1 Introduction

We consider the following family of equations referred to as the hyperelastic rod equation

$$\begin{aligned} u_t-u_{txx} + 3uu_x = \gamma ( 2 u_x u_{xx} + uu_{xxx}),\quad t \in {\mathbb {R}}, x \in {\mathbb {R}}\end{aligned}$$
(1)

where \(\gamma \ne 0\). The initial value problem for (1) is locally well-posed in the Sobolev spaces \(H^s, s > 3/2\)—see [6, 7]. For the corresponding solution map it was shown in [5] in the periodic case that it has not the property to be uniformly continuous on bounded sets, whereas in [3] the same was shown for both cases (periodic and nonperiodic) with an improvement for the s range. Our aim here is to prove that the solution map for the range \(s > 3/2\) has even less regularity. But before we state the main theorem we have to introduce some notation. Note that (1) has the property that for a solution u the scaled quantity

$$\begin{aligned} u_\lambda :=\lambda u(\lambda t,x),\quad \lambda > 0 \end{aligned}$$
(2)

is also a solution. Let \(s > 3/2\) and \(T > 0\). Denote by \(U_T\) the initial values \(u_0 \in H^s({\mathbb {R}})\) for which (1), starting from \(u(0)=u_0\), has a solution which exists longer than T. By the local well-posedness of (1) in \(H^s({\mathbb {R}})\) (see also Theorem 2.1) and scaling (2) we know that \(U_T \subseteq H^s({\mathbb {R}})\) is an open star-shaped neighborhood of \(0 \in H^s({\mathbb {R}})\) in \(H^s({\mathbb {R}})\). Again by the local well-posedness we know that the time T solution map

$$\begin{aligned} \varPhi _T:U_T \subseteq H^s({\mathbb {R}}) \rightarrow H^s({\mathbb {R}}),\quad u_0 \mapsto u(T) \end{aligned}$$

is continuous. With this our main theorem reads as

Theorem 1.1

Let \(s > 3/2\) and \(T > 0\). Denote by \(\varPhi _T\) the time T solution map of the initial value problem for (1) defined on \(U_T \subseteq H^s({\mathbb {R}})\). Then

$$\begin{aligned} \varPhi _T:U_T \rightarrow H^s({\mathbb {R}}),\quad u(0) \mapsto u(T) \end{aligned}$$

is nowhere locally uniformly continuous.

We will rewrite (1) by doing the transformation \(v(t,x)=u(t,\gamma x)\). This gives

$$\begin{aligned} v_t - \frac{1}{\gamma ^2} v_{txx} + \frac{3}{\gamma } v v_x = \frac{1}{\gamma ^2} ( 2v_x v_{xx} + v v_{xxx}) \end{aligned}$$

or rewritten

$$\begin{aligned} \left( 1-\frac{1}{\gamma ^2} \partial _x^2\right) (v_t + vv_x) = \frac{\gamma -3}{\gamma } v v_x - \frac{1}{\gamma ^2} v_x v_{xx} \end{aligned}$$

and equivalently

$$\begin{aligned} v_t + vv_x = \left( 1-\frac{1}{\gamma ^2} \partial _x^2\right) ^{-1}\left( \frac{\gamma -3}{\gamma } v v_x - \frac{1}{\gamma ^2} v_x v_{xx}\right) =:B(v,v) \end{aligned}$$
(3)

Note that B is a continuous quadratic form on \(H^s({\mathbb {R}})\) for \(s > 3/2\). We will establish Theorem 1.1 by showing the corresponding statement for the solution map of v. This is clearly sufficient. The advantage of (3) is that it is convenient for the geometric framework introduced in the next section.

