Global, Local and Dense Non-mixing of the 3D Euler Equation

Abstract

We prove a non-mixing property of the flow of the 3D Euler equation which has a local nature: in any neighbourhood of a “typical” steady solution there is a generic set of initial conditions, such that the corresponding Euler flows will never enter a vicinity of the original steady one. More precisely, we establish that there exist stationary solutions $u_0$ of the Euler equation on ${\mathbb {S}}^3$ and divergence-free vector fields $v_0$ arbitrarily close to $u_0$, whose (non-steady) evolution by the Euler flow cannot converge in the $C^k$ Hölder norm ($k>10$ non-integer) to any stationary state in a small (but fixed a priori) $C^k$-neighbourhood of $u_0$. The set of such initial conditions $v_0$ is open and dense in the vicinity of $u_0$. A similar (but weaker) statement also holds for the Euler flow on ${\mathbb {T}}^3$. Two essential ingredients in the proof of this result are a geometric description of all steady states near certain nondegenerate stationary solutions, and a KAM-type argument to generate knotted invariant tori from elliptic orbits.

Appendices

Appendix A: The Bernoulli Formulation of the Stationary Euler Equation

In Euclidean space, it is standard (see e.g. [2, Chapter II.1]) that the stationary Euler equation $\nabla _u u =-\nabla p$ can be written in the following equivalent way:

$$\begin{aligned} u\times \omega =\nabla {\mathcal {B}}\,, \;\; \mathrm{div}\,\, u=0\,, \end{aligned}$$

(A.1)

where ${\mathcal {B}}:= p+\frac{1}{2}|u|^2$ is the Bernoulli function. The same formulation holds for steady fluid flows on a general Riemannian 3-manifold (M, g). This fact is well known to experts, but difficult to find in the literature, so let us provide a proof.

First, we recall the definition of vector product in a Riemannian 3-manifold. If X and Y are two vector fields on M, its vector product $X\times Y$ is the unique vector field defined as

$$\begin{aligned} i_{X\times Y}g=i_Yi_X\mu \,, \end{aligned}$$

where $\mu$ is the volume form on M, and $i_Zg$ denotes the 1-form $Z^\flat$ dual to the vector field Z by means of the metric g.

The vorticity $\omega =\mathrm{rot}u$ of a field u is defined as the only vector field on M satisfying

$$\begin{aligned} i_\omega \mu =d\alpha \,, \end{aligned}$$

where the 1-form $\alpha :=u^\flat$ is metric-dual to the vector field u.

The first observation to prove equation (A.1) is that the 1-form $\left( \nabla _u u\right) ^\flat$ dual to the covariant derivative $\nabla _u u$ is [2, Chapter IV.1.D]

$$\begin{aligned} \left( \nabla _u u\right) ^\flat =L_u\alpha - \frac{1}{2} d(\alpha (u))\,. \end{aligned}$$

Using Cartan’s formula for the Lie derivative, this differential form can also be written as

$$\begin{aligned} i_ud\alpha +\frac{1}{2} d(\alpha (u))=i_ui_\omega \mu +\frac{1}{2} d(\alpha (u))\,, \end{aligned}$$

where we have used the definition of the vorticity to write the second expression. Accordingly, the stationary Euler equation $\nabla _u u=-\nabla p$ reads in terms of differential forms as

$$\begin{aligned} i_ui_\omega \mu +\frac{1}{2} d(\alpha (u))=-dp\,\,\,\,\,\Leftrightarrow \,\,\,\,\,i_\omega i_u\mu = d(p+\frac{1}{2} \alpha (u))\,. \end{aligned}$$

Using the definition of the vector product on manifolds, and setting ${\mathcal {B}}:=p+\frac{1}{2}\alpha (u)$ as the Bernoulli function, the (dual) vector formulation of the stationary equation reads as

$$\begin{aligned} u\times \omega =\nabla {\mathcal {B}}\,, \end{aligned}$$

as required.

Appendix B: Some Computations for the Shear Steady States

In this section we shall use the notations introduced in Sect. 3 without further mention. The dual 1-forms (using the Euclidean metric) of the vector fields $u_1$ and $u_2$ are

$$\begin{aligned} \alpha _1=-ydx+xdy+\xi dz -zd\xi \qquad \alpha _2=-ydx+xdy-\xi dz +zd\xi \,. \end{aligned}$$

