Phil. Trans. R. Soc

Fluid Mechanics, Differential Equations, Applied Mathematics

Miguel D. Bustamante

# On the role of continuous symmetries in the solution of the 3D Euler fluid
equations and related models

Miguel D. Bustamante  School of Mathematics and Statistics, University College
Dublin, Belfield, Dublin 4, Ireland<EMAIL_ADDRESS>
###### Abstract

We review the continuous symmetry approach and apply it to find the solution,
via the construction of constants of motion and infinitesimal symmetries, of
the 3D Euler fluid equations in several instances of interest, without
recourse to Noether’s theorem. We show that the vorticity field is a symmetry
of the flow and therefore one can construct a Lie algebra of symmetries if the
flow admits another symmetry. For steady Euler flows this leads directly to
the distinction of (non-)Beltrami flows: an example is given where the
topology of the spatial manifold determines whether the flow admits extra
symmetries. Next, we study the stagnation-point-type exact solution of the 3D
Euler fluid equations introduced by Gibbon _et al._ (_Physica D_ , vol. 132,
1999, pp. 497–510) along with a one-parameter generalisation of it introduced
by Mulungye _et al._ (_J. Fluid Mech._ , vol. 771, 2015, pp. 468–502).
Applying the symmetry approach to these models allows for the explicit
integration of the fields along pathlines, revealing a fine structure of
blowup for the vorticity, its stretching rate, and the back-to-labels map,
depending on the value of the free parameter and on the initial conditions.
Finally, we produce explicit blowup exponents and prefactors for a generic
type of initial conditions.

###### keywords:

3D Euler fluid equations, Exact solutions, Finite-time singularities, Lie
algebras, Infinitesimal symmetries

††journal: rsta

##  1 Introduction

Consider a smooth (i.e., of class $C^{\infty}$) 3D vector field

$\displaystyle{\mathbf{u}}:\Omega\times[0,T)\\!\\!\\!\\!$
$\displaystyle\longrightarrow$ $\displaystyle\\!\\!\\!\\!\mathbb{R}^{3}$
$\displaystyle({\mathbf{x}},t)\\!\\!\\!\\!$ $\displaystyle\longmapsto$
$\displaystyle\\!\\!\\!\\!{\mathbf{u}}({\mathbf{x}},t):=(u^{1}({\mathbf{x}},t),u^{2}({\mathbf{x}},t),u^{3}({\mathbf{x}},t)),$

where $T>0$ and ${\mathbf{x}}:=(x^{1},x^{2},x^{3})$ denote local Cartesian
coordinates of a point in the spatial domain $\Omega.$ It is customary to use
superscripts for the components of coordinates and vector fields, as small
displacements

transform as components of contravariant vectors under general coordinate
transformations. In this work we will set either $\Omega=\mathbb{T}^{3}$, or
$\Omega=\mathbb{R}^{2}\times\mathbb{T}$, or
$\Omega=\mathbb{T}^{2}\times\mathbb{R}$, where $\mathbb{T}:=[0,2\pi)$ is the
torus. In what follows, we will assume for simplicity that ${\mathbf{u}}$ is
divergence-free:

$\nabla\cdot{\mathbf{u}}:=\frac{\partial u^{1}}{\partial x^{1}}+\frac{\partial
u^{2}}{\partial x^{2}}+\frac{\partial u^{3}}{\partial
x^{3}}=0\,,\quad\text{for \,\, all}\quad{\mathbf{x}}\in\Omega,\quad
t\in[0,T).$

The given vector field ${\mathbf{u}}$ defines a 3D flow in $\Omega$ by the
following non-linear, non-autonomous system of ordinary differential equations
for the variables ${\mathbf{X}}(t):=(X^{1}(t),X^{2}(t),X^{3}(t))$:

$\dot{X}^{1}=u^{1}({\mathbf{X}},t)\,,\qquad\quad\dot{X}^{2}=u^{2}({\mathbf{X}},t)\,,\qquad\quad\dot{X}^{3}=u^{3}({\mathbf{X}},t)\,,\qquad\quad
t\in[0,T)\,.$ (1)

Solutions to this system subject to initial conditions
$X^{1}(0)=X_{0},X^{2}(0)=Y_{0},X^{3}(0)=Z_{0}$ are known as pathlines, or
characteristics, of ${\mathbf{u}}$. If the zeroes of ${\mathbf{u}}$ at any
fixed time $t\in[0,T)$ are simple (a generic condition), pathlines with
different initial conditions do not cross, but this may be violated at some
time $t\geq T$ if the velocity field ever becomes singular then.

In physical terms one could interpret the field ${\mathbf{u}}$ as a velocity
field for some incompressible fluid. The pathlines would then be the
trajectories of massless particles.

In this paper we will apply infinitesimal symmetry methods, valid for an
arbitrary smooth vector field ${\mathbf{u}}$, for solving system (1), in order
to establish non-trivial results when ${\mathbf{u}}$ is a fluid velocity field
satisfying the 3D Euler fluid equations or related models.

Section 2 is devoted to a review of the infinitesimal symmetry approach for
general 3D vector fields, and the construction of constants of motion starting
from symmetries, without using Noether’s theorem. Section 3 is devoted to the
application of this general work to the solution of the 3D Euler fluid
equations. In it, we will look at general results, then provide an application
to steady flows and finally an application to the stagnation-point like
solution introduced in [1]. Section 4 is devoted to the main application,
namely the $1$-parameter family of models introduced in [2] that generalise
the case of [1], containing it as a particular case. We construct constants of
motion and use them to solve for the main quantities of interest along
pathlines, establishing a formula for the blowup time, and conditions on the
blowup of the main fields, depending on the initial data and on the value of
the model’s parameter. Finally, conclusions are given in section 5.

##  2 The continuous symmetry approach

In this section we will present a simple exposition of the main tools of
interest in the solution of the ODE system (1). These tools are ultimately
based on the formalism of tensor bundles, where tensorial objects of various
ranks are transported by the vector field ${\mathbf{u}}$. We will avoid the
use of this formalism in this work, as it tends to obscure the exposition of
the applications. However, we will introduce (at a formal level) the main
concept pertaining to the fundamental approach: the Lie derivative, which is
extremely useful when proving the most relevant results.

###  2.1 Basic tools: Constants of motion and Symmetries

The most basic tool for solving the ODE system (1) is the constant of motion.

######  Definition 2.1 (Constant of motion).

A smooth scalar function of position and time
$C:\Omega\times[0,T)\rightarrow\mathbb{R}$ is a constant of motion for system
(1) if it satisfies the linear partial differential equation

$\frac{\partial C}{\partial t}+({\mathbf{u}}\cdot\nabla)C=0\,,\quad\text{for
\,\, all}\quad{\mathbf{x}}\in\Omega,\quad t\in[0,T),$ (2)

where ${\mathbf{u}}\cdot\nabla:=u^{1}\frac{\partial}{\partial
x^{1}}+u^{2}\frac{\partial}{\partial x^{2}}+u^{3}\frac{\partial}{\partial
x^{3}}$. In words, a constant of motion takes a fixed numerical value along a
given pathline of the vector field ${\mathbf{u}}$.

######  Remark 2.2.

Notice that if one finds a constant of motion with a nontrivial dependence on
the space and time variables, then in principle the number of independent
variables of system (1) can be reduced by one unit via the formula
$C({\mathbf{X}}(t),t)=C({\mathbf{X}}(0),0)$, where ${\mathbf{X}}(t)$ is a
pathline. By iterating this process, when enough functionally independent
solutions to this equation exist, one can in principle solve the system (1)
explicitly. The difficulty now lies of course in solving the linear partial
differential equation (2).

The next basic tool in terms of complexity is the infinitesimal symmetry. Let
us first recall Sophus Lie’s transformation theory [3], restricting the
discussion to the so-called contemporaneous case, namely when the time is not
transformed [4, Chart A.6, Definition 3].

######  Definition 2.3 (Finite contemporaneous symmetry).

An invertible mapping defined continuously for each $t\in[0,T)$

$\displaystyle f_{t}:\Omega$ $\displaystyle\longrightarrow$
$\displaystyle\Omega\,,\qquad t\in[0,T),$ $\displaystyle(x^{1},x^{2},x^{3})$
$\displaystyle\longmapsto$
$\displaystyle(\widetilde{x}^{1},\widetilde{x}^{2},\widetilde{x}^{3}):=f_{t}(x^{1},x^{2},x^{3})\,,$

is called a finite contemporaneous symmetry for system (1) if, under the
mapping
${\mathbf{\widetilde{X}}}(t):=(\widetilde{X}^{1}(t),\widetilde{X}^{2}(t),\widetilde{X}^{3}(t))=f_{t}(X^{1}(t),X^{2}(t),X^{3}(t))$,
system (1) maps to the system in the same form,

$\dot{\widetilde{X}}{}^{1}=u^{1}({\mathbf{\widetilde{X}}},t)\,,\qquad\quad\dot{\widetilde{X}}{}^{2}=u^{2}({\mathbf{\widetilde{X}}},t)\,,\qquad\quad\dot{\widetilde{X}}{}^{3}=u^{3}({\mathbf{\widetilde{X}}},t)\,,\qquad\quad
t\in[0,T)\,.$ (3)

It is crucial in this definition that the form be maintained, namely that the
exact same functions $u^{1},u^{2},u^{3}$ are used in equations (1) and (3).

In words, a finite contemporaneous symmetry maps pathlines of ${\mathbf{u}}$
into pathlines of ${\mathbf{u}}$.

The latter word definition is key to the utility of finite symmetries for
solving the ODE system (1): once a solution is known, one can use the finite
symmetry to construct new solutions.

It is clear from this definition that the set of all finite symmetry
transformations for system (1) define a group $G$, whose product rule is
induced by the composition operation of functions:

$\displaystyle G\times G$ $\displaystyle\longrightarrow$ $\displaystyle G$
$\displaystyle(f,g)$ $\displaystyle\longmapsto$ $\displaystyle
h\qquad\text{such \,\,that}\qquad h_{t}:=f_{t}\circ g_{t}\,,\qquad
t\in[0,T)\,.$

Infinitesimal symmetry transformations are simply near-identity versions of
finite symmetry transformations (up to first order), and correspondingly they
form an algebra, known as Lie algebra. While Lie introduced the main
techniques to construct infinitesimal symmetry transformations, he did not
focus on the problem of finding infinitesimal symmetries for an arbitrary
vector field ${\mathbf{u}}$, which is what we will do now. The main equations
were first introduced in [5]. A modern account is given by Hojman in [6] (see
also [7]). Because of its importance in what follows, we re-derive the
equations determining contemporaneous infinitesimal symmetries of system (1).

######  Definition 2.4 (Infinitesimal contemporaneous symmetry).

A smooth divergence-free vector field

$\displaystyle\mbox{\boldmath{$\eta$}}:\Omega\times[0,T)$
$\displaystyle\longrightarrow$ $\displaystyle\mathbb{R}^{3}$
$\displaystyle({\mathbf{x}},t)$ $\displaystyle\longmapsto$
$\displaystyle\mbox{\boldmath{$\eta$}}({\mathbf{x}},t):=(\eta^{1}({\mathbf{x}},t),\eta^{2}({\mathbf{x}},t),\eta^{3}({\mathbf{x}},t))\,,$
(5) $\displaystyle\nabla\cdot\mbox{\boldmath{$\eta$}}=0\quad\text{for \,\,
all}\quad{\mathbf{x}}\in\Omega,\quad t\in[0,T),$

is called an infinitesimal contemporaneous symmetry for system (1) if the
functions
${\mathbf{\widetilde{X}}}(t)=(\widetilde{X}^{1}(t),\widetilde{X}^{2}(t),\widetilde{X}^{3}(t))$
defined by

${\mathbf{\widetilde{X}}}(t)={\mathbf{X}}(t)+\epsilon\,\mbox{\boldmath{$\eta$}}({\mathbf{X}}(t),t)$
(6)

satisfy (3) up to and including terms of order ${\mathcal{O}}(\epsilon)$
whenever ${\mathbf{X}}(t)=(X^{1}(t),X^{2}(t),X^{3}(t))$ satisfy (1).

In words, an infinitesimal contemporaneous symmetry maps pathlines of
${\mathbf{u}}$ into nearby pathlines of ${\mathbf{u}}$.

Notice that the divergence-free condition
$\nabla\cdot\mbox{\boldmath{$\eta$}}=0$ is not essential to the general
definition of symmetry. Rather, it is due to our original restriction to flows
based on divergence-free fields.

We enunciate and prove our main theorem regarding infinitesimal symmetries:

######  Theorem 2.5.

A smooth divergence-free vector field $\eta$ defined as in equations (5)–(5)
is an infinitesimal contemporaneous symmetry if, and only if, it satisfies the
following linear partial differential equation:

$\frac{\partial\mbox{\boldmath{$\eta$}}}{\partial
t}+({\mathbf{u}}\cdot\nabla)\mbox{\boldmath{$\eta$}}-(\mbox{\boldmath{$\eta$}}\cdot\nabla){\mathbf{u}}={\mathbf{0}}\,,\quad\text{for
\,\, all}\quad{\mathbf{x}}\in\Omega,\quad t\in[0,T),$ (7)

where ${\mathbf{0}}$ denotes the vector whose components are all equal to
zero.