2 The geometric framework

We will formulate (3) in a geometric way as was done in [2] for the b-family of equations. Consider the flow map of v, i.e.,

$$\begin{aligned} \varphi _t(t,x) = v(t,\varphi (t,x)),\quad \varphi (0,x)=x \end{aligned}$$

The functional space for the \(\varphi \) variable is for \(s > 3/2\) the diffeomorphism group

$$\begin{aligned} {\mathcal {D}}^s({\mathbb {R}})=\{ \varphi :{\mathbb {R}}\rightarrow {\mathbb {R}}\;|\; \varphi -\text{ id } \in H^s({\mathbb {R}}),\quad \varphi _x(x) > 0 \text{ for } \text{ all } x \in {\mathbb {R}}\} \end{aligned}$$

where \(\text{ id }\) is the identity map in \({\mathbb {R}}\). It is a topological group under composition of maps and consists of \(C^1\)-diffeomorphisms. For details on this space, see [1]. We can write (3) in the \(\varphi \) variable as

$$\begin{aligned} \varphi _{tt} = B\left( \varphi _t \circ \varphi ^{-1},\varphi _t \circ \varphi ^{-1}\right) \circ \varphi \end{aligned}$$
(4)

The computations in [2] show that right side is a real analytic map

$$\begin{aligned} {\mathcal {D}}^s({\mathbb {R}}) \rightarrow P_2(H^s({\mathbb {R}});H^s({\mathbb {R}})), \quad \varphi \mapsto \left[ v \mapsto B\left( v \circ \varphi ^{-1},v \circ \varphi ^{-1}\right) \circ \varphi \right] \end{aligned}$$

where we denote by \(P_2(H^s({\mathbb {R}});H^s({\mathbb {R}}))\) the space of continuous quadratic forms on \(H^s({\mathbb {R}})\) with values in \(H^s({\mathbb {R}})\). We can write the second-order Eq. (4) as a first-order equation on the tangent space \(T{\mathcal {D}}^s({\mathbb {R}})={\mathcal {D}}^s({\mathbb {R}}) \times H^s({\mathbb {R}})\)

$$\begin{aligned} \partial _t \left( \begin{array}{c} \varphi \\ v \end{array} \right) = \left( \begin{array}{c} v \\ B\left( v \circ \varphi ^{-1},v \circ \varphi ^{-1}\right) \circ \varphi \end{array}\right) \end{aligned}$$
(5)

The quadratic nature of the second component makes it to a so-called Spray—see [4]. It has in particular an exponential map. To define this map consider the ODE (5) with initial values \(\varphi (0)=\text{ id }\) and \(v(0)=v_0\). Denote by \(V \subseteq H^s({\mathbb {R}})\) those initial values \(v_0\) for which we have existence beyond time 1. With this we define

$$\begin{aligned} \exp :V \subseteq H^s({\mathbb {R}}) \rightarrow {\mathcal {D}}^s({\mathbb {R}}),\quad v_0 \mapsto \varphi (1;v_0) \end{aligned}$$

where \(\varphi (1;v_0)\) is the time 1 value of the \(\varphi \)-component. Because of analytic dependence on initial values \(\exp \) is real analytic. Furthermore for any \(v_0 \in H^s({\mathbb {R}})\) the curve \(\varphi (t)=\exp (t v_0)\) is the \(\varphi \)-component of the solution to (5) with initial values \(\varphi (0)=\text{ id }\) and \(v(0)=v_0\). In particular the solution exists as long as \(tv_0 \in V\).

With this we can construct solutions to (3). So consider (3) with initial condition \(v(0)=v_0 \in H^s({\mathbb {R}})\). For \(\varphi (t)=\exp (tv_0)\) we define

$$\begin{aligned} v(t)=\varphi _t(t) \circ \varphi (t)^{-1} \end{aligned}$$

It turns out that v is a solution to (3)—see [2], where this was established for the b-family of equations—and with this that \(V \subseteq H^s({\mathbb {R}})\) is the set of initial values for which (3) has a solution beyond time \(T=1\). By the local well-posedness for ODEs we immediately recover the local well-posedness result of [6, 7].

Theorem 2.1

The initial value problem for (1) is locally well-posed in the Sobolev spaces \(H^s({\mathbb {R}}), s > 3/2\).

3 Nonuniform dependence

In this section we establish our main result Theorem 1.1. As mentioned already it will be enough to prove this for the modified Eq. (3). We can further simplify this by considering the theorem just for the time \(T=1\) situation, as we have for v a solution to (3) that

$$\begin{aligned} {\tilde{v}}(t,x):=\lambda v(\lambda t,x) \end{aligned}$$

is also a solution to (3).

We proceed as in [2]. In [2] we used a conserved quantity to establish the result. For equation (3) we have something similar.

