1 Introduction

The mechanics of fluids was founded by Euler more than 250 years ago. Euler described the evolution in time of an incompressible three-dimensional velocity field in three space dimensions. The fluid velocity changes according to Newton’s second law, by the action of fluid pressure. Euler equations are unique because fluids obey the basic physical principles of classical mechanics, such as Galilean invariance, Newton’s law, energy conservation, time-reversal symmetry, etc.

Although easy to write, these equations are hard to solve because they are partial differential equations that involve both nonlinearities and nonlocalities: The velocity changes because of pressure which itself depends on the velocity gradients in the whole space.

As a result, the general comprehension of the true nature of solutions to the Euler equations remains an open question. In particular, we have the so-called regularity problem or that of the possible existence of finite-time singularity solutions: Does a finite energy and smooth initial condition for the velocity field remain regular for all times as the velocity flow evolves in accordance with Euler equations for an inviscid and incompressible fluid? By smooth, we mean that the initial condition is differentiable everywhere. Since the early twentieth century, both mathematicians and physicists intend to answer this cumbersome problem [1].

The recent numerical observation of a potential singularity in a three-dimensional axisymetrical flow by Luo and Hou [2] on the boundary of the domain boosted intensive research from physical [3] and mathematical points of view [4]. More recently, Chen and Hou [5] have succeeded to prove the previously observed numerical singularity on the boundary domain [2].

As suggested by Pomeau and collaborators over the years [6–10], the potential connection of the putative finite-time singular solutions of Euler equations with the understanding of the intermittent behavior of turbulent motion makes the question on the existence of singular solutions of Euler equations of fundamental importance. In particular, in the framework of the focusing nonlinear Schödinger equation with small dissipation, it has been observed a scenario of potential singularities being the cause of intermittent behavior and of the anomaly of dissipation in one and two space dimensions [11, 12]. In the case of Luo and Hou, the singularity is raised on the boundaries, thus it may be a key element on the boundary layer dynamics. In a different context, by assuming a space anisotropy, one of the authors has shown that advection may cause a shock-like singularity on a circular rim of finite radius [13].

The main difficulty of Euler equations arises from the nonlocal character, however, recently, Tarek Elgindi [14] proposes a new endeavor: for a bounded vorticity distribution, the velocity field scales proportionally to spatial coordinates and the general nonlocality is reduced only to the coefficient in front. Within this approximation, Elgindi was able to find a self-similar finite-time singularity as a consequence of a nonsmooth initial condition without swirl. Contrary to Refs. [2, 5, 13], Elgindi’s solution arises at a single point in the 3D space [14].

In this paper, we study a point-like finite-time singularity in the frame of the Euler equations in the axisymmetric configuration. Following Elgindi, we approximate the nonlocal dependence of the Biot–Savart law, but keep both the transport and the swirl velocity in the original Euler equations for an axisymmetric flow. Next, by expanding both the vorticity and the swirl velocity in an asymptotic series of the polar angular variable, we derive an infinite hierarchy of partial differential equations for the amplitude coefficients. Assuming a particular self-similar Ansatz, then the hierarchy is reduced to an autonomous set of infinitely many ordinary differential equations, in which a parameter, named ν, remains unknown. This is a “nonlinear eigenvalue” problem [15] which may be tackled via a dynamical system approach. The solution of the infinite hierarchy, together with the right boundary conditions, selects the value of ν. Finally, some special solutions are provided numerically. The numerical existence of such a solution indicates strongly the possibility of a positive existence of point-like finite-time singularities in Euler equations in an infinite domain.

The manuscript is structured as follows: in Sect. 2, by using the Elgindi approximation, the Euler equations for the axisymmetric configuration are simplified to basic model equations (7)–(10). Next, this basic model is written in spherical coordinates and projected into an infinite set of partial differential equations for the vorticity and swirl velocity amplitudes. In Sect. 3, the search for self-similar solutions transforms the original set of partial differential equations into a set of an autonomous hierarchy of ordinary differential equations, and a qualitative description of the resulting dynamical system is done. Finally, in Sect. 4, assisted by a numerical scheme of the search of solutions, the selection of the nonlinear eigenvalue ν is illustrated. To do that, the infinite hierarchy is truncated up to some order N, and a finite set of 3N differential equations is solved by a shooting-like method. An example of a solution is sketched for \(N=2\). Finally, we conclude.

2 Basic model

2.1 Euler equations for the axisymmetric flow

In an axisymmetric geometry, the fluid equations are simpler than in the general case. The flow is described by the three components of velocity, but no dependence is considered in the azimuthal variable ϕ. We can assume indistinctly either cylindrical or spherical coordinates. Initially, we use the cylindrical coordinates, thus \(\boldsymbol{v}= ( v_{r}(r,z,t), v_{\phi}(r,z,t), v_{z}(r,z,t) )\).