It is easy to check that in terms of the coordinates $(\theta _1,\theta _2,\rho )$ on ${\mathbb {S}}^3$, these forms read as

$$\begin{aligned} \alpha _1=\rho d\theta _1-(1-\rho )d\theta _2\qquad \alpha _2=\rho d\theta _1+(1-\rho )d\theta _2\,, \end{aligned}$$

and their exterior derivatives are given by

$$\begin{aligned} d\alpha _1=-d\theta _1\wedge d\rho -d\theta _2\wedge d\rho \qquad d\alpha _2=-d\theta _1\wedge d\rho +d\theta _2\wedge d\rho \,. \end{aligned}$$

Recalling that the volume form in these coordinates is $\mu =\frac{1}{2} d\theta _1\wedge d\theta _2\wedge d\rho$, these expressions and the fact that $i_{\mathrm{rot}u_i}\mu = d\alpha _i$ imply that $\mathrm{rot}u_1=-2u_1$ and $\mathrm{rot}u_2=2u_2$.

Now we are ready to compute the rot of the vector field $u=f_1(\rho )u_1+f_2(\rho )u_2$. Its dual 1-form and exterior derivative are

$$\begin{aligned} \alpha&=\rho (f_1+f_2)d\theta _1+(1-\rho )(-f_1+f_2)d\theta _2\,,\\ d\alpha&=\Big (-\rho f'_1-\rho f'_2-f_1-f_2\Big )d\theta _1\wedge d\rho \\&\quad +\Big ((1-\rho )f'_1-(1-\rho )f'_2-f_1+f_2\Big )d\theta _2\wedge d\rho \,. \end{aligned}$$

Since $\mathrm{rot}u$ satisfies that $i_{\mathrm{rot}u}\mu =d\alpha$, a straightforward computation yields

$$\begin{aligned} \mathrm{rot}u&=\Big ((2\rho -1)(f'_2-f'_1)+2(f_2-f_1)+f'_1-f'_2\Big )\partial _{\theta _1}\\&\quad +\Big (2\rho (f'_1+f'_2)+2(f_1+f_2)\Big )\partial _{\theta _2}\,, \end{aligned}$$

which is equal to the expression given in Sect. 3.

To compute the Bernoulli function ${\mathcal {B}}$, we simply use the fact (see Appendix A) that

$$\begin{aligned} d{\mathcal {B}}=-i_ud\alpha \,, \end{aligned}$$

which, after a few computations, gives

$$\begin{aligned} d{\mathcal {B}}=\Big (f_1f_1+f_2f_2'+4f_1f_2+(2\rho -1)(f_1f_2'+f_2f_1')\Big )d\rho \,. \end{aligned}$$

Integrating this expression, we obtain that ${\mathcal {B}}$ is a function of $\rho$ given by the formula (3.5).

Appendix C: KAM Stability of Generic Elliptic Points

The goal of this appendix is to show that a generic elliptic point (in the sense of Lemma 4.3) of a $C^4$ area-preserving diffeomorphism of a disk is accumulated by continuous invariant curves with dense orbits, jointly forming a set of positive measure. This fact was stated by J. Moser [19, Theorem 2.12] with an idea of the proof given, its full implementation is apparently new. We learned the implementation of this idea, presented below, from D. Turaev, to whom we are grateful for the explanation.

Let $\Pi$ be a $C^4$ area-preserving diffeomorphism of a disk with an elliptic fixed point at the origin. Using the complex notation $z\in {\mathbb {C}}$, it can be written in the form

$$\begin{aligned} \Pi (z)=e^{i\omega }z+H(z,{\overline{z}})\,, \qquad z\in D_{r_0}\,, \end{aligned}$$

where H is a complex-valued function of class $C^4$, $\omega \in {\mathbb {R}}$ is the rotation number of the elliptic point $z=0$, and $D_{r_0}:=\{z: |z|<r_0\}$. We assume that the map is non-resonant in the sense that

$$\begin{aligned} e^{i\omega k} \ne 1\ \text { for } \ k=1,2,3,4 \,. \end{aligned}$$

(C.1)

Then, Birkhoff’s normal form theorem [19, Theorem 2.12] ensures that if $r_0$ is small enough, there exists an analytic area-preserving change of coordinates taking $\Pi$ to the form

$$\begin{aligned} \Pi (z)=e^{i(\omega +\alpha |z|^2)}z+{\widetilde{H}}(z,{\overline{z}})\,, \end{aligned}$$

where ${\widetilde{H}}$ is a $C^4$ function such that $\partial _{z,{\overline{z}}}^\beta {\widetilde{H}}(0,0)=0$ for $|\beta |\le 3$, which implies that

$$\begin{aligned} |\partial _{z,{\overline{z}}}^\beta {\widetilde{H}}|\le C|z|^{4-|\beta |}\ \text { in } \ D_{r_0},\ \text { if } \ |\beta |\le 4\,. \end{aligned}$$

(C.2)

We assume that the first Birkhoff constant $\alpha$ does not vanish; this is a generic condition.