Lemma 3.1

Let \(s >3/2\). For v a solution to (3) with initial value \(v(0)=v_0 \in H^s({\mathbb {R}})\) we have

$$\begin{aligned} \left( \left( 1-\frac{1}{\gamma ^2}\partial _x^2\right) v(t)\right) \circ \varphi (t) \cdot \varphi _x(t)^2 = \left( 1-\frac{1}{\gamma ^2}\partial _x^2\right) v_0 + \int _0^t \frac{3\gamma -3}{\gamma } \frac{\varphi _t(s) \varphi _{tx}(s)}{\varphi _x(s)} \;\mathrm{d}s \end{aligned}$$
(6)

where \(\varphi (t)=\exp (tv_0)\).

The essential thing here is that the “remainder” term, the integral term which is in \(H^{s-1}({\mathbb {R}})\), is more regular than the first term which is in \(H^{s-2}({\mathbb {R}})\).

Proof

We differentiate the expression on the left in the lemma with respect to t and use the equation for v. We have by the chain rule

$$\begin{aligned} \frac{d}{\mathrm{d}t} \left( \left( \left( 1-\frac{1}{\gamma ^2}\partial _x^2\right) v\right) \circ \varphi \right)= & {} \left( \left( 1-\frac{1}{\gamma ^2}\partial _x^2\right) v_t\right) \circ \varphi + \left( \left( 1-\frac{1}{\gamma ^2}\partial _x^2\right) v_x\right) \circ \varphi \cdot \varphi _t\\= & {} \left( \left( 1-\frac{1}{\gamma ^2}\partial _x^2\right) v_t + \left( \left( 1-\frac{1}{\gamma ^2}\partial _x^2\right) v_x\right) \cdot v\right) \circ \varphi \\= & {} \left( \left( 1-\frac{1}{\gamma ^2}\partial _x^2\right) v_t + \left( 1-\frac{1}{\gamma ^2}\partial _x^2\right) (v_x \cdot v) + \frac{3}{\gamma ^2} v_x v_{xx}\right) \circ \varphi \\= & {} \left( \frac{\gamma -3}{\gamma } vv_x + \frac{2}{\gamma ^2} v_x v_{xx}\right) \circ \varphi \end{aligned}$$

where we used Eq. (3) in the last equality. Therefore we have

$$\begin{aligned} \frac{d}{\mathrm{d}t} \left( \left( \left( 1-\frac{1}{\gamma ^2}\partial _x^2\right) v\right) \circ \varphi \cdot \varphi _x^2\right)= & {} \left( vv_x + \frac{2}{\gamma ^2} v_x v_{xx}\right) \circ \varphi \cdot \varphi _x^2 +\left( v-\frac{1}{\gamma ^2} v_{xx}\right) 2 \varphi _x \varphi _{tx} \\= & {} \frac{3\gamma -3}{\gamma } (v v_x) \circ \varphi = \frac{3\gamma -3}{\gamma } \frac{\varphi _t \varphi _{tx}}{\varphi _x} \end{aligned}$$

where we used \(\varphi _{tx} \varphi _x^{-1} = v_x \circ \varphi \). As \(\varphi (0)=\text{ id }\) integrating gives the result in the case where we work with regular solutions. But as long as \(\Vert v_x\Vert _{L^\infty }\) is controlled (similar to the Beale–Majda–Kato criterium) one has continuation of the solution—see [7]. Thus by approximation by regular solutions one has (6) for all \(s > 3/2\). \(\square \)

In the following we will use the notation

$$\begin{aligned} y(t):=\left( 1-\frac{1}{\gamma ^2} \partial _x^2\right) v(t) \quad \text{ and } \quad \varPsi (t)=\int _0^t \frac{3\gamma -3}{\gamma } \frac{\varphi _t(s) \varphi _{tx}(s)}{\varphi _x(s)} \;\mathrm{d}s \end{aligned}$$

Hence from (6)

$$\begin{aligned} y(1) = \left( \frac{y(0)}{\varphi _x(1)^2}\right) \circ \varphi (1)^{-1} + \left( \frac{\varPsi (1)}{\varphi _x(1)^2} \right) \circ \varphi (1)^{-1} \end{aligned}$$

In the following we will use also \(\varPsi _{v_0}:=\varPsi (1)\) for the corresponding initial value \(v_0\). Theorem 1.1 will follow from

Proposition 3.2

Let \(V \subseteq H^s({\mathbb {R}})\) be the domain of definition of \(\exp \) (which is also the domain of definition for the time \(T=1\) solution map of (3)). We denote by v(t) solutions to (3). Then the map

$$\begin{aligned} \varPhi :V \subseteq H^s({\mathbb {R}}) \rightarrow H^s({\mathbb {R}}),\quad v(0) \mapsto v(1) \end{aligned}$$

is nowhere locally uniformly continuous.