The axial vorticity, \(\omega _{\phi}\), and the axial velocity, \(v_{\phi}\), obey the equations [16]:

$$\begin{aligned} &\biggl( \frac{\partial }{\partial t} + v_{r} \frac{\partial }{\partial r} + v_{z} \frac{\partial }{\partial z} \biggr) \omega _{\phi } = \frac{v_{r} \omega _{\phi}}{r}+ \frac{1}{r} \frac{\partial v_{\phi}^{2} }{\partial z} , \end{aligned}$$
(1)
$$\begin{aligned} &\biggl( \frac{\partial }{\partial t} + v_{r} \frac{\partial }{\partial r} + v_{z} \frac{\partial }{\partial z} \biggr) v_{\phi }= - \frac{v_{r} v_{\phi}}{r}. \end{aligned}$$
(2)

These coupled partial differential equations become self-contained if a stream function is introduced:

$$ v_{r} = -\frac{1}{r} \frac{\partial \psi }{\partial z}, \qquad v_{z} = \frac{1}{r} \frac{\partial \psi }{\partial r}, $$
(3)

such that the velocity flow remains incompressible, and the axial vorticity component depends on terms of the stream function:

$$\begin{aligned} \omega _{\phi }= \frac{\partial v_{r}}{\partial z} - \frac{\partial v_{z}}{\partial r}= - \frac{1}{r} \biggl( \frac{\partial ^{2} \psi }{\partial r^{2}} -\frac{1}{r} \frac{\partial \psi }{\partial r} + \frac{\partial ^{2} \psi }{\partial z^{2}} \biggr). \end{aligned}$$
(4)

This Poisson-like equation introduces a nonlocal dependence of the velocities, \(v_{r}\) and \(v_{z}\), on \(\omega _{\phi}\).

In conclusion, equations (1) and (2), together with (3) and (4), are formally a set of two time-dependent partial differential equations for the fields \(\psi (r,z,t)\) and \(v_{\phi}(r,z,t)\). The previous system was already numerically studied in the early 1990s [17, 18] and, more recently, but in a finite domain in [2–4].

The boundary conditions for the vorticity and the velocity are such that [2]

$$\begin{aligned} \omega _{\phi }(r=0,z,t)=v_{\phi}(r=0,z,t) = 0 , \end{aligned}$$
(5)

and, because of the condition of finite energy flow, both fields, \(\omega _{\phi }(r=0,z,t)\) and \(v_{\phi}(r=0,z,t) \), decrease to zero as \(z\to \pm \infty \). Equation (2) conserves \(rv_{\phi}\) thus, remarkably, if \(v_{\phi }\geq 0\) initially, then \(v_{\phi}\geq 0\) for all times.

2.2 Axisymmetric flow with the Elgindi approximation

By analogy with the magneto-statics, the solution of (4) is

$$\begin{aligned} \psi (r,z,t) = \frac{r}{4 \pi} \int \frac{\cos \phi ' }{ ( {r'}^{2}+ r^{2} - 2 {r'} r \cos \phi '+ ( z-z')^{2} )^{1/2} } \omega _{\phi}\bigl(r',z',t' \bigr) r' \,dr' \,d\phi ' \,d z' . \end{aligned}$$
(6)

Essentially, the Elgindi approximation of the Biot–Savart law considers a punctured domain excluding the inner core, \(r^{\prime 2}+z^{\prime 2} <{ \eta (r^{2}+z^{2})} \), contribution to the integral which is finite because \(|\omega _{\phi}({r'},z',t)|\) remains bounded. Then the inner behavior of the integrals for the velocities is regular.Footnote 1 Finally, the contribution to the Biot–Savart integral (6) inside the punctured domain may be approximated via a multipolar expansion for \({r'}\gg r\) [14]. The inclusion of \(\eta >1 \) ensures that it is always possible to use this multipolar approximation. Nevertheless, it results that the final singularity solution does not depend on this parameter η.

From now on, as Elgindi, we assume up–down-symmetric vorticity, \(\omega _{\phi}(r,-z,t) = -\omega _{\phi}(r,z,t)\), which implies that the vertical velocity is also an odd function, \(v_{z}(r,-z,t) =- v_{z}(r,z,t)\), while the radial velocity \(v_{r}(r,-z,t) = v_{r}(r,z,t)\) is an even function of z. Lastly, as \(\omega _{\phi}\), the swirl velocity is an odd function, \(v_{\phi}(r,-z,t) = -v_{\phi}(r,z,t)\).

Following the Elgindi approximation, the radial, \(v_{r}\), and vertical, \(v_{z}\), velocity fields (3) become

$$\begin{aligned} v_{r}(r,z)\approx - r {\mathcal {L}} [\omega _{\phi}] (r,z,t) \quad \text{and} \quad v_{z}(r,z)\approx 2z {\mathcal {L}} [\omega _{\phi}] (r,z,t) , \end{aligned}$$

where the nonlocal kernel reads [14]

$$\begin{aligned} \mathcal {L} [\omega _{\phi}] (r,z,t)= \frac{3}{4 } \int _{ r^{\prime 2}+z^{\prime 2} > \eta (r^{2}+z^{2}) } \frac{{r'}^{2} {z'} \omega _{\phi}({r'},z',t) }{ ( {r'}^{2}+{z'}^{2} )^{5/2} } \,d{r'} \,d z' . \end{aligned}$$
(7)

Therefore, the final axisymmetric Euler equations simplify to

$$\begin{aligned} &\frac{\partial \omega _{\phi }}{\partial t} + {\mathcal {L}} [\omega _{ \phi}] {\mathcal {D}} \omega _{\phi }= - \omega _{\phi }{\mathcal {L}} [ \omega _{\phi}] + \frac{1}{r} \frac{\partial v_{\phi}^{2} }{\partial z}, \end{aligned}$$
(8)
$$\begin{aligned} &\frac{\partial v_{\phi }}{\partial t} + {\mathcal {L}} [\omega _{\phi}] { \mathcal {D}} v_{\phi }= v_{\phi }{\mathcal {L}} [\omega _{\phi}] , \end{aligned}$$
(9)

where \({\mathcal {D}}\) is a free-scale advection or transport defined by

$$\begin{aligned} {\mathcal {D}} \equiv \biggl( - r \frac{\partial }{\partial r} +2 z \frac{\partial }{\partial z} \biggr) . \end{aligned}$$
(10)