Passing to the polar coordinates $z=re^{i\varphi }$, we rewrite $\Pi$ as

$$\begin{aligned} \Pi _r(r,\varphi )=r+f(r,\varphi )\,, \qquad \Pi _{\varphi }(r,\varphi )=\varphi +\omega +\alpha r^2+ {g(r,\varphi )} \,, \end{aligned}$$

where f and g are $C^4$ functions in the punctured disk $D_{r_0}\backslash \{0\}$. Rescaling the radius as $r=\epsilon R$, $\epsilon \ll 1$, we write $\Pi$ in the form

$$\begin{aligned} \Pi _R(R,\varphi )=R+F(R,\varphi )\,, \quad \Pi _{\varphi }(R,\varphi )=\varphi +\omega +\alpha \epsilon ^2 R^2+G(R,\varphi )\,, \end{aligned}$$

(C.3)

where

$$\begin{aligned} (R, \varphi ) \in {\mathbb {A}}:=\{1<R<2, \varphi \in [0,2\pi ] \}\,, \end{aligned}$$

and $F(R,\varphi )= \epsilon ^{-1} f(\epsilon R, \varphi )$, $G(R, \varphi ) = g(\epsilon R, \varphi )$.

Now we will estimate the functions G and F. To estimate G, we define the auxiliary function

$$\begin{aligned} \Phi (R, \varphi ) := {{\widetilde{H}}(z,{\overline{z}})}{z^{-1}} e^{-i(\omega +\alpha |z|^2)}\;,\text { with} \; z= \epsilon Re^{i\varphi }\,, \end{aligned}$$

so that $G(R,\varphi )=\text {Im}\ln \Big (1+\Phi (R, \varphi )\Big )$. If $(R, \varphi ) \in {\mathbb {A}}$, then $\epsilon<r<2\epsilon$. So by (C.2) with $\beta =0$, for small $\epsilon$ we have that $|\Phi | <1/2$, and accordingly $\ln \big (1+\Phi (R,\varphi ) \big )$ is a $C^4$ function of $(R, \varphi )$ in ${\mathbb {A}}$. Noticing that $\partial ^k/\partial R^k = \epsilon ^k\partial ^k/\partial r^k$, we derive from (C.2) the estimates

$$\begin{aligned} \Big | \frac{\partial ^k}{\partial R^k} {{\widetilde{H}}(R, \varphi )} \Big | =\epsilon ^k \Big | \frac{\partial ^k}{\partial r^k} {{\widetilde{H}}} \Big | \le C\epsilon ^4,\quad \Big | \frac{\partial ^k}{\partial \varphi ^k} {{\widetilde{H}}(R, \varphi )} \Big | \le C\epsilon ^4\,, \end{aligned}$$

(C.4)

for $0\le k\le 4$ and $(R,\varphi )\in {\mathbb {A}}$. It then follows that $\partial ^\beta _{R, \varphi } \ln \big (1+\Phi (R, \varphi ) \big ) =O(\epsilon ^3)$ for $(R,\varphi ) \in {\mathbb {A}}$ and $|\beta |\le 4$. Therefore $\Vert G\Vert _{C^{4}({\mathbb {A}})}\le C\epsilon ^3\,.$

To estimate F we consider the function $(\Pi _R)^2= R^2 +2RF +F^2 =: R^2+ J(R, \varphi ).$ Then $F= (\Pi _R^2)^{1/2} -R = R\big ( \sqrt{1+R^{-2} J(R,\varphi })-1\big )$. Since $\epsilon ^2\Pi _R^2(R,\varphi ) = |\Pi (z)|^2$ with $z= \epsilon Re^{i\varphi }$, then

$$\begin{aligned} J(R,\varphi ) = 2\epsilon ^{-1} \text {Re} \big ( e^{-i(\omega +\epsilon ^2 \alpha R^2) }R e^{-i\varphi } {{\widetilde{H}}(R, \varphi )} \big ) +\epsilon ^{-2} {{\widetilde{H}}} \overline{{\widetilde{H}}}(R, \varphi )\,, \end{aligned}$$

and in view of the estimates (C.4), $\Vert J\Vert _{C^{4}({\mathbb {A}})}<C\epsilon ^3$. We then conclude that $\Vert F \Vert _{C^{4}({\mathbb {A}})}<C\epsilon ^3$.

