Abstract
We consider the smooth, compactly supported solutions of the steady 3D Euler equations of incompressible fluids constructed by Gavrilov (Geom Funct Anal (GAFA) 29(1):190–197, 2019), and we study the corresponding fluid particle dynamics. This is an ode analysis, which contributes to the description of Gavrilov’s vector field.
Similar content being viewed by others
Avoid common mistakes on your manuscript.
1 Introduction and Main Result
In the remarkable paper [10], Gavrilov proved the existence of a nontrivial solution of \(C^\infty \) class, with compact support, of the steady Euler equations of incompressible fluids in \({{\mathbb {R}}}^3\). The result in [10] is important and surprising, because previously, on the basis of some negative partial results, it was conjectured that compactly supported, nontrivial, smooth solutions of the 3D steady Euler equations cannot exist: see the clear explanation at the beginning of [5] and the general discussion about existence of compactly supported smooth solutions in [13]. In addition, another reason of interest for the fruitful construction of [10] is that recently it has been used as a building block to produce other interesting solutions, both stationary and time-dependent, of the Euler equations of fluid dynamics, see Sect. 1.3 below.
Now suppose that a fluid moves according to the Gavrilov solution of the Euler equations, that is, suppose that the fluid particles are driven by Gavrilov’s velocity vector field. Which movement of the fluid do we observe? Of course the particles outside the support of the vector field do not move at all, but how do they move in the region where the field is nonzero?
In the present paper we deal with this question. It turns out that every fluid particle travels along a trajectory that lies on a nearly toroidal surface, which is a level set of the pressure. The motion of every fluid particle is periodic or quasi-periodic in time; we prove that, in fact, there are both periodic and quasi-periodic motions, and the value of the pressure determines whether the trajectories on its level set are all periodic or all quasi-periodic.
In fact, the system of differential equations in \({{\mathbb {R}}}^3\) describing the motion of the fluid particles turns out to be integrable, as it can be transformed into the system of a Hamiltonian system of one degree of freedom and a third equation that can be directly solved by integration. We write the Hamiltonian system in angle-action coordinates \((\sigma , I)\), and prove that there exists a change of variables on a neighborhood of the support of Gavrilov’s vector field such that the equations of motion in the new coordinates \((\sigma , \beta , I)\) becomes
where \(\sigma \) and \(\beta \) are angle variables rotating with constant angular velocities \(\Omega _1(I)\), \(\Omega _2(I)\), and I is a constant action variable, which is, in fact, a reparametrization of the pressure. The full statement is in Theorem 1.1.
1.1 The Gavrilov Solutions of the Steady Euler Equations
In the main part of the construction in [10], given any \(R > 0\), the circle
in \({{\mathbb {R}}}^3\) is considered, and, in an open neighborhood \(\mathcal {N}\) of \(\mathcal {C}\), two functions U, P are defined, \(U: \mathcal {N}\rightarrow {{\mathbb {R}}}^3\) and \(P: \mathcal {N}\rightarrow {{\mathbb {R}}}\), both in \(C^\infty (\mathcal {N}{\setminus } \mathcal {C})\), solving the steady Euler equations
in \(\mathcal {N}\setminus \mathcal {C}\), together with the fundamental “localizability condition”
As a final step of the proof, the functions U, P are multiplied by smooth cut-off functions to obtain \(C^\infty ({{\mathbb {R}}}^3)\) functions \({\tilde{U}}, {\tilde{P}}\), where \({\tilde{U}}\) and \(\nabla {\tilde{P}}\) have compact support contained in \(\mathcal {N}\setminus \mathcal {C}\), solving (1.1) (and also (1.2)) in \({{\mathbb {R}}}^3\).
Let us be more precise. Denote \(\rho := \sqrt{x^2 + y^2}\). For \(\delta \in (0,R)\), let
In \(\mathcal {N}\), the solution (U, P) of [10] is given by
where
and \(\alpha ,H\) are functions defined in [10] in terms of solutions of certain differential equations; \(H(0) = 0\), and H is analytic in a neighborhood of 0; \(\alpha \) has a strict local minimum at (1, 0), with \(\alpha (1,0) = 0\), and it is analytic in a neighborhood of (1, 0). Hence \(\alpha \) and \(H \circ \alpha \) are both well-defined and analytic in a disc of \({{\mathbb {R}}}^2\) of center (1, 0) and radius \(r_0\), for some universal constant \(r_0 > 0\) (where “universal” means that \(r_0\) does not depend on anything). If \(\delta \) in (1.3) satisfies
then \(a(\rho ,z)\) and \(H(a(\rho ,z))\), where \(\rho = \sqrt{x^2 + y^2}\), are well-defined and analytic in \(\mathcal {N}\) (note that, in \(\mathcal {N}\), one has \(0< R-\delta< \rho < R+\delta \); in particular, \(\rho \) is bounded away from zero). Also, \(b(\rho ,z)\) is well-defined and continuous in \(\mathcal {N}\), and (because of the square root \(\sqrt{H}\)) it is analytic in \(\mathcal {N}\setminus \mathcal {C}\). Hence P is analytic in \(\mathcal {N}\), while U is continuous in \(\mathcal {N}\) and analytic in \(\mathcal {N}{\setminus } \mathcal {C}\).
The solution \(({\tilde{U}}, {\tilde{P}})\) in [10] is defined in \(\mathcal {N}\) as
where \(\omega : {{\mathbb {R}}}\rightarrow {{\mathbb {R}}}\) is any \(C^\infty \) function vanishing outside the interval \([\varepsilon , 2 \varepsilon ]\), with \(\varepsilon > 0\) small enough, and \(W: {{\mathbb {R}}}\rightarrow {{\mathbb {R}}}\) is a primitive of \(\omega ^2\); for example,
Then \(({\tilde{U}}, {\tilde{P}})\) is extended to \({{\mathbb {R}}}^3\) by defining \(({\tilde{U}}, {\tilde{P}}) = (0, W(2\varepsilon ))\) in \({{\mathbb {R}}}^3 \setminus \mathcal {N}\).
Note that \({\tilde{U}}\) and \(\nabla {\tilde{P}}\) can be nonzero only in the set
and, if \(\omega (s)\) is nonzero at some \(s \in (\varepsilon , 2\varepsilon )\), then both \({\tilde{U}}\) and \(\nabla {\tilde{P}}\) are actually nonzero at the corresponding points in \(\mathcal {S}\). Moreover, \(P = 0\) on the circle \(\mathcal {C}\), and, for \(\varepsilon \) small enough, \(P > 3 \varepsilon \) at all points of \(\mathcal {N}\) sufficiently far from \(\mathcal {C}\); more precisely, to fix the details of the construction, we introduce a parameter \(\tau > 0\) and assume that
and \(\tau \ge 3 \varepsilon \), where
Thus, the closure of \(\mathcal {S}\) is contained in the open set
and \(\mathcal {S}^* \subseteq (\mathcal {N}' {\setminus } \mathcal {C}) \subseteq (\mathcal {N}{\setminus } \mathcal {C})\).
1.2 Main Result: Description of the Fluid Particle Dynamics
A preliminary, basic observation regarding the solutions of Sect. 1.1 is that any such solution with a given \(R>0\) can be obtained, by rescaling, from another one having \(R=1\). Even more, we show that Gavrilov’s solutions are a 1-parameter subset of a larger 2-parameter family of solutions, where R plays a dual role related to two different scaling invariances of the Euler equations, both preserving the localizability condition (1.2). These basic observations are in Sect. 2 (see Lemma 2.1). Thanks to these properties, we study the motion of the fluid particles in the normalized case \(R=1\); the motion for any other \(R>0\) is immediately obtained by rescaling the amplitude and the time variable, as explained in Lemma 2.2.
To study the motion of the fluid particles driven by the Gavrilov vector field \({\tilde{U}}\) means to study the solutions \({{\mathbb {R}}}\rightarrow {{\mathbb {R}}}^3\), \(t \mapsto (x(t), y(t), z(t))\) of the system
which is an autonomous ode in \({{\mathbb {R}}}^3\). The dot above a function denotes its time derivative. Before dealing with system (1.11), we recall some definitions about quasi-periodic functions.
A vector \(\Omega = (\Omega _1, \ldots , \Omega _n) \in {{\mathbb {R}}}^n\), \(n \ge 1\), is said to be rationally independent if \(\Omega \cdot k = \Omega _1 k_1 + \cdots + \Omega _n k_n\) is nonzero for all integer vectors \(k = (k_1, \ldots , k_n) \in {\mathbb {Z}}^n {\setminus } \{ 0 \}\).
Given a set X, a function \(v: {{\mathbb {R}}}\rightarrow X\), \(t \mapsto v(t)\) is said to be quasi-periodic with frequency vector \(\Omega = (\Omega _1, \ldots , \Omega _n) \in {{\mathbb {R}}}^n\), \(n \ge 2\), if \(\Omega \) is rationally independent and there exists a function \(w: {{\mathbb {R}}}^n \rightarrow X\), \(2\pi \)-periodic in each real variable, such that \(v(t) = w(\Omega _1 t, \ldots , \Omega _n t)\) for all \(t \in {{\mathbb {R}}}\). Moreover, to ensure that the number n is not higher than necessary, we add the condition that there does not exist any vector \({\tilde{\Omega }} = ({\tilde{\Omega }}_1, \ldots , {\tilde{\Omega }}_m) \in {{\mathbb {R}}}^m\), with \(m < n\), and any function \({\tilde{w}}: {{\mathbb {R}}}^m \rightarrow X\), \(2\pi \)-periodic in each real variable, such that \(v(t) = {\tilde{w}} ({\tilde{\Omega }}_1 t, \ldots , {\tilde{\Omega }}_m t)\) for all \(t \in {{\mathbb {R}}}\).
For example, for \(X={{\mathbb {R}}}\), \(n=3\), and \(\Omega = (\Omega _1, \Omega _2, \Omega _3) \in {{\mathbb {R}}}^3\) a rationally independent vector, if \(w( \vartheta _1, \vartheta _2, \vartheta _3) = \cos (\vartheta _1 + \vartheta _2) + \cos (\vartheta _3)\), then \(n=3\) is not minimal, as it can be reduced to \(n=2\) by taking \(\tilde{w}(\vartheta _1, \vartheta _2) = \cos (\vartheta _1) + \cos (\vartheta _2)\), \(\tilde{\Omega }= ({\tilde{\Omega }}_1, {\tilde{\Omega }}_2) \in {{\mathbb {R}}}^2\), with \({\tilde{\Omega }}_1 = \Omega _1 + \Omega _2\) and \({\tilde{\Omega }}_2 = \Omega _3\), while \(n=2\) cannot be further reduced. Hence the function \(v(t) = {\tilde{w}}({\tilde{\Omega }}_1 t, {\tilde{\Omega }}_2 t) = \cos ({\tilde{\Omega }}_1 t) + \cos ({\tilde{\Omega }}_2 t)\) is quasi-periodic with frequency vector \({\tilde{\Omega }} \in {{\mathbb {R}}}^2\).
For \(n=2\), a vector \(\Omega = (\Omega _1, \Omega _2) \in {{\mathbb {R}}}^2\) is rationally independent if and only if \(\Omega _1\) is nonzero and the ratio \(\Omega _2 / \Omega _1\) is irrational. Hence a function \(v(t) = w(\Omega _1 t, \Omega _2 t)\) is quasi-periodic with frequency vector \(\Omega = (\Omega _1, \Omega _2)\) if \(w(\vartheta _1, \vartheta _2)\) is \(2\pi \)-periodic in \(\vartheta _1\) and in \(\vartheta _2\), \(\Omega _1\) is nonzero, \(\Omega _2 / \Omega _1\) is irrational, and v(t) is not a periodic function.
The main result of this paper is the following description of Gavrilov’s fluid particle dynamics.
Theorem 1.1
There exist universal positive constants \(\delta _0, \tau _0, \varepsilon _0, I^*\) with the following properties. Let
be the sets, constants, and functions defined in Sect. 1.1 for \(R=1\), with \(\delta = \delta _0\), \(\tau = \tau _0\), and \(0 < \varepsilon \le \varepsilon _0\). In particular, \(\omega \) is any smooth cut-off function of the Gavrilov construction, so that the composition \(\omega \circ P\), and therefore the functions \({\tilde{U}}\) and \(\nabla {\tilde{P}}\), vanish outside the set \(\mathcal {S}\); moreover, the set \(\mathcal {S}^*\) is an open neighborhood of the closure of \(\mathcal {S}\), and it is independent of \(\omega , \mathcal {S}, \varepsilon \).
(i) There exists an analytic diffeomorphism
where \({{\mathbb {T}}}:= {{\mathbb {R}}}/ 2 \pi {\mathbb {Z}}\), such that the change of variable \((x,y,z) = \Phi (\sigma ,\beta ,I)\) transforms system (1.11) into a system of the form
As a consequence, the solution (x(t), y(t), z(t)) of the Cauchy problem (1.11) with initial datum
is the function
defined for all \(t \in {{\mathbb {R}}}\), where
The angle variables \(\sigma (t), \beta (t) \in {{\mathbb {T}}}\) rotate with constant (possibly zero) angular frequencies \(\Omega _1(I_0)\), \(\Omega _2(I_0)\) respectively, and the variable \(I(t) = I_0\) is constant in time.
(ii) The first and third equations of the transformed system (1.12) form a Hamiltonian system
with Hamiltonian \(\mathcal {H}(\sigma , I) = \mathcal {H}(I)\) independent of the angle variable \(\sigma \); hence (1.16) is a Hamiltonian system in angle-action variables.