To prove Proposition 3.2 we will show that \(y(0) \mapsto y(1)\) is nowhere locally uniformly continuous. This is clearly enough. Before doing this we state some facts—see [2] for the proofs.

For \(\varphi _\bullet \in {\mathcal {D}}^s({\mathbb {R}})\) there is \(C > 0\) with

$$\begin{aligned} \frac{1}{C} \left\| \left( \frac{y}{{\widetilde{\varphi }}_x^2}\right) \circ \varphi ^{-1}\right\| _{s-2} \le \Vert y\Vert _{s-2} \le C \left\| \left( \frac{y}{{\widetilde{\varphi }}_x^2}\right) \circ \varphi ^{-1}\right\| _{s-2} \end{aligned}$$

for all \(y \in H^{s-2}({\mathbb {R}})\) and for all \({\widetilde{\varphi }}, \varphi \) in some neighborhood of \(\varphi _\bullet \).

For \(\varphi _\bullet \in {\mathcal {D}}^s({\mathbb {R}})\) there is \(C > 0\) with

$$\begin{aligned} \left\| f \circ \varphi _1^{-1} - f \circ \varphi _2^{-1}\right\| _{s-2} \le C \Vert f\Vert _{s-1} \left\| \varphi _1^{-1}-\varphi _2^{-1}\right\| _{s-2} \end{aligned}$$

for all \(f \in H^{s-1}({\mathbb {R}})\) and for all \(\varphi _1,\varphi _2\) in a neighborhood of \(\varphi _\bullet \).

Further we construct a dense subset \(S \subseteq V\) with \(S \subseteq H^{s+1}({\mathbb {R}})\) and \(d_v \exp \ne 0\) for all \(v \in S\). Here \(d_v \exp \) is the differential of the exponential map at v. Take an arbitrary \(v \in V \cap H^{s+1}({\mathbb {R}})\) and \(w \in H^s({\mathbb {R}}), x \in {\mathbb {R}}\) with \(w(x) \ne 0\). Consider the analytic map

$$\begin{aligned} {\mathbb {R}}\rightarrow {\mathbb {R}},\quad t \mapsto \left( d_{tv}\exp (w)\right) (x) \end{aligned}$$

which at \(t=0\) is w(x) (see [4] for the fact that \(d_0 \exp \) is the identity map), in particular nonzero. Thus there is a sequence \(t_n \uparrow 1\) with \(\left( d_{t_nv}\exp (w)\right) (x) \ne 0\). So putting \(t_n v\) to S gives the construction we need.

With this preparation we can proceed to the proof of Proposition 3.2. It is essentially the same proof as in [2] established for the b-family of equations.

Proof of Proposition 3.2

We take \(v_0 \in S \subseteq H^{s+1}({\mathbb {R}})\) in the dense subset and show that \(\varPhi \) is not uniformly continuous on any ball \(B_R(v_0) \subseteq V\) of radius \(R >0\) with center \(v_0\). By the construction of S we can fix \(g \in H^s({\mathbb {R}})\) and \(x_0 \in {\mathbb {R}}\) with

$$\begin{aligned} \left( d_{v_0} \exp (g)\right) (x_0) > m \Vert g\Vert _s \end{aligned}$$

for some \(m > 0\). Denote by \(\varphi _\bullet =\exp (v_0)\). We choose \(R_1 > 0\) in such a way that we have

$$\begin{aligned} \frac{1}{C_1} \left\| \left( \frac{y}{{\widetilde{\varphi }}_x^2}\right) \circ \varphi ^{-1}\right\| _{s-2} \le \Vert y\Vert _{s-2} \le C_1 \left\| \left( \frac{y}{{\widetilde{\varphi }}_x^2}\right) \circ \varphi ^{-1}\right\| _{s-2} \end{aligned}$$