To study the invariant curves of the map (C.3), we consider its Nth iterates with $N\le \epsilon ^{-2}$. Writing the map $\Pi ^N$ as

$$\begin{aligned} \Pi ^N_R(R,\varphi )=R+F_N(R,\varphi )\,, \quad \Pi ^N_{\varphi }(R,\varphi )=\varphi +\omega _N+\alpha N\epsilon ^2 R^2+G_N(R,\varphi )\,, \end{aligned}$$

where $\omega _N:=N\omega \, (\text {mod }2\pi )$, we check by induction that $F_N$ and $G_N$ are bounded as $\Vert F_N\Vert _{C^4({\mathbb {A}})}<C\epsilon$ and $\Vert G_N\Vert _{C^{4}({\mathbb {A}})}<C\epsilon ,$ for $1\le N \le \epsilon ^{-2}$.

Denote by $N_\epsilon$ the integer part of $\epsilon ^{-2}$. Then $\alpha N_\epsilon \epsilon ^2 R^2 = \alpha R^2 +O(\epsilon ^2)$. Next let us choose a sequence $\epsilon _j\rightarrow 0$ such that $\omega _{N_j}\rightarrow \Omega \, (\text {mod }2\pi )$, where $N_j$ stands for $N_{\epsilon _j}$. We set $\ \delta _j = \max ( |\omega _{N_j} - \Omega |, \epsilon _j),$ and consider the twist-map

$$\begin{aligned} {\widetilde{\Pi }}(R,\varphi )=(R,\varphi +\Omega + \alpha R^2), \qquad \alpha \ne 0. \end{aligned}$$

Then $\ \Vert \Pi ^{N_j}-{\widetilde{\Pi }}\Vert _{C^{4}({\mathbb {A}})}<C\delta _j\,.$ Now Herman’s twist theorem [14] implies that for j large enough the area–preserving map $\Pi ^{N_j}$ of the annulus ${\mathbb {A}}$ has a positive measure set of continuous quasi-periodic invariant curves with Diophantine rotation numbers.

These curves also are invariant for the map $\Pi$. This follows from the uniqueness of Diophantine invariant curves in Herman’s twist theorem [14, Section 5.10] (if $\gamma$ is a Diophantine invariant curve of $\Pi ^{N_j}$, since $\Pi (\gamma )$ is also an invariant curve of $\Pi ^{N_j}$ with the same rotation number, then $\Pi (\gamma )=\gamma$). However let us provide a self-contained proof of this fact for the sake of completeness. Indeed, let $\gamma$ be one of the curves, invariant for $\Pi ^{N_j}$, and let $\gamma '$ be this curve, written in the z-variable. It is invariant for the original diffeomorphism $\Pi (z)$. To prove this, denote by $D'$ the domain bounded by $\gamma '$. Since the map $\Pi (z)$ is area-preserving, then the boundary of $\Pi (D')$ intersects $\gamma '$, i.e. $\Pi (\gamma ')$ intersects $\gamma '$, and $\Pi (\gamma )$ intersects $\gamma$. Take any $p\in \gamma$ such that $\Pi (p)\in \gamma$. Considering its orbit under the mapping $\Pi ^{N_j}$, $\Gamma :=\cup _{m\in {\mathbb {Z}}}\Pi ^{mN_j}(p)\subset \gamma$, we observe that $\Pi (\Gamma )\subset \gamma$. Indeed, denoting by $\gamma _m:=\Pi ^{mN_j}(p)$ and $\gamma _m':=\Pi (\gamma _m)$ , we have that $\gamma _0'=\Pi (p)\in \gamma$, $\gamma _1'=\Pi ^{N_j+1}(p)=\Pi ^{N_j}(\gamma _0')\in \gamma$, etc. Since $\Gamma$ is dense in $\gamma$ (it is a quasi-periodic trajectory of the map $\Pi ^{N_j}$), we conclude that $\Pi (\gamma )\subset \gamma$, and in fact $\Pi (\gamma )=\gamma$ because $\Pi$ is a diffeomorphism, thus showing that $\gamma$ is an invariant curve of $\Pi$. Clearly an orbit of each point in $\gamma$ is dense in $\gamma$.

Summarising, we conclude that the map $\Pi (z)$ has a positive measure set of continuous invariant curves with dense orbits; written in the complex Birkhoff’s coordinate z they lie in the annulus $\{\epsilon _j<|z|<2\epsilon _j\}$. Since the above argument can be repeated for infinitely many values of the small parameter $\epsilon _j\rightarrow 0$, we deduce the following:

If the area-preserving map $\Pi (z)$ satisfies (C.1) and its first Birkhoff constant $\alpha \ne 0$, then $\Pi$ exhibits a positive measure set of continuous invariant curves with dense trajectories, contained in a sequence of disjoint annuli accumulating at the origin, so the KAM-stability of the elliptic fixed point follows.