(iii) The frequency \(\Omega _1(I)\) is given by the product
where \(\omega \) is the \(C^\infty \) cut-off function in the assumptions, and \(\mathcal {K}(I)\) is the restriction to the interval \((0, I^*)\) of a function defined and analytic in the interval \((-I^*, I^*)\), with Taylor expansion
around \(I=0\), and strictly increasing in \((-I^*, I^*)\). The frequency \(\Omega _2(I)\) is given by the product
where \(\mathcal {R}(I)\) is the restriction to the interval \((0, I^*)\) of a function defined and analytic in the interval \((-I^*, I^*)\), with Taylor expansion
around \(I=0\). If \(\omega ( \mathcal {K}(I)) \ne 0\), then both \(\Omega _1(I)\) and \(\Omega _2(I)\) are nonzero, with ratio
The quantity \(\sqrt{I} \, \mathcal {R}(I)\) is well-defined also if \(\omega ( \mathcal {K}(I)) = 0\), because \(\mathcal {R}(I)\) is independent of \(\omega \). The function \(I \mapsto \sqrt{I} \, \mathcal {R}(I)\) is strictly increasing and analytic in \((0, I^*)\). Therefore it is rational for infinitely many values of I, and irrational for infinitely many other values of I. More precisely, the set \(\{ I \in (0, I^*): \sqrt{I} \, \mathcal {R}(I) \in {\mathbb {Q}}\}\) is a countable set, while the set \(\{ I \in (0,I^*): \sqrt{I} \, \mathcal {R}(I) \notin {\mathbb {Q}}\}\) has full Lebesgue measure.
(iv) For \(\omega ( \mathcal {K}(I_0) ) \ne 0\), the solution (1.14) of the Cauchy problem (1.11), (1.13) is periodic in time if \(\sqrt{I_0}\, \mathcal {R}(I_0)\) is rational, and it is quasi-periodic in time with frequency vector \((\Omega _1(I_0), \Omega _2(I_0))\) if \(\sqrt{I_0}\, \mathcal {R}(I_0)\) is irrational. For \(\omega ( \mathcal {K}(I_0) ) = 0\), the solution (1.14) of the Cauchy problem (1.11), (1.13) is constant in time, and it coincides with the initial datum.
(v) The map \(\Phi (\sigma ,\beta ,I)\) admits a converging expansion in powers of \(\sqrt{I}\) around \(I=0\); more precisely, there exists a map \(\Psi (\sigma , \beta , \mu )\), defined and analytic in \({{\mathbb {T}}}^2 \times (- \mu _0, \mu _0)\), where \(\mu _0 = \sqrt{I^*}\), such that \(\Phi (\sigma , \beta , I) = \Psi (\sigma , \beta , \sqrt{I})\) for all \((\sigma , \beta , I) \in {{\mathbb {T}}}^2 \times (0, I^*)\). The map \(\Phi (\sigma , \beta , I)\) has the form
where the functions \(\varrho (\sigma ,I), \eta (\sigma ,I), \zeta (\sigma ,I)\) have expansion
The map \(\Phi \) can be continuously extended to \(I=0\), with \(\Phi (\sigma , \beta , 0) = (\cos \beta , \sin \beta , 0)\), so that \(\Phi ({{\mathbb {T}}}^2 \times \{ 0 \}) = \mathcal {C}\).
(vi) The action variable I and the pressure P in (1.4) are related by the identity
The pressure level and the action are in bijective correspondence; thus, the action is a reparametrization of the pressure P. The frequencies \(\Omega _1(I), \Omega _2(I)\) could also be expressed in terms of the pressure P. The pressure \({\tilde{P}}\) in (1.6) satisfies
where W is defined in (1.7).
(vii) The orbit \(\{ (x(t), y(t), z(t)): t \in {{\mathbb {R}}}\}\) of the solution (1.14) lies in the level set
of the pressure P, where the value \(\ell = P(x_0, y_0, z_0) \in (0, \tau _0)\) is determined by the initial datum \((x_0, y_0, z_0)\) in (1.13). The level set \(\mathcal {T}_\ell \) has a nearly toroidal shape, that is, \(\mathcal {T}_\ell \) is close to an axisymmetric torus with circular cross section, because P(x, y, z) is given by (1.4), (1.5), and
around \((\rho ,z) = (1,0)\). In fact, the surface \(\mathcal {T}_\ell \) is the diffeomorphic image
of the 2-dimensional torus \({{\mathbb {T}}}^2 \times \{ I \} = \{ (\sigma , \beta , I): (\sigma , \beta ) \in {{\mathbb {T}}}^2 \}\), where I is determined by the identity
The pressure level \(\ell = P(x_0, y_0, z_0)\) of a point \((x_0, y_0, z_0) \in \mathcal {S}^*\) determines whether the solution (1.14), (1.15) of the Cauchy problem (1.11) with initial datum (1.13) on the surface \(\mathcal {T}_\ell \) is constant, periodic or quasi-periodic: if \(\omega (\ell ) = 0\), then the solution is constant in time; if \(\omega (\ell ) \ne 0\) and \(\Omega _2(I) / \Omega _1(I) \in {\mathbb {Q}}\), where I and \(\ell \) are related by (1.18), then the solution is periodic in time; if \(\omega (\ell ) \ne 0\) and \(\Omega _2(I) / \Omega _1(I) \notin {\mathbb {Q}}\), then the solution is quasi-periodic in time, with frequency vector \((\Omega _1(I), \Omega _2(I))\). Different solutions of system (1.11) lying on the same surface \(\mathcal {T}_\ell \), i.e., having the same level of pressure P, share the same frequencies \(\Omega _1(I)\), \(\Omega _2(I)\).
If \(\omega (\ell ) \ne 0\), or if \(\ell \) is an isolated zero of \(\omega \), then the function W in (1.7) is strictly increasing in a neighborhood of \(\ell \), and the surface \(\mathcal {T}_\ell \) is also the level set \(\{ (x,y,z) \in \mathcal {N}: {\tilde{P}}(x,y,z) = W(\ell ) \}\) of the pressure \({\tilde{P}}\) in (1.6).
(viii) The Jacobian determinant of \(\Phi \) is
Hence the map \(\Phi \) is volume-preserving (where the volume of any subset of \({{\mathbb {T}}}^2 \times (0, I^*)\) is defined as the 3-dimensional Lebesgue measure on the box \([0, 2\pi ]^2 \times (0,I^*)\); in other words, the integral of any periodic function is its Lebesgue integral over one period).
The relation (1.18) between the pressure and the action has the following geometrical property: for all \(\ell \in (0, \tau _0)\), the bounded region \(E_\ell \) enclosed by the surface \(\mathcal {T}_\ell \), namely the open set
satisfies
where I and \(\ell \) are related by (1.18), and
where \({\textrm{vol}}\) is the Lebesgue measure in \({{\mathbb {R}}}^3\).
(ix) The constants \(\delta _0, \tau _0, \varepsilon _0, I^*\), the set \(\mathcal {S}^*\), and the functions \(\Phi , \mathcal {K}, \mathcal {R}, \varrho , \eta , \zeta \) are all universal, in the sense that they do not depend on any parameter; in particular, they are independent of \(\omega , \varepsilon , \mathcal {S}\).
Remark 1.2
(Support of the cut-off function). Theorem 1.1 is stated for cut-off functions \(\omega \) supported in \([\varepsilon , 2\varepsilon ]\), like those in [10]; however, Theorem 1.1, as well as the result of [10], also holds for \(\omega \) supported in any interval \([\varepsilon _1, \varepsilon _2]\) with \(0< \varepsilon _1 < \varepsilon _2 \le \varepsilon _0\), without changing anything in the proof.
Remark 1.3
(3-dimensional invariant set of a Hamiltonian system with 2 degrees of freedom). By (1.16), system (1.12) is equivalent to the integrable Hamiltonian system with 2 degrees of freedom in angle-action variables
with Hamiltonian
depending only on the action variables I, K, where \(\mathcal {H}(I)\) and \(\Omega _2(I)\) are in Theorem 1.1, restricted to the 3-dimensional invariant set \(K = 0\).
Remark 1.4
(Proper invariant subsets). By Theorem 1.1, the dynamics of system (1.11) on each level set \(\mathcal {T}_\ell \) of the pressure is conjugated to a Kronecker flow on \({{\mathbb {T}}}^2\) (i.e., a linear flow mod \(2\pi \))
see (1.12). As an immediate consequence, regarding the orbits of the system, we deduce that
-
(i)
the orbits are closed subsets of \(\mathcal {T}_\ell \) if \(\Omega _2 / \Omega _1 \in {\mathbb {Q}}\), and dense subsets of \(\mathcal {T}_\ell \) if \(\Omega _2 / \Omega _1 \notin {\mathbb {Q}}\);
-
(ii)
each orbit in \(\mathcal {T}_\ell \) is a proper invariant subset of \(\mathcal {T}_\ell \), i.e. it is nonempty and it cannot fill the whole surface \(\mathcal {T}_\ell \). The last property is simple to check for (1.20) on \({{\mathbb {T}}}^2\): if \(\Omega _1\) is nonzero, then the intersection of any orbit with the set \(\{ 0 \} \times {{\mathbb {T}}}\) of the pairs \((\sigma ,\beta )\) with \(\sigma = 0\) has at most countably many points, and therefore infinitely many points on \(\{ 0 \} \times {{\mathbb {T}}}\) are outside the orbit; if \(\Omega _1\) is zero, then also \(\Omega _2\) is zero, and any orbit is a single point;
-
(iii)
the orbits are minimal invariant sets, i.e. any proper subset of an orbit is not an invariant set;
-
(iv)
any arbitrary union of orbits is also an invariant set.
We conclude that there are no invariant sets other than the orbits themselves (and, trivially, their arbitrary unions).
Remark 1.5
(Physical interpretation of the action variable). Item (viii) in Theorem 1.1 gives a physical interpretation of the action variable: up to a constant multiplicative factor, the action I corresponding to a value \(\ell \) of the pressure by (1.18) gives the mass of the fluid that occupies the region \(E_\ell \) enclosed by the pressure level set \(\mathcal {T}_\ell \).
Remark 1.6
(Physical interpretation of the angle variables). The action variable I has a geometrical, and therefore physical, meaning, see item (viii) in Theorem 1.1 and Remark 1.5. Can we say something similar also for the angle variables \(\sigma , \beta \)?
Regarding the angle \(\sigma \), its physical interpretation is strictly related to the construction of the angle-action coordinates. The full details of the construction are in Sect. 3.7; here we just add some comment.
After passing to cylindrical coordinates \(x = \rho \cos \varphi \), \(y = \rho \sin \varphi \) in Sect. 3.1, changing the vertical coordinate \(z = {\tilde{z}} / \rho \) (to obtain a canonical Hamiltonian structure) in Sect. 3.2, and introducing symplectic polar coordinates in the radial–vertical plane \(\rho = 1 + \sqrt{2\xi } \sin \vartheta \), \({\tilde{z}} = \sqrt{2\xi } \cos \vartheta \) in Sect. 3.3, we end up with variables \((\vartheta , \varphi , \xi )\) with a rather clear geometrical meaning: a point (x, y, z) in a neighborhood of the circle \(\mathcal {C}\) is expressed in terms of \((\vartheta , \varphi , \xi )\) as
The equations for the variables \((\vartheta , \xi )\) form the Hamiltonian system (3.30), which is equivalent to the single equation (3.41) for the angle variable \(\vartheta \) on the energy level c; (3.41) is an autonomous scalar equation, and its integration is elementary (Sect. 3.5).
In fact, to construct the angle-action variables, we solve (3.41) with \(\chi (c)\) replaced by 1 (the reason is explained in Remark 1.7). After this, and after proving that the solutions \(\vartheta (t)\) are periodic functions of time, one realizes that the pair “(time, energy)”, namely \(s_1^*\) and c in the lines between (3.49) and (3.51), could be used as new symplectic coordinates: the energy level c determines the orbit, and the time \(s_1^* \in [0, T_c^*)\) (where \(T_c^*\) is the period, see (3.50)) determines the point on that orbit. Note that here the “energy” is the value of the Hamiltonian \(\alpha _3(\vartheta ,\xi )\), which is the pressure of the fluid expressed in terms of the coordinates \((\vartheta ,\xi )\). However, as is well-known, using the (time, energy) coordinates has two drawbacks:
-
(i)
even if the time \(s_1^*\) can be considered as a circular variable, in fact it is not an angle variable, because it makes a round after a time interval \(T_c^*\), while, for angle variables, a complete round should be \(2\pi \);
-
(ii)
the frequency \(2 \pi / T_c^*\) is hidden in the definition of the new coordinates (time, energy), and it does not appear any longer in the transformed system: in fact, the Hamiltonian system in the (time, energy) \(= (s, c)\) coordinates becomes \({\dot{s}} = 1\), \({\dot{c}} = 0\).
These problems can be easily solved by introducing the normalization factor \(2 \pi / T_c^*\) multiplying the time variable \(s_1^*\), obtaining a genuine angle variable: this is where the angle variable \(\sigma \) comes from (see the lines between (3.50) and (3.51)). After the circular time variable has been normalized to obtain the geometric angle variable \(\sigma \), the energy c must also be modified, in order to preserve the canonical Hamiltonian structure of the system: this is the reason for introducing the new variable I, the action, in place of the energy c.
In conclusion, we can say that the physical interpretation of the angle variable \(\sigma \) is this: \(\sigma \) is the product of the time variable of a periodic solution of equation (3.48) (which is the same as equation (3.41) but with \(\chi (c)\) replaced by 1) multiplied by the frequency of that solution, so that \(\sigma \) corresponds to a rescaled, normalized version of the time variable of a solution in which the solution takes \(2\pi \) units to make a complete round and arriving at its starting point.
The other angle, \(\beta \), is introduced in Sect. 3.8 as a translation of the original horizontal angle \(\varphi \) of the cylindrical coordinates, namely \(\varphi = \beta + \eta (\sigma , I)\). From (1.15) and (1.17), the solution (1.14) of the Cauchy problem (1.11), (1.13) expressed in cylindrical coordinates is
where
and, in short, we have written \(\Omega _1, \Omega _2\) instead of \(\Omega _1(I_0), \Omega _2(I_0)\). Thus \(\rho (t)\) and z(t) are periodic functions of time with period \(2 \pi / \Omega _1\), whereas the horizontal angle \(\varphi (t)\) is the sum \(\beta (t) + {\tilde{\eta }}(t)\) of the two components
If \(\Omega _2 \ne 0\), then \(\beta (t)\), as a function taking values in \({{\mathbb {R}}}\), is a linear function of time, and it is not periodic. For this reason, \(\beta (t)\), as a function taking values in \({{\mathbb {T}}}\), is a rotation. On the contrary, \({\tilde{\eta }}(t)\) is periodic also as a function taking values in \({{\mathbb {R}}}\), because \(\eta \) is \(2\pi \)-periodic in its first variable, see (3.72). Hence, borrowing a terminology often employed when speaking about the orbits of the pendulum, we could say that \({\tilde{\eta }}(t)\) is a libration.