Because (8) depends explicitly on terms of \(v_{\phi}^{2}(r,z,t)\), and (9) preserves the sign of the initial condition, namely, if initially for \(z>0\), \(v_{\phi}(r,z,t=0)\) is nonnegative, then, \(v_{\phi}(r,z,t)\) will be nonnegative for all times. Therefore, one defines the nonnegative function \(W(r,z,t)\) by

$$\begin{aligned} v_{\phi}^{2}(r,z,t) = \bigl(r^{2}+z^{2} \bigr) W(r,z,t). \end{aligned}$$
(11)

Moreover, as already noticed by Elgindi, the Biot–Savart approximation (7) becomes simple in spherical coordinates, that is, by taking \(z =\rho \cos \theta \) and \(r =\rho \sin \theta \). Replacing (11) into (8) and (9), and after doing the change of variables to spherical coordinates, the equations for an axisymmetric flow become

$$\begin{aligned} &\frac{\partial \omega _{\phi }}{\partial t} - {\mathcal {L}} [\omega _{ \phi}]{\mathcal {D}} \omega _{\phi }= - {\mathcal {L}} [\omega _{\phi}] \omega _{\phi }- \frac{\partial W}{\partial \theta } + \frac{ \cos \theta}{\sin \theta} \biggl( 2 W + \rho \frac{\partial W}{\partial \rho } \biggr) , \end{aligned}$$
(12)
$$\begin{aligned} &\frac{\partial W}{\partial t} - {\mathcal {L}} [\omega _{\phi}] { \mathcal {D}} W= 2 \bigl( 2 \sin ^{2} \theta - \cos ^{2}\theta \bigr) {\mathcal {L}} [\omega _{\phi}] W, \end{aligned}$$
(13)

where \({\mathcal {D}}\) corresponds to a free scale advection linear operator defined in (10), which in spherical coordinates becomes

$$ {\mathcal {D}} \equiv \biggl( 3 \sin \theta \cos \theta \frac{\partial}{\partial \theta } + \bigl(1-3 \cos ^{2}\theta \bigr)\rho \frac{\partial}{\partial \rho } \biggr). $$

To these equations, one needs to add the nonlocal term in spherical coordinates,

$$\begin{aligned} \mathcal {L} [\omega _{\phi}] = \frac{3}{4 } \int _{{ \eta} \rho} ^{ \infty }\frac{d\rho ' }{\rho '} \int _{0}^{\pi }\sin ^{2}\theta ' \cos \theta ' \omega _{\phi}\bigl(\rho ',\theta ',t\bigr) \,d \theta ' . \end{aligned}$$
(14)

Equations (12) and (13), with the nonlocal term (14), provide the basic framework for our study of the Euler equations in the axisymmetric configuration.

An important consequence of these equations is that they do not have an a priori intrinsic length scale. This has an important consequence, instead of ρ, the relevant variable will be logρ, therefore any dilation would be a translation in this log variable. Nonetheless important, neither the constant \(\eta >1\) in (14) is relevant because it is just a translation in log variables.

2.3 Series expansion in the polar angular variable in spherical coordinates

Because \(\mathcal {L} [\omega _{\phi}] \) in (14) has no dependence on the angular variable, equations (8) and (9) are “linear” in this variable, therefore the fields \(\omega _{\phi}\) and \(v_{\phi}\) are expanded in a series of the form

$$\begin{aligned} \omega _{\phi }(\rho ,\theta ,t)= \sum_{m=1}^{\infty} \Omega _{m}( \rho ,t) \varphi _{m}(\theta ) \quad \text{and} \quad W(\rho ,\theta ,t)= \sum_{m=1} ^{\infty }W_{m}(\rho ,t) \varphi _{m}(\theta ) , \end{aligned}$$
(15)

where the amplitudes \(\Omega _{n}(\rho ,t)\) and \(W_{n}(\rho ,t)\) are functions of ρ and t, which follow a coupled set of PDEs. The special choice of the orthonormal basis \(\varphi _{m} (\theta ) \) depends on the boundary condition at the axis \(r=0\). Indeed, the axisymmetric flow requires that both original variables, \(\omega _{\phi}(r,z,t)\) and \(W(r,z,t)\), vanish on the axis of rotation \(r=0\), that is, at \(\theta =0\) and \(\theta =\pi \). Moreover, because of the up–down asymmetry of the flow, only the odd modes for \(\theta =\pi /2\) are considered.

We define the inner product:Footnote 2

$$\begin{aligned} \langle f,g \rangle = \int _{0}^{\pi /2} f(\theta ) g(\theta ) \,d \theta , \end{aligned}$$
(16)

such that the basis \(\{\varphi _{n}\}\) is orthonormal, \(\langle \varphi _{n}, \varphi _{m} \rangle = \delta _{nm}\).