Appendix D: The Vey Theorem in ${\mathbb {R}}^2$

Consider the plane ${\mathbb {R}}^2=\{ x=(x_1,x_2)\}$, endowed with the standard area-form $\omega _0= dx_1\wedge dx_2$. By $D_\rho$ we denote the disc $\{|x|<\rho \},\ \rho >0$, and we define the action variable $I:=r^2/2$, where $(r,\phi )$ denote the polar coordinates in the plane.

Theorem D.1

Let $H\in C^k(D_\rho ), \ k\ge 4$, such that $H(0)=0$, $dH(0)=0$ and $d^2 H(0) >0$. Then there exists a $C^{k-3}$ area-preserving diffeomorphism $\Psi ^+: D_\rho \rightarrow D_{\rho '}$, $\Psi ^+(0) =0$, and a $C^{[k/2]-1}$ function h, $h(0)=0$, $h'(0)\ne 0$, such that $H(x) = h\big ( I(\Psi ^+(x))\big )$.

An analytic version of this result is due to Vey and is well known [25]. We were not able to find in the literature a finite-smoothness version of this theorem and instead give below its proof; it is based on ideas of Eliasson’s work [9], where a much more complicated multi-dimensional version of this result is established without explicit control on how the smoothness of $\Psi ^+$ depends on k. More precisely, we follow the interpretation of Eliasson’s proof, given in [16, pp. 10-15], skipping the infinite-dimensional technicalities. Our notation mostly agrees with that of [16], and we refer the reader to that work for missing details.

Proof

Multiplying H by a positive constant and making a linear area-preserving change of coordinates we can safely assume that $D^2H(0) = \tfrac{1}{2}$Id. Accordingly, all changes of variables we are performing below are of the form $x\mapsto x+O(x^2)$. By “a germ" we mean “a germ at the origin" of a function, or of a vector-field, etc..

Step 1 (Morse lemma). Applying Morse lemma we find a germ of diffeomorphism³ of class $C^{k-1}$ $\Psi$ such that

$$\begin{aligned} i) \qquad H(x) = \tfrac{1}{2} |\Psi (x)|^2, \qquad \Psi (x) = x+ O(x^2)\,. \qquad \qquad \qquad \qquad \qquad \end{aligned}$$

We denote by G the germ of $\Psi ^{-1}$ and set

$$\begin{aligned}&\omega _1 :=G^* \omega _0, \; \omega _\Delta := \omega _1 - \omega _0,\; \alpha _0:= \tfrac{1}{2}( x_1dx_2 - x_2dx_1),\;\\&\alpha _1 := G^*\alpha _0, \; \alpha _\Delta := \alpha _1-\alpha _0\,. \end{aligned}$$

Then $d\alpha _j = \omega _j$ and $d\alpha _\Delta = \omega _\Delta$. Write $\alpha _\Delta$ as W(x)dx; then $W= O(x^2)$. Clearly all introduced objects are $C^{k-2}$-smooth.

We denote by $(\cdot , \cdot )$ the standard scalar product in ${\mathbb {R}}^2$, and by $\langle \cdot , \cdot \rangle$ the usual pairing between 1-forms and vector fields, and write 2-forms in ${\mathbb {R}}^2$ as (J(x)dx, dx), where J(x) is an antisymmetric operator in ${\mathbb {R}}^2$ (i.e., a $2\times 2$ antisymmetric matrix), and $(J(x) dx, dx)(\xi ,\eta ) = (J(x) \xi , \eta )$ for any two vector fields $\xi ,\eta$. Then $\omega _0 = (J_0 dx, dx)$, where $J_0(x_1,x_2) = (-x_2/2, x_1/2)$, and $\omega _1(x) = ({\bar{J}}(x) dx, dx)$ with ${\bar{J}}(0) = J_0$.

Step 2 (Averaging in angle). Denote by $\Phi _\theta , \ \theta \in {\mathbb {R}}$, the operator of rotation by angle $\theta$, $\Phi _\theta (r,\phi ) := (r, \phi +\theta )$. Then $\Phi ^*_\theta \alpha _0 =\alpha _0$, $\Phi ^*_\theta \omega _0 =\omega _0$ and

$$\begin{aligned} \Phi ^*_\theta (V(x)dx) = \Phi _{-\theta } V(\Phi _\theta x) dx, \quad \Phi ^*_\theta (J(x)dx, dx) = (\Phi _{-\theta } J(\Phi _\theta x)\Phi _\theta dx, dx). \end{aligned}$$