###### Proof.

$(\Rightarrow)$ Replacing ${\mathbf{\widetilde{X}}}(t)$ from (6) into the LHS
of (3) gives, using the chain rule,

$\dot{{\mathbf{\widetilde{X}}}}(t)=\dot{{\mathbf{X}}}(t)+\epsilon\,({\mathbf{\dot{{X}}}}(t)\cdot\nabla)\mbox{\boldmath{$\eta$}}({\mathbf{X}}(t),t)+\epsilon\,\frac{\partial\mbox{\boldmath{$\eta$}}}{\partial
t}({\mathbf{X}}(t),t)\,,$

where $\frac{\partial\mbox{\boldmath{$\eta$}}}{\partial t}$ denotes the
partial derivative of $\eta$ with respect to the last argument only (namely
the time argument). Performing the same replacement into the RHS of (3) gives

${\mathbf{u}}({\mathbf{\widetilde{X}}}(t),t)={\mathbf{u}}({\mathbf{X}}(t)+\epsilon\,\mbox{\boldmath{$\eta$}}({\mathbf{X}}(t),t),t)={\mathbf{u}}({\mathbf{X}}(t),t)+\epsilon\,(\mbox{\boldmath{$\eta$}}({\mathbf{X}}(t),t)\cdot\nabla){\mathbf{u}}({\mathbf{X}}(t),t)+\mathcal{O}(\epsilon^{2})\,,$

where the last equality was obtained after Taylor expanding about
$\epsilon=0$. Equating these obtained approximations to the LHS and RHS of (3)
gives

$\dot{{\mathbf{X}}}(t)+\epsilon\,({\mathbf{\dot{{X}}}}(t)\cdot\nabla)\mbox{\boldmath{$\eta$}}({\mathbf{X}}(t),t)+\epsilon\,\frac{\partial\mbox{\boldmath{$\eta$}}}{\partial
t}({\mathbf{X}}(t),t)={\mathbf{u}}({\mathbf{X}}(t),t)+\epsilon\,(\mbox{\boldmath{$\eta$}}({\mathbf{X}}(t),t)\cdot\nabla){\mathbf{u}}({\mathbf{X}}(t),t)+\mathcal{O}(\epsilon^{2})\,,$

and using (1) to eliminate
$\dot{{\mathbf{X}}}(t)={\mathbf{u}}({\mathbf{X}}(t),t)$ we obtain

$\epsilon\,({\mathbf{u}}({\mathbf{X}}(t),t)\cdot\nabla)\mbox{\boldmath{$\eta$}}({\mathbf{X}}(t),t)+\epsilon\,\frac{\partial\mbox{\boldmath{$\eta$}}}{\partial
t}({\mathbf{X}}(t),t)=\epsilon\,(\mbox{\boldmath{$\eta$}}({\mathbf{X}}(t),t)\cdot\nabla){\mathbf{u}}({\mathbf{X}}(t),t)+\mathcal{O}(\epsilon^{2})\,,$

valid for all times $t\in[0,T)$. As this must be true for any small
$\epsilon$, we conclude that the linear term in $\epsilon$ must be equal to
zero. Finally, recalling that ${\mathbf{X}}(t)$ can be replaced by an
arbitrary ${\mathbf{x}}\in\Omega$ by virtue of the fact that one can choose
the initial condition arbitrarily, we obtain the desired result:
$({\mathbf{u}}({\mathbf{x}},t)\cdot\nabla)\mbox{\boldmath{$\eta$}}({\mathbf{x}},t)+\frac{\partial\mbox{\boldmath{$\eta$}}}{\partial
t}({\mathbf{x}},t)=(\mbox{\boldmath{$\eta$}}({\mathbf{x}},t)\cdot\nabla){\mathbf{u}}({\mathbf{x}},t)$.
The proof of the converse is obvious. ∎

###  2.2 Lie derivatives

Given a vector field ${\mathbf{u}}$, the equations that determine constants of
motion (2) and infinitesimal contemporaneous symmetries (7) are best written
in terms of the linear operator
$\mathop{\mathcal{L}}_{\scriptsize{{\mathbf{u}}}}$, called the Lie derivative
along the vector field ${\mathbf{u}}$, which maps tensors into tensors of the
same rank and satisfies the Leibniz product rule of differentiation. Without
going into the details of the Lie derivative’s natural definition in terms of
transport on tensor bundles (see [4, 8] for details, and [9, 10, 11] for
applications in fluids) we simply define the Lie derivative in the two cases
of interest:

######  Definition 2.6.

Let ${\mathbf{V}},{\mathbf{W}}:\Omega\times[0,T)\to\mathbb{R}^{3}$ be two
smooth divergence-free vector fields and let
$B:\Omega\times[0,T)\to\mathbb{R}$ be a smooth scalar function. The Lie
derivative of $B$ along ${\mathbf{V}}$ is defined as the smooth scalar
function

$\mathop{\mathcal{L}}_{\scriptsize{{\mathbf{V}}}}B:={\mathbf{V}}\cdot\nabla
B\,,\quad\text{for \,\, all}\quad{\mathbf{x}}\in\Omega,\quad t\in[0,T)\,.$

Similarly, the Lie derivative of ${\mathbf{W}}$ along ${\mathbf{V}}$ is
defined as the smooth divergence-free vector field

$\mathop{\mathcal{L}}_{\scriptsize{{\mathbf{V}}}}{\mathbf{W}}:={\mathbf{V}}\cdot\nabla{\mathbf{W}}-{\mathbf{W}}\cdot\nabla{\mathbf{V}}\,,\quad\text{for
\,\, all}\quad{\mathbf{x}}\in\Omega,\quad t\in[0,T)\,.$ (8)

We say that two vector fields ${\mathbf{V}},{\mathbf{W}}$ commute if
$\mathop{\mathcal{L}}_{\scriptsize{{\mathbf{V}}}}{\mathbf{W}}={\mathbf{0}}$.

######  Remark 2.7.

In the case of vector fields, it follows from (8) that the Lie derivative
defines a Lie algebra of divergence-free vector fields via the bilinear binary
operation known as the Lie bracket,
$[{\mathbf{V}},{\mathbf{W}}]:=\mathop{\mathcal{L}}_{\scriptsize{{\mathbf{V}}}}{\mathbf{W}}$,
satisfying

$\displaystyle\mathop{\mathcal{L}}_{\scriptsize{{\mathbf{W}}}}{\mathbf{V}}+\mathop{\mathcal{L}}_{\scriptsize{{\mathbf{V}}}}{\mathbf{W}}={\mathbf{0}}$
$\displaystyle\text{(anticommutativity)}\,,$
$\displaystyle\mathop{\mathcal{L}}_{\scriptsize{{\mathbf{U}}}}\mathop{\mathcal{L}}_{\scriptsize{{\mathbf{V}}}}{\mathbf{W}}+\mathop{\mathcal{L}}_{\scriptsize{{\mathbf{V}}}}\mathop{\mathcal{L}}_{\scriptsize{{\mathbf{W}}}}{\mathbf{U}}+\mathop{\mathcal{L}}_{\scriptsize{{\mathbf{W}}}}\mathop{\mathcal{L}}_{\scriptsize{{\mathbf{U}}}}{\mathbf{V}}={\mathbf{0}}$
$\displaystyle\text{(Jacobi \,\, identity)}\,,$ (9)

where ${\mathbf{U}},{\mathbf{V}},{\mathbf{W}}$ are any divergence-free vector
fields.

As a result of these definitions, the equations defining constants of motion
and infinitesimal symmetries become extremely simple:

######  Corollary 2.8.

Under the same hypotheses of definitions 2.1 and 2.4, a smooth scalar $C$ is a
constant of the motion and a smooth divergence-free vector field $\eta$ is an
infinitesimal contemporaneous symmetry if and only if

$\displaystyle\frac{\partial C}{\partial
t}+\mathop{\mathcal{L}}_{\scriptsize{{\mathbf{u}}}}C$ $\displaystyle=$
$\displaystyle 0\,,\quad\text{for \,\,
all}\quad{\mathbf{x}}\in\Omega,\,t\in[0,T)\,,$ (10)
$\displaystyle\frac{\partial\mbox{\boldmath{$\eta$}}}{\partial
t}+\mathop{\mathcal{L}}_{\scriptsize{{\mathbf{u}}}}\mbox{\boldmath{$\eta$}}$
$\displaystyle=$ $\displaystyle{\mathbf{0}}\,,\quad\text{for \,\,
all}\quad{\mathbf{x}}\in\Omega,\,t\in[0,T)\,.$ (11)

###### Proof.

The proof follows directly from the respective definitions of Lie derivative.
∎

###  2.3 The gradient of a constant of motion

Taking the gradient of equation (2) gives an equation for the gradient of a
constant of motion:

$\frac{\partial\nabla C}{\partial t}+({\mathbf{u}}\cdot\nabla)\nabla
C+(\nabla{\mathbf{u}})^{T}\nabla C={\mathbf{0}}\,,\quad\text{for \,\,
all}\quad{\mathbf{x}}\in\Omega,\,t\in[0,T)\,,$ (12)

where $(\nabla{\mathbf{u}})^{T}$ is the square matrix with components
$[(\nabla{\mathbf{u}})^{T}]_{i}^{\,\,j}:=\frac{\partial u^{j}}{\partial
x^{i}}$. One has the result:

######  Lemma 2.9.

Suppose the gradient of a scalar function $C$ satisfies equation (12). Then
the scalar function $C$ is a constant of motion for system (1), up to an
additive function of time that can be found.

###### Proof.

Retracing the steps leading to (12) from (2) we find
$\nabla\left(\frac{\partial C}{\partial
t}+({{\mathbf{u}}}\cdot\nabla)C\right)={\mathbf{0}}$, implying $\frac{\partial
C}{\partial t}+({{\mathbf{u}}}\cdot\nabla)C=f(t)$ for some scalar function
$f$. Thus, the scalar ${C}^{\prime}({\mathbf{x}},t):=C({\mathbf{x}},t)-\int
f(t)\mathrm{d}t$ clearly satisfies $\frac{\partial C^{\prime}}{\partial
t}+({{\mathbf{u}}}\cdot\nabla)C^{\prime}=0$, which means that
${C}^{\prime}({\mathbf{x}},t)$ is a constant of motion. ∎

Unsurprinsingly, equation (12) is yet another instance of a transport equation
involving Lie derivatives, this time for dual vector fields, of which the
gradient is a particular case. Solutions of (12) where $\nabla C$ is replaced
by a dual vector that is not a gradient, are called Lagrangian $1$-forms [12,
13], and are used in the construction of action principles for system (1), in
what is known as the inverse problem of the calculus of variations. See [5, 6,
7] and references therein for details, including Lie-advected higher-order
tensors such as $2$-forms (called symplectic forms), antisymmetric $(2,0)$
tensors (called Poisson structures) and $(1,1)$ tensors (called recursion
operators). Also, notice that Cauchy invariants (and generalised versions
thereof) correspond to Lie-transported $p$-forms that are exact, such as the
gradient, which is an exact $1$-form. See [14, 15, 10] and references therein
for details.

###  2.4 Zeroes of an infinitesimal symmetry and regularity of the flow

Generic smooth vector fields ${\mathbf{F}}:\Omega\times[0,T)\to\mathbb{R}^{3}$
have single zeros in $\Omega$ for each $t\in[0,T)$, because the number of
equations in ${\mathbf{F}}({\mathbf{x}},t)={\mathbf{0}}$ is equal to the
number of unknowns ${\mathbf{x}}$. We have the following important result
regarding the time evolution of the zeros of infinitesimal symmetries:

######  Theorem 2.10 (Conservation of zeroes of an infinitesimal symmetry and
regularity of the flow).

Let ${\mathbf{u}}:\Omega\times[0,T)\to\mathbb{R}^{3}$ be a smooth vector field
and let ${\mathbf{X}}(t)$ be a solution of the corresponding system (1) for
$t\in[0,T)$, with initial position ${\mathbf{X}}(0)=(X_{0},Y_{0},Z_{0}).$ Let
$\mbox{\boldmath{$\eta$}}({\mathbf{x}},t)$ be a smooth infinitesimal
contemporaneous symmetry of system (1). Then, for any $t\in[0,T)$ we have
$\mbox{\boldmath{$\eta$}}({\mathbf{X}}(t),t)={\mathbf{0}}$ if and only if
$\mbox{\boldmath{$\eta$}}(X_{0},Y_{0},Z_{0},0)={\mathbf{0}}.$

In words, if the governing flow is differentiable then the zeroes of any
infinitesimal symmetry follow the pathlines, and no new zeroes can be
generated at any time $t\in(0,T)$.

