface. Let us denote this by $v_{\nu }$, so that

$$\begin{aligned} v_{\nu }=T_{\Phi }z, \end{aligned}$$

with $\Phi$ defined in (52). In view of Lemma 3.2$v_{\nu }\in C^2([0,T];C^{\alpha }_*(\mathbb {R}))$ and therefore (51),(53) amount to the expansions

$$\begin{aligned} \begin{aligned} u^{(i)}_{\nu }&=v_{\nu }+O(t^{\alpha }),\\ \tfrac{1}{2N}\sum _iu^{(i)}_{\nu }&=v_{\nu }+t{\bar{c}}\sigma (\partial _sz_0)\partial _s^2z_0+O(t^{1+\alpha }). \end{aligned} \end{aligned}$$

4 Construction of the curve z

In this section, we construct a function $z=z(s,t)$ satisfying the conditions of Proposition 2.1.

Proposition 4.1

Let $z_0(s)=\beta s+{\tilde{z}}_0(s)$ with ${\tilde{z}}_0\in C^{3,\alpha }_*(\mathbb {R})$ for some $0<\alpha <1$ and $\beta \in \mathbb {R}$. Let $c=c(s)>0$ with $\partial _sc\in C^{\alpha }_*(\mathbb {R})$. If $\inf _sc(s)=0$, assume in addition that $\beta =0$ and there exists $c_\mathrm{min}>0$ such that

$$\begin{aligned} c(s)\ge c_\mathrm{min}(1+|s|^{2\alpha /3})^{-1}. \end{aligned}$$

(55)

For any $T>0$ there exists ${{\tilde{z}}}\in C^2([0,T];C^{1,\alpha }_*(\mathbb {R}))$ such that the conditions (20)–(22) of Proposition 2.1 are satisfied.

Proof

First of all, let us fix $f_1,f_2\in C_c^\infty (\mathbb {R})$ such that $\int _{I_k}f_l(s)\,\mathrm{d}s=\delta _{kl}$, where $I_1=(-\infty ,0)$ and $I_2=(0,\infty )$. We define

$$\begin{aligned} z(s,t)=z_0(s)+tz_1(s)+\tfrac{1}{2}t^2z_2(s)+\sum _{k=1}^2\psi _k(t)f_k(s), \end{aligned}$$

(56)

where

$$\begin{aligned} z_1&:=T_{\Phi _0}{\tilde{z}}_0,\\ z_2&:=T_{\Phi _0}z_1+T_{\Psi _0}{\tilde{z}}_0+\bar{c}\sigma (\partial _sz_0)\partial _s^2z_0, \end{aligned}$$

with $\Phi _0$, $\Psi _0$ and ${\bar{c}}$ as defined in Proposition 3.8, and $\psi _k\in C^2([0,T])$ are functions of time still to be fixed, such that $\psi _k(0)=\psi _k'(0)=\psi _k''(0)=0$ for $k=1,2$.

Let us check that z satisfies the conditions of Proposition 3.8. Since ${\tilde{z}}_0\in C^{3,\alpha }_*(\mathbb {R})$, Lemma 3.2 (applied with $z(s,t)=z_0(s)$) implies that $\Phi _0\in {\mathcal {W}}^{2,\alpha }$. Theorem 3.1 then implies that $z_1\in C^{2,\alpha }_*(\mathbb {R})$. Consequently, from the expression for $\Psi _0$ in Proposition 3.8 and using Lemma 3.2 again we deduce that $\Phi _1\in {\mathcal {W}}^{1,\alpha }$. Hence $z_2\in C^{1,\alpha }_*(\mathbb {R})$, using once more Theorem 3.1. Thus we have shown that $z\in C^2([0,T];C^{1,\alpha }_*(\mathbb {R}))$. It follows now from Proposition 3.8 that

$$\begin{aligned} \lim _{t\rightarrow 0}\sup _{s}(1+|s|^{1+\alpha })\left| \partial _tz(s,t)-u_{\nu }^{(i)}(s,t)\right|&= 0, \end{aligned}$$

(57)

$$\begin{aligned} \lim _{t\rightarrow 0}\sup _s\frac{1}{t}(1+|s|^{1+\alpha })\left| \partial _tz(s,t)-\frac{1}{2N}\sum _{i}u_\nu ^{(i)}(s,t)\right|&=0. \end{aligned}$$

(58)

In the case $\inf _s c(s)>0$ conditions (20)–(22) follow directly from (57)–(58). In particular, in this case we can take $\psi _k\equiv 0$ for $k=1,2$.

In what follows, let us then assume $\inf _s c(s)=0$, so that also $\beta =0$ and (55) holds. Observe first of all that (55) together with (57) directly implies (20). Furthermore,

$$\begin{aligned} |\partial _sz|,|\partial _sc|\lesssim (1+|s|^{1+\alpha })^{-1}\lesssim c^{1/2}, \end{aligned}$$

so that also (22) holds. It remains to verify (21). To this end, our aim is to choose $\psi _k$ in such a way that

$$\begin{aligned} \int _{I_k}\partial _tz(s,t)-\frac{1}{2N}\sum _{i}u_\nu ^{(i)}(s,t)\,\mathrm{d}s=0\quad \forall \,t\in [0,T], k=1,2. \end{aligned}$$

(59)

Indeed, assume for the moment that (59) holds. Then (58) implies for $s>0$

$$\begin{aligned} \int _0^s\partial _tz-\frac{1}{2N}\sum _{i}u_\nu ^{(i)}\,\mathrm{d}s'=-\int _s^\infty \partial _tz-\frac{1}{2N}\sum _{i}u_\nu ^{(i)}\,\mathrm{d}s'=\frac{o(t)}{1+|s|^{\alpha }}\sim o(t)c^{3/2}(s), \end{aligned}$$

and similarly for $s<0$. Thus in this case also (21) is verified.

To complete the proof, it therefore remains to choose $\psi _k$ to ensure (59). Let

$$\begin{aligned} h_k(t,\psi (t)):=\int _{I_k}\frac{1}{2N}\sum _{i}u_\nu ^{(i)}(s',t)-z_1(s')-tz_2(s')\,\mathrm{d}s', \end{aligned}$$

where dependence on $\psi =(\psi _1,\psi _2)$ appears in the implicit dependence of $u_\nu ^{(i)}$ on z defined in (56) via (30)–(32). Then, recalling the choice of $f_k$, (59) is equivalent to the ODE system

$$\begin{aligned} \psi _k'(t)=h_k(t,\psi (t)),\quad k=1,2 \end{aligned}$$

with initial condition $\psi _k(0)=0$. Using estimate (43) in Lemma 3.3, we verify that $x\mapsto h(t,x)$ is uniformly Lipschitz continuous. Therefore, the Cauchy–Lipschitz theorem is applicable and yields a unique solution $\psi :[0,T]\rightarrow \mathbb {R}^2$.

Then, by recalling our choice for $z_1$, $z_2$ and the expansion (53), we see that $h(t,0)=o(t)$. By differentiating the ODE, we also obtain $\tfrac{\mathrm{d}}{\mathrm{d}t}\psi '(t)=\partial _th+\partial _xh\psi '$. Since $\psi (0)=0$ and $h(0,0)=0$, we obtain $\psi '(0)=0$ and hence $\psi (t)=o(t)$. Furthermore, since $\partial _t h(0,0)=0$ also, we deduce $\psi ''(0)=0$, and furthermore, from the equation we deduce $\psi '(t)=o(t)$. This completes the proof.

$\square$