for some \(C_1 > 0\) for all \(y \in H^{s-2}({\mathbb {R}})\) and \({\tilde{\varphi }}, \varphi \in \exp (B_{R_1}(v_0))\) which is possible due to the continuity properties of the composition—see [1]. Taking \(0 < R_2 \le R_1\) we can guarantee again by the continuity properties of the composition that

$$\begin{aligned} \left\| y \circ \varphi ^{-1}\right\| _{s-2} \le C_2 \Vert y\Vert _{s-2} \end{aligned}$$

for some \(C_2\) and for all \(y \in H^{s-2}({\mathbb {R}})\) and \(\varphi \in \exp (B_{R_2}(v_0))\). Choosing \(0 < R_3 \le R_2\) we can ensure (see [1])

$$\begin{aligned} \left\| f \circ \varphi _1^{-1}-f \circ \varphi _2^{-1}\right\| _{s-2} \le {\tilde{C}}_3 \Vert f\Vert _{s-1} \left\| \varphi _1^{-1}-\varphi _2^{-1}\right\| _{s-2} \le C_3 \Vert f\Vert _{s-1} \left\| \varphi _1-\varphi _2\right\| _s \end{aligned}$$

for some \(C_3 > 0\) and for all \(f \in H^{s-1}({\mathbb {R}})\) and \(\varphi _1,\varphi _2 \in \exp (B_{R_3}(v_0))\). Furthermore we denote by \(C > 0\) the constant in the Sobolev imbedding

$$\begin{aligned} \Vert f\Vert _{L^\infty } \le C \Vert f\Vert _s \end{aligned}$$

Consider the Taylor expansion for the exponential map \(\exp :V \rightarrow H^s({\mathbb {R}})\)

$$\begin{aligned} \exp (w+h) = \exp (w) + d_w \exp (h) + \int _0^1 (1-t) d_{w+th}\exp (h,h) \;\mathrm{d}t \end{aligned}$$

We choose \(0 < R_4 \le R_3\) in such a way that we have

$$\begin{aligned} \left\| d_w^2 \exp (h_1,h_2)\right\| _s \le K \Vert h_1\Vert _s \Vert h_2\Vert _s \end{aligned}$$

and

$$\begin{aligned} \left\| d_{w_1}^2 \exp (h_1,h_2)-d_{w_2}^2 \exp (h_1,h_2)\right\| _s \le K \Vert w_1-w_2\Vert _s \Vert h_1\Vert _s \Vert h_2\Vert _s \end{aligned}$$

for some \(K > 0\) and for all \(w,w_1,w_2 \in \exp (B_{R_4}(v_0))\) and for all \(h_1,h_2 \in H^s({\mathbb {R}})\) which is possible by the smoothness of the exponential map. By taking \(0 < R_5 \le R_4\) small enough we have

$$\begin{aligned} \max \left\{ C \cdot K \cdot R_5,C \cdot K \cdot R_5^2\right\} < m/2 \end{aligned}$$

By the final choice \(0 < R_*\le R_5\) we can make

$$\begin{aligned} |\varphi (x)-\varphi (y)| \le L |x-y| \text{ and } \Vert \varPsi _v\Vert _s \le M \text{ and } \left\| \exp (v)-\exp ({\tilde{v}})\right\| _s \le L \Vert v-{\tilde{v}}\Vert _s \end{aligned}$$

to hold for all \(\varphi \in \exp (B_{R_*})\) and \(v,{\tilde{v}} \in B_{R_*}(v_0)\) due to the Sobolev imbedding and the smoothness of the exponential map. The goal is to prove that \(\varPhi \) is not uniformly continuous on \(B_R(v_0)\) for any \(0 < R \le R_*\). So we fix \(0 < R \le R_*\). We define the sequence of radii

$$\begin{aligned} r_n = \frac{m}{8n} \Vert g\Vert _s, \quad n \ge 1 \end{aligned}$$

and take an arbitrary smooth \(w_n\) with support in \((x_0-\frac{r_n}{L},x_0+\frac{r_n}{L})\) and constant mass \(\Vert w_n\Vert _s = R/4\). Further we define \(g_n=g/n\), which tends to zero in \(H^s({\mathbb {R}})\). With this we introduce two sequences

$$\begin{aligned} z_n=v_0 + w_n \quad \text{ and } \quad {\tilde{z}}_n= z_n + g_n=v_0 + w_n + g_n \end{aligned}$$