###### Proof.

Consider the vector function ${\mathbf{f}}:[0,T)\to\mathbb{R}^{3}$ defined by
${\mathbf{f}}(t):=\mbox{\boldmath{$\eta$}}({\mathbf{X}}(t),t)$. Using the
chain rule, its total time derivative satisfies
$\dot{{\mathbf{f}}}(t)=\frac{\partial\mbox{\boldmath{$\eta$}}}{\partial
t}({\mathbf{X}}(t),t)+(\dot{{\mathbf{X}}}(t)\cdot\nabla)\mbox{\boldmath{$\eta$}}({\mathbf{X}}(t),t)$.
By virtue of (1) we have
$\dot{{\mathbf{X}}}(t)={\mathbf{u}}({\mathbf{X}}(t),t)$ and thus
$\dot{{\mathbf{f}}}(t)=\frac{\partial\mbox{\boldmath{$\eta$}}}{\partial
t}({\mathbf{X}}(t),t)+({\mathbf{u}}({\mathbf{X}}(t),t)\cdot\nabla)\mbox{\boldmath{$\eta$}}({\mathbf{X}}(t),t)$.
As $\eta$ is an infinitesimal symmetry, from (7) we get
$\dot{{\mathbf{f}}}(t)=(\mbox{\boldmath{$\eta$}}({\mathbf{X}}(t),t)\cdot\nabla){\mathbf{u}}({\mathbf{X}}(t),t)=({\mathbf{f}}(t)\cdot\nabla){\mathbf{u}}({\mathbf{X}}(t),t)$.
The latter is a non-autonomous linear homogeneous first-order differential
equation for the vector ${\mathbf{f}}(t)$. If the vector field ${\mathbf{u}}$
is smooth then so are the matrix components of this equation,
$\nabla{\mathbf{u}}({\mathbf{X}}(t),t)$. From uniqueness of solutions to
first-order ODEs we get ${\mathbf{f}}(t)={\mathbf{0}}$ if and only if
${\mathbf{f}}(0)={\mathbf{0}}$, which gives the desired result. ∎

This result is quite relevant to the scenario of finite-time singularity in
fluid equations when we interpret ${\mathbf{u}}$ as the velocity field. It
implies that new zeroes of infinitesimal symmetries may be created at the
singularity time of the velocity field (see remark 4.12 for an application). A
similar result can be obtained for the gradient of a constant of motion $C$:
the zeroes of $\nabla C$ follow the pathlines, and no new zeroes can be
generated at any time $t\in(0,T)$.

###  2.5 Constructing new symmmetries and constants of motion from known
symmetries

The general Lie algebra of divergence-free vector fields mentioned in remark
2.7 has an important sub-algebra induced by the group structure inherent to
the finite symmetry transformations for system (1), definition 2.3. This is
the Lie algebra of infinitesimal symmetries for system (1).

######  Theorem 2.11 (Construction of new symmetries starting from known
symmetries).

Let $\mbox{\boldmath{$\eta$}}_{1}$, $\mbox{\boldmath{$\eta$}}_{2}$ be two
smooth divergence-free infinitesimal contemporaneous symmetries for system
(1). Namely, $\mbox{\boldmath{$\eta$}}_{1}$ and $\mbox{\boldmath{$\eta$}}_{2}$
satisfy the PDE system (11). Then the smooth divergence-free vector field
defined by

$\mbox{\boldmath{$\eta$}}_{3}:=\mathop{\mathcal{L}}_{\scriptsize{\mbox{\boldmath{$\eta$}}_{1}}}\mbox{\boldmath{$\eta$}}_{2}=(\mbox{\boldmath{$\eta$}}_{1}\cdot\nabla)\mbox{\boldmath{$\eta$}}_{2}-(\mbox{\boldmath{$\eta$}}_{2}\cdot\nabla)\mbox{\boldmath{$\eta$}}_{1}$

is an infinitesimal contemporaneous symmetry for system (1).

###### Proof.

By hypothesis, $\mbox{\boldmath{$\eta$}}_{1}$ and
$\mbox{\boldmath{$\eta$}}_{2}$ are smooth vector fields that satisfy, for all
${\mathbf{x}}\in\Omega,\,t\in[0,T)\,,$

$\frac{\partial\mbox{\boldmath{$\eta$}}_{1}}{\partial
t}+\mathop{\mathcal{L}}_{\scriptsize{{\mathbf{u}}}}\mbox{\boldmath{$\eta$}}_{1}={\mathbf{0}}\,,\qquad\qquad\frac{\partial\mbox{\boldmath{$\eta$}}_{2}}{\partial
t}+\mathop{\mathcal{L}}_{\scriptsize{{\mathbf{u}}}}\mbox{\boldmath{$\eta$}}_{2}={\mathbf{0}}\,.$
(13)

Consider the smooth vector field $\mbox{\boldmath{$\eta$}}_{3}$ defined above.
From bilinearity of the Lie bracket, one has

$\frac{\partial\mbox{\boldmath{$\eta$}}_{3}}{\partial
t}=\frac{\partial}{\partial
t}\left(\mathop{\mathcal{L}}_{\scriptsize{\mbox{\boldmath{$\eta$}}_{1}}}\mbox{\boldmath{$\eta$}}_{2}\right)=\mathop{\mathcal{L}}_{\scriptsize{\mbox{\boldmath{$\eta$}}_{1}}}\frac{\partial\mbox{\boldmath{$\eta$}}_{2}}{\partial
t}+\mathop{\mathcal{L}}_{\scriptsize{\frac{\partial\mbox{\boldmath{$\eta$}}_{1}}{\partial
t}}}\mbox{\boldmath{$\eta$}}_{2}=\mathop{\mathcal{L}}_{\scriptsize{\mbox{\boldmath{$\eta$}}_{1}}}\frac{\partial\mbox{\boldmath{$\eta$}}_{2}}{\partial
t}-\mathop{\mathcal{L}}_{\scriptsize{\mbox{\boldmath{$\eta$}}_{2}}}{\frac{\partial\mbox{\boldmath{$\eta$}}_{1}}{\partial
t}},$

where the last equality is obtained from the anticommutativity of the Lie
bracket. Similarly, the Jacobi identity (9) gives

$\mathop{\mathcal{L}}_{\scriptsize{{\mathbf{u}}}}\mbox{\boldmath{$\eta$}}_{3}=\mathop{\mathcal{L}}_{\scriptsize{{\mathbf{u}}}}\mathop{\mathcal{L}}_{\scriptsize{\mbox{\boldmath{$\eta$}}_{1}}}\mbox{\boldmath{$\eta$}}_{2}=\mathop{\mathcal{L}}_{\scriptsize{\mbox{\boldmath{$\eta$}}_{1}}}\mathop{\mathcal{L}}_{\scriptsize{{\mathbf{u}}}}\mbox{\boldmath{$\eta$}}_{2}+\mathop{\mathcal{L}}_{\scriptsize{\mbox{\boldmath{$\eta$}}_{2}}}\mathop{\mathcal{L}}_{\scriptsize{\mbox{\boldmath{$\eta$}}_{1}}}{{\mathbf{u}}}=\mathop{\mathcal{L}}_{\scriptsize{\mbox{\boldmath{$\eta$}}_{1}}}\mathop{\mathcal{L}}_{\scriptsize{{\mathbf{u}}}}\mbox{\boldmath{$\eta$}}_{2}-\mathop{\mathcal{L}}_{\scriptsize{\mbox{\boldmath{$\eta$}}_{2}}}\mathop{\mathcal{L}}_{\scriptsize{{\mathbf{u}}}}\mbox{\boldmath{$\eta$}}_{1}\,,$

where again the last equality is obtained from the anticommutativity of the
Lie bracket. Combining the last two results we get

$\frac{\partial\mbox{\boldmath{$\eta$}}_{3}}{\partial
t}+\mathop{\mathcal{L}}_{\scriptsize{{\mathbf{u}}}}\mbox{\boldmath{$\eta$}}_{3}=\mathop{\mathcal{L}}_{\scriptsize{\mbox{\boldmath{$\eta$}}_{1}}}\frac{\partial\mbox{\boldmath{$\eta$}}_{2}}{\partial
t}-\mathop{\mathcal{L}}_{\scriptsize{\mbox{\boldmath{$\eta$}}_{2}}}{\frac{\partial\mbox{\boldmath{$\eta$}}_{1}}{\partial
t}}+\mathop{\mathcal{L}}_{\scriptsize{\mbox{\boldmath{$\eta$}}_{1}}}\mathop{\mathcal{L}}_{\scriptsize{{\mathbf{u}}}}\mbox{\boldmath{$\eta$}}_{2}-\mathop{\mathcal{L}}_{\scriptsize{\mbox{\boldmath{$\eta$}}_{2}}}\mathop{\mathcal{L}}_{\scriptsize{{\mathbf{u}}}}\mbox{\boldmath{$\eta$}}_{1}=\mathop{\mathcal{L}}_{\scriptsize{\mbox{\boldmath{$\eta$}}_{1}}}\left(\frac{\partial\mbox{\boldmath{$\eta$}}_{2}}{\partial
t}+\mathop{\mathcal{L}}_{\scriptsize{{\mathbf{u}}}}\mbox{\boldmath{$\eta$}}_{2}\right)-\mathop{\mathcal{L}}_{\scriptsize{\mbox{\boldmath{$\eta$}}_{2}}}\left(\frac{\partial\mbox{\boldmath{$\eta$}}_{1}}{\partial
t}+\mathop{\mathcal{L}}_{\scriptsize{{\mathbf{u}}}}\mbox{\boldmath{$\eta$}}_{1}\right)\,,$

where the last equality follows from linearity of the Lie derivative. By
virtue of equations (13) the desired result follows. ∎

Theorem 2.11 is very important in practice as it allows for the construction
of a succession of infinitesimal symmetries for system (1) starting from two
or more infinitesimal symmetries, providing thus a Lie algebra that can be
finite- or infinite-dimensional. The only case when this construction is not
possible is when the two starting infinitesimal symmetries commute:
$\mathop{\mathcal{L}}_{\scriptsize{\mbox{\boldmath{$\eta$}}_{1}}}\mbox{\boldmath{$\eta$}}_{2}={\mathbf{0}}$.
For this case we have an even better result:

######  Theorem 2.12 (Construction of constants of motion starting from known
commuting symmetries).

Let $\mbox{\boldmath{$\eta$}}_{1}$, $\mbox{\boldmath{$\eta$}}_{2}$ be two
smooth divergence-free infinitesimal contemporaneous symmetries for system (1)
such that they are not linearly dependent (i.e., such that they are not
proportional via a simple numerical factor). Assume these symmetries commute:
$\mathop{\mathcal{L}}_{\scriptsize{\mbox{\boldmath{$\eta$}}_{1}}}\mbox{\boldmath{$\eta$}}_{2}={\mathbf{0}}$.
Then there are two cases:

  1. 1.

If the vector product
$\mbox{\boldmath{$\eta$}}_{1}\times\mbox{\boldmath{$\eta$}}_{2}$ is not zero
everywhere on $\Omega$ for all times, then
$\mbox{\boldmath{$\eta$}}_{1}\times\mbox{\boldmath{$\eta$}}_{2}=\nabla K$
where $K$ is a non-trivial constant of motion for system (1), defined at least
locally in $\Omega\times[0,T)$.

  2. 2.

If the vector product
$\mbox{\boldmath{$\eta$}}_{1}\times\mbox{\boldmath{$\eta$}}_{2}$ is zero
everywhere on $\Omega$ for all times, then
$\mbox{\boldmath{$\eta$}}_{2}=K\mbox{\boldmath{$\eta$}}_{1}$, where $K$ is a
constant of motion for system (1) whose gradient $\nabla K$ is not zero
everywhere and is orthogonal to $\mbox{\boldmath{$\eta$}}_{1}$ and
$\mbox{\boldmath{$\eta$}}_{2}$.

###### Proof.

(i) If $\mbox{\boldmath{$\eta$}}_{1}\times\mbox{\boldmath{$\eta$}}_{2}$ is not
zero everywhere on $\Omega$ for all times, we first want to find a scalar
function $K:\Omega\times[0,T)\to\mathbb{R}$ such that
$\mbox{\boldmath{$\eta$}}_{1}\times\mbox{\boldmath{$\eta$}}_{2}=\nabla K$. The
compatibility condition for finding $K$ is
$\nabla\times(\mbox{\boldmath{$\eta$}}_{1}\times\mbox{\boldmath{$\eta$}}_{2})={\mathbf{0}}$
on $\Omega\times[0,T)$. The function $K$ may not be globally defined, but at
least it can be found to be smooth on some neighbourhood
$B_{\epsilon}({\mathbf{x}})\times[0,T)$, where
$B_{\epsilon}({\mathbf{x}})\subset\Omega$ is the ball of radius $\epsilon$ and
centered at ${\mathbf{x}}\in\Omega$. From elementary vector calculus
identities we have
$\nabla\times(\mbox{\boldmath{$\eta$}}_{1}\times\mbox{\boldmath{$\eta$}}_{2})=\mbox{\boldmath{$\eta$}}_{1}(\nabla\cdot\mbox{\boldmath{$\eta$}}_{2})-\mbox{\boldmath{$\eta$}}_{2}(\nabla\cdot\mbox{\boldmath{$\eta$}}_{1})+(\mbox{\boldmath{$\eta$}}_{2}\cdot\nabla)\mbox{\boldmath{$\eta$}}_{1}-(\mbox{\boldmath{$\eta$}}_{1}\cdot\nabla)\mbox{\boldmath{$\eta$}}_{2}$.
But
$\nabla\cdot\mbox{\boldmath{$\eta$}}_{1}=\nabla\cdot\mbox{\boldmath{$\eta$}}_{2}=0$
since the fields are divergence-free. Therefore
$\nabla\times(\mbox{\boldmath{$\eta$}}_{1}\times\mbox{\boldmath{$\eta$}}_{2})=-\mathop{\mathcal{L}}_{\scriptsize{\mbox{\boldmath{$\eta$}}_{1}}}\mbox{\boldmath{$\eta$}}_{2}={\mathbf{0}}$.
So $K$ exists locally.