Introducing \(\omega _{\phi}\) into the integral (7), we obtain

$$\begin{aligned} &{\mathcal {L}}(\rho ,t)= \sum_{m=1}^{\infty } \kappa _{m} \Lambda _{m}( \rho ,t) \quad \text{with} \end{aligned}$$
(17)
$$\begin{aligned} &\kappa _{n}= \frac{3}{2 } \int _{0}^{\pi /2} \sin ^{2}\theta '\cos \theta ' \varphi _{n}\bigl(\theta '\bigr) \,d \theta ' , \quad \text{and} \end{aligned}$$
(18)
$$\begin{aligned} &\Lambda _{n}(\rho ,t)= \int _{ \eta \rho}^{\infty } \frac{ \Omega _{n}({\rho '},t) }{\rho ' } \,d{\rho '} \quad \Leftrightarrow \quad \Omega _{n} = - \rho \frac{\partial \Lambda _{n}}{\partial \rho} \quad \text{and}\quad \lim_{\rho \to \infty} \Lambda _{n}(\rho ,t) \to 0. \end{aligned}$$
(19)

The equations for the amplitudes, \(\Omega _{n}(\rho ,t)\) and \(W_{n}(\rho ,t)\), are obtained projecting equations (12) and (13) in spherical coordinates into the basis \(\varphi _{n}(\theta ) \). By multiplying (12) and (13) by \(\varphi _{n}(\theta ) \) and integrating over \(\theta \in [0,\pi /2]\), one obtains an infinite matrix representation

$$\begin{aligned}& \begin{aligned}[b] &\frac{\partial }{\partial t} \Omega _{n}(\rho ,t) + {\mathcal {L}} ( \rho ,t) { \mathcal {D}}_{nm} \Omega _{m} \\ &\quad = - {\mathcal {L}} (\rho ,t) \Omega _{n} - \biggl( c_{nm} W_{m} + d_{nm} \biggl( 2 W_{m} + \rho \frac{\partial W_{m}}{\partial \rho } \biggr) \biggr) , \end{aligned} \end{aligned}$$
(20)
$$\begin{aligned}& \frac{\partial}{\partial t} W_{n}(\rho ,t) + {\mathcal {L}} (\rho ,t) { \mathcal {D}}_{nm} W_{m} = {\mathcal {L}} (\rho ,t) e_{nm}W_{m} , \end{aligned}$$
(21)
$$\begin{aligned}& \rho \frac{\partial \Lambda _{n}}{\partial \rho} = - \Omega _{n}( \rho ,t) , \end{aligned}$$
(22)

where \({\mathcal {L}}(\rho ,t) \) is defined in (17); moreover, we define the matrix and differential operator \({\mathcal {D}}_{nm} \) by

$$ {\mathcal {D}}_{nm} = \biggl( a_{nm} + b_{nm} \rho \frac{\partial }{\partial \rho } \biggr). $$

There, the repeated indices, m, stand for the Einstein summation convention. The vector \(\kappa _{n}\) is given by (18) and the matrix elements \(a_{nm}\), \(b_{nm}\), \(c_{nm}\), \(d_{nm}\), and \(e_{nm}\) are given by

$$ \begin{aligned} &a_{nm}= -3 \biggl\langle \varphi _{n}, \sin \theta \cos \theta \frac{\partial \varphi _{m}}{\partial \theta } \biggr\rangle ,\qquad b_{nm} = - \bigl\langle \varphi _{n}, \bigl(1-3 \cos ^{2}\theta \bigr) \varphi _{m} \bigr\rangle , \\ &c_{nm} = \biggl\langle \varphi _{n}, \frac{\partial \varphi _{m}}{\partial \theta } \biggr\rangle , \qquad d_{nm} = - \biggl\langle \varphi _{n}, \frac{ \cos \theta}{\sin \theta} \varphi _{m} \biggr\rangle , \\ &e_{nm}= 2 \bigl\langle \varphi _{n}, \bigl( 2 \sin ^{2} \theta - \cos ^{2} \theta \bigr) \varphi _{m} \bigr\rangle = 2 (\delta _{nm} - b_{nm} ). \end{aligned} $$
(23)

All these matrices may be computed easily for a given orthonormal basis, their elements are just numbers. Among them, \(b_{nm}\), \(d_{nm}\), and \(e_{nm}\) are symmetric, and \(c_{nm}\) is fully antisymmetric. Finally, \(a_{nm}\) is neither symmetric nor antisymmetric. On the other hand, \(a_{nm}\), \(b_{nm}\), and \(e_{nm}\) are tridiagonal, but \(c_{nm}\) and \(d_{nm}\) are fully completed by numbers. The explicit coefficients are given in the Appendix for the Fourier basis (see Eq. (41)).

3 Finite-time self-similar solution

3.1 Self-similar Ansatz

In the following, we look for a particular class of possible solutions of (20) and (21). Set the self-similar dependence:

$$\begin{aligned} &\Omega _{n} (\rho ,t) =\frac{1}{t_{c}-t} { F} _{n} \biggl( \frac{\rho }{(t_{c}-t)^{\nu }} \biggr) , \end{aligned}$$
(24)
$$\begin{aligned} &{ W}_{n} (\rho ,t) =\frac{1}{(t_{c}-t)^{2}} { G} _{n} \biggl( \frac {\rho }{(t_{c}-t)^{\nu }} \biggr) , \end{aligned}$$
(25)
$$\begin{aligned} &{ \Lambda}_{n} (\rho ,t) = \frac{1}{t_{c}-t} { H} _{n} \biggl( \frac {\rho }{(t_{c}-t)^{\nu }} \biggr), \end{aligned}$$
(26)

where the self-similar rescaled variable reads

$$ \xi = \frac{\rho }{(t_{c}-t)^{\nu }}, $$

where \(\nu >0\) is a critical exponent to be found [15]. Furthermore, the Elgindi kernel (17) scales as

$$\begin{aligned} {\mathcal {L}} (\rho ,t) =& \frac{1}{t_{c}-t} L(\xi ) \quad \text{with} \quad L(\xi )= \sum_{p=1}^{\infty }\kappa _{p} H_{p}( \xi ) . \end{aligned}$$
(27)

Here the exponent ν is unknown. As it will be clarified later on, the parameter ν is a kind of a “nonlinear eigenvalue”, in the sense that the solutions for \({ F} _{n}(\xi )\), \({ G} _{n}(\xi )\), and \({ H} _{n}(\xi )\) are determined for a precise set of values for ν. This kind of self-similarity is called “2nd-kind self-similarity” [15].