In conclusion, we can say that the angle \(\beta \) (or, more precisely, the solution \(\beta (t)\), which is a function of time) has this physical interpretation: \(\beta (t)\) is the rotation component of the motion of the horizontal angle \(\varphi (t)\), net of its libration component.
Remark 1.7
(The map \(\Phi \) is independent of the cut-off function \(\omega \)). In the definition of the transformations \(\Phi _4\) in Sect. 3.7 and \(\Phi _5\) in Sect. 3.8, we use a simple, convenient trick: we keep the Gavrilov cut-off function \(\omega \) (which, after the changes of variables, becomes the factor \(\chi (c)\) in (3.41) and \(\chi (h(I))\) in (3.70), see (3.7)) out of the construction, using the solutions of the equations one obtains by replacing \(\omega \) with the constant 1. The gain is that, at the end of the entire construction,
-
(i)
the map \(\Phi \) is independent of the cut-off \(\omega \) and of its support \([\varepsilon , 2\varepsilon ]\);
-
(ii)
the map \(\Phi \) is defined on the open set \(\mathcal {S}^*\), independent on \(\omega , \varepsilon \), surrounding the circle \(\mathcal {C}\), in the sense that \(\mathcal {S}^* \cup \mathcal {C}\) is an open neighborhood of \(\mathcal {C}\); the set \(\mathcal {S}\) (see (1.8)) containing the support of the composition \(\omega \circ P\) and of the smooth Gavrilov vector field \({\tilde{U}}\) (see (1.6)) can be placed in an arbitrary location inside \(\mathcal {S}^*\), without affecting the definition and the properties of \(\Phi \);
-
(iii)
the analytic regularity of the functions \(\Psi (\sigma , \beta , \mu )\), \(\mathcal {K}(I)\), \(\mathcal {R}(I)\), which appear in the formulas of \(\Phi (\sigma , \beta , I)\), \(\Omega _1(I)\), \(\Omega _2(I)\) in Theorem 1.1, has nothing to do with the \(C^\infty \) regularity of the cut-off function \(\omega \). The analyticity is only limited by the “natural” singularities of the symplectic polar coordinates, i.e., the presence of the square root and the loss of injectivity at the origin. Note that Gavrilov’s functions U, P in (1.4) (the ones before introducing the cut-off \(\omega \)) are analytic in \(\mathcal {N}\setminus \mathcal {C}\), and contain a square root singularity at \(\mathcal {C}\): our construction does not deteriorate this regularity, because also the various functions described in Theorem 1.1 are analytic outside the circle \(\mathcal {C}\), and have only \(\mathcal {C}\) (corresponding to the value \(I=0\) of the action variable) as a possible singular set. \(\square \)
Remark 1.8
(Relation with Arnold’s structure theorem). By Theorem 1.1, the domain \(\mathcal {S}^*\) is foliated by the 2-dimensional tori \(\mathcal {T}_\ell \), invariant for the vector field \({\tilde{U}}\), on which the motion is periodic or quasi-periodic. This is true not only for the Gavrilov vector field, but also for all steady 3D Euler flows whose velocity and vorticity are linearly independent at each point, by an important structure theorem of Arnold, see the papers [1, 2] and the book [3], Chapter II, Section 1.
Thus, Arnold’s result gives the existence of the angle-angle-action variables. On the other hand, since we restrict our study to the specific case of Gavrilov’s vector field, we can go much beyond showing the existence of the variables \((\sigma , \beta , I)\): Theorem 1.1 gives an explicit description of the frequencies \(\Omega _1(I)\), \(\Omega _2(I)\) close to \(I=0\) (i.e., close to the circle \(\mathcal {C}\)), proving the existence of both periodic and quasi-periodic motions of the fluid particles.
We remark that, thanks to our detailed knowledge of the Gavrilov vector field (axisymmetric structure and sufficiently high order Taylor expansions), in the proof of Theorem 1.1 we replace Arnold’s general topological argument with the explicit construction of the transformation \(\Phi \), leading to a more detailed knowledge about the particle dynamics than in the much more general case covered by Arnold’s structure theorem. We also observe that our proof does not use the vorticity vector field, which, on the contrary, is one of the main ingredients in Arnold’s proof.
Theorem 1.1 is proved in Sects. 3 and 4, splitting the proof into several short simple steps. The proof uses only basic tools from the classical theory of odes and dynamical systems, including (Sect. 4.3) the application of the implicit function theorem in a situation of degeneracy—the kind of degeneracy one can expect to find in the construction of the angle-action variables around a strict minimum of the Hamiltonian.
1.3 Related Literature
A general discussion about the existence of compactly supported smooth solutions of pdes, also in comparison with the result of Gavrilov for the steady Euler equations, can be found in the recent preprint [13]; in particular, for Navier-Stokes equations, see [12].
The nice, explicit construction of Gavrilov’s original paper [10] has been revisited and generalized by Constantin et al. [5]. The paper [5] also uses the result by Gavrilov as a building block to prove the existence of compactly supported solutions of the steady Euler equations that have a given Hölder regularity \(C^\alpha \) and are not in \(C^\beta \) for any \(\beta > \alpha \). The proof employs the invariances of the Euler equations and the fact that the sum of compactly supported solutions with disjoint supports is itself a solution.
The result by Gavrilov has also been used recently by Enciso, Peralta-Salas and Torres de Lizaur in [8] to produce time-quasi-periodic solutions of the 3D Euler equations. In [8] the authors extend to the 3D case, and to the n-dimensional case for all n even, the construction by Crouseilles and Faou [6] of time-quasi-periodic solutions of the 2D Euler equations. Both [6, 8] use in a clever way the compactly supported solutions of the steady equations as the main ingredients of the construction.
The time-quasi-periodic solutions in [6, 8] are not of kam type, that is, they are constructed outside the context of the Kolmogorov–Arnold–Moser perturbation theory of nearly integrable dynamical systems, where small divisor problems typically appear. Time-quasi-periodic solutions of the 3D Euler equations of kam type are obtained in [4] by Montalto and the author in presence of a forcing term, using pseudo-differential calculus and techniques of kam theory for pdes.
As is mentioned in Remark 1.8, the foliation of a region of \({{\mathbb {R}}}^3\) by a family of invariant 2-dimensional tori, labelled by the action variable, on which the particle motion is periodic or quasi-periodic, occurs not only for the specific case of the Gavrilov vector field, but also for much more general situations, by Arnold’s structure theorem; see the papers [1, 2] and the book [3], Chapter II, Section 1; see also Remark 1.8 for further comments. Arnold’s result, and its role in the study of the mixing property for Euler flows, is discussed by Khesin et al. [11].
For the existence of periodic motions of fluid particles in steady 3D Euler flows and some related interesting questions, see [9] and Remark 1.13 in Chapter II, Section 1, of [3].
It is reasonable to expect that the proof of Theorem 1.1 can be adapted to obtain an analogous result also for the steady Euler vector fields constructed in [7], which are piecewise smooth, axisymmetric, with compact support in \({{\mathbb {R}}}^3\).
2 Dual Role of the Parameter R in Gavrilov’s Solutions
Gavrilov’s construction depends on one parameter, R, but in fact the natural free parameters in his construction are two, not only one. These two natural parameters, called \(\lambda \) and \(\mu \) in the present section, correspond to two different invariances of the steady Euler equations. The paper [10], without explicitly mentioning it, fixes \(\lambda = R\), \(\mu = R^2\), namely it considers parameters constrained to stay on the half parabola \((\lambda ,\mu ) = (\lambda , \lambda ^2)\), \(\lambda > 0\), instead of pairs of parameters free to vary in the quarter of plane \((\lambda ,\mu )\), \(\lambda , \mu > 0\).
In other words, two different roles are played simultaneously by the parameter R in Gavrilov’s solutions, because R is
-
both a rescaling factor for the independent variable \((x,y,z) \in {{\mathbb {R}}}^3\), appearing as \(R^{-1}\) in the argument of \(\alpha \) in (1.5),
-
and an amplitude factor multiplying the vector fields \(U, {\tilde{U}}\) and the pressures \(P, {\tilde{P}}\), appearing as powers \(R^3\) and \(R^4\) in the definition of b and p in (1.5).
In fact, there exists a family of solutions described by two real parameters \((\lambda , \mu )\) such that the solutions of Sect. 1.1 are obtained in the special case \((\lambda , \mu ) = (R, R^2)\). This means that, regarding the parameter R, Gavrilov’s solutions form a 1-parameter subset of a 2-parameter family of solutions. Each element of the family is obtained from any other element of the family by two rescalings, which correspond to two basic invariances of the Euler equations, also preserving the localizability condition. This allows us to study only one element of the family (in particular, a normalized one), obtaining directly a description of all the other elements.
Given \(R>0\), let
denote the list of the elements (sets, constants, and functions) defined in Sect. 1.1. For every \(\lambda > 0\) and \(\mu > 0\), we define a rescaled version of each element of the list \(\mathcal {U}\) in the following way. We define
where \(\rho = \sqrt{x^2 + y^2}\). We denote by \(\mathcal {A}_{\lambda ,\mu } \mathcal {U}:= (\mathcal {A}_{\lambda ,\mu } \mathcal {C}, \ldots , \mathcal {A}_{\lambda ,\mu } \mathcal {S}^*)\) the list of the rescaled elements. The properties of \(\mathcal {U}\) in Sect. 1.1 become the following properties for \(\mathcal {A}_{\lambda ,\mu } \mathcal {U}\).
-
The constant \(\mathcal {A}_{\lambda ,\mu } \delta = \lambda \delta \) satisfies \(0< \lambda \delta < \lambda R\) and \(\lambda \delta \le \lambda r_0 R\).
-
The rescaled pressure \(\mathcal {A}_{\lambda ,\mu } P\) is analytic in \(\mathcal {A}_{\lambda ,\mu } \mathcal {N}\).
-
The rescaled vector field \(\mathcal {A}_{\lambda ,\mu } U\) is continuous in \(\mathcal {A}_{\lambda ,\mu } \mathcal {N}\) and analytic in \((\mathcal {A}_{\lambda ,\mu } \mathcal {N}) {\setminus } (\mathcal {A}_{\lambda ,\mu } \mathcal {C})\).
-
The pair \((\mathcal {A}_{\lambda ,\mu } U, \, \mathcal {A}_{\lambda ,\mu } P)\) satisfies the Euler equations and the localizability condition in \((\mathcal {A}_{\lambda ,\mu } \mathcal {N}) \)\( {\setminus } (\mathcal {A}_{\lambda ,\mu } \mathcal {C})\).
-
The function \(\mathcal {A}_{\lambda ,\mu } \omega \) is \(C^\infty ({{\mathbb {R}}},{{\mathbb {R}}})\) with support contained in \([\mathcal {A}_{\lambda ,\mu } \varepsilon , 2 \mathcal {A}_{\lambda ,\mu } \varepsilon ] = [\mu ^2 \varepsilon , 2 \mu ^2 \varepsilon ]\).
-
The function \(\mathcal {A}_{\lambda ,\mu } W\) satisfies
$$\begin{aligned} (\mathcal {A}_{\lambda ,\mu } W)(s) = \int _0^s (\mathcal {A}_{\lambda ,\mu } \omega )^2(\sigma ) \, d\sigma \quad \ \forall s \in {{\mathbb {R}}}. \end{aligned}$$ -
The vector field \(\mathcal {A}_{\lambda , \mu } {\tilde{U}}\) satisfies
$$\begin{aligned} \begin{aligned} (\mathcal {A}_{\lambda ,\mu } {\tilde{U}})(x,y,z)&= \mu \omega \Big ( P \Big ( \frac{x}{\lambda }, \frac{y}{\lambda }, \frac{z}{\lambda } \Big ) \Big ) U \Big ( \frac{x}{\lambda }, \frac{y}{\lambda }, \frac{z}{\lambda } \Big ) \\ {}&= (\mathcal {A}_{\lambda ,\mu } \omega ) \big ( (\mathcal {A}_{\lambda ,\mu } P)(x,y,z) \big ) \cdot (\mathcal {A}_{\lambda ,\mu } U)(x,y,z) \quad \ \forall (x,y,z) \in \mathcal {A}_{\lambda ,\mu } \mathcal {N}\end{aligned} \end{aligned}$$and \((\mathcal {A}_{\lambda ,\mu } {\tilde{U}})(x,y,z) = 0\) for all \((x,y,z) \in {{\mathbb {R}}}^3 {\setminus } (\mathcal {A}_{\lambda ,\mu } \mathcal {N})\).
-
The pressure \(\mathcal {A}_{\lambda ,\mu } {\tilde{P}}\) satisfies
$$\begin{aligned} (\mathcal {A}_{\lambda ,\mu } {\tilde{P}})(x,y,z)&= \mu ^2 W \Big ( P \Big ( \frac{x}{\lambda }, \frac{y}{\lambda }, \frac{z}{\lambda } \Big ) \Big ) = (\mathcal {A}_{\lambda ,\mu } W) \big ( (\mathcal {A}_{\lambda ,\mu } P)(x,y,z) \big ) \quad \ \forall (x,y,z) \in \mathcal {A}_{\lambda ,\mu } \mathcal {N}\end{aligned}$$and \((\mathcal {A}_{\lambda ,\mu } {\tilde{P}})(x,y,z) = \mu ^2 W(2\varepsilon ) = (\mathcal {A}_{\lambda ,\mu } W)(2 \mathcal {A}_{\lambda ,\mu } \varepsilon )\) for all \((x,y,z) \in {{\mathbb {R}}}^3 {\setminus } (\mathcal {A}_{\lambda ,\mu } \mathcal {N})\).