Second, we want to show that $\nabla C:=\nabla K$ satisfies (12). The LHS of
this equation is:

${\mathbf{F}}:=\frac{\partial\nabla K}{\partial
t}+({\mathbf{u}}\cdot\nabla)\nabla K+(\nabla{\mathbf{u}})^{T}\nabla
K=\frac{\partial}{\partial
t}(\mbox{\boldmath{$\eta$}}_{1}\times\mbox{\boldmath{$\eta$}}_{2})+({\mathbf{u}}\cdot\nabla)(\mbox{\boldmath{$\eta$}}_{1}\times\mbox{\boldmath{$\eta$}}_{2})+(\nabla{\mathbf{u}})^{T}(\mbox{\boldmath{$\eta$}}_{1}\times\mbox{\boldmath{$\eta$}}_{2}).$

Looking at the first two terms of the last member, both the time derivative
and the gradient satisfy the Leibniz product rule, so

${\mathbf{F}}=\left(\frac{\partial\mbox{\boldmath{$\eta$}}_{1}}{\partial
t}+({\mathbf{u}}\cdot\nabla)\mbox{\boldmath{$\eta$}}_{1}\right)\times\mbox{\boldmath{$\eta$}}_{2}+\mbox{\boldmath{$\eta$}}_{1}\times\left(\frac{\partial\mbox{\boldmath{$\eta$}}_{2}}{\partial
t}+({\mathbf{u}}\cdot\nabla)\mbox{\boldmath{$\eta$}}_{2}\right)+(\nabla{\mathbf{u}})^{T}(\mbox{\boldmath{$\eta$}}_{1}\times\mbox{\boldmath{$\eta$}}_{2})\,.$

Now, as $\mbox{\boldmath{$\eta$}}_{1}$ and $\mbox{\boldmath{$\eta$}}_{2}$ are
infinitesimal symmetries they satisfy (7), so we get

${\mathbf{F}}=\left((\mbox{\boldmath{$\eta$}}_{1}\cdot\nabla){\mathbf{u}}\right)\times\mbox{\boldmath{$\eta$}}_{2}+\mbox{\boldmath{$\eta$}}_{1}\times\left((\mbox{\boldmath{$\eta$}}_{2}\cdot\nabla){\mathbf{u}}\right)+(\nabla{\mathbf{u}})^{T}(\mbox{\boldmath{$\eta$}}_{1}\times\mbox{\boldmath{$\eta$}}_{2})\,.$

By inspection, the vector ${\mathbf{F}}$ is an anticommutative algebraic
bilinear function of the vectors
$\mbox{\boldmath{$\eta$}}_{1},\mbox{\boldmath{$\eta$}}_{2}$, such that
${\mathbf{F}}\cdot\mbox{\boldmath{$\eta$}}_{1}={\mathbf{F}}\cdot\mbox{\boldmath{$\eta$}}_{2}=0$.
Therefore,
${\mathbf{F}}=M\mbox{\boldmath{$\eta$}}_{1}\times\mbox{\boldmath{$\eta$}}_{2}$,
where $M$ is some scalar function. By taking
$\mbox{\boldmath{$\eta$}}_{1}=(1,0,0)$ and
$\mbox{\boldmath{$\eta$}}_{2}=(0,1,0)$ we get readily
$M=\nabla\cdot{\mathbf{u}}$. As ${\mathbf{u}}$ is divergence-free, we get
$M=0$ and thus ${\mathbf{F}}={\mathbf{0}}$. Therefore $\nabla K$ satisfies
(12).

Third and last, lemma 2.9 establishes that $K$ is a constant of motion up to
an additive function of time that can be found.

(ii) If the vector product
$\mbox{\boldmath{$\eta$}}_{1}\times\mbox{\boldmath{$\eta$}}_{2}$ is zero
everywhere on $\Omega$ for all times, then
$\mbox{\boldmath{$\eta$}}_{2}=K\mbox{\boldmath{$\eta$}}_{1}$ on $\Omega$ for
all times, where by hypothesis $K$ is a scalar function that is not a simple
numerical constant, thus $\nabla K$ is not zero everywhere. Let us apply the
operator $\frac{\partial}{\partial
t}+\mathop{\mathcal{L}}_{\scriptsize{{\mathbf{u}}}}$ to the equation
$\mbox{\boldmath{$\eta$}}_{2}=K\mbox{\boldmath{$\eta$}}_{1}$. Using the
Leibniz product rule for the Lie derivative we get

$\frac{\partial\mbox{\boldmath{$\eta$}}_{2}}{\partial
t}+\mathop{\mathcal{L}}_{\scriptsize{{\mathbf{u}}}}\mbox{\boldmath{$\eta$}}_{2}=K\left(\frac{\partial\mbox{\boldmath{$\eta$}}_{1}}{\partial
t}+\mathop{\mathcal{L}}_{\scriptsize{{\mathbf{u}}}}\mbox{\boldmath{$\eta$}}_{1}\right)+\left(\frac{\partial
K}{\partial
t}+\mathop{\mathcal{L}}_{\scriptsize{{\mathbf{u}}}}K\right)\mbox{\boldmath{$\eta$}}_{1}\,.$

Recalling now that $\mbox{\boldmath{$\eta$}}_{1}$ and
$\mbox{\boldmath{$\eta$}}_{2}$ are infinitesimal symmetries for system (1),
namely that they satisfy (7), we readily obtain
${\mathbf{0}}=\left(\frac{\partial K}{\partial
t}+\mathop{\mathcal{L}}_{\scriptsize{{\mathbf{u}}}}K\right)\mbox{\boldmath{$\eta$}}_{1}$
which, at the positions where $\mbox{\boldmath{$\eta$}}_{1}$ is nonzero
(generically this is almost everywhere), implies $\frac{\partial K}{\partial
t}+\mathop{\mathcal{L}}_{\scriptsize{{\mathbf{u}}}}K=0$ and thus $K$ is a
constant of motion for system (1). Now,
$\nabla\cdot\mbox{\boldmath{$\eta$}}_{2}=\nabla\cdot\mbox{\boldmath{$\eta$}}_{1}=0$
imply $\mbox{\boldmath{$\eta$}}_{1}\cdot\nabla
K=\mbox{\boldmath{$\eta$}}_{2}\cdot\nabla K=0$. ∎

######  Remark 2.13.

A dual version of theorem 2.12, stating in our notation that the cross product
between the gradients of two constants of motion must be a symmetry, has been
applied to fluid models in [16, 17, 18, 19]. Also, a result analogous to
theorem 2.11, stating that the Lie derivative of a constant of motion along a
symmetry is a constant of motion, is well known [6] and has been applied to
fluid equations recently [20].

Theorem 2.12 requires solely the existence of two commuting infinitesimal
symmetries in order to construct a constant of motion, without recourse to
Noether’s theorem, namely without the need that system (1) be derived from a
Lagrangian or action principle that is invariant under some symmetry. See [6]
for a description of these so-called “non-Noetherian” symmetries.

##  3 The 3D Euler fluid equations; symmetries of the velocity field

Consider an inviscid, incompressible fluid, of unit mass density, moving in
three spatial dimensions with smooth velocity field
${\mathbf{u}}:\Omega\times[0,T)\to\mathbb{R}^{3}$. The notations
${\mathbf{x}}:=(x,y,z)=(x^{1},x^{2},x^{3})\in\Omega$ and
${\mathbf{u}}({\mathbf{x}},t):=(u_{\mathrm{x}}({\mathbf{x}},t),u_{\mathrm{y}}({\mathbf{x}},t),u_{\mathrm{z}}({\mathbf{x}},t))=(u^{1}({\mathbf{x}},t),u^{2}({\mathbf{x}},t),u^{3}({\mathbf{x}},t))$
will be used intechangeably. We will set either $\Omega=\mathbb{T}^{3}$, or
$\Omega=\mathbb{R}^{2}\times\mathbb{T}$, or
$\Omega=\mathbb{T}^{2}\times\mathbb{R}$, where $\mathbb{T}:=[0,2\pi)$ is the
torus.

The velocity field is thus divergence-free ($\nabla\cdot{\mathbf{u}}=0$) and
satisfies the 3D Euler fluid equations:

$\frac{\partial{\mathbf{u}}}{\partial
t}+({\mathbf{u}}\cdot\nabla){\mathbf{u}}=-\nabla p\,,\qquad\text{for \,\,
all}\quad{\mathbf{x}}\in\Omega,\,t\in[0,T)\,,$ (14)

where the pressure $p$ is determined from the condition
$\nabla\cdot{\mathbf{u}}=0$. We will be interested in the solution of this
nonlinear partial differential equation for the field
${\mathbf{u}}({\mathbf{x}},t)$, under known smooth initial data
${\mathbf{u}}({\mathbf{x}},0)={\mathbf{u}}_{0}({\mathbf{x}})$. At the same
time we will be interested in the formalism of the previous sections, in terms
of the solutions of the ODE system (1), namely the pathlines of the velocity
field ${\mathbf{u}}$. This velocity field may admit constants of motion and
infinitesimal symmetries, namely solutions of equations (10)–(11).

The vorticity field $\mbox{\boldmath{$\omega$}}:=\nabla\times{\mathbf{u}}$ is
clearly divergence-free and taking the curl of equation (14) one obtains the
well-known Helmholtz vorticity formulation [21, 22, 9]:

$\frac{\partial\mbox{\boldmath{$\omega$}}}{\partial
t}+({\mathbf{u}}\cdot\nabla)\mbox{\boldmath{$\omega$}}-(\mbox{\boldmath{$\omega$}}\cdot\nabla){\mathbf{u}}={\mathbf{0}}\,,\qquad\text{for
\,\, all}\quad{\mathbf{x}}\in\Omega,\,t\in[0,T)\,.$ (15)

From theorem 2.5 it is then direct to prove:

######  Lemma 3.1 (The vorticity is an infinitesimal symmetry of the flow).

Let ${\mathbf{u}}:\Omega\times[0,T)\to\mathbb{R}^{3}$ be a smooth divergence-
free velocity field satisfying the 3D Euler fluid equations (14). Then the
smooth divergence-free vorticity field
$\mbox{\boldmath{$\omega$}}=\nabla\times{\mathbf{u}}$ is an infinitesimal
contemporaneous symmetry for the system (1). Namely, the vorticity vector
field satisfies (7) or, equivalently, (11).

######  Remark 3.2.

Historically, equation (15) was first obtained in its $2$-form formulation in
[21]. Similar transport equations for $3$-forms were found: helicity in 3D
Euler equations [22], potential vorticity in atmospheric models [23, 24, 25].
Of course, global helicity was shown to be conserved in 3D Euler equations
much earlier [26, 27]. Further, the relation between Lie-advected $p$-forms
and Cauchy invariants along with a historical account of how these have been
rediscovered over and over again is documented in [10].

If the flow admits another symmetry different from the vorticity, then it is
possible to generate new symmetries and constants of motion, via theorems 2.12
and 2.11. In the following subsections we will consider in detail two examples
where this construction is possible.

###  3.1 Symmetries of steady 3D Euler flows

By definition, steady 3D Euler flows are time-independent solutions
${\mathbf{u}}({\mathbf{x}})$ to the 3D Euler fluid equations (14). Of course,
the initial data must be carefully selected so that the solution remains
static in time. The ODE system (1) becomes autonomous. The vorticity $\omega$
is thus time-independent, and from equation (15) and the definition of Lie
derivative we get
$\mathop{\mathcal{L}}_{\scriptsize{{\mathbf{u}}}}\mbox{\boldmath{$\omega$}}={\mathbf{0}}$,
for all ${\mathbf{x}}\in\Omega$. On the other hand, since ${\mathbf{u}}$
itself is time-independent and
$\mathop{\mathcal{L}}_{\scriptsize{{\mathbf{u}}}}{\mathbf{u}}={\mathbf{0}}$,
we deduce that ${\mathbf{u}}$ is an infinitesimal symmetry for the ODE system
(1). In conclusion, steady Euler flows possess two commuting symmetries, given
by the static vector fields $\mbox{\boldmath{$\eta$}}_{1}={\mathbf{u}}$ and
$\mbox{\boldmath{$\eta$}}_{2}=\mbox{\boldmath{$\omega$}}.$ Theorem 2.12 gives:

######  Corollary 3.3.

Let ${\mathbf{u}}$ be a steady 3D Euler flow, with vorticity field
$\mbox{\boldmath{$\omega$}}=\nabla\times{\mathbf{u}}$. Then there exists a
constant of motion $\alpha$ satisfying
${\mathbf{u}}\times\mbox{\boldmath{$\omega$}}=\nabla\alpha$.

###### Proof.

This is a direct application of theorem 2.12 regarding the symmetries
$\mbox{\boldmath{$\eta$}}_{1}={\mathbf{u}}$ and
$\mbox{\boldmath{$\eta$}}_{2}=\mbox{\boldmath{$\omega$}}$. ∎

######  Remark 3.4.

The statement in this corollary is well known. In fact, $\alpha$ is the
Bernoulli function, equal to
$\alpha=p+\frac{1}{2}{\mathbf{u}}\cdot{\mathbf{u}}$, where $p$ is the pressure
field. Moreover, when $\nabla\times{\mathbf{u}}=k{\mathbf{u}}$ for some
$k\in\mathbb{R}$ this corollary does not apply, in the sense that $\alpha$
becomes a numerical constant. In this case the velocity field is known as a
Beltrami field [28, 29]. As a result, Beltrami fields can be nonintegrable
[9].