Replacing the self-similar Ansatz into (20) and (21), one obtains

$$\begin{aligned} & \bigl( F_{n}+ \nu \xi F_{n} ' \bigr) + L( \xi ) \bigl( ( a_{nm} + { \delta }_{nm} ) F_{m}+ \xi b _{nm} F_{m}' \bigr) \\ &\quad {} + \bigl( { c } _{nm} G_{m}+ { d}_{nm} \bigl( 2 G_{m} + \xi {G}_{m}' \bigr) \bigr) =0, \end{aligned}$$
(28)
$$\begin{aligned} & \bigl( 2 G_{n}+ \nu \xi G_{n}' \bigr)+ L( \xi ) \bigl( ( a_{nm} - e_{nm} ) { G}_{m} + \xi b_{nm} { G}_{m} ' \bigr) = 0 , \end{aligned}$$
(29)
$$\begin{aligned} &\xi { H}_{n} ' = - F_{n}, \end{aligned}$$
(30)
$$\begin{aligned} &L(\xi ) = \sum_{p=1}^{\infty }\kappa _{p} H_{p}(\xi ). \end{aligned}$$
(31)

This dynamical system must be complemented with the following boundary conditions. The outer field becomes asymptotically stationary for ρ finite and \(t\to t_{c}\) [15]. More precisely, because of the structure of the self-similar solutions (24) and (25), the dynamics are such that both \(\partial _{t} \Omega _{n}\to 0 \) and \(\partial _{t} W_{n}\to 0 \), as \(\xi \to \infty \). Therefore, the outer boundary conditions read:

$$\begin{aligned} \bigl( F_{n}+ \nu \xi F_{n} ' \bigr) \to 0 \quad \text{and} \quad \bigl( 2 G_{n}+ \nu \xi G_{n}' \bigr)\to 0, \quad \text{as } \xi \to \infty , \end{aligned}$$
(32)

whence

$$\begin{aligned} F_{n}(\xi ) \approx \frac{ F_{n}^{(\infty )}}{\xi ^{1/\nu} } + {\mathrm{h.o.t.}}, \qquad G_{n}(\xi ) \approx \frac{G_{n}^{(\infty )}}{\xi ^{2/\nu} } + { \mathrm{h.o.t.}}, \qquad { H}_{n} (\xi ) \approx \frac{\nu F_{n}^{(\infty )}}{\xi ^{1/\nu} } + {\mathrm{h.o.t}}. \end{aligned}$$
(33)

In conclusion, in the limit \({\xi \to \infty} \) all variables vanish, \(F_{n} (\xi ) , G_{n} (\xi ), H_{n} (\xi ) \to 0\).

On the other hand, for the inner region, a differentiability condition must be imposed at \(\xi =0\). Therefore, because \(F_{n} ' (0) \) and \(G_{n} ' (0) \) are finite, from the equations (28) and (29), it follows that \(F_{n} (0) =G_{n} (0) =0\) and \(H_{n} (0)= H_{n}^{0}\neq 0\).

3.2 Solution of the equations via dynamical system theory

As already noticed, equations (28)–(31) are free of any scale, therefore, using the variable \(s= \log \xi \), one obtains an autonomous dynamical system for the hierarchy of differential equations:

$$\begin{aligned} &\begin{aligned}[b] &\bigl( \nu \delta _{nm} + L(s) b _{nm} \bigr)F_{m} ' (s) + \bigl( \delta _{nm} + L(s) ( a_{nm} + { \delta }_{nm} ) \bigr) F_{m} \\ &\quad {} + ( c_{nm} + 2 d_{nm} ) G_{m} +d_{nm} {G}_{m}'(s) = 0 , \end{aligned} \end{aligned}$$
(34)
$$\begin{aligned} &\bigl( \nu \delta _{nm} + L(s) b _{nm} \bigr) G_{m}'(s) + \bigl( 2 \delta _{nm} + L(s) ( a_{nm} - e_{nm} ) \bigr) { G}_{m} = 0 , \end{aligned}$$
(35)
$$\begin{aligned} &{ H}_{n} ' (s) = - F_{n}, \end{aligned}$$
(36)
$$\begin{aligned} &L(s) = \sum_{p=1}^{\infty }\kappa _{p} H_{p}(s). \end{aligned}$$
(37)

This dynamical system must be solved in view of satisfying the previously mentioned boundary conditions. Indeed, these boundary conditions correspond exactly to the fixed points of the dynamical system. On the other hand, as usual, a dynamical system as well possesses a number of singular points. Both the fixed points and the singular points will be discussed next.