It is easy to see that $\Phi _{-\theta } J(\Phi _\theta x)\Phi _\theta =J(\Phi _\theta x)$. Now consider the averaging operator M, where for a function f, $Mf(x) :=\frac{1}{2\pi }\int _0^{2\pi }f(\Phi _tx)\,dt$, while for a form $\beta$, $M\beta (x) :=\frac{1}{2\pi }\int _0^{2\pi }(\Phi _t^*\beta )(x)\,dt$. Then

$$\begin{aligned} M(J(x) dx, dx) = (MJ(x) dx, dx),\qquad MJ(x) =\frac{1}{2\pi }\int _0^{2\pi }\Phi _{-\theta } J(\Phi _\theta x) \Phi _\theta \,d\theta \,. \end{aligned}$$

Accordingly, the form $M\omega _1$ is $(M{\bar{J}}(x) dx, dx)$. Writing the operator ${\bar{J}}(x)$ as

$$\begin{aligned} {\bar{J}}(x) = J_0 +(x, \nabla _x {\bar{J}}(0)) + J_2(x), \qquad J_2 =O(x^2)\,, \end{aligned}$$

we easily see that $M{\bar{J}} = J_0 + MJ_2$, where $C^{k-2}\ni MJ_2(x) =O(x^2)$. Now consider the linear homotopy of $J_0$ and $M{\bar{J}}$:

$$\begin{aligned} (M{\bar{J}})^\tau := (1-\tau )J_0 +\tau M{\bar{J}} = J_0 + \tau MJ_2\,, \end{aligned}$$

with $\tau \in [0,1]$, and set ${\hat{J}}^\tau (x) := - \big ((M{\bar{J}})^\tau (x)\big )^{-1}= -\big ( J_0 +\tau M J_2(x)\big )^{-1}$. This is a germ of a operator-valued map (an antisymmetric $2\times 2$ matrix) of class $C^{k-2}$.

Let us return to the 1-form $\alpha _\Delta =W(x) dx$, consider MW(x), which is given by

$$\begin{aligned} MW(x)=\frac{1}{2\pi }\int _0^{2\pi }\Phi _{-\theta }W(\Phi _\theta x)\,d\theta \,, \end{aligned}$$

and define the $C^{k-2}$ non-autonomous vector-field $V^\tau (x) := {\hat{J}}^\tau (x) (MW)(x)$; obviously $V^\tau = O(x^2)\,.$ Consider the differential equation

$$\begin{aligned} \dot{x}(\tau ) = V^\tau (x(\tau ))\,, \end{aligned}$$

and denote by $\varphi _\tau$, $0\le \tau \le 1$, the germ of its flow-map. Then $\varphi _0=$id, $\varphi _\tau (x) = x+ O(x^2)$ and all germs $\varphi _\tau$ commute with rotations because the vector field $V^\tau$ does. By direct calculation, using Cartan’s formula and that $d (M\alpha _\Delta )=M\omega _1-\omega _0$, we verify that $\varphi _\tau ^* {\hat{\omega ^\tau }}= \mathrm{const},$ where ${\hat{\omega ^\tau }} = \big ( (M{\bar{J}}(x))^\tau dx, dx\big ).$ Therefore

$$\begin{aligned} \varphi _1^* {\hat{\omega }}^1 = \varphi _1^* M\omega _1 = {\hat{\omega }}^0 = \omega _0\,. \end{aligned}$$

(D.1)

Now let us set ${\bar{\Psi }} :=\varphi _1^{-1} \circ \Psi$. This is a germ of a $C^{k-2}$ diffeomorphism, satisfying ${\bar{\Psi }} (x) = x+O(x^2)$. Since $\varphi _1^{-1}$ commutes with rotations, then $2I( \varphi _1^{-1}(y)) = {\tilde{h}}(2I(y))$. To see which kind of function ${\tilde{h}}$ is, let us denote $2I= r^2$ and take $y=(r,0)$. Then $2I(y) = r^2$, so ${\tilde{h}}(r^2) = 2I( \varphi _1^{-1}(r,0)) =: f(r)$. The function f is of class $C^{k-2}$ and even, so by Whitney’s theorem [26] ${\tilde{h}}\in C^{[k/2] -1}$. Since $\varphi _1^{-1} (r,0) = (r,0) +O(r^2)$, then ${\tilde{h}}(2I) = 2I +o(I)$, and ${\tilde{h}}'(0) =1$. Thus relation i) implies that

$$\begin{aligned} i') \qquad H(x) = h(I({\bar{\Psi }}(x)), \qquad h(0)=0,\ h'(0) =1, \ h\in C^{[k/2] -1} \,. \qquad \qquad \qquad \end{aligned}$$

Now we re-denote ${\bar{\Psi }}$ to $\Psi$. We have arrived at the same situation as in Step 1, but with $\Psi \in C^{k-2}$, i) is replaced by $i')$ and, in view of (D.1),

$$\begin{aligned} M\omega _1 = \omega _0\,, \end{aligned}$$

(D.2)

where we recall that $\omega _1=(\Psi ^{-1})^*\omega _0$.