As an example, consider a velocity field defined in Cartesian coordinates by
${\mathbf{u}}(x,y,z)=(\cos z,a\sin z,0)$, where $a\in\mathbb{R}$ is a
constant, and where either $\Omega=\mathbb{T}^{3}$ or
$\Omega=\mathbb{R}^{2}\times\mathbb{T}$. The vorticity is
$\mbox{\boldmath{$\omega$}}=-(a\cos z,\sin z,0)$ and we find
${\mathbf{u}}\times\mbox{\boldmath{$\omega$}}=\nabla\alpha\,,$ where
$\alpha=\frac{1}{4}(1-a^{2})\cos 2z$ is a constant of motion (nontrivial when
$|a|\neq 1$). Of course, when $|a|=1$ the flow is Beltrami, and a particular
case of an ABC flow [28, 29]. The pathlines of the velocity field satisfy the
equations

$\dot{X}=\cos Z,\qquad\dot{Y}=a\sin Z,\qquad\dot{Z}=0\,,$ (16)

and are simply straight lines (i.e., geodesics) on the sub-manifold
$\Omega_{Z}$ defined by the equation $Z=\mathrm{const}$. For this simple flow,
a general type of infinitesimal symmetry is thus given by

$\mbox{\boldmath{$\eta$}}_{\mathrm{gen}}=f(\alpha){\mathbf{u}}+g(\alpha)\mbox{\boldmath{$\omega$}}\,,$
(17)

where $f,g$ are arbitrary scalar functions. Notice that this general symmetry
has no $z$-component. Is it possible to find a new infinitesimal symmetry with
a non-zero $z$-component? The answer depends on the topology of the spatial
domain $\Omega$:

(i) If $\Omega=\mathbb{T}^{3}$ then $\Omega_{Z}=\mathbb{T}^{2}$. Notice that
the values of $Z$ for which the pair $(\cos Z,a\sin Z)$ is incommensurate,
cover densely the region $Z\in[0,2\pi).$ In these cases, the pathlines are
infinitely long and cover densely the $2$-torus $\Omega_{Z}$ parametrised by
$(x,y).$ Let us call the sub-manifolds $\Omega_{Z}$ ‘ergodic’ in such cases.
On the other hand, embedded in the region $Z\in[0,2\pi)$ are an infinite
number of points for which the pair $(\cos Z,a\sin Z)$ is commensurate, with
closed pathlines on the $2$-torus $\Omega_{Z}$. We call the sub-manifolds
$\Omega_{Z}$ regular in these cases. Let $\Omega_{Z_{0}}$ be an arbitrary
regular sub-manifold, where $Z_{0}\in(0,2\pi)$. Then, the open region
${\mathcal{B}}_{\epsilon}=\bigcup_{Z\in(Z_{0}-\epsilon,Z_{0}+\epsilon)}\Omega_{Z}$
contains, for arbitrary small enough $\epsilon>0$, ergodic sub-manifolds
$\Omega_{Z}$. This is due to the fact that irrational numbers are uniformly
distributed on the real line (equidistribution theorem).

We claim that it is impossible to find an infinitesimal symmetry continuous on
${\mathcal{B}}_{\epsilon}$, and defined globally there, with a non-zero
component along the $z$-direction. To see this, suppose that such a symmetry
exists, call it $\mbox{\boldmath{$\eta$}}_{0}(x,y,z,t)$, and assume without
loss of generality that its $z$-component is equal to $1$ at
$t=0,x=0,y=0,z=Z_{0}:$ $\eta_{0}^{z}(0,0,Z_{0},0)=1\,.$ Let $\delta$ be such
that $0<\delta<\epsilon$ and such that
$\Omega_{Z_{0}+\delta}\in{\mathcal{B}}_{\epsilon}$ be ergodic. Recall that
$\delta$ is arbitrarily small. Thus, the symmetry vector field
$\delta\mbox{\boldmath{$\eta$}}_{0}(x,y,z,t)$ connects the point
$P:=(0,0,Z_{0})\in\Omega_{Z_{0}}$ with the point
$\widetilde{P}:=(\delta\eta_{0}^{x}(0,0,Z_{0},0),\delta\eta_{0}^{y}(0,0,Z_{0},0),Z_{0}+\delta)\in\Omega_{Z_{0}+\delta}\,.$
Using equation (16), let us denote the pathline passing through $P$ by
$(X(r),Y(r),Z_{0})\in\Omega_{Z_{0}}\quad\forall r\in[0,1],$ with
$(X(0),Y(0))=(X(1),Y(1))=(0,0)$ since $\Omega_{Z_{0}}$ is regular. Now, let us
denote the pathline passing through $\widetilde{P}$ by
$(\widetilde{X}(s),\widetilde{Y}(s),Z_{0}+\delta)\in\Omega_{Z_{0}+\delta},\quad
s\in(-\infty,\infty)$, which is not closed because $\Omega_{Z_{0}+\delta}$ is
ergodic. But by definition the symmetry $\mbox{\boldmath{$\eta$}}_{0}$ maps
the closed pathline (of finite length) to the non-closed one (of infinite
length). This is a contradiction, as it is not possible to continuously map
these two pathlines. Therefore we conclude that the symmetry
$\mbox{\boldmath{$\eta$}}_{0}(x,y,z,t)$ is not continuous on
${\mathcal{B}}_{\epsilon}$.

(ii) If $\Omega=\mathbb{R}^{2}\times\mathbb{T}$ then
$\Omega_{Z}=\mathbb{R}^{2}$ and the pathlines are just straight lines which
extend without bound. All sub-manifolds are regular, and therefore the
limitation of the previous case disappears. The following is an infinitesimal
symmetry: $\mbox{\boldmath{$\eta$}}_{1}(x,y,z,t)=(-y,x\,a^{2},a)$. The
commutator of the vorticity with this symmetry is
$\mbox{\boldmath{$\eta$}}_{2}:=\mathop{\mathcal{L}}_{\scriptsize{\mbox{\boldmath{$\omega$}}}}{\mbox{\boldmath{$\eta$}}_{1}}=(1-a^{2})(\sin
z,a\cos z,0)$. As this is zero when $|a|=1$, we get two sub-cases:
(ii.1) When $|a|=1$, i.e. when the flow is Beltrami, we can write
$\mbox{\boldmath{$\omega$}}\times\mbox{\boldmath{$\eta$}}_{1}=\nabla C,$
giving the constant of motion $C=-ax\sin z+y\cos z$, which provides an
equation for the pathlines on $\Omega_{z}$.
(ii.2) When $|a|\neq 1$, the symmetry $\mbox{\boldmath{$\eta$}}_{2}$ is of the
form (17) so there is no new information. In this case the following time-
dependent symmetry does commute with the vorticity:
$\mbox{\boldmath{$\eta$}}_{3}(x,y,z,t)=(-y\,a^{2}-t\,(1-a^{2})\,a\sin
z,x-t\,(1-a^{2})\cos z,a)$. Thus, we can write
$\mbox{\boldmath{$\omega$}}\times\mbox{\boldmath{$\eta$}}_{3}=\nabla
C^{\prime},$ giving the constant of motion $\frac{1}{a}C^{\prime}=x\sin
z-ay\cos z-(1-a^{2})t\sin z\cos z$ which provides an explicit time
parameterisation for the pathlines on $\Omega_{z}$.

A complete Lie algebra of symmetries can be produced, which we omit for the
sake of brevity.

###  3.2 Symmetries of non-steady 3D Euler flows of stagnation-point type

We consider flows defined over the spatial domain
$\Omega=\mathbb{T}^{2}\times\mathbb{R}$ and use $(x,y)$ to parametrise
$\mathbb{T}^{2}:=[0,2\pi)^{2}$ and $z$ to parametrise $\mathbb{R}.$ An exact
solution of the 3D Euler equation (14) is [1]:

${\mathbf{u}}(x,y,z,t)=(u_{\mathrm{x}}(x,y,t),u_{\mathrm{y}}(x,y,t),z\,\gamma(x,y,t))$
(18)

where, with reference to the plane $z=0$, the “horizontal” velocity
${\mathbf{u}}_{\mathrm{h}}:=(u_{\mathrm{x}},u_{\mathrm{y}})$ is determined
uniquely via the Helmholtz decomposition from the equations

$\nabla_{\mathrm{h}}\cdot{\mathbf{u}}_{\mathrm{h}}:=\frac{\partial
u_{\mathrm{x}}}{\partial x}+\frac{\partial u_{\mathrm{y}}}{\partial
y}=-\gamma(x,y,t)\,,\qquad\qquad\nabla_{\mathrm{h}}\times{\mathbf{u}}_{\mathrm{h}}:=\frac{\partial
u_{\mathrm{y}}}{\partial x}-\frac{\partial u_{\mathrm{x}}}{\partial
y}=\omega(x,y,t),$ (19)

where the scalar functions $\omega$ and $\gamma$ are, respectively, the out-
of-plane vorticity and its vorticity stretching rate. These two scalars are
the fundamental fields of this solution and satisfy the system

$\displaystyle\frac{\partial}{\partial
t}\omega+{\mathbf{u}}_{\mathrm{h}}\cdot\nabla_{\mathrm{h}}\omega$
$\displaystyle=$ $\displaystyle\gamma\,\omega\,,$ (20)
$\displaystyle\frac{\partial}{\partial
t}\gamma+{\mathbf{u}}_{\mathrm{h}}\cdot\nabla_{\mathrm{h}}\gamma$
$\displaystyle=$ $\displaystyle 2\langle\gamma^{2}\rangle-\gamma^{2}\,,$ (21)

where $\langle f(\cdot,t)\rangle$ denotes the average of $f(x,y,t)$ over
$(x,y)\in\mathbb{T}^{2}$ and is thus a function of time only. Equations (19)
imply $\langle\gamma\rangle=\langle\omega\rangle=0$ at all times, which is
consistent with (20)–(21).

Because (18) is an exact solution of (14), the full 3D vorticity

$\mbox{\boldmath{$\omega$}}=\nabla\times{\mathbf{u}}=\left(z\frac{\partial\gamma}{\partial
y},-z\frac{\partial\gamma}{\partial x},\omega\right)\,,$ (22)

is an infinitesimal symmetry for the non-autonomous system (1). Let us try to
find another symmetry for this system, of the form
$\mbox{\boldmath{$\eta$}}=(0,0,\eta^{z}(x,y,t))\,,$ where $\eta^{z}$ is
unknown. Plugging this into (11) and using (18) we get

$\partial_{t}\eta^{z}+{\mathbf{u}}_{\mathrm{h}}\cdot\nabla_{\mathrm{h}}\eta^{\mathrm{z}}-\gamma\,\eta^{\mathrm{z}}=0\,.$
(23)

By inspection and comparing with (20) we see that a solution is
$\eta^{\mathrm{z}}=\omega$, giving the symmetry

$\displaystyle\mbox{\boldmath{$\eta$}}_{1}(x,y,t):=(0,0,\omega(x,y,t))\,.$
(24)

This is the out-of-plane vorticity. Thus, by linearity, the in-plane vorticity
is a symmetry:

$\displaystyle\mbox{\boldmath{$\eta$}}_{2}(x,y,z,t):=\mbox{\boldmath{$\omega$}}(x,y,z,t)-\mbox{\boldmath{$\eta$}}_{1}(x,y,t)=\left(z\frac{\partial\gamma}{\partial
y},-z\frac{\partial\gamma}{\partial x},0\right)\,.$ (25)

From theorem 2.11, successive Lie derivatives of
$\mbox{\boldmath{$\eta$}}_{1}$ along $\mbox{\boldmath{$\eta$}}_{2}$ produces
new symmetries:

$\mbox{\boldmath{$\eta$}}_{3}:=\mathop{\mathcal{L}}_{\scriptsize{\mbox{\boldmath{$\eta$}}_{1}}}\mbox{\boldmath{$\eta$}}_{2}=\left(\omega\,\frac{\partial\gamma}{\partial
y},-\omega\,\frac{\partial\gamma}{\partial
x},{z}\,\nabla_{\mathrm{h}}\gamma\times\nabla_{\mathrm{h}}\omega\right),\qquad\mbox{\boldmath{$\eta$}}_{4}:={\mathop{\mathcal{L}}_{\scriptsize{\mbox{\boldmath{$\eta$}}_{1}}}}\mbox{\boldmath{$\eta$}}_{3}=\left(0,0,2\,\omega\,\nabla_{\mathrm{h}}\gamma\times\nabla_{\mathrm{h}}\omega\right),$
(26)

with
$\mathop{\mathcal{L}}_{\scriptsize{\mbox{\boldmath{$\eta$}}_{1}}}\mbox{\boldmath{$\eta$}}_{4}={\mathbf{0}}\,,$
so ${\mbox{\boldmath{$\eta$}}_{1}}$ and $\mbox{\boldmath{$\eta$}}_{4}$ commute
and are collinear (see equations (26) and (24)), which by virtue of theorem
2.12 gives the constant of motion
$C_{1}(x,y,t):=\nabla_{\mathrm{h}}\gamma\times\nabla_{\mathrm{h}}\omega\,.$

These results relate the fields $\gamma$ and $\omega$. In order to solve the
evolution equations, it would be desirable to find an infinitesimal symmetry
depending on the field $\gamma$ only. Let us try to find again a symmetry
$\mbox{\boldmath{$\eta$}}=(0,0,\eta^{\mathrm{z}}(x,y,t))\,,$ but this time
look for solutions of equation (23) of the form

$\eta^{\mathrm{z}}(x,y,t)=\left(A(t)\,\gamma(x,y,t)+B(t)\right)^{k}\,,$ (27)

where $k$ is a constant to be determined and $A(t),B(t)$ are functions of time
to be determined. Replacing we get:
$k\,\dot{A}\,\gamma+k\,\dot{B}+\left(\partial_{t}\gamma+{\mathbf{u}}_{\mathrm{h}}\cdot\nabla_{\mathrm{h}}\gamma\right)\,k\,A-\left(A\,\gamma+B\right)\,\gamma=0\,,$
and using equation (21) we obtain
$k\,\dot{A}\,\gamma+k\,\dot{B}+(2\langle\gamma^{2}\rangle-\gamma^{2})\,k\,A-\left(A\,\gamma+B\right)\,\gamma=0$,
which we can solve by eliminating the coefficients of $1$, $\gamma$ and
$\gamma^{2}$ to get the system