3.3 Characterization of the fixed and singular points of the dynamical system

As usually happens in a dynamical system, a qualitative understanding comes from the knowledge of the attractors, in the current case the fixed points, and their stability. The fixed points of this dynamical system are characterized by a continuous family,

$$ F_{n}^{*}= G_{n}^{*}=0 \quad \text{and} \quad {H}_{n}^{*}\quad \text{arbitrary}. $$

A simple stability analysis indicates that the fixed point

$$\begin{aligned} F_{n}^{*}= G_{n}^{*}= H_{n}^{*}=0 \end{aligned}$$
(38)

is stable, whereas the fixed point

$$\begin{aligned} F_{n}^{(0)}= G_{n}^{(0)}=0 \quad \text{and} \quad {H}_{n}^{(0)} \neq 0, \end{aligned}$$
(39)

may be unstable for, at least, one direction.

It results that the linear stability of the stable fixed point (38) is consistent with the condition \(\nu >0\) and with the asymptotic behavior (33). On the other hand, the unstable manifold of the fixed point (39) is characterized by an eigenvalue \(\sigma ^{*}>0\) and a nonzero perturbation along a \(G_{n}\) direction given by an eigenvector of the matrix \(( a_{nm}-e_{nm}+ \sigma ^{*} b_{nm} ) \) with an eigenvalue, −ϕ, satisfying the condition \(\phi = \frac{(2+\nu \sigma ^{*} )}{ L^{(0)}} \).

Moreover, because of the differentiability of the self-similar functions at the origin \(\rho =0\) (or \(\xi =0\)) [15], a supplementary condition on the most unstable eigenvalue, \(\sigma ^{*}\) is required. Indeed, the asymptotic solution of (34)–(37) for \(s\to -\infty \) is governed by the local behavior:

$$ F_{n} = e^{\sigma ^{*} s} \delta f_{n} , \qquad G_{n}(s) = e^{\sigma ^{*} s} \delta g_{n} , \quad \text{and} \quad H_{n}(s) = H_{n}^{(0)} + e^{ \sigma ^{*} s} \delta h_{n}. $$

Therefore, in the original variable, near \(\xi \approx 0\), all functions become power laws: \(F_{n} = \xi ^{\sigma ^{*}} \delta f_{n}\), etc., thus the local behavior may not be differentiable at \(\xi =0\), unless \(\sigma ^{*}\) is a positive integer,Footnote 3 that is, \(\sigma ^{*} = q\), with \(q=1,2,3,\ldots \) .

In conclusion, the pertinent solution is a heteroclinic orbit that connects, from the limit \(s\to -\infty \), the unstable fixed point with the stable fixed point \(F_{n}^{*}= G_{n}^{*}= H_{n}^{*}=0\), in the limit \(s\to \infty \). Up to this level, the values of the constants \(H_{n}^{(0)}\), as well as the value of ν, must be selected by the full solution of the problem. This is illustrated in the Fig. 1 which comes from a solution for the special case to be discussed in the next section.

Figure 1
figure 1

Plot of a 3D projection of the six-dimensional heteroclinic orbit using the axes \(F_{1}(s)\), \(G_{1}(s)\), and \(L(s)\). This solution corresponds to the case labeled as \(2^{\S}\) in the Table 1. The two planes correspond to the two singular conditions (40)

Full size image

Besides the fixed points, the dynamical system (34) and (35) admits singular points. These are given by the points, \(s_{*}\), such that

$$ \det \bigl[ \nu \delta _{nm} + L(s_{*}) b_{nm} \bigr]=0. $$

In other words, if \(\{ \lambda ^{(b)} \}\) denotes the set for the spectrum of the matrix \(b_{nm}\), then the critical points are given for \(s_{*}\) such that \(-\frac{\nu}{ L(s_{*})} \) belongs to the set \(\{ \lambda ^{(b)} \}\). Because the matrix \(b_{nm} \) is symmetric, its eigenvalues are real; moreover, let \(\lambda ^{(b)}_{\mathrm{max}}>0\) be the maximum and \(\lambda ^{(b)}_{\mathrm{min}}<0\) the minimum eigenvalue. In general, if

$$\begin{aligned} -\frac{1}{\lambda ^{(b)}_{\mathrm{max}}} < \frac{ L(s)}{\nu} < \frac{1}{ \vert \lambda ^{(b)}_{\mathrm{min}} \vert },\quad \forall s \in \mathbb{R}, \end{aligned}$$
(40)

then the trajectory of the solutions of the dynamical system will never cross any singular point, as sketched in Fig. 1.

3.4 Selection mechanism for the nonlinear eigenvalue ν

The linear stability of the inner solution imposes a condition for ν,

$$ \phi = \frac{(2+\nu q )}{ L^{(0)}} , $$

where \(L^{(0)}= \sum_{n=1}^{\infty }\kappa _{n} H_{n}^{(0)}\), q is an integer, and −ϕ is an eigenvalue of the matrix

$$ ( a_{nm}-e_{nm}+ q b_{nm} ). $$

However, the other amplitudes \(\{ H_{n}^{(0)}\}\) remain to be selected. The selection happens because of the existence of a heteroclinic orbit as a solution to the dynamical system. The search for such a heteroclinic solution of the ordinary differential equations (34)–(37) passes by a shooting-like method for the unknowns \(\{ H_{n}^{(0)}\}\), that is, by tuning the parameters \(\{ H_{n}^{(0)}\}\), to select the right solution that satisfies the right boundary conditions (32). Additionally, the function \(L(s)\), defined through equation (37), has to satisfy condition (40).

Naturally, because this hierarchy is infinite, this is not an easy task. As we see in the next section, this problem may be cumbersome even for a truncation keeping a moderate number of variables. Nevertheless, some solutions do exist. In the following section, we provide a numerical example of a solution.