-
Both \(\mathcal {A}_{\lambda ,\mu } {\tilde{U}}\) and \(\mathcal {A}_{\lambda ,\mu } {\tilde{P}}\) are \(C^\infty ({{\mathbb {R}}}^3)\) and satisfy the Euler equations and the localizability condition in \({{\mathbb {R}}}^3\).
-
Both \(\mathcal {A}_{\lambda ,\mu } {\tilde{U}}\) and \(\nabla (\mathcal {A}_{\lambda ,\mu } {\tilde{P}})\) vanish outside the bounded set \(\mathcal {A}_{\lambda ,\mu } \mathcal {S}\).
-
One has \(\mathcal {A}_{\lambda ,\mu } P > \mathcal {A}_{\lambda ,\mu } \tau \) in \((\mathcal {A}_{\lambda ,\mu } \mathcal {N}) {\setminus } (\mathcal {A}_{\lambda ,\mu } \mathcal {N}')\).
-
One has \(\mathcal {A}_{\lambda ,\mu } \tau \ge 3 \mathcal {A}_{\lambda ,\mu } \varepsilon \).
Thus, we have obtained the 2-parameter family \(\{ \mathcal {A}_{\lambda ,\mu } \mathcal {U}\}_{\lambda , \mu }\). One has the group property
for all \(\lambda _1, \lambda _2, \lambda , \mu _1, \mu _2, \mu \in (0,\infty )\). The check of (2.3) is straightforward.
Lemma 2.1
Given \(R>0\), let \(\mathcal {U}_R\) be the list (2.1) of the elements (sets, constants, and functions) defined in Sect. 1.1. Then
where \(\mathcal {U}_1\) is a list of elements constructed in Sect. 1.1 for \(R=1\), and \(\mathcal {A}_{R, R^2}\) is the rescaling operator \(\mathcal {A}_{\lambda ,\mu }\), defined in (2.2), at \(\lambda = R\), \(\mu = R^2\).
Proof
Let \(\mathcal {U}_R\) be the list (2.1) of elements constructed in Sect. 1.1 for a given \(R>0\). We observe that the list \(\mathcal {A}_{\lambda , \mu } \mathcal {U}_R\) with \(\lambda = 1/R\) and \(\mu = 1/R^2\) coincides with a list of elements that one obtains by choosing \(R=1\) in Sect. 1.1, which we call \(\mathcal {U}_1\). The check is elementary; for example, regarding the vector field U and the pressure P, by (1.4) and (1.5) one has
Then, by (2.3), \(\mathcal {U}_R = \mathcal {A}_{R, R^2} ( \mathcal {A}_{\frac{1}{R}, \frac{1}{R^2}} \, \mathcal {U}_R ) = \mathcal {A}_{R, R^2} \, \mathcal {U}_1\). \(\square \)
Regarding the fluid particle system (1.11), the consequence of Lemma 2.1 is the following lemma, whose proof is trivial.
Lemma 2.2
Let \({\tilde{U}}_R = \mathcal {A}_{R, R^2} {\tilde{U}}_1\), where \({\tilde{U}}_R\) is given by Sect. 1.1 for some \(R>0\) and \(\tilde{U}_1\) is given by Sect. 1.1 for \(R=1\). Then a function (x(t), y(t), z(t)) solves the fluid particle system (1.11) with velocity field \({\tilde{U}} = {\tilde{U}}_R\) if and only if
where \((x_1(t), y_1(t), z_1(t))\) solves
3 Conjugation to a Linear Flow on \({{\mathbb {T}}}^2\)
In this section and in Sect. 4 we prove Theorem 1.1. Thus, assume that \(\mathcal {U}\) in (2.1) is given by Sect. 1.1 for \(R=1\). Hence, in particular,
where \(\rho = \sqrt{x^2 + y^2}\). The constant \(\delta \) satisfies \(0< \delta < 1\) and \(\delta \le r_0\). The vector field U and the pressure P in \(\mathcal {N}\) are given by (1.4), (1.5), where \(a(\rho ,z) = \alpha (\rho ,z)\) and
The constants \(\tau \) and \(\varepsilon \) satisfy (1.9) and \(\tau \ge 3 \varepsilon \). The function \(\omega \) is supported in \([\varepsilon , 2 \varepsilon ]\), and W is in (1.7). The vector field \({\tilde{U}}\) and the pressure \({\tilde{P}}\) are given by (1.6) in \(\mathcal {N}\), and \(({\tilde{U}}, {\tilde{P}}) = (0, W(2\varepsilon ))\) in \({{\mathbb {R}}}^3 \setminus \mathcal {N}\). The sets \(\mathcal {S}, \mathcal {S}^*\) are given by (1.8), (1.10).
Outside the set \(\mathcal {S}\), the vector field \({\tilde{U}}\) is zero, and the solutions of system (1.11) are constant in time. Hence, by the uniqueness property of the solution of Cauchy problems, any solution of (1.11) that is in \(\mathcal {S}\) at some time \(t_0\) remains in \(\mathcal {S}\) for its entire lifespan; in other words, \(\mathcal {S}\) is an invariant set for (1.11). Moreover \(\mathcal {S}\) is bounded, and therefore, by basic ode theory, the solutions of (1.11) in \(\mathcal {S}\) are all global in time.
Trivially, any subset of \({{\mathbb {R}}}^3 \setminus \mathcal {S}\) is also an invariant set for (1.11). Hence any subset of \({{\mathbb {R}}}^3\) containing \(\mathcal {S}\) is invariant for (1.11). In particular, \(\mathcal {S}^*\) and \(\mathcal {N}\) are invariant for (1.11).
3.1 Cylindrical Coordinates
To study system (1.11) in \(\mathcal {N}\), first of all we move on to cylindrical coordinates, that is, we consider the diffeomorphism
and the change of variables \({\tilde{v}} = \Phi _1(v)\), with \({\tilde{v}} = (x,y,z)\), \(v = (\rho , \varphi , z)\), i.e., \(x = \rho \cos \varphi \), \(y = \rho \sin \varphi \). Using cylindrical coordinates is a natural choice because the quantities \(u_\rho , u_\varphi , u_z, p\) in (1.5) are already expressed in terms of \(\rho ,z\). Now a function \({\tilde{v}}(t) = \Phi _1(v(t))\) solves (1.11) in \(\mathcal {N}\) if and only if v(t) solves
in \(\mathcal {N}_1\), where \(V_1\) is the vector field
The Jacobian matrix \(D \Phi _1(v)\) and its inverse matrix are
the composition \({\tilde{U}} (\Phi _1(v))\) is
and therefore
Since \(\alpha \) and H are the functions constructed and studied in [10], it is convenient to express the other quantities in terms of them. Hence, we define
and, recalling (3.2), we rewrite (3.6) as
The curve \(\mathcal {C}\) and the sets \(\mathcal {S}^*, \mathcal {N}'\) become
Moreover, the map \(\Phi _1\) is analytic in \(\mathcal {N}_1\).
The vector field U satisfies the localizability condition (1.2) in \(\mathcal {N}\setminus \mathcal {C}\), and therefore \({\tilde{U}} \cdot \nabla P = 0\) in \(\mathcal {N}\). Hence the pressure P is a prime integral of system (1.11) in \(\mathcal {N}\). In cylindrical coordinates, this means that \(p(\rho ,z)\), and therefore \(\alpha (\rho ,z)\) too, are prime integrals of (3.4) in \(\mathcal {N}_1\). This can also be verified directly: by (3.4) and (3.8),
Hence every trajectory \(\{ v(t) = (\rho (t), \varphi (t), z(t)): t \in {{\mathbb {R}}}\}\) of system (3.4) in \(\mathcal {N}_1\) lies in a level set
In particular, every trajectory of (3.4) in \(\mathcal {S}_1^*\) lies in a level set \(\mathcal {P}_c\) with \(0< c < 4 \tau \). In fact, the set \(\mathcal {S}_1^*\) is exactly the union of all the level sets \(\mathcal {P}_c\) with \(c \in (0, 4 \tau )\).
3.2 Elimination of the Factor \(1/\rho \) and Canonical Hamiltonian Structure
We want to remove the factor \(1/\rho \) appearing the first and third component of \(V_1\) in (3.6). We consider the diffeomorphism
and the change of variables \({\tilde{v}} = \Phi _2(v)\), with \({\tilde{v}} = ({\tilde{\rho }}, {\tilde{\varphi }}, {\tilde{z}})\), \(v = (\rho ,\varphi ,z)\). A function \({\tilde{v}}(t) = \Phi _2( v(t))\) solves (3.4) in \(\mathcal {N}_1\) if and only if v(t) solves
in \(\mathcal {N}_2\), where
The Jacobian matrix \(D \Phi _2(v)\) and its inverse are
We define
for all \((\rho , z) \in \mathcal {D}_2\). Hence
and, recalling the second identity in (3.2),
Then the vector field \(V_2\) in (3.13) is
The sets \(\mathcal {C}_1, \mathcal {S}_1^*, \mathcal {N}_1'\) become
Note that \(\Phi _2\) leaves \(\mathcal {C}_2 = \mathcal {C}_1\) invariant, because \(z/\rho = z\) at \(\rho = 1\). By (3.9), one has
The level set \(\mathcal {P}_c\) of \(\alpha \) in (3.10) becomes the level set
of \(\alpha _2\). The set \(\mathcal {S}_2^*\) is the union of the level sets \(\mathcal {P}_{2,c}\) with \(c \in (0, 4 \tau )\). The map \(\Phi _2\) is analytic in \(\mathcal {N}_2\). The function \(\alpha _2(\rho ,z)\) is a prime integral of system (3.12), and it is also analytic; its Taylor expansion around (1, 0) is in (4.14). The first and third equation of system (3.12) are the Hamiltonian system
where
3.3 Symplectic Polar Coordinates in the Radial–Vertical Plane
Now we move on to polar coordinates (in their symplectic version) to describe the pair \((\rho ,z)\). The pairs \((\rho ,z)\) such that \((\rho , \varphi ,z) \in \mathcal {N}_2\) do not form a disc; thus, it is convenient to consider a subset of \(\mathcal {N}_2\) that fits with polar coordinates better than how \(\mathcal {N}_2\) does. We consider the open sets
where \(\delta _2 = C \delta \), \(C>0\), and we observe that
if \(C(1+\delta ) \le 1\) and \(1 + (\delta /4) \le 2C\). This holds, for example, for \(C = 2/3\) and all \(0 < \delta \le 1/2\).
Proof
(Proof of (3.21)) Let \(\delta _2 = C \delta \), let \(C(1+\delta ) \le 1\), and \((\rho ,\varphi ,z) \in \mathcal {B}_2\). Then \((\rho -1)^2 < \delta _2^2\), whence \(\rho > 1 - \delta _2\). Moreover \(1 - \delta _2 = 1 - C \delta \ge C > 0\). Hence
where the last inequality holds because \(C(1+\delta ) \le 1\). Therefore \((\rho ,\varphi , z) \in \mathcal {N}_2\). Moreover \(0 < (\rho -1)^2 + z^2\). This proves that \(\mathcal {B}_2 \subseteq (\mathcal {N}_2 {\setminus } \mathcal {C}_2)\).
Now let \(\delta _2 = C \delta \), let \(1 + (\delta /4) \le 2C\), and \((\rho ,\varphi ,z) \in \mathcal {N}_2' \setminus \mathcal {C}_2\). Since \((\rho ,\varphi ,z) \in \mathcal {N}_2'\), one has \((\rho -1)^2 < (\delta /4)^2\), whence \(\rho < 1 + (\delta /4)\), and
where the last inequality holds because \(1 + (\delta /4) \le 2 C\). Also, \(0 < (\rho -1)^2 + z^2\) because \((\rho ,\varphi ,z) \notin \mathcal {C}_2\). Therefore \((\rho ,\varphi ,z) \in \mathcal {B}_2'\). This proves that \((\mathcal {N}_2' {\setminus } \mathcal {C}_2) \subseteq \mathcal {B}_2'\).
The last inclusion in (3.21) holds because \(\mathcal {C}_2 \subseteq \mathcal {N}_2' \subseteq \mathcal {N}_2\) and \((\mathcal {N}_2 {\setminus } \mathcal {C}_2) {\setminus } (\mathcal {N}_2' {\setminus } \mathcal {C}_2) = (\mathcal {N}_2 {\setminus } \mathcal {N}_2')\). \(\square \)
Thus \(\mathcal {S}_2^* \subseteq \mathcal {B}_2' \subseteq \mathcal {B}_2\), and they are invariant sets for system (3.12).
We consider the diffeomorphism
where \(\xi _3:= \delta _2^2 / 2 = 2 \delta ^2 / 9\), and the change of variables \({\tilde{v}} = \Phi _3(v)\), with \({\tilde{v}} = ({\tilde{\rho }}, {\tilde{\varphi }}, {\tilde{z}})\), \(v = (\vartheta , \varphi , \xi )\). A function \(\tilde{v}(t) = \Phi _3( v(t))\) solves (3.12) in \(\mathcal {B}_2\) if and only if v(t) solves
in \(\mathcal {B}_3\), where
The Jacobian matrix \(D \Phi _3(v)\) and its inverse are
We define
for all \((\vartheta ,\xi ) \in {{\mathbb {T}}}\times [0, \xi _3)\). In fact, in the set \(\mathcal {B}_3\), \(\xi \) varies in the interval \((0, \xi _3)\), but it is convenient to consider \(\alpha _3\) also for \(\xi =0\); one has \(\alpha _3(\vartheta ,0) = \alpha _2(1,0) = \alpha (1,0) = 0\) (see (3.15)). From (3.26) it follows that
for all \((\vartheta ,\xi ) \in {{\mathbb {T}}}\times (0, \xi _3)\), where \((\rho ,z) = ( 1 + \sqrt{2\xi } \sin \vartheta , \sqrt{2\xi } \cos \vartheta )\). Hence
The set \(\mathcal {C}_2\) is out of \(\mathcal {B}_2\); the sets \(\mathcal {S}_2^*\), \(\mathcal {B}_2'\) become
By (3.22), one has
Thus \(\mathcal {S}_3^* \subset \mathcal {B}_3'\). The level set \(\mathcal {P}_{2,c}\) of \(\alpha _2\) in (3.17) becomes the level set
of \(\alpha _3\). The set \(\mathcal {S}_3^*\) is the union of the level sets \(\mathcal {P}_{3,c}\) with \(c \in (0, 4 \tau )\). The map \(\Phi _3\) is analytic in \(\mathcal {B}_3\). The function \(\alpha _3(\vartheta ,\xi )\) is a prime integral of system (3.24); its behavior near \(\xi =0\) is studied in Sects. 3.4 and 4.3. The first and third equation of system (3.24) are the Hamiltonian system
where
where \(\mathcal {H}_2\) and \(\Gamma \) are defined in (3.19). Moreover \(\vartheta \) (as well as \(\varphi \)) is an angle variable, i.e., it varies in \({{\mathbb {T}}}= {{\mathbb {R}}}/ 2 \pi {\mathbb {Z}}\).