$-(k+1)\,A=0\,,\qquad k\,\dot{A}-B=0\,,\qquad
k\,\dot{B}+2\,k\langle\gamma^{2}\rangle\,A=0\,,$

which gives the solution $k=-1,\quad B=-\dot{A},$ and
$\ddot{A}-2\langle\gamma^{2}\rangle\,A=0$, so the new symmetry is

$\mbox{\boldmath{$\eta$}}_{5}(x,y,t)=\left(0,0,\frac{1}{A(t)\,\gamma(x,y,t)-\dot{A}(t)}\right)\,,\qquad\text{where}\qquad\ddot{A}-2\langle\gamma^{2}\rangle\,A=0\,.$
(28)

As $\mbox{\boldmath{$\eta$}}_{5}$ and $\mbox{\boldmath{$\eta$}}_{1}$ are
collinear, theorem 2.12 gives the constant of motion
$C_{2}(x,y,t):=\omega(x,y,t)(A(t)\,\gamma(x,y,t)-\dot{A}(t))$. We want to find
a constant of motion involving the field $\gamma$ only. So far the only two
symmetries that depend just on $\gamma$ are $\mbox{\boldmath{$\eta$}}_{2}$ and
$\mbox{\boldmath{$\eta$}}_{5}$.Their Lie bracket is

$\mbox{\boldmath{$\eta$}}_{6}:=\mathop{\mathcal{L}}_{\scriptsize{\mbox{\boldmath{$\eta$}}_{5}}}\mbox{\boldmath{$\eta$}}_{2}=\frac{1}{A(t)\,\gamma(x,y,t)-\dot{A}(t)}\left(\frac{\partial\gamma}{\partial
y},-\frac{\partial\gamma}{\partial x},0\right),$ (29)

proportional to $\mbox{\boldmath{$\eta$}}_{2}$. Thus, theorem 2.12 gives the
constant of motion $C_{3}(x,y,z,t):=z(A(t)\,\gamma(x,y,t)-\dot{A}(t))$. This
constant determines the out-of-plane pathlines. Finally, notice that
$\mbox{\boldmath{$\eta$}}_{6}$ as well as $\mbox{\boldmath{$\eta$}}_{5}$ do
not depend on $z$. It is direct to check that they commute. As these
symmetries are not proportional, theorem 2.12(i) allows for the construction
of a new constant of motion such that
$\mbox{\boldmath{$\eta$}}_{6}\times\mbox{\boldmath{$\eta$}}_{5}=\nabla C_{4}$.
We get, preliminarily, $\nabla
C_{4}=-{(A(t)\,\gamma(x,y,t)-\dot{A}(t))^{-2}}\left(\frac{\partial\gamma}{\partial
x},\frac{\partial\gamma}{\partial y},0\right)$, which is integrated to give
$C_{4}(x,y,t)=f(t)+(A(t)[A(t)\,\gamma(x,y,t)-\dot{A}(t)])^{-1}$, where $f(t)$
is to be determined. Using (2) and the second equation in (28) we get
$f^{\prime}(t)=-[A(t)]^{-2}$. Thus

$C_{4}(x,y,t)=\frac{1}{A(t)[A(t)\,\gamma(x,y,t)-\dot{A}(t)]}-\int_{0}^{t}\frac{1}{[A(s)]^{2}}~{}\mathrm{d}s\,.$

This constant of motion allows one to solve for $\gamma$ along the pathlines
and establish conditions for blowup, in a result that is equivalent to the
solution presented in [30] (see also [31]). The constant of motion $C_{2}$ is
then used to solve for $\omega$ along the pathlines. We will not go further in
the solution here because this problem is a particular case of a more general
model that we will study in detail in the next section.

##  4 A new family of models of 3D Euler flows of stagnation-point type:
continuous symmetries and proof of finite-time blowup

The exact solution of the 3D Euler fluid equation considered in the previous
subsection is a limited model for a general solution that has a symmetry plane
at $z=0$: in a more general solution such as the one obtained from a numerical
simulation on a domain $\Omega=\mathbb{T}^{3}$ [32, 33, 34, 35], due to
periodicity the velocity is not a linear function of $z$ anymore, and so the
ansatz (18)–(21) will fail at reproducing the actual behaviour of the fields
at the symmetry plane. In particular, it fails to predict modulations of the
curvature of the pressure field, resulting in that
$\frac{\partial^{2}p}{\partial z^{2}}$ evaluated at the symmetry plane is a
function of time only, not of the in-plane coordinates $(x,y)$. For this
reason, a minimal generalisation of this model was introduced by [2], so that
this pressure curvature would show some modulation on the plane, while
maintaining the Lie structure of infinitesimal symmetries for the model. The
new model has a free parameter $\lambda\in\mathbb{R}$, the case $\lambda=0$
corresponding to the old 3D Euler equations:

$\frac{\partial}{\partial
t}{\mathbf{u}}+{\mathbf{u}}\cdot\nabla{\mathbf{u}}=-\nabla
p-\hat{{\mathbf{e}}}_{z}\,\lambda\,u_{z}\frac{\partial u_{z}}{\partial
z}\,,\qquad\nabla\cdot{\mathbf{u}}=0\,,$ (30)

where $\hat{{\mathbf{e}}}_{z}$ is the unit vector in direction $z$. The new
term can be considered as a continuous deformation of the convective
derivative term, a familiar procedure in more tractable 1D fluid models [36,
37, 38, 39]. An exact solution of this new model is given again by ansatz
(18)–(19), where the scalar function $\omega$ again satisfies (20) while the
scalar function $\gamma$ satisfies the modified equation

$\displaystyle\frac{\partial}{\partial
t}\gamma+{\mathbf{u}}_{\mathrm{h}}\cdot\nabla_{\mathrm{h}}\gamma$
$\displaystyle=$ $\displaystyle
2\langle\gamma^{2}\rangle-\gamma^{2}+\lambda\left(\langle\gamma^{2}\rangle-\gamma^{2}\right)$
(31)

where, as usual, $\langle f(\cdot,t)\rangle$ denotes the average of $f(x,y,t)$
over $(x,y)\in\mathbb{T}^{2}$, and
$\langle\gamma\rangle=\langle\omega\rangle=0$.

To see that this model with the modified equation (31) gives a modulated
force, we need to compute the $z$-component of the force $f_{\mathrm{z}}$
acting on the particles, directly from equation (30). We obtain, for our
solution, $f_{\mathrm{z}}(x,y,z,t)=-\frac{\partial p}{\partial
z}-\lambda\,z\,\gamma^{2}=z\,\left[(2+\lambda)\langle\gamma^{2}\rangle-\lambda\,\gamma^{2}\right]$,
which depends on $(x,y)$ via $\gamma$ as long as $\lambda\neq 0\,,$ and is
very small near the plane $z=0$. Therefore the model equations are very close
to the original 3D Euler equations near the plane $z=0$ but they include the
required modulation in $(x,y)$. It is worth mentioning that the spatial
average of $f_{\mathrm{z}}$ over $(x,y)$ on the plane $z=$ constant is
independent of $\lambda:$ $\langle
f_{\mathrm{z}}(\cdot,z,t)\rangle=2\,z\langle\gamma^{2}(\cdot,t)\rangle$, a
robust feature of the model.

###  4.1 Symmetries and constants of motion for the new model

The new model comes at a price: the full vorticity
$\mbox{\boldmath{$\omega$}}=\nabla\times{\mathbf{u}}$ is not an infinitesimal
symmetry anymore because the extra forcing term is not a pure gradient. But
all other symmetries found in the previous subsection persist in the new
model, perhaps with a dependence on the parameter $\lambda$, and the method
for finding these symmetries is quite similar. The relevant symmetries are:

$\displaystyle\mbox{\boldmath{$\eta$}}_{1}(x,y,t)=(0,0,\omega(x,y,t)),\qquad\mbox{\boldmath{$\eta$}}_{2}(x,y,z,t)=\left(z^{2\,\lambda+1}\,\frac{\partial\gamma}{\partial
y},-z^{2\,\lambda+1}\,\frac{\partial\gamma}{\partial x},0\right)\,,$ (32)
$\displaystyle\mbox{\boldmath{$\eta$}}_{5}(x,y,t)=\left(0,0,\left(A(t)\,\gamma(x,y,t)-\frac{\dot{A}(t)}{1+\lambda}\right)^{-\frac{1}{1+\lambda}}\right)\,,\qquad\lambda\neq-1\,,$
(33)
$\displaystyle\ddot{A}(t)-(\lambda+1)(\lambda+2)\langle\gamma^{2}\rangle\,A(t)=0\,.$
(34)

From here on we assume $\lambda\neq-1$. The case $\lambda=-1$ has an
elementary solution and will be omitted.

As $\mbox{\boldmath{$\eta$}}_{1}$ and $\mbox{\boldmath{$\eta$}}_{5}$ are
collinear, we obtain the constant of motion

$C_{2}(x,y,t):=\omega(x,y,t)\left(A(t)\,\gamma(x,y,t)-\frac{\dot{A}(t)}{1+\lambda}\right)^{\frac{1}{1+\lambda}}\,,\qquad\lambda\neq-1\,.$

Also, the Lie derivative of $\mbox{\boldmath{$\eta$}}_{2}$ along
$\mbox{\boldmath{$\eta$}}_{5}$ is another symmetry (from theorem 2.11) and
satisfies
$\mathop{\mathcal{L}}_{\scriptsize{\mbox{\boldmath{$\eta$}}_{5}}}\mbox{\boldmath{$\eta$}}_{2}=(2\,\lambda+1)\,z^{-1}\left(A(t)\,\gamma(x,y,t)-\frac{\dot{A}(t)}{1+\lambda}\right)^{-\frac{1}{1+\lambda}}\,\mbox{\boldmath{$\eta$}}_{2}$.
As this is proportional to $\mbox{\boldmath{$\eta$}}_{2}$, from theorem
2.12(ii) we get the constant of motion (also valid for $\lambda=-1/2$)

$C_{3}(x,y,z,t):=z\,\left(A(t)\,\gamma(x,y,t)-\frac{\dot{A}(t)}{1+\lambda}\right)^{\frac{1}{1+\lambda}}\,,\qquad\lambda\neq-1\,.$

Defining now the re-scaled symmetry
${\mbox{\boldmath{$\eta$}}}_{6}(x,y,z,t):=\left[C_{3}(x,y,z,t)\right]^{-1-2\,\lambda}\,\mbox{\boldmath{$\eta$}}_{2}(x,y,z,t)$,
we obtain
$\mathop{\mathcal{L}}_{\scriptsize{\mbox{\boldmath{$\eta$}}_{5}}}\mbox{\boldmath{$\eta$}}_{6}={\mathbf{0}}$.
As $\mbox{\boldmath{$\eta$}}_{5}$ and $\mbox{\boldmath{$\eta$}}_{6}$ are not
collinear, theorem 2.12(i) gives
$\mbox{\boldmath{$\eta$}}_{6}\times\mbox{\boldmath{$\eta$}}_{5}=\nabla
C_{4}\,,$ where

$C_{4}(x,y,t)=\frac{1}{A(t)\left(A(t)\,\gamma(x,y,t)-\frac{\dot{A}(t)}{1+\lambda}\right)}+f(t)\,,$

and $f(t)$ is to be determined. To determine it we simply compute the
convective time derivative of $C_{4}$ using (2), (18) and (34), giving
$\dot{f}(t)+(\lambda+1)/(A(t))^{2}=0$, which is integrated to yield the final
form of the constant of motion $C_{4}$:

$C_{4}(x,y,t)=\frac{1}{A(t)\left(A(t)\,\gamma(x,y,t)-\frac{\dot{A}(t)}{1+\lambda}\right)}-(\lambda+1)\int_{0}^{t}\frac{1}{[A(s)]^{2}}~{}\mathrm{d}s\,,\qquad\lambda\neq-1\,.$
(35)

###  4.2 Solving for the fields along the pathlines

The constant of motion (35) allows one to solve for $\gamma$ along pathlines
$(X(t),Y(t),Z(t))$, solutions of the system (1). Setting initial conditions at
$t=0$ and noting that $A$ satisfies the second-order ODE (34), we set for
simplicity $A(0)=1,\quad\dot{A}(0)=0\,.$ Defining
$\gamma_{0}(x,y)=\gamma(x,y,0)$ we get:
$C_{4}(X(t),Y(t),t)=C_{4}(X_{0},Y_{0},0)=1/\gamma_{0}(X_{0},Y_{0})\,,$ and
solving for $\gamma(X(t),Y(t),t)$ we get

$\gamma(X(t),Y(t),t)=\frac{\gamma_{0}(X_{0},Y_{0})}{A(t)^{2}\left(1+(\lambda+1)\,\gamma_{0}(X_{0},Y_{0})\int_{0}^{t}\frac{1}{A(s)^{2}}\mathrm{d}s\right)}+\frac{\dot{A}(t)}{(\lambda+1)\,A(t)}\,,\quad\lambda\neq-1\,.$
(36)

This solution either blows up in a finite time or remains regular forever,
depending on $\lambda$ and $\gamma_{0}$.