For N large enough we clearly have \(z_n, {\tilde{z}}_n \in B_R(v_0)\) for \(n \ge N\) and \(\Vert z_n-{\tilde{z}}_n\Vert _s \rightarrow 0\) as \(n \rightarrow \infty \). Further we introduce the corresponding diffeomorphisms

$$\begin{aligned} \varphi _n=\exp (z_n) \quad \text{ and } \quad {\widetilde{\varphi }}_n=\exp ({\tilde{z}}_n) \end{aligned}$$

The result will follow from \(\limsup _{n \rightarrow \infty } \Vert \varPhi (z_n)-\varPhi ({\tilde{z}}_n)\Vert _s > 0\). Reexpressing \(\varPhi \) with (6) and using the notation \(y_n=\left( 1-\frac{1}{\gamma ^2}\partial _x^2\right) z_n\) and \({\tilde{y}}_n=\left( 1-\frac{1}{\gamma ^2}\partial _x^2\right) {\tilde{z}}_n\) and \(\varPsi _{z_n},\varPsi _{{\tilde{z}}_n}\) for the “remainder” terms this is equivalent to

$$\begin{aligned} \limsup _{n \rightarrow \infty } \left\| \frac{y_n}{(\varphi _n)_x^2} \circ \varphi _n^{-1} - \frac{{\tilde{y}}_n}{({\widetilde{\varphi }}_n)_x^2} \circ {\widetilde{\varphi }}^{-1} + \frac{\varPsi _{z_n}}{(\varphi _n)_x^2} \circ \varphi _n^{-1} - \frac{\varPsi _{{\tilde{z}}_n}}{({\widetilde{\varphi }}_n)_x^2} \circ {\widetilde{\varphi }}_n^{-1}\right\| _{s-2} > 0 \end{aligned}$$

As the \(\varPsi \) terms are more regular than \(H^{s-2}\), namely in \(H^{s-1}\), we have

$$\begin{aligned}&\left\| \frac{\varPsi _{z_n}}{(\varphi _n)_x^2} \circ \varphi _n^{-1} - \frac{\varPsi _{{\tilde{z}}_n}}{({\widetilde{\varphi }}_n)_x^2} \circ {\widetilde{\varphi }}_n^{-1}\right\| _{s-2} \\&\quad \le \left\| \frac{\varPsi _{z_n}}{(\varphi _n)_x^2} \circ \varphi _n^{-1} - \frac{\varPsi _{z_n}}{(\varphi _n)_x^2} \circ {\widetilde{\varphi }}_n^{-1}\right\| _{s-2} + \left\| \frac{\varPsi _{z_n}}{(\varphi _n)_x^2} \circ {\widetilde{\varphi }}_n^{-1} - \frac{\varPsi _{{\tilde{z}}_n}}{({\widetilde{\varphi }}_n)_x^2} \circ {\widetilde{\varphi }}_n^{-1}\right\| _{s-2} \\&\quad \le C_3 \left\| \frac{\varPsi _{z_n}}{(\varphi _n)_x^2} \right\| _{s-1} \left\| \varphi _n-{\widetilde{\varphi }}_n\right\| _s + C_2 \left\| \frac{\varPsi _{z_n}}{(\varphi _n)_x^2}- \frac{\varPsi _{{\tilde{z}}_n}}{({\widetilde{\varphi }}_n)_x^2}\right\| _{s-2} \rightarrow 0 \end{aligned}$$

as \(n \rightarrow \infty \) since \(z \mapsto \varPsi _z/(\partial _x \exp (z))^2\) is smooth. Thus it remains to establish

$$\begin{aligned} \limsup _{n \rightarrow \infty } \left\| \frac{y_n}{(\varphi _n)_x^2} \circ \varphi _n^{-1} - \frac{{\tilde{y}}_n}{({\widetilde{\varphi }}_n)_x^2} \circ {\widetilde{\varphi }}_n^{-1}\right\| _{s-2} > 0 \end{aligned}$$