Step 3 (End of the proof). By  (D.2), $dM\alpha _\Delta = M d\alpha _\Delta = M( \omega _1 -\omega _0) =0$. So $M\alpha _\Delta = dg$, $g(0)=0$, where g is a $C^{k-1}$-germ. Since $dMg = Mdg = M\alpha _\Delta$, then by replacing g with Mg we achieve that $dg =M\alpha _\Delta$ and $Mg=g$. Accordingly, $\nabla g(0)=0$ and $g(x) = O(x^2)$. Denote by $\chi$ the vector field of rotations $-x_2 \partial _{x_1} + x_1\partial _{x_2}$. Since g is rotationally invariant, $\langle dg, \chi \rangle =0$, so

$$\begin{aligned} M\langle \alpha _\Delta , \chi \rangle = \langle M\alpha _\Delta , \chi \rangle = \langle dg, \chi \rangle =0\,. \end{aligned}$$

(D.3)

We set $T(x) = \langle \alpha _\Delta , \chi \rangle$. This is the germ of a $C^{k-2}$-function, satisfying $MT=0$. Since $\alpha _\Delta = O(x^2)$ and $\chi = O(x)$, then $T=O(x^3)$. Let us consider the differential equation for a germ of a function f:

$$\begin{aligned} \chi (f) = T(x). \end{aligned}$$

(D.4)

Since $MT=0$, it is easy to solve it in polar coordinates for a germ f, satisfying $Mf=0$: $f(r,\phi ) = \frac{1}{2\pi }\int _0^{2\pi }t \, T(r, \phi +t)\,dt.$ In Cartesian coordinates the solution f reads as

$$\begin{aligned} f(x) = \frac{1}{2\pi }\int _0^{2\pi }t \, T(\Phi _t(x))\,dt\,. \end{aligned}$$

Similarly to T, $f\in C^{k-2}$ and $f=O(x^3)$.

Recalling that $(\Psi ^*)^{-1} \omega _0 =: \omega _1 = ({\bar{J}}(x) dx, dx)$, where ${\bar{J}}\in C^{k-2}$ with ${\bar{J}}(0) = J_0$, we interpolate $J_0$ and ${\bar{J}}$ by setting ${\bar{J}}^\tau := (1-\tau ) J_0 + \tau {\bar{J}},$ and define $J^\tau (x) := -({\bar{J}}^\tau (x))^{-1}$. Then ${\bar{J}}^\tau$ and $J^\tau$ are germs of antisymmetric operators of class $C^{k-2}$. Denote

$$\begin{aligned} \omega ^\tau := (1-\tau ) \omega _0 + \tau \omega _1 = ( {\bar{J}}^\tau (x) dx, dx)\,, \end{aligned}$$

and set $V^\tau (x) := J^\tau (x) (W(x) -\nabla f(x)).$ Then $C^{k-3} \ni V^\tau = O(x^2)$. Consider the ODE

$$\begin{aligned} \dot{x} = V^\tau (x), \quad 0\le \tau \le 1\,, \end{aligned}$$

(D.5)

and denote by $\varphi _\tau$ the germ of its flow-maps. Then $\varphi _\tau (x) = x+O(x^2)$, and another simple calculation shows that $\varphi _\tau ^* \omega ^\tau = \mathrm{const}.$ Thus $\varphi _1^* \omega _1 = \omega _0$. That is, the germ of the diffeomorphism

$$\begin{aligned} \Psi ^+ := \varphi _1^{-1} \circ \Psi \in C^{k-3}, \qquad \Psi ^+(x) = x+O(x^2) \end{aligned}$$

satisfies $(\Psi ^+)^*\omega _0 =\omega _0$, i.e. $\Psi ^+$ is an area-preserving diffeomorphism.

Finally, notice that

$$\begin{aligned} \omega ^\tau (V^\tau (x), J_0x)= & {} ( {\bar{J}}^\tau (x) V^\tau (x), J_0x) = -(W(x) -\nabla f(x), J_0x) \\= & {} - \frac{1}{2}\langle \alpha _\Delta , \chi \rangle + \frac{1}{2}\langle df, \chi \rangle =0, \end{aligned}$$

by Eq. (D.4) (where $T= \langle \alpha _\Delta , \chi \rangle$). Since $\omega ^\tau (V^\tau , V^\tau )=0$, then $V^\tau (x) \parallel J_0x$, i.e. the vector field $V^\tau$ is tangent to the foliation defined by $\chi$. Therefore, the solutions of Eq. (D.5) satisfy $(d/ d\tau ) |x(\tau )|^2 = 2 (V^\tau (x), x) =0.$ That is, $|\varphi _\tau (x)|^2= |x|^2$ for all $\tau$. Then, $I(\varphi _\tau (x)) =I(x)$, so the germ $\Psi ^+$ still satisfies $i')$, which completes the proof of the theorem. $\square$

The theorem above implies Vey’s theorem for local area preserving diffeomorphisms of class $C^k$ in ${\mathbb {R}}^2$.