Evaluating $C_{2}(x,y,t)$ at the pathline at $t=0$ we get
$C_{2}(x,y,t)=\omega_{0}(X_{0},Y_{0})\left[\gamma_{0}(X_{0},Y_{0})\right]^{\frac{1}{1+\lambda}}$
and using equation (35) to eliminate $\gamma$ we obtain

$\omega(X(t),Y(t),t)=\omega_{0}(X_{0},Y_{0})\left[A(t)\,\left(1+(\lambda+1)\,\gamma_{0}(X_{0},Y_{0})\int_{0}^{t}\frac{1}{A(s)^{2}}\mathrm{d}s\right)\right]^{\frac{1}{\lambda+1}}\,.$
(37)

A crucial quantity is the Jacobian of the back-to-labels transformation in the
plane $z=0$ parameterised by $(x,y)\in\mathbb{T}^{2}.$ We define

$J(t;X_{0},Y_{0})=\det\left(\frac{\partial(X(t),Y(t))}{\partial(X_{0},Y_{0})}\right)\,,$

where $X(t),Y(t)$ are the $x$\- and $y$-components of the characteristic with
initial positions $X_{0},Y_{0}.$ A computation of the area of the 2-torus
$\mathbb{T}^{2}$ gives an identity to be satisfied by the Jacobian:
$\int_{\mathbb{T}^{2}}\mathrm{d}x\mathrm{d}y=4\pi^{2}=\int_{\mathbb{T}^{2}}J(t;X_{0},Y_{0})\mathrm{d}X_{0}\mathrm{d}Y_{0}\,,$
or in terms of averages:

$\langle J(t;\cdot)\rangle_{0}=1\,,$ (38)

where $\langle f\rangle_{0}$ denotes an average over the ‘label’ variables
$X_{0},Y_{0}$. Noticing that $J$ is positive as long as there is no blowup, we
can interpret $J(t;X_{0},Y_{0})$ as a probability density on the 2-torus.

From the fact that $\nabla_{\mathrm{h}}\cdot{\mathbf{u}}_{\mathrm{h}}=-\gamma$
we readily obtain an evolution equation for $J$:

$\frac{\dot{J}(t;X_{0},Y_{0})}{J(t;X_{0},Y_{0})}=-\gamma(X(t),Y(t),t)\quad\Longrightarrow\quad
J(t;X_{0},Y_{0})=\exp\left({-\int_{0}^{t}\gamma(X(s),Y(s),s)\mathrm{d}s}\right)\,,$

and using equation (36) we get
$J(t;X_{0},Y_{0})=\left[A(t)\,\left(1+(\lambda+1)\,\gamma_{0}(X_{0},Y_{0})\int_{0}^{t}\frac{1}{A(s)^{2}}ds\right)\right]^{-\frac{1}{\lambda+1}}$.
Finally, replacing this solution into condition (38) and defining the
increasing function

$S(t)=\int_{0}^{t}\frac{1}{A(s)^{2}}\mathrm{d}s\qquad\Longrightarrow\qquad
A(t)=[\dot{S}(t)]^{-1/2}\,,$ (39)

we get a remarkable first-order ODE for $S(t)$:

$\dot{S}(t)=\left\langle\bigg{[}1+(\lambda+1)\,\gamma_{0}(\cdot)\,S(t)\bigg{]}^{-\frac{1}{\lambda+1}}\right\rangle_{0}^{-2(\lambda+1)}\,,\quad
S(0)=0\,.$ (40)

Once the solution of this ODE is found, then $A(t)$ can be obtained and so the
basic fields $\gamma,\omega$ can be found explicitly along the pathlines. This
ODE contains all the needed information to determine whether a blowup occurs
in a finite time. The initial data for $\omega$ plays no role.

###  4.3 Singularity time and blowup asymptotics

Combining Eqs. (36)–(37) with Eq. (39) we obtain $\gamma$ and $\omega$ along
the pathlines in terms of $S(t)$:

$\displaystyle\gamma(X(t),Y(t),t)$ $\displaystyle=$
$\displaystyle\frac{\dot{S}(t)\,\gamma_{0}(X_{0},Y_{0})}{1+(\lambda+1)\,\gamma_{0}(X_{0},Y_{0})\,S(t)}-\frac{1}{2(\lambda+1)}\frac{\ddot{S}(t)}{\dot{S}(t)}\,,$
(41) $\displaystyle\omega(X(t),Y(t),t)$ $\displaystyle=$
$\displaystyle\omega_{0}(X_{0},Y_{0})\left[\frac{1+(\lambda+1)\,\gamma_{0}(X_{0},Y_{0})\,S(t)}{\dot{S}(t)^{1/2}}\right]^{\frac{1}{\lambda+1}}\,.$
(42)

A singularity occurs at the first time $T_{*}$ when
$1+(\lambda+1)\,\gamma_{0}(X_{0},Y_{0})\,S(T_{*})=0$ for some
$(X_{0},Y_{0})\in\mathbb{T}^{2}$, because $\dot{S}\geq 0$. The behaviour of
$\dot{S}$ and $\ddot{S}$ at $T_{*}$ may affect the blowup asymptotics. From
(41) we get $d\gamma/d\gamma_{0}>0$, and thus the level sets of $\gamma$ keep
the same ordering at all times. Notice that system (40)–(42) combined with
(19) to solve for the in-plane velocity, allows for general (not necessarily
continuous) initial distributions $\gamma_{0},\omega_{0}$: bounded and of zero
mean on $\mathbb{T}^{2}$.

######  Theorem 4.1 (Singularity time).

Let $\gamma_{0}$ be nonzero, bounded on $\mathbb{T}^{2}$ and of zero mean.
Then there is a first time $T_{\mathrm{*}}\in(0,\infty)\cup\\{\infty\\}$ at
which a singularity occurs, defined by the condition

$S(T_{*})=S_{*}\,,\qquad\text{where}\qquad
0<\frac{1}{S_{*}}:=-(\lambda+1)\begin{cases}\displaystyle{\sup_{(X_{0},Y_{0})\in\mathbb{T}^{2}}{\gamma_{0}(X_{0},Y_{0})}},&\lambda<-1\,,\\\
\displaystyle\inf_{(X_{0},Y_{0})\in\mathbb{T}^{2}}{\gamma_{0}(X_{0},Y_{0})},&-1<\lambda\,.\end{cases}$
(43)

Moreover, an explicit formula for this singularity time (or blowup time) is

$T_{*}=\int_{0}^{S_{*}}\left\langle\bigg{[}1+(\lambda+1)\,\gamma_{0}(\cdot)\,S\bigg{]}^{-\frac{1}{\lambda+1}}\right\rangle_{0}^{2(\lambda+1)}\,\mathrm{d}S\,,$
(44)

###### Proof.

Because $\gamma_{0}$ is nonzero, bounded and of zero mean, it has a positive
global maximum and a negative global minimum on $\mathbb{T}^{2}.$ Noting that
$S(0)=0$, it follows from (40)–(43) that $\dot{S}(t)>0$ when $S(t)<S_{*}$ and
$\lambda\neq-1\,.$ Thus, eventually $S(t)=S_{*}$ for some $t=T_{*}>0$, which
we call the singularity time. Integrating the ODE (40) using separation of
variables we obtain (44). ∎

######  Remark 4.2.

We have in general $S_{*}<\infty$ when $\lambda\neq-1,$ but we do not know
whether $T_{*}$ is finite or infinite: this will depend on the type of initial
condition as well as on the value of $\lambda.$ The singularity time
$T_{\mathrm{*}}$ is finite if and only if the integral (44) converges. Thus,
in order to determine the singular behaviour of the system, we need to look at
the behaviour of the solution of the ODE (40) near $S=S_{*}$.

The following results will help us identify and classify some finite-time
singularity cases:

######  Lemma 4.3.

If $S(t)$ satisfies $\displaystyle\lim_{t\to T_{*}}\dot{S}(t)>0$ then
$T_{*}<\infty$, i.e. the singularity occurs at a finite time.

###### Proof.

The function $\dot{S}(t)$ is nothing but the multiplicative inverse of the
integrand in (44). By definition, $\dot{S}(t)>0$ if $0\leq t<T_{*}$. Thus if
$\displaystyle\lim_{t\to T_{*}}\dot{S}(t)>0$ then the integral converges. ∎

######  Lemma 4.4 (Formula for $\ddot{S}(t)$).

The second time derivative of the function $S(t)$, solution of (40), is

$\ddot{S}(t)=2(\lambda+1)\left[\dot{S}(t)\right]^{2+\frac{1}{2(\lambda+1)}}\left\langle\gamma_{0}(\cdot)\bigg{[}1+(\lambda+1)\,\gamma_{0}(\cdot)\,S(t)\bigg{]}^{-\frac{\lambda+2}{\lambda+1}}\right\rangle_{0}$
(45)

###### Proof.

The result follows directly after time differentiation of equation (40). ∎

######  Definition 4.5 (Positions of supremum and infimum of $\gamma_{0}$).

Let the initial condition $\gamma_{0}$ be nonzero, bounded on $\mathbb{T}^{2}$
and of zero mean. We define this function’s supremum $\gamma_{+}$ and infimum
$\gamma_{-}$, and their respective positions $(X_{+},Y_{+})$ and
$(X_{-},Y_{-})$, as the (not necessarily unique) points on $\mathbb{T}^{2}$
such that

$\gamma_{+}:=\gamma_{0}(X_{+},Y_{+})=\sup_{(X_{0},Y_{0})\in\mathbb{T}^{2}}{\gamma_{0}(X_{0},Y_{0})}\,,\qquad\qquad\gamma_{-}:=\gamma_{0}(X_{-},Y_{-})=\inf_{(X_{0},Y_{0})\in\mathbb{T}^{2}}{\gamma_{0}(X_{0},Y_{0})}\,.$

As the nonnegative terms $[1+(\lambda+1)\,\gamma_{0}(\cdot)\,S(t)]$, appearing
in the averages over $\mathbb{T}^{2}$ in equations (40), (44) and (45), are
close to zero near the singularity time and near the position of
supremum/infimum of $\gamma_{0}$, the key idea to understand singularity time
and blowup asymptotics is to look at whether the relevant exponents are
negative or not. In what follows, we call “unconditional results” those
results that do not depend on the specific profile of $\gamma_{0}$ near the
position of its supremum or infimum. We call “conditional results” everything
else.

###  4.4 Unconditional blowup results: singularity time and asymptotics

######  Theorem 4.6 (Unconditional finite-time blowup for $\lambda<-1$).

Let $\lambda<-1$. Let the initial condition $\gamma_{0}$ be nonzero, bounded
on $\mathbb{T}^{2}$ and of zero mean. Then the singularity occurs at a finite
time, i.e., $T_{*}<\infty.$

###### Proof.

If $\lambda<-1$ and $\gamma_{0}$ is bounded then the integrand of the area
integral in equation (40), at $S=S_{*}$, is bounded and non-negative. As
$\gamma_{0}$ is of zero mean and bounded, this integrand is strictly positive
in a subset of $\mathbb{T}^{2}$ of nonzero measure. Therefore the area
integral is positive and bounded and thus $0<\displaystyle\lim_{t\to
T_{*}}\dot{S}(t)$. From lemma 4.3 this implies $T_{*}<\infty$. ∎

######  Theorem 4.7 (Unconditional blowup asymptotics for $\lambda<-1$).

Let $\lambda<-1$. Let the initial condition $\gamma_{0}$ be nonzero, bounded
on $\mathbb{T}^{2}$ and of zero mean. Let $(X_{+},Y_{+})\in\mathbb{T}^{2}$ be
any position of the supremum of $\gamma_{0}$. Then the stretching rate
$\gamma$ blows up at time $T_{*}$ at the point
$(X(T_{*}),Y(T_{*}))\in\mathbb{T}^{2},$ corresponding to the pathline with
initial position $(X(0),Y(0))=(X_{+},Y_{+})$. Explicitly we have:

$\displaystyle\sup_{(x,y)\in\mathbb{T}^{2}}\gamma(x,y,t)\sim\gamma(X(t),Y(t))\sim\frac{1}{|\lambda+1|}\left(T_{*}-t\right)^{-1}\,,\qquad\text{as}\qquad
t\to T_{*}\,.$

As for the out-of-plane vorticity $\omega,$ it blows up at time $T_{*}$ if and
only if the initial vorticity satisfies $\omega_{0}(X_{+},Y_{+})\neq 0.$ The
blowup occurs at the same point at which $\gamma$ blows up and we have,
explicitly:

$\displaystyle\sup_{(x,y)\in\mathbb{T}^{2}}|\omega(x,y,t)|\sim|\omega(X(t),Y(t))|\sim
K_{+}\,\left(T_{*}-t\right)^{-\frac{1}{|\lambda+1|}}\,,\qquad\text{as}\qquad
t\to T_{*}\,,$

where
$K_{+}=\frac{{|\omega_{0}(X_{+},Y_{+})|}}{|\lambda+1|^{\frac{1}{|\lambda+1|}}}\left\langle\left[{\gamma_{0}(X_{+},Y_{+})}-{\gamma_{0}(\cdot)}\right]^{\frac{1}{|\lambda+1|}}\right\rangle_{0}^{-1}.$
At any other pathline not starting at the position of the supremum of
$\gamma_{0}$, the stretching may blow up when $\lambda<-2$, but it would do so
asymptotically more slowly than its supremum, while the vorticity at any other
pathline is finite at all times.