4 Numerical search of a solution and for the second kind similarity exponent

To solve in practice the infinite ordinary differential equations system (34)–(37), it is necessary to truncate these equations up to some order N, that is, the matrices indices run from \(n=1\) up to \(n=N \). Under this assumption, \(a_{nm}\), \(b_{nm}\), etc., are \(N\times N\) matrices, and the self-similar functions are arrays of order N. Thus, \(( F_{1}(s) ,F_{2}(s),\dots ,F_{N}(s))\), \(( G_{1}(s) , G_{2}(s),\dots ,G_{N}(s) )\), and \(( H_{1}(s) , H_{2}(s),\dots ,H_{N}(s) )\).

For a given integer \(q =1,2,3 \dots \), it is required to solve a 3N-dimensional dynamical system that needs N independently selected parameters.

A shooting method with N parameters can be done easily for \(N=1\), and perhaps for \(N=2\), however, for higher values of N it may be a cumbersome scheme. Therefore, we use a differential evolution approach [19] to estimate the shooting parameters for the dynamical set of equations (34)–(37), together with the boundary conditions (32) as an error estimate [20].

In the following, we use a Fourier orthonormal basis with the inner product (16),

$$\begin{aligned} \varphi _{n}(\theta )= { \frac{2 }{\sqrt {\pi}}} \sin (2n \theta ). \end{aligned}$$
(41)

In the Appendix, the specific values of the coefficients \(\kappa _{n}\) and the matrices a, b, etc., are provided. Within this basis, we have found a large number of solutions of the problem which are listed in Table 1.

Table 1 Summary of some numerically obtained solutions
Full size table

Among these solutions, we notice first that for \(N=1\) the only existing solution is for \(q=1\) and \(\nu =0.89005\). Although in this case the vorticity, \(\omega _{\phi}\), is differentiable, since \(\omega _{\phi}\sim \rho \sin (2 \theta ) \), the swirl velocity, \(v_{\phi}\sim \rho ^{3/2} \vert \sin (2 \theta ) \vert \), is not differentiable more than once at the origin. Nevertheless, it is possible to relax the condition admitting to the space of solutions the initial nonnegative swirl velocities which are smooth in \(v_{\phi}^{2}(r,z,t=0)\).

For \(N=2\) we have observed the existence of multiple possible solutions for all \(q\geq 1\). For \(N=3\) and \(q=1\), it follows that there is no suitable solution. In general, for \(q=1\) and N odd and greater than 3, we have not found any suitable solution, therefore these solutions either do not exist or are rare. A large number of solutions are obtained for \(N=4\).

Graphically we provide the particular case of \(N=2\) and \(q=2\). Figure 2(a) shows the solutions for the six amplitudes, \(F_{1}(s)\), \(F_{2}(s)\), \(G_{1}(s)\), \(G_{2}(s)\), and \(H_{1}(s)\), \(H_{2}(s)\), as functions of the variable, s.

Figure 2
figure 2

Numerical solutions of the system for the case \(N=2\). (a) Plots of the Fourier amplitudes \(F_{1}(s)\) and \(F_{2}(s)\) (red and red-dashed), \(G_{1}(s)\) and \(G_{2}(s)\) (blue and blue-dashed), and \(H_{1}(s)\) and \(H_{2}(s)\) (green and green-dashed). The solution selects \(\nu = 1.10091\), \(H_{1}^{(0)}= 3.13151 \), and \(H_{2}^{(0)} = 0.0993008\). (Inset) The plot of the function \(L(s)\) shows that it never crosses the critical points defined by condition (40). From \(L(s)\), \(F_{1}(s)\), and \(G_{1}(s)\), we built the 3D heteroclinic orbit of Fig. 1. (b) Same plot as (a) but in semilog scale, \(\log |F_{n}(s)|\), \(\log |G_{n}(s)|\), and \(\log |H_{n}(s)|\) vs. s. One notices readily the exponential behavior for the inner behavior of the amplitudes \(F_{n}(s) \sim e^{2s}\) and \(G_{n}(s) \sim e^{2s} \). Additionally, we observe the right slopes \(1/\nu \) and \(2/\nu \) in the large s limit

Full size image

Figure 2(b) shows the same plot in a semilog scale, that is, in a log–log scale if considering the original variable \(\xi =\log s\). This plot clearly shows the power law behaviors as \(\xi \to 0\) and \(\xi \to \infty \).

As it can be seen in Fig. 2(a), the amplitudes \(F_{1}(s)\), \(G_{1}(s)\), and \(H_{1}(s)\) have no nodes (zeroes), therefore, by analogy with the spectral theory of linear eigensystems, we call these solutions as the “ground states” (see the parameter solutions labeled with a § in Table 1. In the same vein, the first and second “excited states” are also obtained.

Finally, to represent the physical flow, Figs. 3(a) and 3(b) show the contour plots of the self-similar vorticity (a) and the self-similar swirl velocity (b), defined by

$$ \Omega _{\phi}(\xi _{r},\xi _{z}) = \sum _{m=1} \varphi _{m}(\theta )F_{m}( \xi ) \quad \text{and} \quad V_{\phi}(\xi _{r},\xi _{z})= \xi \biggl( \sum_{m=1} \varphi _{m}(\theta ) G_{m}(\xi ) \biggr)^{1/2}, $$

where the self-similar variables are defined in the cylindrical rescaled variables,

$$ \xi _{r} = \xi \sin \theta \quad \text{and}\quad \xi _{z} = \xi \cos \theta . $$
Figure 3
figure 3

(a) The self-similar function \(\Omega _{\phi}(\xi _{r},\xi _{z})\) representing the finite-time singular behavior of the vorticity field. (b) The self-similar function \(V_{\phi}(\xi _{r},\xi _{z})\) representing the finite-time singular behavior of the swirl velocity

Full size image

The self-similar vorticity field, Fig. 3(a), as well as the swirl velocity, Fig. 3(b), show a global circulation in the first quadrant, in the self-similar variables \((\xi _{r},\xi _{z})\). After a reconstruction of the full three-dimensional space, the main flow realized by the self-similar solution has a quadrupolar structure.