Remark 3.1
-
(i)
We use the symplectic transformation \((\rho , z) = (1 + \sqrt{2\xi } \sin \vartheta , \sqrt{2\xi } \cos \vartheta )\), instead of the simpler polar coordinates \((\rho ,z) = (1 + r \cos \vartheta , r \sin \vartheta )\) or \((\rho ,z) = (1 + r \sin \vartheta , r \cos \vartheta )\), in order to preserve the canonical Hamiltonian structure of system (3.18).
-
(ii)
Using \(\cos \vartheta \) for z and \(\sin \vartheta \) for \(\rho \) in the definition (3.23) of \(\Phi _3\), instead of vice versa, is just a matter of convenience: in this way, we get a positive angular velocity for the solution \(\vartheta (t)\).
3.4 Level Sets
Since the square root function \(\xi \mapsto \sqrt{\xi }\) is analytic in \((0, \infty )\), the function \(\alpha _3(\vartheta ,\xi )\) defined in (3.26) is analytic in \((\vartheta ,\xi ) \in {{\mathbb {T}}}\times (0, \xi _3)\). By the Taylor expansion (4.14) of \(\alpha _2(\rho ,z)\), by (3.26) and (3.27), one has
uniformly in \(\vartheta \in {{\mathbb {T}}}\). Note that \(\alpha _3(\vartheta ,\xi )\) is not analytic around \(\xi =0\), namely \(\alpha _3(\vartheta ,\xi )\), as a function of \(\xi \), is not a power series of \(\xi \) centered at zero; in fact, (3.32) is deduced from the analyticity of \(\alpha _2(\rho ,z)\) (and of its partial derivatives \(\partial _\rho \alpha _2(\rho ,z)\), \(\partial _z \alpha _2(\rho ,z)\)) around (1, 0), and then from the evaluation at \((\rho ,z) = (1 + \sqrt{2\xi } \sin \vartheta , \sqrt{2\xi } \cos \vartheta )\). See Sect. 4.3 for more details.
Moreover, \(\alpha _3(\vartheta ,0) = \alpha _2(1,0) = 0\), and, by (3.32),
Thus \(\alpha _3\) is \(C^1\) in \({{\mathbb {T}}}\times [0, \xi _3)\). Taking \(\xi _3\) smaller (i.e., \(\delta \) smaller) if necessary, we have \(\partial _\xi \alpha _3(\vartheta ,\xi ) > 0\) for all \((\vartheta ,\xi ) \in {{\mathbb {T}}}\times [0,\xi _3)\). Hence the function \(\xi \mapsto \alpha _3(\vartheta ,\xi )\) is strictly increasing on \([0, \xi _3)\). Moreover \(\alpha _3(\vartheta ,0) = 0\) and, by (3.29), \(\alpha _3(\vartheta , \xi _3/4) > 4 \tau \) for all \(\vartheta \in {{\mathbb {T}}}\).
As a consequence, for every \(c \in [0, 4 \tau ]\), for every \(\vartheta \in {{\mathbb {T}}}\), there exists a unique \(\xi \in [0, \xi _3/4)\) such that \(\alpha _3(\vartheta ,\xi ) = c\). We denote \(\gamma _c(\vartheta )\) the unique solution \(\xi \) of the equation \(\alpha _3(\vartheta ,\xi ) = c\). Thus,
Moreover, \(\gamma _0(\vartheta ) = 0\). Since \(\alpha _3\) is analytic in \((\vartheta ,\xi ) \in {{\mathbb {T}}}\times (0, \xi _3)\), by the implicit function theorem, the function \(\gamma _c(\vartheta )\) is analytic in \((\vartheta ,c) \in {{\mathbb {T}}}\times (0,4 \tau )\). The behavior of \(\gamma _c(\vartheta )\) around \(c=0\) is studied in Sect. 4.3. From (3.34),
for all \((\vartheta ,c) \in {{\mathbb {T}}}\times (0, 4 \tau )\). Since \(\partial _\xi \alpha _3(\vartheta ,\xi )\) is positive for all \((\vartheta ,\xi ) \in {{\mathbb {T}}}\times [0,\xi _3)\), by (3.36) it follows that
Thus, for all \(c \in [0, 4 \tau ]\), the level set \(\mathcal {P}_{3,c}\) is globally described as the graph
3.5 Integration of the System in the Radial–Vertical Plane
We consider the Cauchy problem of system (3.24) with initial datum \((\vartheta _0, \varphi _0, \xi _0) \in \mathcal {S}_3^*\) at the initial time \(t=0\). The components \(\vartheta _0\) and \(\xi _0\) of the initial datum determine the level \(c = \alpha _3(\vartheta _0,\xi _0)\); then the solution \((\vartheta (t), \varphi (t), \xi (t))\) of the Cauchy problem is in the level set \(\mathcal {P}_{3,c}\) for all \(t \in {{\mathbb {R}}}\), and, by (3.38), \(\xi (t) = \gamma _c(\vartheta (t))\) for all \(t \in {{\mathbb {R}}}\).
The first and third equations of system (3.24) are the Hamiltonian system (3.30), i.e.,
Since \(\xi (t) = \gamma _c(\vartheta (t))\), (3.39) becomes
By (3.35), the second equation in (3.40) is equal to the first equation in (3.40) multiplied by \(\partial _\vartheta \gamma _c(\vartheta )\), and therefore system (3.40) is equivalent to its first equation alone. Moreover, by (3.36), the first equation in (3.40) is also
where the denominator is nonzero by (3.36) or (3.37). Equation (3.41) with initial datum \(\vartheta (0) = \vartheta _0 \in {{\mathbb {T}}}\) is an autonomous Cauchy problem for the function \(\vartheta (t)\) taking values in \({{\mathbb {T}}}\), and it can be integrated by basic calculus.
The function \(\gamma _c(\vartheta )\) and its partial derivative \(\partial _c \gamma _c(\vartheta )\) are defined for \(\vartheta \in {{\mathbb {T}}}\), and hence they can also be considered as functions defined for \(\vartheta \in {{\mathbb {R}}}\) that are \(2\pi \)-periodic in \(\vartheta \). Thus, we first solve (3.41) considered as an equation for a function \(\vartheta ^r(t)\) taking values in \({{\mathbb {R}}}\); then the equivalence class of \(\vartheta ^r(t)\) mod \(2\pi \) will be a function \(\vartheta (t)\) taking values in \({{\mathbb {T}}}\) and solving (3.41).
For all \(c \in (0, 4 \tau )\), we define
By (3.37), \(F_c\) is a diffeomorphism of \({{\mathbb {R}}}\). Let \(\vartheta ^r_0 \in {{\mathbb {R}}}\) be a representative of the equivalence class \(\vartheta _0 \in {{\mathbb {T}}}\). If \(\vartheta ^r: {{\mathbb {R}}}\rightarrow {{\mathbb {R}}}\), \(t \mapsto \vartheta ^r(t)\) solves (3.41) with \(\vartheta ^r(0) = \vartheta _0^r\), then
and
Hence \(\vartheta ^r(t)\) in (3.44) is the unique solution of equation (3.41) with initial datum \(\vartheta ^r(0) = \vartheta ^r_0\). Now let \(\vartheta (t)\) be the equivalence class mod \(2\pi \) of \(\vartheta ^r(t)\), i.e.,
for all \(t \in {{\mathbb {R}}}\). Then the function \(\vartheta (t)\) in (3.45) is a function of time taking values in \({{\mathbb {T}}}\), and it solves (3.41) with \(\vartheta (0) = \vartheta _0\).
3.6 Rotation Period in the Radial–Vertical Plane
We show that the function \(\vartheta (t)\) in (3.45) is periodic, and we calculate its period. We begin with observing that the function \(F_c\) defined in (3.42) satisfies
Proof of (3.46)
Consider the definition (3.42), split the integral over \([0, 2 \pi + \vartheta ]\) into the sum of (i) the integral over \([0, 2\pi ]\) and (ii) the integral over \([2\pi , 2 \pi + \vartheta ]\); then \((i) = F(2\pi )\) by definition, while \((ii) = F(\vartheta )\) because the function \(\partial _c \gamma (\sigma ,c)\) is \(2\pi \)-periodic in \(\sigma \) and therefore (ii) is equal to the integral over \([0,\vartheta ]\). \(\square \)
Suppose that \(\chi (c)\) is nonzero. Then the function \(\vartheta ^r(t)\) defined in (3.44) satisfies
Proof of (3.47)
Applying (3.43) twice, one has
By the definition of \(T_c\) in (3.47) and by identity (3.46),
Hence \(F_c(\vartheta ^r(t + T_c)) = F_c( \vartheta ^r(t) + 2\pi )\), and, since \(F_c\) is invertible, we obtain (3.47). \(\square \)
From (3.47) it follows that \(\vartheta (t)\) defined in (3.45) is periodic in time with period \(T_c\).
3.7 Angle-Action Variables in the Radial–Vertical Plane
The Hamiltonian system (3.30) has one degree of freedom, and therefore, by the classical theory of Hamiltonian systems, it is completely integrable both in the sense that the differential equations can be solved by quadrature (i.e., they can be transformed into a problem of finding primitives of functions), and in the sense that they admit “angle-action variables” (this is the one-dimensional, simplest case of the Liouville-Arnold theorem). System (3.30) has been integrated in Sect. 3.5; now we construct its angle-action variables.
Given a point \((\vartheta ^*,\varphi ^*,\xi ^*) \in \mathcal {S}_3^*\), we calculate its level \(c = \alpha _3(\vartheta ^*,\xi ^*)\), which is in the interval \((0, 4 \tau )\), and we consider the Cauchy problem
The first equation in (3.48) is equation (3.41) in which \(\chi (c)\) is replaced by 1. Hence, following Sect. 3.5 line by line with 1 instead of \(\chi (c)\), the \({{\mathbb {R}}}\)-valued solution of (3.48) is given by (3.44) in which \(\chi (c)\) is replaced by 1; note that the functions \(\alpha _3(\vartheta ,\xi )\), \(\gamma _c(\vartheta )\), \(F_c(\vartheta )\) are all independent of \(\chi (c)\). We indicate \(\vartheta ^r_c(t)\) the \({{\mathbb {R}}}\)-valued solution of (3.48). Thus, since \(F_c(0) = 0\),
Moreover, following Sect. 3.6 with 1 instead of \(\chi (c)\), one has
We fix the representative \(\vartheta _1^*\) of the class \(\vartheta ^* \in {{\mathbb {T}}}\) such that \(\vartheta ^*_1 \in [0,2\pi )\), and we take the unique real number \(s_1^* \in [0,T_c^*)\) such that \(\vartheta ^r_c (s_1^*) = \vartheta ^*_1\), i.e., by (3.49), \(s_1^* = F_c(\vartheta _1^*)\). Then we define \(\sigma _1^* = s_1^* 2 \pi /T_c^*\), and we note that \(\sigma _1^* \in [0,2\pi )\) because \(s_1^* \in [0,T_c^*)\). By the definition of \(\sigma _1^*, s_1^*, T_c^*\), one has \(\sigma _1^* = f_c(\vartheta _1^*)\), where
Also, \(\vartheta _1^* = \vartheta _c^r(s_1^*)\) and \(s_1^* = \sigma _1^* T_c^* / 2 \pi \), whence \(\vartheta _1^* = g_c(\sigma _1^*)\), where
The function \(f_c\) is a diffeomorphism of \({{\mathbb {R}}}\) because it is a multiple of \(F_c\), and it satisfies
because \(F_c\) satisfies (3.46). The function \(g_c\) also satisfies
because \(\vartheta ^r_c\) satisfies (3.50). By (3.49), (3.51), (3.52), \(g_c(f_c(\vartheta )) = \vartheta \) for all \(\vartheta \in {{\mathbb {R}}}\) and \(f_c(g_c(\sigma )) = \sigma \) for all \(\sigma \in {{\mathbb {R}}}\), i.e., \(g_c\) is the inverse diffeomorphism of \(f_c\).
By (3.53) and (3.54), \(f_c\) and \(g_c\) induce diffeomorphisms of the torus
(with a common little abuse, we use the same notation for the diffeomorphisms of \({{\mathbb {R}}}\) and for the corresponding induced diffeomorphism of \({{\mathbb {T}}}\)). Now \(\sigma _1^*\) and \(\vartheta _1^*\) are related by the identities \(\vartheta _1^* = g_c( \sigma _1^* )\), \(\sigma _1^* = f_c(\vartheta _1^*)\). By the definition of \(\vartheta _1^*\), \(\vartheta ^* \in {{\mathbb {T}}}\) is the equivalence class of \(\vartheta _1^*\) mod \(2\pi \); let \(\sigma ^* \in {{\mathbb {T}}}\) be the equivalence class of \(\sigma _1^*\) mod \(2\pi \). Then
By (3.34), \(\xi ^* = \gamma _c(\vartheta ^*)\). Thus, we have expressed \((\vartheta ^*, \xi ^*)\) in terms of \((\sigma ^*,c)\). From now on, we write \(\sigma \) instead of \(\sigma ^*\).