We split

$$\begin{aligned} y_n = \left( 1-\frac{1}{\gamma ^2}\partial _x^2\right) (v_0 + w_n) \text{ resp. } {\tilde{y}}_n = \left( 1-\frac{1}{\gamma ^2}\partial _x^2\right) (v_0 + w_n + g_n) \end{aligned}$$

As \(v_0 \in H^{s+1}\) we can treat the \(v_0\) terms in the same way as the \(\varPsi \) terms and get

$$\begin{aligned} \lim _{n \rightarrow \infty } \left\| \frac{\left( 1-\frac{1}{\gamma ^2}\partial _x^2\right) v_0}{(\varphi _n)_x^2} \circ \varphi _n^{-1} - \frac{\left( 1-\frac{1}{\gamma ^2}\partial _x^2\right) v_0}{\left( {\widetilde{\varphi }}_n\right) _x^2} \circ {\widetilde{\varphi }}_n^{-1}\right\| _{s-2} = 0 \end{aligned}$$

For the \(g_n\) term we have trivially

$$\begin{aligned} \left\| \frac{\left( 1-\frac{1}{\gamma ^2}\partial _x^2\right) g_n}{\left( {\widetilde{\varphi }}_n\right) _x^2} \circ {\widetilde{\varphi }}_n^{-1}\right\| _{s-2} \le C_1 \left\| \left( 1-\frac{1}{\gamma ^2}\partial _x^2\right) g_n\right\| _{s-2} \rightarrow 0 \end{aligned}$$

The only remaining thing is to consider

$$\begin{aligned} \limsup _{n \rightarrow \infty } \left\| \frac{\left( 1-\frac{1}{\gamma ^2}\partial _x^2\right) w_n}{(\varphi _n)_x^2} \circ \varphi _n^{-1} - \frac{\left( 1-\frac{1}{\gamma ^2}\partial _x^2\right) w_n}{({\widetilde{\varphi }}_n)_x^2} \circ {\widetilde{\varphi }}_n^{-1}\right\| _{s-2} \end{aligned}$$

In order to estimate this from below we will establish that the two terms have disjoint support. This we do by estimating the distance \(|\varphi _n(x_0)-{\widetilde{\varphi }}_n(x_0)|\). By the Taylor expansion we have

$$\begin{aligned} \varphi _n = \exp (v_0) + d_{v_0} \exp (w_n) + \int _0^1 (1-t) d_{v_0+tw_n}^2 \exp (w_n,w_n) \;\mathrm{d}t \end{aligned}$$

resp.

$$\begin{aligned} {\widetilde{\varphi }}_n = \exp (v_0) + d_{v_0} \exp (w_n+g_n) + \int _0^1 (1-t) d_{v_0 + t(w_n+g_n)}^2 \exp (w_n+g_n,w_n+g_n) \;\mathrm{d}t \end{aligned}$$

Taking the difference we can write

$$\begin{aligned} {\widetilde{\varphi }}_n - \varphi _n = d_{v_0}\exp (g_n) + {\mathcal {R}}_1 + {\mathcal {R}}_2 + {\mathcal {R}}_3 \end{aligned}$$

where

$$\begin{aligned} {\mathcal {R}}_1 = \int _0^1 (1-t) (d_{v_0+t(w_n+g_n)}^2(w_n,w_n)-d_{v_0+tw_n}^2(w_n,w_n))\;\mathrm{d}t \end{aligned}$$

and

$$\begin{aligned} {\mathcal {R}}_2 = \int _0^1 (1-t) d_{v_0+t(w_n+g_n)}^2(g_n,g_n) \;\mathrm{d}t \end{aligned}$$

and

$$\begin{aligned} {\mathcal {R}}_2 = 2 \int _0^1 (1-t) d_{v_0+t(w_n+g_n)}^2(w_n,g_n) \;\mathrm{d}t \end{aligned}$$