5 Conclusions and discussion

By approximating the nonlocal Biot–Savart law and projecting the axial vorticity, \(\omega _{\phi}\), as well as the swirl velocity, \(v_{\phi}\), in an adequate basis, the original coupled set of partial differential equations for an axisymmetric flow is mapped into an infinite set of partial differential equations for the corresponding vorticity and swirl velocity amplitudes. The final set of nonlinear partial differential equations admits a possible finite-time self-similar behavior. The existence of this finite-time singularity is possible by the solution of an infinite system of ordinary differential equations with adequate boundary conditions.

Although the formal proof of an existing solution for the full infinite system seems to be a great challenge, the existence of such a solution may be mapped into a dynamical system problem, in which the required solution is selected by a high-dimensional shooting argument. This scheme is corroborated via the numerical solution of the original infinite dynamical system truncated up to a finite order N. We provide numerical solutions of the truncated system, indicating strong evidence of the existence of a finite-time self-similar solution of the axisymmetric Euler equations for incompressible fluids.

We end this paper with the following remarks:

  1. 1

    The validity of the approximate equations (8) and (9) depends on the Elgindi approximation, that is, the contribution to the Biot–Savart integral from the inner core must be negligible regarding the full integral. In other words, the following inequality must be satisfied:

    $$ \begin{aligned}& \biggl\vert \int _{\eta \rho}^{\infty } \frac{d{\rho '}}{{\rho '}} \int _{0}^{\pi /2} \sin ^{2}\theta ' \cos \theta ' \omega _{\phi}\bigl(\rho ',\theta ',t\bigr) \,d \theta ' \biggr\vert \\ &\quad \gg \biggl\vert \int _{0}^{\eta \rho} \frac{d{\rho '}}{{\rho '}} \int _{0}^{\pi /2} \sin ^{2}\theta ' \cos \theta ' \omega _{\phi}\bigl( \rho ',\theta ',t\bigr) \,d \theta ' \biggr\vert . \end{aligned} $$

    Here the left-hand side corresponds to Elgindi approximation and the right-hand side to the inner contribution, assumed negligible. By using the series expansion (15), the inequality becomes

    $$ \Biggl\vert \sum_{n=1}^{\infty }\kappa _{n} \int _{\eta \xi} ^{\infty }F_{n}\bigl( \xi '\bigr) \frac{d{\xi '}}{\xi '} \Biggr\vert \gg \Biggl\vert \sum _{n=1}^{ \infty }\kappa _{n} \int _{0}^{\eta \xi}F_{n}\bigl(\xi '\bigr) \frac{d{\xi '}}{\xi '} \Biggr\vert . $$

    By the results of Sect. 3.3, the value of the inner core solution of the self-similar vorticity becomes \(F_{n}(\xi ) \sim \xi ^{q} \), therefore we conclude

    $$ \Biggl\vert \sum_{n=1}^{\infty }\kappa _{n} \int _{\eta \xi} ^{\infty }F_{n}\bigl( \xi '\bigr) \frac{d{\xi '}}{\xi '} \Biggr\vert \gg \frac{(\eta \xi ) ^{q} }{q} \sum _{n=1}^{\infty }\kappa _{n} . $$

    Because \(\kappa _{n} \sim 1/n^{3}\), the series converges,Footnote 4 the right-hand side may be as small as desired as \(\xi \to 0\), therefore, negligible regarding the Elgindi approximation.Footnote 5

  2. 2

    Neglecting advective terms, that is, whenever the matrix elements \(a_{nm}\), \(b_{nm}\) are zero, and also setting the swirl velocity to zero (\(G_{n}=0\)) in equations (28) and (29), makes possible finding an exact solution of the resulting self-similar equation for the vorticity (28). The resulting dynamical system simplifies to

    $$ \begin{aligned}& F_{n}+ \nu \xi F_{n} ' + L(\xi ) F_{n}= 0 , \quad \text{and} \quad \xi L'(\xi )=- \sum_{p=1}^{\infty } \kappa _{p} F_{p}(s) . \end{aligned} $$
    (42)

    This infinite set of ordinary differential equations corresponds to Elgindi solution [14]. It can be integrated directly

    $$ L(\xi ) = -\frac{2}{K \xi ^{1/\nu} +1}, $$
    (43)

    which is not smooth at \(\xi =0\), unless \(1/\nu \) is a positive integer, whereas Elgindi showed that advection may be discarded if and only if \(1/\nu \to 0\). Nevertheless, an interesting feature of this solution is that \(L(\xi =0)<0\), contrary to the numerically obtained solutions for the full ODE system in Sect. 4.

  3. 3

    Additionally, setting the swirl velocity to zero, i.e., \(G_{n}\equiv 0\ \forall n\), the system (34)–(37) cannot be integrated analytically, but it can be solved numerically. As expected, the possible numerical solutions do not satisfy the outer boundary conditions, therefore there is no possible solution in the case of zero swirl, in agreement with Ukhovskii–Yudovich [21].