Now we introduce a variable I and a function h to express the level c in terms of I, i.e., \(c = h(I)\); the function h is a diffeomorphism (to be determined) of some interval \((0, I^*)\) (to be determined) onto the interval \((0, 4 \tau )\) to which c belongs. We consider the map
defined on the set
and we want to calculate its Jacobian determinant. One has
where \(c = h(I)\). Hence the Jacobian matrix is
and its determinant is
where \(c = h(I)\). By (3.52) and (3.48),
and \(T_c^* = F_c(2\pi )\), see (3.50). Hence
where \(c = h(I)\). We want the determinant (3.57) to be \(= 1\), so that the transformation \(\Phi _4\) is symplectic and the Hamiltonian structure is preserved. The average
is a strictly increasing function of \(c \in [0, 4 \tau ]\) because its derivative \(h_1'(c) = F_c(2\pi ) / 2\pi \) is positive for all \(c \in (0, 4 \tau )\) by (3.37) and (3.42). Since \(\gamma _0(\vartheta ) = 0\), one has \(h_1(0) = 0\). We define
and we note that, by the definition of \(\gamma _c(\vartheta )\) in Sect. 3.4, one has \(0< I^* < \xi _3/4\). Thus, the interval \([0, I^*]\) is the image \(\{ h_1(c): c \in [0, 4 \tau ] \}\) of the interval \([0, 4 \tau ]\). We define \(h: [0, I^*] \rightarrow [0, 4 \tau ]\) as the inverse of \(h_1: [0, 4 \tau ] \rightarrow [0, I^*]\). Hence \(h(0) = 0\), \(h(I^*) = 4 \tau \),
and
This concludes the definition of the map \(\Phi _4\) in (3.55), which is a symplectic diffeomorphism of \(\mathcal {S}_4^*\) onto \(\mathcal {S}_3^*\).
Now that the transformation \(\Phi _4\) has been defined, we use it to transform system (3.24). We consider the change of variables \({\tilde{v}} = \Phi _4(v)\), where \({\tilde{v}} = (\vartheta , \varphi , \xi ) \in \mathcal {S}_3^*\), \(v = (\sigma ,\varphi ,I) \in \mathcal {S}_4^*\). This means that \(\vartheta = g_{h(I)}(\sigma )\) and \(\xi = \gamma _{h(I)}(g_{h(I)}(\sigma ))\), that is, \(\vartheta = g_c(\sigma )\) and \(\xi = \gamma _c(g_c(\sigma ))\) where \(c = h(I)\). A function \({\tilde{v}}(t) = \Phi _4(v(t))\) solves (3.24) in \(\mathcal {S}_3^*\) if and only if v(t) solves
in \(\mathcal {S}_4^*\), where
The second row of the inverse matrix \((D \Phi _4(v))^{-1}\) is (0, 1, 0), and therefore the second component of the vector field \(V_4\) is simply the second component of \(V_3\) (see (3.28)) evaluated at \(\Phi _4(\sigma ,\varphi ,I)\), which is \(\chi (h(I)) Q(\sigma ,I)\), where
Since the first and third equations of (3.62) are the Hamiltonian system (3.30) and \(\Phi _4\) is symplectic in the \((\sigma ,I)\) variables, the first and third components of the vector field \(V_4\) are \(\partial _I \mathcal {H}_4(\sigma ,I)\) and \(- \partial _\sigma \mathcal {H}_4(\sigma ,I)\) respectively, where \(\mathcal {H}_4:= \mathcal {H}_3 \circ \Phi _4\) (of course this can also be checked directly, without using the properties of the symplectic transformations). Now \(\mathcal {H}_3\) is defined in (3.31), with \(\Gamma \) defined in (3.19). Hence
and system (3.62) is
in \(\mathcal {S}_4^*\), with \(Q(\sigma ,I)\) defined in (3.64). The action I is constant in time. The angle \(\sigma \) rotates with constant angular velocity \(\partial _I \mathcal {H}_4(I) = \chi (h(I)) h'(I)\).
3.8 Reduction to a Constant Rotation in the Tangential Direction
Now that the equations of the motion in the radial–vertical plane have been written in angle-action variables \((\sigma , I)\) in (3.66), we want to obtain a similar simplification for the equation of the motion in the tangential direction \({\dot{\varphi }}\). Recalling the definition (3.56) of \(\mathcal {S}_4^*\), we consider the diffeomorphism
where \(\eta : {{\mathbb {T}}}\times (0, I^*) \rightarrow {{\mathbb {R}}}\) is a function to be determined. We consider the change of variables \({\tilde{v}} = \Phi _5(v)\), where \({\tilde{v}} = (\sigma , \varphi , I) \in \mathcal {S}_4^*\), \(v = (\sigma ,\beta ,I)\) \(\in \mathcal {S}_4^*\). A function \({\tilde{v}}(t) = \Phi _5(v(t))\) solves (3.62) in \(\mathcal {S}_4^*\) if and only if v(t) solves
in \(\mathcal {S}_4^*\), where
The Jacobian matrix and its inverse are
By (3.65) and (3.66), the vector field \(V_4\) in (3.63) is
Hence
We decompose \(Q(\sigma ,I)\) as the sum of its average in \(\sigma \in {{\mathbb {T}}}\) and its zero-average remainder, that is,
and define
Note that \(h'(I)\) is nonzero by (3.60). The function \(\eta \) is \(2\pi \)-periodic in \(\sigma \) because \({\tilde{Q}}\) is \(2\pi \)-periodic in \(\sigma \) with zero average on \({{\mathbb {T}}}\). By (3.72) and (3.71),
Thus, the vector field \(V_5\) depends only on the action variable I, and it is
where
Then system (3.68) is
3.9 Ratio of the Two Rotation Periods
Let \((\sigma _0, \beta _0, I_0) \in \mathcal {S}_4^* = {{\mathbb {T}}}^2 \times (0, I^*)\). Let \(\sigma _0^r \in {{\mathbb {R}}}\) be a representative of the equivalence class \(\sigma _0 \in {{\mathbb {T}}}\), and let \(\beta _0^r \in {{\mathbb {R}}}\) be a representative of \(\beta _0 \in {{\mathbb {T}}}\). The solution of the Cauchy problem for (3.74) in \({{\mathbb {R}}}^2 \times (0, I^*)\) with initial data \((\sigma _0^r, \beta _0^r, I_0)\) — i.e., system (3.74) where the first two equations are considered as equations for functions \(\sigma ^r(t)\), \(\beta ^r(t)\) taking values in \({{\mathbb {R}}}\) with initial data \(\sigma _0^r, \beta _0^r\) — is
Let \(\sigma (t), \beta (t)\) be the equivalence classes of \(\sigma ^r(t), \beta ^r(t)\) mod \(2\pi \), i.e.,
Then \((\sigma (t), \beta (t), I(t))\) is a function of time, taking values in \(\mathcal {S}_4^* = {{\mathbb {T}}}^2 \times (0, I^*)\), solving (3.74) with initial datum \((\sigma _0, \beta _0, I_0)\).
If \(\chi (h(I_0))\) is zero, then both \(\Omega _1(I_0)\) and \(\Omega _2(I_0)\) are zero by (3.73), and the solution \((\sigma (t),\beta (t)\), I(t)) is constant in time. If, instead, \(\chi (h(I_0))\) is nonzero, then \(\Omega _1(I_0)\) is also nonzero by (3.73) and (3.60), and the function \(\sigma (t)\) is periodic in time, with frequency \(\Omega _1(I_0)\) and period \(T_1(I_0):= 2 \pi / \Omega _1(I_0)\). By (3.73), and recalling that the determinant in (3.57) is \(=1\), the period \(T_1(I_0)\) is equal to the period \(T_c\) defined in (3.47) with \(c = h(I_0)\). If \(\Omega _2(I_0)\) is nonzero, then the function \(\beta (t)\) is also periodic in time, with frequency \(\Omega _2(I_0)\) and period \(T_2(I_0):= 2 \pi / \Omega _2(I_0)\).
We observe that the factor \(\chi (h(I))\) cancels out in the frequency ratio
and \(Q_0(I) / h'(I)\) is well-defined even if \(\chi (h(I))\) is zero. Hence \(Q_0(I) / h'(I)\) is well-defined for all \(I \in (0, I^*)\). By (3.71) and (3.64),
We make the change of variable \(g_{h(I)}(\sigma ) = \vartheta \) in the integral, that is, \(\sigma = f_c(\vartheta )\), where \(c = h(I)\) and \(f_c\) is defined in (3.51). Then \(d\sigma = f_c'(\vartheta ) d\vartheta \), and, by (3.51) and (3.42), \(f_c'(\vartheta ) = \partial _c \gamma _c(\vartheta ) 2 \pi / F_c(2\pi )\). Hence
where \(c = h(I)\). Moreover \(h'(I) F_c(2\pi ) = 2\pi \) because \(h'(I) F_c(2\pi ) / 2\pi \) is the determinant in (3.57), which is 1. Therefore
3.10 Motion of the Fluid Particles
We consider the composition
of the transformations defined in (3.3), (3.11), (3.23), (3.55), (3.67). The map \(\Phi \) is defined on \(\mathcal {S}_4^* = {{\mathbb {T}}}^2 \times (0, I^*)\), its image \(\Phi (\mathcal {S}^*_4)\) is the set \(\mathcal {S}^*\) in (1.10), and \(\Phi \) is a diffeomorphism of \(\mathcal {S}_4^*\) onto \(\mathcal {S}^*\). The image \((x,y,z) = \Phi (\sigma ,\beta ,I) \in \mathcal {S}^*\) of a point \((\sigma ,\beta ,I) \in \mathcal {S}_4^*\) is
with
where \(c = h(I)\). The analytic regularity of the map \(\Phi \) and its expansion around \(I=0\) are studied in Sect. 4.8.
By construction, a function \({\tilde{v}}(t) = \Phi (v(t))\) is the solution of the Cauchy problem (1.11) with initial datum \({\tilde{v}}_0 = \Phi (v_0) \in \mathcal {S}^*\) if and only if the function v(t) is the solution of (3.68), i.e., (3.74), with initial datum \(v_0 \in \mathcal {S}_4^*\). As a consequence, recalling (3.75), (3.76), the solution \(\tilde{v}(t)\) of the Cauchy problem (1.11) with initial datum \({\tilde{v}}_0 = (x_0, y_0, z_0) = \Phi (\sigma _0, \beta _0, I_0)\) is the function
The function in (3.81) has the form \({\tilde{v}}(t) = w(\Omega _1 t, \Omega _2 t)\) where \(w: {{\mathbb {T}}}^2 \rightarrow \mathcal {S}^*\) is the function \(w(\vartheta _1, \vartheta _2) = \Phi (\sigma _0 + \vartheta _1, \beta _0 + \vartheta _2, I_0)\) and \(\Omega _i = \Omega _i(I_0)\), \(i=1,2\). Hence \({\tilde{v}}(t)\) is quasi-periodic with frequency vector \((\Omega _1, \Omega _2)\) if \(\Omega _1 \ne 0\), the ratio \(\Omega _2 / \Omega _1\) is irrational, and the number of frequencies cannot be reduced, i.e., if \({\tilde{v}}(t)\) is not a periodic function.
Now suppose that \(\Omega _1\) is nonzero, that \(\Omega _2 / \Omega _1\) is irrational, and that \({\tilde{v}}(t)\) in (3.81) is periodic with a certain period \(T > 0\). Then, by (3.81),
Hence \(\sigma _0 + \Omega _1 (t + T)\) and \(\sigma _0 + \Omega _1 t\) are the same element of \({{\mathbb {T}}}\). This means that \(\Omega _1 T = 2 \pi n\) for some \(n \in {\mathbb {Z}}\). Similarly, \(\beta _0 + \Omega _2 (t + T) = \beta _0 + \Omega _2 t\) in \({{\mathbb {T}}}\), and \(\Omega _2 T = 2 \pi m\) for some \(m \in {\mathbb {Z}}\). Since \(\Omega _1\) is nonzero, n is also nonzero, and \(\Omega _2 / \Omega _1 = m/n\) is rational, a contradiction. This proves that, for \(\Omega _1\) nonzero and \(\Omega _2 / \Omega _1\) irrational, the function \({\tilde{v}}(t)\) in (3.81) is not periodic, and therefore it is quasi-periodic with frequency vector \((\Omega _1, \Omega _2)\).
3.11 The Pressure in Terms of the Action
The pressure and the action are related in the following way. By (1.4), (3.2) and (3.3), one has \(P (\Phi _1 (\rho , \varphi , z)) = p(\rho ,z) = \frac{1}{4} \alpha (\rho ,z)\). By (3.11), (3.15),
For notation convenience, we define
so that \(P(\Phi (\sigma ,\beta ,I)) = \mathcal {K}(I)\).
As a consequence, given any \(\ell \in (0, \tau )\), the level set \(\mathcal {T}_\ell = \{ (x,y,z) \in \mathcal {S}^*: P(x,y,z) = \ell \}\) of the pressure is the imagine
of the set \({{\mathbb {T}}}^2 \times \{ I \} \subseteq \mathcal {S}^*_4\), where \(I \in (0, I^*)\) satisfies \(\mathcal {K}(I) = \ell \). Moreover, \(\mathcal {K}(I^*) = \frac{1}{4} h(I^*) = \tau \), see (3.59) and the lines just after it.
3.12 Volume of the Region Enclosed by a Level Set
By the definition of \(\Phi _2\) in (3.11), for all \((\rho , \varphi , z) \in \mathcal {N}_2\) one has
and therefore
By (3.25), (3.61) and (3.69), one has
Hence, by the definition (3.79) of \(\Phi \), using repeatedly the chain rule and the formula for the determinant of the product of matrices, we deduce that \(\det D \Phi (\sigma , \beta , I) = 1\) for all \((\sigma , \beta , I) \in \mathcal {S}_4^*\). As a consequence, the map \(\Phi \) is volume-preserving.