Corollary D.2

Let $\Pi$ be a $C^k$ area preserving diffeomorphism of $D_{\rho _1}$ onto its image, $\Pi (0)=0$. Assume that it admits a first integral $H\in C^k(D_{\rho _1})$ (i.e. $H\circ \Pi =H$) such that $H(0)=0$, $dH(0)=0$ and $d^2 H(0) >0$. Then there exists $\rho _0>0$ and an area preserving $C^{k-3}$-smooth change of variables $\Psi : D_{\rho _0} \rightarrow \Psi ( D_{\rho _0} ) \subset D_{\rho _1}$, such that $\Psi (0)=0$ and the transformed diffeomorphism ${\hat{\Pi }} = \Psi ^{-1}\circ \Pi \circ \Psi$ in polar coordinates reads as ${\hat{\Pi }} (r,\phi ) =(r, \phi + {\hat{W}}(r^2)),$ for some $C^{[(k-3)/2]}$ function ${\hat{W}}$.

Proof

By Theorem D.1 there exists an area preserving local $C^{k-3}$–diffeomorphism $\Psi$ such that $\Psi (0)=0$ and $H\circ \Psi = {\hat{H}}(x)$, where ${\hat{H}}(x) =h(x_1^2+ x_2^2)$ for some $C^{[k/2]-1}$ function h. Since H is a first integral of $\Pi$, then the transformed map ${\hat{\Pi }} = \Psi ^{-1}\circ \Pi \circ \Psi$ in polar coordinates reads as ${\hat{\Pi }}(r,\phi )=(r,\phi +V(r,\phi ))$. The fact that ${\hat{\Pi }}$ preserves the standard area form $rdr\wedge d\phi$ implies that $V(r,\phi )\equiv V(r)$. Since V is a $C^{k-3}$ even function, then Whitney’s theorem ensures that $V(r)={\hat{W}}(r^2)$ for some $C^{[(k-3)/2]}$ function ${\hat{W}}$, thus proving the desired result. $\square$

Remark D.3

We note that, obviously, the function ${\hat{W}}(t)$ in Corollary D.2 is of class $C^{k-3}$ for $t>0$.

Appendix E: Suspension of Area-Preserving Diffeomorphisms

Let $A(a,b):={\mathbb {S}}^1\times (a,b)$ be an annular domain with $0<a<b<1$, and consider the toroidal manifold $M:=A(0,1)\times {\mathbb {S}}^1$. We can endow it with coordinates $(\theta _1,\rho ,\theta _2)\in {\mathbb {S}}^1\times (0,1)\times {\mathbb {S}}^1$ and with the canonical volume form $d\theta _1\wedge d\rho \wedge d\theta _2$. Assume that w is a $C^k$ divergence-free vector field on M that is transverse to the section $\{\theta _2=0\}$. Its first return map at this section defines a $C^k$ diffeomorphism (onto its image) ${\mathcal {P}}^w:A(a,b)\rightarrow A(0,1)$ that preserves an area form $\mu _2$. The following suspension result is due to D. Treschev [24]:

Theorem E.1

Let $\Pi : A(a,b)\rightarrow A(0,1)$ be a $C^k$ map that preserves the area $\mu _2$. We assume that $\Pi$ is $C^k$-close to ${\mathcal {P}}^w$, i.e. $\Vert \Pi -{\mathcal {P}}^w\Vert _{C^k(A(a,b))}<\delta$, and that $\Pi ={\mathcal {P}}^w$ in a neighborhood of $\partial A(a,b)$. Then if $\delta$ is sufficiently small, there exists a $C^k$ divergence-free vector field ${\hat{w}}$ on M transverse to the section $\{\theta _2=0\}$ whose Poincaré map is ${\mathcal {P}}^{{\hat{w}}}=\Pi$, $\delta$-close to w, that is $\Vert w-{\hat{w}}\Vert _{C^{k}(M)}<C\delta$, and such that ${\hat{w}}=w$ in a neighborhood of $\partial A(a,b)\times {\mathbb {S}}^1$.