For these we have

$$\begin{aligned} \Vert {\mathcal {R}}_1\Vert _\infty \le C \Vert {\mathcal {R}}_1\Vert _s \le C K \Vert g_n\Vert _s \Vert w_n\Vert _s^2 \le \frac{1}{n} C K \Vert g\Vert _s (R/4)^2 \le \frac{1}{4n} C K R^2 \Vert g\Vert _s \end{aligned}$$

and

$$\begin{aligned} \Vert {\mathcal {R}}_2\Vert _\infty \le C \Vert {\mathcal {R}}_2\Vert _s \le 2 C K \Vert g_n\Vert _s \Vert w_n\Vert _s \le \frac{1}{n} C K \Vert g\Vert _s (R/4) \le \frac{2}{4n} C K R \Vert g\Vert _s \end{aligned}$$

and

$$\begin{aligned} \Vert {\mathcal {R}}_3\Vert _\infty \le C \Vert {\mathcal {R}}_3\Vert _s \le C K \Vert g_n\Vert _s^2 \le \frac{1}{n} C K \Vert g\Vert _s (R/4) \le \frac{1}{4n} C K R \Vert g\Vert _s \end{aligned}$$

Therefore

$$\begin{aligned} |\varphi _n(x_0)-{\widetilde{\varphi }}_n(x_0)|\ge & {} |d_{v_0}\exp (g_n)| - \Vert {\mathcal {R}}_1\Vert _\infty - \Vert {\mathcal {R}}_2\Vert _\infty - \Vert {\mathcal {R}}_3\Vert _\infty \\\ge & {} \frac{1}{n} m \Vert g\Vert _s - \frac{1}{n} \frac{m}{2} \Vert g\Vert _s = \frac{m}{2n} \Vert g\Vert _s \end{aligned}$$

The support of \(\frac{\left( 1-\frac{1}{\gamma ^2}\partial _x^2\right) w_n}{(\varphi _n)_x^2} \circ \varphi _n^{-1}\) is contained in \((\varphi _n(x_0)-r_n,\varphi _n(x_0)+r_n)\) taking into account the Lipschitz property of \(\varphi _n\) with Lipschitz constant L and the definition of \(w_n\). Analogously the support of \(\frac{\left( 1-\frac{1}{\gamma ^2}\partial _x^2\right) w_n}{({\widetilde{\varphi }}_n)_x^2} \circ {\widetilde{\varphi }}_n^{-1}\) is contained in \(({\widetilde{\varphi }}_n(x_0)-r_n,{\widetilde{\varphi }}_n(x_0)+r_n)\). As we have

$$\begin{aligned} r_n \le |\varphi _n(x_0)-{\tilde{\varphi }}_n(x_0)|/4 \end{aligned}$$

we can “separate” the disjointly supported terms (see also [2]). Thus we have

$$\begin{aligned}&\limsup _{n \rightarrow \infty } \left\| \frac{\left( 1-\frac{1}{\gamma ^2}\partial _x^2\right) w_n}{(\varphi _n)_x^2} \circ \varphi _n^{-1}-\frac{\left( 1-\frac{1}{\gamma ^2}\partial _x^2\right) w_n}{({\widetilde{\varphi }}_n)_x^2} \circ {\widetilde{\varphi }}_n^{-1}\right\| _{s-2}^2 \\&\quad \ge \limsup _{n \rightarrow \infty } {\tilde{C}} \left( \left\| \frac{\left( 1-\frac{1}{\gamma ^2}\partial _x^2\right) w_n}{(\varphi _n)_x^2} \circ \varphi _n^{-1}\right\| _{s-2}^2+\left\| \frac{\left( 1-\frac{1}{\gamma ^2}\partial _x^2\right) w_n}{({\widetilde{\varphi }}_n)_x^2} \circ {\widetilde{\varphi }}_n^{-1}\right\| _{s-2}^2\right) \\&\quad \ge \limsup _{n \rightarrow \infty } {\tilde{C}} \frac{2}{C^2} \left\| \left( 1-\frac{1}{\gamma ^2}\partial _x^2\right) w_n\right\| _{s-2}^2 \ge \limsup _{n \rightarrow \infty } {\tilde{K}} \Vert w_n\Vert _s^2 = {\tilde{K}} R^2/4 \end{aligned}$$

So for any \(R \le R_*\) we have constructed \((z_n)_{n \ge N},({\tilde{z}}_n)_{n \ge N} \subseteq B_R(u_0)\) with \(\lim _{n \rightarrow \infty } \Vert z_n-{\tilde{z}}_n\Vert _s=0\) and \(\limsup _{n \rightarrow \infty } \Vert \varPhi (z_n)-\varPhi ({\tilde{z}}_n)\Vert _s \ge C \cdot R\) for some constant \(C > 0\) independent of R showing the claim. \(\square \)