Now fix any \(\ell _0 \in (0, \tau )\), and consider the set \(E_{\ell _0} = \{ (x,y,z) \in \mathcal {N}: P(x,y,z) < \ell _0 \}\). Since the circle \(\mathcal {C}\) in (3.1) has zero volume, the open sets \(E_{\ell _0}\) and \(E_{\ell _0} \setminus \mathcal {C}\) have the same volume. One has
and \(E_{\ell _0} \setminus \mathcal {C}\) is the union of the pressure level sets \(\mathcal {T}_\ell \) over \(\ell \in (0, \ell _0)\). Hence, by (3.84),
where \(I_0 \in (0, I^*)\) satisfies \(\mathcal {K}(I_0) = \ell _0\). Since the Jacobian determinant of \(\Phi \) is 1, one has
4 Taylor Expansions and Transversality
In this section we prove that the frequency \(\Omega _1(I)\) and the ratio \(\Omega _2(I) / \Omega _1(I)\), see (3.73) and (3.77), admit a Taylor expansion around \(I=0\) (which, in principle, is not obvious because of the square roots in the construction), and we calculate the first nonzero coefficient in their expansion after the constant term. This forces us to expand the function \(\alpha \) to degree 6. As a consequence, we obtain that \(\Omega _1(I)\) and \(\Omega _2(I) / \Omega _1(I)\) really change as I changes, and they vary in a smooth, strictly monotonic way, passing across every value of an interval exactly one time. This can be geometrically viewed as a transversality property. We also calculate the expansion of \(\Phi (\sigma , \beta , I)\) around \(I=0\).
4.1 Expansion of \(\alpha \)
The function \(\alpha \) is constructed in [10] by solving the pde system (4.4), where the functions F and G are expressed in terms of a function \(\psi \), see (4.2) and (4.3). The function \(\psi \) is defined in [10] as the solution of a degenerate ode problem. In Section 2.1 of [10] one finds the Taylor expansion of \(\psi \) of order 5 around zero; here we only use its expansion of order 3, which is
In Section 2.1 of [10] the functions H, F, G are defined in terms of \(\psi \) as
In Lemma 3 of [10] the analytic function \(\alpha (x,y)\) is defined as the unique solution of the system
in a neighborhood of \((x,y) = (1,0)\) such that \(\alpha (1,0) = 0\), with \(\partial _y \alpha \) not identically zero. In Remark 2 of [10] it is observed that \(\alpha \) is even in y, i.e., \(\alpha (x,y) = \alpha (x, -y)\). The coefficients of the monomials of degree 2 and 3 in the Taylor series
are given in Remark 4 of [10]; here we want to calculate the coefficients of the monomials of degree 4, 5 and 6. Monomials with odd exponent \(2j+1\) are not present in (4.5) because \(\alpha \) is even in y. One has
where \(O_n\) denotes terms with homogeneity \(\ge n\) in \((x-1, y)\). Since \(\alpha (x,y) = O_2\), by (4.1) and (4.5) one has
Expanding x and \(x^3\) around \(x=1\), and recalling the definition (4.2) of F, we calculate
From (4.6), (4.7) and the identity of each monomial in the differential equation \(\partial _x \alpha = F(x,\alpha )\) we get
To find the value of \(\alpha _{04}\) and \(\alpha _{06}\), we consider \(\partial _y \alpha (x,y)\) and \(G(x, \alpha (x,y))\) at \(x=1\) and we expand them around \(y=0\). By (4.5), since \(\alpha \) is even in y, one has
whence
By (4.7),
(it is to obtain (4.10) that we use the coefficient of \(s^3\) in (4.1)). Therefore, by (4.8),
Hence
By (4.9), (4.11) and the identity of each monomial in the differential equation \((\partial _y \alpha )^2 = G(x,\alpha )\) at \(x=1\) we get \(\alpha _{04} = 11/8\) and \(\alpha _{06} = 113/160\). Hence \(\alpha _{14} = 15/32\), \(\alpha _{24} = 249/128\), and
4.2 Expansion of \(\alpha _2\)
The function \(\alpha _2(\rho ,z) = \alpha (\rho , z/\rho )\) defined in (3.15) is even in z, it is analytic around \((\rho ,z) = (1,0)\), and its Taylor series
matches with the expansion in (4.12) in which \(x=\rho \) and \(y = z / \rho \). We expand
and we obtain
4.3 Expansion of \(\gamma _c(\vartheta )\)
Recalling the definition (3.20) of \(\mathcal {B}_2\) and the first inclusion in (3.21), the function \(\alpha _2(\rho ,z)\) in (3.15) is defined and analytic in the disc \((\rho - 1)^2 + z^2\) \(< \delta _2^2\). We define
The function \({\widetilde{\alpha }}_3\) is well-posed and analytic in \((\vartheta ,r) \in {{\mathbb {T}}}\times (- \delta _2, \delta _2)\), and, by (4.13), it is the power series
All \(P_n(\vartheta )\) are trigonometric polynomials, and \(P_n(-\vartheta ) = (-1)^n P_n(\vartheta )\), i.e., \(P_n\) is even for n even and \(P_n\) is odd for n odd, because \((-1)^k = (-1)^n\) for \(k+2j=n\). Hence
namely \({\widetilde{\alpha }}_3\) is an even function of the pair \((\vartheta ,r)\). In fact, (4.17) is the symmetry property \(\alpha _2(\rho ,-z) = \alpha _2(\rho ,z)\) expressed in terms of the function \({\widetilde{\alpha }}_3(\vartheta ,r)\). By (4.14),
Hence
The Fourier coefficients \((P_5)_5\) in (4.21) and \((P_6)_k\), \(k=2,4,6\), in (4.22) are not involved in the calculations we are going to make, and therefore we avoid to calculate their numerical value.
When \({\widetilde{\alpha }}_3(\vartheta ,r)\) is evaluated at \(r = \sqrt{2 \xi }\), we obtain the function \(\alpha _3(\vartheta ,\xi )\) defined in (3.26), i.e.,
The function \(\gamma _c(\vartheta )\) is defined in (3.34) as the unique solution \(\xi \) of the equation \(\alpha _3(\vartheta ,\xi ) = c\). Because of the square root in the construction, \(\gamma _c(\vartheta )\), as a function of c, is not analytic around \(c=0\), i.e., it is not a power series of the form \(\sum Q_n(\vartheta ) c^n\) for some analytic functions \(Q_n(\vartheta )\). However, \(\gamma _c(\vartheta )\) is a power series of the form \(\sum Q_n(\vartheta ) c^{n/2}\), namely there exists a function \({\widetilde{\gamma }}(\vartheta ,\mu )\), analytic around \(\mu = 0\), such that \(\gamma _c(\vartheta )\) is \({\widetilde{\gamma }}(\vartheta ,\mu )\) evaluated at \(\mu = \sqrt{c}\). To prove it, we use the implicit function theorem for analytic functions, taking into account the degeneracy of the problem.
We define
The function \(\mathcal {F}\) in (4.24) is well-defined and analytic in \({{\mathbb {T}}}\times (- \mu _0, \mu _0) \times (-w_0, w_0)\), for some \(\mu _0, w_0 > 0\) small enough, and, by (4.16),
Moreover, by (4.17) and (4.24),
For every \(\vartheta \in {{\mathbb {T}}}\) one has
Hence there exist two constants \(\mu _1, w_1\), with \(0 < \mu _1 \le \mu _0\), \(0 < w_1 \le w_0\), and a function \(w(\vartheta ,\mu )\), defined and analytic in \({{\mathbb {T}}}\times (-\mu _1, \mu _1)\), taking values in \((- w_1, w_1)\), such that
and such that if a point \((\vartheta , \mu , a) \in {{\mathbb {T}}}\times (- \mu _1, \mu _1) \times (- w_1, w_1)\) is a zero of \(\mathcal {F}\), then \(a = w(\vartheta ,\mu )\).
By (4.26) and (4.25), for all \((\vartheta ,\mu )\) one has
Hence the point \((\vartheta , \mu , w(-\vartheta , -\mu )) \in {{\mathbb {T}}}\times (- \mu _1, \mu _1) \times (- w_1, w_1)\) is a zero of \(\mathcal {F}\), and therefore it belongs to the graph of the implicit function, i.e.,
From the second identity in (4.26) and formula (4.16) it follows that \(w(\vartheta ,r)\) is the power series
where the functions \(W_n(\vartheta )\) are determined by the identity \(\sum _{n=2}^\infty P_n(\vartheta ) w^n(\vartheta ,\mu ) \mu ^{n-2} = 1\), i.e., \(W_0(\vartheta ) = 2^{-\frac{1}{2}}\) and \(W_n(\vartheta )\) are trigonometric polynomials recursively determined by the system
By (4.27) and (4.28), \(W_n(-\vartheta ) = (-1)^n W_n(\vartheta )\), i.e., \(W_n\) is even for n even and \(W_n\) is odd for n odd. We will make use of equations (4.29) for \(m=1,2,3,4\), which are
By the definition (4.24) of \(\mathcal {F}\), the second identity in (4.26) implies that
with \(\mu \ne 0\). Identity (4.34) also holds for \(\mu =0\) because \({\widetilde{\alpha }}_3(\vartheta ,0) = \alpha _2(\phi (\vartheta ,0)) = \alpha _2(1,0) = 0\).
By the first identity in (4.26), taking \(\mu _1\) smaller if necessary, one has \(w(\vartheta ,\mu ) > 0 \) for all \((\vartheta , \mu ) \in {{\mathbb {T}}}\times (- \mu _1, \mu _1)\). Given any \(c \in [0, \mu _1^2)\), there exists a unique \(\mu \in [0, \mu _1)\) such that \(c = \mu ^2\), that is, \(\mu = \sqrt{c}\). Also, \(\mu w(\vartheta ,\mu ) \ge 0\), and there exists a unique \(\xi \ge 0\) such that \(\mu w(\vartheta ,\mu ) = \sqrt{2 \xi }\), that is, \(\xi = \frac{1}{2} \mu ^2 w^2(\vartheta ,\mu )\). As a consequence, by (4.23) and (4.34),
Hence \(\xi = \gamma _c(\vartheta )\), and therefore \(\gamma _c(\vartheta ) = \frac{1}{2} \mu ^2 w^2(\vartheta ,\mu )\) where \(\mu = \sqrt{c}\). In other words, we have proved that
where \({\widetilde{\gamma }}\) is the analytic function
By (4.27),
By (4.38) and (4.39), one has \(Q_n(-\vartheta ) = (-1)^n Q_n(\vartheta )\).
4.4 Expansion of the Average of \(\gamma _c(\vartheta )\) and \(\partial _c \gamma _c(\vartheta )\)
We study the average of \({\widetilde{\gamma }}(\vartheta ,\mu )\), \(\gamma _c(\vartheta )\) and \(\partial _c \gamma _c(\vartheta )\) over \(\vartheta \in [0,2\pi ]\). To shorten the notation, given any \(2\pi \)-periodic function \(f(\vartheta )\), we denote \(\langle f \rangle \) its average over the period, i.e.,
For n odd, the trigonometric polynomial \(Q_n(\vartheta )\) in (4.39) is \(2\pi \)-periodic and odd, and therefore \(\langle Q_n \rangle = 0\). As a consequence, by (4.39),
By (3.58), (4.36) and (4.40), one has
Hence \(h_1(c)\) is analytic around \(c=0\), in the sense that \(h_1(c)\), which is defined for \(c \in [0, \mu _1^2)\), coincides in \([0, \mu _1^2)\) with the power series in (4.41), which is a function defined for \(c \in (-\mu _1^2, \mu _1^2)\) and analytic in that interval. Note that the average of \(\gamma _c(\vartheta )\) is analytic around \(c=0\) even if the function \(\gamma _c(\vartheta )\) itself is not analytic in c around \(c=0\).
Now we calculate the averages \(\langle Q_n \rangle \) for \(n=2,4,6\). By (4.39),
Hence \(\langle Q_2 \rangle = 1/4\). Since \(P_2 = 2\) and \(W_0 = 2^{-\frac{1}{2}}\), from (4.30) we get
Using (4.31) to substitute \((2 W_0 W_2 + W_1^2)\), and using also (4.43), we calculate
By (4.19) and (4.20), we obtain
Using (4.33) to substitute \((2 W_0 W_4 + 2 W_1 W_3 + W_2^2)\), and also (4.43), we calculate
From (4.31) we obtain
The Fourier coefficients \((W_3)_k\), \(k=5,7,9\), are not involved in the calculation of \(\langle Q_6 \rangle \), and therefore we do not calculate them. By (4.19), ..., (4.22), (4.46), (4.47), we calculate
Hence, integrating (4.45), we obtain
(where \(65536 = 2^{16}\)), and, by (4.41),
Since \(\langle Q_6 \rangle \) in (4.48) is nonzero, \(h_1(c)\) is a nonlinear, analytic function of c. Its derivative is
which is a nonconstant, analytic function of c.
4.5 Expansion of J(c)
The average J(c) is defined in (3.78). By (4.36), the partial derivative \(\partial _c \gamma _c(\vartheta )\) satisfies
where \(\nu (\vartheta ,\mu )\) is the analytic function
with \(Q_n(\vartheta )\) defined in (4.39). By (4.38), one has \(\nu (-\vartheta ,-\mu ) = \nu (\vartheta ,\mu )\). Moreover
and \(Q_2 = 1/4\) by (4.42), \(Q_3 = W_0 W_1\) by (4.39), \(\langle Q_3 \rangle = 0\) because \(Q_3\) is odd, and \(\langle Q_4 \rangle = 0\) by (4.44). Regarding the denominator in the definition of J(c), one has \(\sqrt{2 \gamma _c(\vartheta )} = \sqrt{2 \xi } = \sqrt{c} \, w(\vartheta , \sqrt{c})\) by construction (see (4.35) and the lines preceding it), and, by (4.28), (4.43), (4.19),
Taking a smaller \(\mu _1\) if necessary, the function
is defined and analytic in \({{\mathbb {T}}}\times (- \mu _1, \mu _1)\), it satisfies
and it has expansion \(m(\vartheta ,\mu ) = \sum _{n=0}^\infty M_n(\vartheta ) \mu ^n\) for some trigonometric polynomials \(M_n(\vartheta )\). From (4.56) it follows that \(M_n(-\vartheta ) = (-1)^n M_n(\vartheta )\). Therefore \(\langle M_n \rangle = 0\) for n odd, and
By (4.43) and (4.19), the average of \(W_1(\vartheta ) \sin \vartheta \) is \(- 1/8\). Hence \(\langle M_2 \rangle = 7/16\), and
Thus, the average J(c) is an analytic function of c around \(c=0\) (even if the integrand function \(m(\vartheta , \sqrt{c})\) is not). Moreover, taking \(\mu _1\) smaller if necessary, J(c) is strictly increasing in \([0, \mu _1^2)\).
4.6 Expansion of h(I) and \(\Omega _1(I)\)
We have already proved that \(h_1(c)\) in (3.58) is analytic around \(c=0\), with expansion (4.49). Hence its inverse function \(h(I) = h_1^{-1}(I)\) is also analytic around \(I=0\), and it satisfies
The frequency \(\Omega _1(I)\) is defined in (3.73), and it is the product of the \(C^\infty \) cut-off function \(\chi (h(I))\) times the analytic function \(h'(I)\) in (4.58).
4.7 Expansion of the Frequency Ratio \(\Omega _2(I) / \Omega _1(I)\)
By (3.77) and (3.78), the frequency ratio \(\Omega _2(I) / \Omega _1(I)\) coincides with the function
By its definition in [10], the function H(c) is analytic around \(c=0\). Hence the composition H(h(I)) is analytic around \(I=0\), and, by (4.10) and (4.58),
We write its square root as the product
where
Since the function \(x \mapsto \sqrt{1 + x}\) is analytic around \(x=0\), the function B(I) is analytic around \(I=0\). The function J(h(I)) is also analytic around \(I=0\), and, by (4.57) and (4.58),
Hence
and the function \(\mathcal {R}(I)\) is analytic around \(I=0\). Taking a smaller \(I^*\) if necessary, both \(\mathcal {R}(I)\) and A(h(I)) are strictly increasing functions of \(I \in [0, I^*)\).
4.8 Expansion of \(\Phi (\sigma , \beta , I)\)
Since \(Q_2 = 1/4\), by (4.36), (4.39), (4.51), (4.52) one has
Hence, by (3.42), \(F_c(\vartheta ) = \frac{\vartheta }{4} + O(c^{\frac{1}{2}})\) and therefore, by (3.51), \(f_c(\vartheta ) = \vartheta + O(c^{\frac{1}{2}})\). As a consequence, \(g_c\) in (3.52), being the inverse of \(f_c\), satisfies
By (4.58), \(h(I) = 4 I + O(I^3)\) and \(h'(I) = 4 + O(I^2)\). Hence, at \(c = h(I)\), one has
and therefore the functions \(\varrho (\sigma ,I), \zeta (\sigma ,I)\) in (3.80) have expansion
By (4.62) and (4.63), the function \(Q(\sigma ,I)\) in (3.64) satisfies
Hence \(Q_0, {\widetilde{Q}}, \eta \) in (3.71), (3.72) satisfy
The map \(\Phi _1\) in (3.3) is analytic in \(\mathcal {N}_1\) because \(\rho \) does not vanish in \(\mathcal {N}_1\); the map \(\Phi _2\) in (3.11) is also analytic in \(\mathcal {N}_2\) because \(\rho \) does not vanish in \(\mathcal {N}_2\).
The map \(\Phi _3\) in (3.23) is analytic in \(\mathcal {B}_3\) because \(\xi \) is positive in \(\mathcal {B}_3\); however, \(\Phi _3\) is not analytic in \(\xi \) around \(\xi =0\). Nonetheless, \(\Phi _3\) can be obtained by evaluating at \(r = \sqrt{2 \xi }\) a map that is analytic around \(r=0\), exactly like \(\alpha _3(\vartheta ,\xi )\) in (4.23), like \(\gamma _c(\vartheta )\) in (4.36), and like \(\partial _c \gamma _c(\vartheta )\) in (4.51).
Similarly, both the map \(\Phi _4\) in (3.55) and the map \(\Phi _5\) in (3.67) are analytic in \(\mathcal {S}_4^*\), they are not analytic in I around \(I=0\), but they can be obtained by evaluating at \(\mu = \sqrt{I}\) some suitable maps that are analytic around \(\mu = 0\), because \(\gamma _c, F_c, f_c, g_c, \sqrt{H(c)}\) are all functions of this type.
As a consequence, the map \(\Phi \) in (3.79) is analytic in \(\mathcal {S}_4^*\), it is not analytic in I around \(I=0\), and it can be obtained by evaluating at \(\mu = \sqrt{I}\) a map that is analytic around \(\mu = 0\). Hence \(\Phi \) admits a converging expansion in powers of \(\sqrt{I}\) around \(I=0\).
4.9 Smallness Conditions
The parameter \(\delta \) in the definition (3.1) of the set \(\mathcal {N}\) is subject to the following smallness conditions. After (3.1) we have taken \(\delta \in (0,1)\) to obtain that \(\mathcal {N}\) is an open neighborhood of the circle \(\mathcal {C}\) in (3.1) with \(\rho> 1-\delta > 0\) in \(\mathcal {N}\), and \(\delta \le r_0\) to obtain that the functions \(\alpha (\sqrt{x^2 + y^2},z)\) and \(H(\alpha (\sqrt{x^2 + y^2},z))\) are analytic in \((x,y,z) \in \mathcal {N}\), where \(r_0\) is a universal constant given by the definition of \(\alpha \) and H, i.e., by Gavrilov’s construction in [10]. After (3.20) we have defined \(\delta _2 = 2 \delta / 3\), and we have assumed \(\delta \le 1/2\) to obtain the inclusion (3.21). After (3.23) we have defined \(\xi _3 = \delta _2^2/2 = 2 \delta ^2 / 9\). In (3.33) we have proved that \(\partial _\xi \alpha _3(\vartheta ,0) = 4\) for all \(\vartheta \in {{\mathbb {T}}}\), and that \(\partial _\xi \alpha _3(\vartheta ,\xi )\) is continuous in \({{\mathbb {T}}}\times [0, \xi _3)\) — in fact, in Sect. 4.3 we have proved more, because \(\alpha _3(\vartheta ,\xi ) = {\widetilde{\alpha }}_3(\vartheta , \sqrt{2\xi })\), see (4.23), and \(\widetilde{\alpha }_3(\vartheta ,r)\) is the analytic function in (4.15), (4.16); hence \(\partial _\xi \alpha _3(\vartheta ,\xi ) = \sum _{n=2}^\infty P_n(\vartheta ) n (2\xi )^{(n-2)/2} = 4 + 3 P_3(\vartheta ) \sqrt{2\xi } + O(\xi )\). Hence, by continuity, there exists a universal constant \(\xi ^* > 0\) such that \(\partial _\xi \alpha _3(\vartheta ,\xi ) > 0\) in \({{\mathbb {T}}}\times [0, \xi ^*)\). Therefore the condition on \(\xi _3\) after (3.33) (where we say “Taking \(\xi _3\) smaller if necessary”) is \(\xi _3 \le \xi ^*\). In terms of \(\delta \), this means \(\delta \le 3 \sqrt{\xi ^*/2}\), which is a universal constant.
The constants \(w_0, \mu _0\) after the definition (4.24) of \(\mathcal {F}\) are universal, and the constants \(w_1, \mu _1\) after the application of the implicit function theorem in (4.26) are universal too. By (4.26), \(w(\vartheta ,0) > 0\), and therefore, by continuity, there exists a universal constant \(\mu _1^* > 0\) such that \(w(\vartheta ,\mu ) > 0\) for all \((\vartheta ,\mu ) \in {{\mathbb {T}}}\times (-\mu _1^*, \mu _1^*)\). Hence the condition on \(\mu _1\) after (4.34) (where we say “taking \(\mu _1\) smaller if necessary”) is \(\mu _1 \le \mu _1^*\). The same happens for the condition on \(\mu _1\) after (4.54), to obtain that the function \(m(\vartheta ,\mu )\) in (4.55) is well-defined and analytic in \({{\mathbb {T}}}\times (-\mu _1, \mu _1)\), and for the condition on \(\mu _1\) after (4.57), to obtain that J(c) is strictly increasing in \([0, \mu _1^2)\). Thus, we fix \(\mu _1\) as the smallest of these three constants, and we obtain that \(\mu _1\) is a universal positive constant.
The parameter \(\tau \) is related to \(\mu _1\) by the inequality \(4 \tau \le \mu _1^2\), because \([0, 4 \tau )\) is the interval where c varies, and the functions \(\gamma _c(\vartheta )\), \(\partial _c \gamma _c(\vartheta )\) and J(c) are obtained by evaluating at \(\mu = \sqrt{c}\) some functions of \((\vartheta ,\mu )\) that are well-defined, analytic and monotonic in \({{\mathbb {T}}}\times (-\mu _1, \mu _1)\), see (4.36), (4.51), (4.57). Thus, we want that, for all \(c \in [0, 4\tau )\), the square root \(\sqrt{c}\) belongs to the domain \((-\mu _1, \mu _1)\) of those analytic functions, and this is true if \(4 \tau \le \mu _1^2\).
Regarding the parameter \(\tau \), there are two other conditions to consider. The first one is (1.9), which holds if \(\tau \) is smaller than the infimum of the pressure P on the set \(\mathcal {N}{\setminus } \mathcal {N}'\). Since \(P(x,y,z) = p(\rho ,z) = \frac{1}{4} \alpha (\rho ,z)\) (see (3.2)), by the definition (3.1) of \(\mathcal {N}\) and \(\mathcal {N}'\) it follows that that infimum depends only on \(\delta \). The last condition for \(\tau \) is after (4.61), where we say “Taking a smaller \(I^*\) if necessary”, to obtain that \(\mathcal {R}(I)\) is strictly increasing in \([0, I^*)\). Since the function \(\mathcal {R}\) does not depend on any parameter, this is a condition of the form \(I^* \le I_0\) for some universal constant \(I_0 > 0\). Moreover, the invertible function \(h_1\) expressing \(I^*\) in terms of \(\tau \) in (3.59) is also independent on any parameter (see the definition of \(h_1\) in (3.58) and its expansion in (4.49)), and therefore this condition for \(\tau \) is satisfied for \(\tau \) smaller than a universal constant.
Regarding \(\varepsilon \), the only condition to consider is that \(0 < \varepsilon \le \tau / 3\), see after (1.9).
In conclusion, the parameters \(\delta , \tau \) and \(\varepsilon \) must satisfy
where \(\delta _0\) is a universal constant, and \(\tau _0(\delta )\) depends only on \(\delta \). We fix \(\delta = \delta _0\) and \(\tau = \tau _0(\delta _0)\). Both \(\delta _0\) and \(\tau _0(\delta _0)\) are universal constants. We rename \(\tau _0:= \tau _0(\delta _0)\) and \(\varepsilon _0:= \tau _0/3\). Since \(\tau = \tau _0\), by (3.59), \(I^*\) is also a universal constant.
For notation convenience, we consider the function \(\mathcal {K}(I) = \frac{1}{4} h(I)\) defined in (3.83). Hence, by (3.73) and (3.7),
For all \(\varepsilon \in (0, \varepsilon _0]\), the proof of Theorem 1.1 is complete.
References
Arnold, V.I.: Sur la topologie des écoulements stationnaires des fluides parfaits. C. R. Acad. Sci. Paris 261, 17–20 (1965)
Arnold, V.I.: Sur la géometrie différentielle des groupes de Lie de dimension infinie et ses application à l’hydrodynamique des fluides parfaits. Ann. Inst. Fourier (Grenoble) 16(1), 319–361 (1966)
Arnold, V.I., Khesin, B.A.: Topological Methods in Hydrodynamics, 2nd edn. Springer, Berlin (2021)
Baldi, P., Montalto, R.: Quasi-periodic incompressible Euler flows in 3D. Adv. Math. 384, 107730 (2021)
Constantin, P., La, J., Vicol, V.: Remarks on a paper by Gavrilov: Grad-Shafranov equations, steady solutions of the three dimensional incompressible Euler equations with compactly supported velocities, and applications. Geom. Funct. Anal. (GAFA) 29(6), 1773–1793 (2019)
Crouseilles, N., Faou, E.: Quasi-periodic solutions of the 2D Euler equation. Asympt. Anal. 81, 31–34 (2013)
Domínguez-Vázquez, M., Enciso, A., Peralta-Salas, D.: Piecewise smooth stationary Euler flows with compact support via overdetermined boundary problems. Arch. Ration. Mech. Anal. 239, 1327–1347 (2021)
Enciso, A., Peralta-Salas, D., de Lizaur, F.T.: Quasi-periodic solutions to the incompressible Euler equations in dimensions two and higher. J. Differential Equations 354, 170–182 (2023)
Etnyre, J., Ghrist, R.: Contact topology and hydrodynamics. II. Solid tori. Ergod. Theory Dyn. Syst. 22(3), 819–833 (2002)
Gavrilov, A.V.: A steady Euler flow with compact support. Geom. Funct. Anal. (GAFA) 29(1), 190–197 (2019)
Khesin, B., Kuksin, S., Peralta-Salas, D.: KAM theory and the 3D Euler equation. Adv. Math. 267, 498–522 (2014)
Koch, G., Nadirashvili, N., Seregin, G.A., Šverák, V.: Liouville theorems for the Navier-Stokes equations and applications. Acta Math. 203(1), 83–105 (2009)
Peralta-Salas, D., Slobodeanu, R.: A Liouville-type theorem for the 3D primitive equations. Phys. D Nonlin. Phenom. 453, 133821 (2023)
Acknowledgements
Supported by the Project PRIN 2020XB3EFL Hamiltonian and dispersive PDEs. The author is grateful to the two anonymous referees for their interesting comments and their stimulating questions.
Funding
Open access funding provided by Università degli Studi di Napoli Federico II within the CRUI-CARE Agreement.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The author states that there is no conflict of interest.
Additional information
Communicated by A. Vasseur.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Baldi, P. Nearly Toroidal, Periodic and Quasi-periodic Motions of Fluid Particles Driven by the Gavrilov Solutions of the Euler Equations. J. Math. Fluid Mech. 26, 1 (2024). https://doi.org/10.1007/s00021-023-00836-1
Accepted:
Published:
DOI: https://doi.org/10.1007/s00021-023-00836-1
Keywords
- 3D Euler equations
- Gavrilov solutions
- Fluid particle dynamics
- Periodic motions
- Quasi-periodic motions
- Frequency function