###### Proof.

We first note that, from the proof of theorem 4.6 we have $T_{*}<\infty$ and
$0<\dot{S}(T_{*})<\infty$. Thus, looking at the formulae for $\gamma$ and
$\omega$ along the pathlines, (41)–(42), we see that the blowup depends on the
factor $1+(\lambda+1)\,\gamma_{0}(X_{+},Y_{+})\,S(t)=1-S(t)/S_{*}$, which has
a simple zero as a function of $t$ at $t=T_{*}$, because $S(t)\sim
S_{*}-\dot{S}(T_{*})(T_{*}-t)$. Thus, this factor will produce a blowup only
if the initial position of the pathline is at a supremum of $\gamma_{0}$.
Regarding the blowup of $\gamma$, the only question remaining is whether the
term $\ddot{S}$ is finite, or at least asymptotically smaller than the first
term in (41). From equation (45) we see that if $-2\leq\lambda<-1$ then the
exponent $-\frac{\lambda+2}{\lambda+1}$ is nonnegative and thus
$\ddot{S}(T_{*})$ is finite. For $\lambda<-2$, the exponent
$-\frac{\lambda+2}{\lambda+1}$ is negative but we can bound:

$\bigg{[}1+(\lambda+1)\,\gamma_{0}(\cdot)\,S(t)\bigg{]}^{-\frac{\lambda+2}{\lambda+1}}\leq\bigg{[}1+(\lambda+1)\,\gamma_{+}\,S(t)\bigg{]}^{-\frac{\lambda+2}{\lambda+1}}=S_{*}^{\frac{\lambda+2}{\lambda+1}}(S_{*}-S(t))^{-\frac{\lambda+2}{\lambda+1}},$

leading to $|\ddot{S}(t)|\leq c(S_{*}-S(t))^{-\frac{\lambda+2}{\lambda+1}}$, a
bound that blows up. But the first term in (41) blows up as
$(S_{*}-S(t))^{-1}$, which is asymptotically faster than $|\ddot{S}(t)|$
because $-\frac{\lambda+2}{\lambda+1}>-1$. Thus, the blowup of $\gamma$ in
(41) is dictated by the first term, with the desired result. As for the blowup
of $\omega$ in (42) at the pathline starting at $(X_{+},Y_{+})$, notice that
$\dot{S}(T_{*})={\gamma_{0}(X_{+},Y_{+})^{-2}}\left\langle\left[{\gamma_{0}(X_{+},Y_{+})}-{\gamma_{0}(\cdot)}\right]^{-\frac{1}{\lambda+1}}\right\rangle_{0}^{-2(\lambda+1)}$.
The desired result follows after direct algebraic manipulations. At any other
pathline, $\omega$ remains finite. ∎

###  4.5 Conditional blowup results for singularity time and asymptotics, in
terms of the type of initial conditions

For $-1<\lambda$ the singularity time and blowup asymptotics depend on the
initial condition for the stretching rate, specifically on the local profile
of $\gamma_{0}$ near the positions $(X_{+},Y_{+})$ and $(X_{-},Y_{-})$ of
definition 4.5. For simplicity we consider only one type of initial condition.

######  Definition 4.8 (Generic type of function).

Let the function $f:\mathbb{T}^{2}\to\mathbb{R}$ be nonzero, bounded and of
zero mean. This function is said to be of generic type if it attains its
supremum at a single point $(X_{+},Y_{+})$ and its infimum at a single point
$(X_{-},Y_{-})$, such that the matrix of second derivatives $D^{2}f$ is
negative definite at $(X_{+},Y_{+})$ and positive definite at $(X_{-},Y_{-})$.
In words, $f$ is of generic type if it has a unique global maximum and a
unique global minimum, and its level contours near these points are ellipses.

Theorem 4.6 already establishes a finite-time blowup if $\lambda<-1.$ For
other values of $\lambda$ we have:

######  Theorem 4.9 (Conditions on singularity time for generic initial
conditions and $-1<\lambda$).

Let the initial condition $\gamma_{0}$ be nonzero, bounded on
$\mathbb{T}^{2}$, of zero mean and of generic type. These statements follow:

  1. 1.

If $-1<\lambda\leq-1/2$ then the flow is regular at all times: $T_{*}=\infty$,
and $\lim_{t\to T_{*}}\dot{S}(t)=0.$

  2. 2.

If $-1/2<\lambda\leq 0$ then a singularity occurs at a finite time:
$T_{*}<\infty$, and $\dot{S}(T_{*})=0.$

  3. 3.

If $0<\lambda$ then $0<\lim_{t\to T_{*}}\dot{S}(t)$ and thus a singularity
occurs at a finite time: $T_{*}<\infty.$

###### Proof.

By looking at the RHS of (40), the exponent inside the area integral is
negative, so the contribution to the singularity when $S\sim S_{*}$, if any,
must come from the region in $\mathbb{T}^{2}$ that is near the global minimum
$(X_{-},Y_{-})$ of definition 4.5. After a rotation and rescaling we get:

$\dot{S}(t)\approx\left[\frac{1}{2\pi\sqrt{\det
D^{2}\gamma_{0}(X_{-},Y_{-})}}\int_{0}^{\epsilon}(1-S(t)/S_{*}+(\lambda+1)S(t)r^{2}/2)^{-1/(\lambda+1)}r\mathrm{d}r\right]^{-2(\lambda+1)}\,,$
(46)

where $\epsilon>0$ is small but fixed and $D^{2}\gamma_{0}(X_{-},Y_{-})$ is
the matrix of second derivatives of $\gamma_{0}$ evaluated at the global
minimum. We first consider the case $0<\lambda$. There, the integral over $r$
converges as $S\to S_{*}$ so we deduce $\dot{S}(T_{*})>0$ which by lemma 4.3
implies $T_{*}<\infty$, thus proving statement _(iii)_. Second, for
$-1<\lambda\leq 0$ the integral is elementary, and keeping the leading term
(at $r=0$) we get these asymptotic expressions as $S\to S_{*}$:

$\dot{S}(t)\sim\begin{cases}\left[4\pi^{2}S_{*}^{2}\lambda^{2}\,\det
D^{2}\gamma_{0}(X_{-},Y_{-})\right]^{\lambda+1}\left(1-S(t)/S_{*}\right)^{-2\,\lambda}\,,&\quad-1<\lambda<0\,,\\\
4\pi^{2}S_{*}^{2}\left[\det
D^{2}\gamma_{0}(X_{-},Y_{-})\right]\left(\log[1-S(t)/S_{*}]\right)^{-2}\,,&\quad\lambda=0\,.\end{cases}$
(47)

In all these cases $\dot{S}=0$ at $S=S_{*}$. Recalling that
$T_{*}=\int_{0}^{S_{*}}\dot{S}^{-1}\mathrm{d}S$, the above expressions can be
used to determine if $T_{*}$ converges or not. Clearly, $T_{*}=\infty$ for
$-1<\lambda\leq-1/2$, so statement _(i)_ is proven. In contrast,
$T_{*}<\infty$ for $-1/2<\lambda<0$ and also for $\lambda=0$, thus statement
_(ii)_ is proven. ∎

######  Remark 4.10.

The statements of this theorem depend sensitively on the profile of
$\gamma_{0}$ near the position of its infimum. For example, if one considers
an initial condition such that $D^{2}\gamma_{0}(X_{-},Y_{-})$ has one positive
eigenvalue and one zero eigenvalue, then by an elementary calculation the
statements of theorem 4.9 get modified by simply changing the respective
ranges of values of $\lambda$: in (i) the new range is $-1<\lambda\leq 0$; in
(ii) the new range is $0<\lambda\leq 1$; in (iii) the new range is
$1<\lambda$. Thus, the case of exact solution of 3D Euler equations,
$\lambda=0$, suddenly becomes regular ($T_{*}=\infty$) for this type of
initial condition.

Theorem 4.7 shows unconditional blowup asymptotics for $\lambda<-1$. Theorem
4.9 shows there is no blow up with generic initial conditions for
$-1<\lambda\leq 1/2$. As for other values of $\lambda$ we have:

######  Theorem 4.11 (Conditions on blowup asymptotics for generic initial
conditions and $-1/2<\lambda$).

Let $-1/2<\lambda$. Let the initial condition $\gamma_{0}$ be nonzero, bounded
on $\mathbb{T}^{2}$, of zero mean and of generic type. Let $(X_{-},Y_{-})$ and
$(X_{+},Y_{+})\in\mathbb{T}^{2}$ be the respective positions of the infimum
$\gamma_{-}$ and supremum $\gamma_{+}$ of $\gamma_{0}$. Let $\Delta:=\det
D^{2}\gamma_{0}(X_{-},Y_{-})>0$ be the determinant of the matrix of second
derivatives of $\gamma_{0}$ at the position of its infimum. Then the
stretching rate $\gamma$ has the following asymptotic behaviour as $t\to
T_{*}$:

(i) At the pathline with initial position $(X(0),Y(0))=(X_{-},Y_{-})$, the
field $\gamma$ blows up as

$\displaystyle\displaystyle\inf_{(x,y)\in\mathbb{T}^{2}}\gamma(x,y,t)$
$\displaystyle\sim$
$\displaystyle\begin{cases}-\frac{1}{2\lambda+1}\left(T_{*}-t\right)^{-1},&-\frac{1}{2}<\lambda\leq
0\,,\\\ -\frac{1}{\lambda+1}\left(T_{*}-t\right)^{-1},&\quad
0<\lambda\,,\end{cases}$ (48)

(ii) At the pathline with initial position $(X(0),Y(0))=(X_{+},Y_{+})$, the
field $\gamma$ blows up more mildly as:

$\displaystyle\displaystyle\sup_{(x,y)\in\mathbb{T}^{2}}\gamma(x,y,t)$
$\displaystyle\sim$
$\displaystyle\begin{cases}\frac{|\lambda|}{(\lambda+1)(2\,\lambda+1)}\left(T_{*}-t\right)^{-1}\,,&-\frac{1}{2}<\lambda<0\,,\\\
\frac{1}{2\,\left|W_{-1}\left(-\pi\sqrt{S_{*}\Delta}\,(T_{*}-t)^{1/2}\right)\right|}\left(T_{*}-t\right)^{-1}\,,&\lambda=0\,,\\\
{\scriptstyle\frac{(\lambda+1)^{\frac{\lambda}{\lambda+1}}}{2\pi\sqrt{\Delta}\left\langle\left({\gamma_{0}(\cdot)}+|\gamma_{-}|\right)^{-\frac{1}{\lambda+1}}\right\rangle_{0}^{2\lambda+1}}}\left(T_{*}-t\right)^{-\frac{1}{\lambda+1}},&0<\lambda\,,\end{cases}$
(49)

where $W_{-1}$ is the negative branch of the Lambert $W$ function.

(iii) At any pathline starting at neither $(X_{-},Y_{-})$ nor $(X_{+},Y_{+})$,
the stretching blows up as in (49).

(iv) The out-of-plane vorticity $\omega$ vanishes at the pathline with initial
position ${\mathbf{X}}(0)=(X_{-},Y_{-})$:

$\displaystyle{\omega({\mathbf{X}}(t),t)}$ $\displaystyle\sim$
$\displaystyle{\omega_{0}(X_{-},Y_{-})}\times\begin{cases}{\scriptstyle\left(2\pi|\lambda|(2\,\lambda+1)\sqrt{\Delta}\right)^{\frac{1}{2\,\lambda+1}}}\left(T_{*}-t\right)^{\frac{1}{2\,\lambda+1}}\,,&-\frac{1}{2}<\lambda<0\,,\\\
\frac{\pi\sqrt{\Delta}}{\left|W_{-1}\left(-\pi\sqrt{S_{*}\Delta}\,(T_{*}-t)^{1/2}\right)\right|}{\left(T_{*}-t\right)}\,,&\lambda=0\,,\\\
\frac{{(\lambda+1)}^{\frac{1}{\lambda+1}}}{\left\langle\left({\gamma_{0}(\cdot)}+|\gamma_{-}|\right)^{-\frac{1}{\lambda+1}}\right\rangle_{0}}\,\left(T_{*}-t\right)^{\frac{1}{\lambda+1}}\,,&0<\lambda\,,\end{cases}$
(50)

while, at any other pathline starting at ${\mathbf{X}}_{0}\neq(X_{-},Y_{-})$,
it remains bounded if $0<\lambda$ and blows up if $-1/2<\lambda\leq 0$:

$\displaystyle\scriptstyle{\omega({\mathbf{X}}(t),t)}\sim\scriptstyle{\omega_{0}({\mathbf{X}}_{0})}\times\begin{cases}\frac{\big{(}(\lambda+1)(2\,\lambda+1)\left(|\gamma_{-}|+\gamma_{0}({\mathbf{X}}_{0})\right)\big{)}^{\frac{1}{\lambda+1}}}{\left(2\pi|\lambda|(2\,\lambda+1)\sqrt{\Delta}\right)^{\frac{1}{2\,\lambda+1}}}\left(T_{*}-t\right)^{-\frac{|\lambda|}{(\lambda+1)(2\,\lambda+1)}}\,,&\scriptstyle-\frac{1}{2}<\lambda<0\,,\\\
\frac{\left(|\gamma_{-}|+\gamma_{0}({\mathbf{X}}_{0})\right)}{\pi\sqrt{\Delta}}{\left|W_{-1}\left(-\pi\sqrt{S_{*}\Delta}\,(T_{*}-t)^{1/2}\right)\right|}\,,&\scriptstyle\lambda=0\,.\end{cases}$
(51)

The proof is straightforward but long and is provided in the appendix.

######  Remark 4.12.

Regarding this theorem, the blowup of the infimum of the stretching rate
$\gamma$ is always accompanied by a collapse of the out-of-plane vorticity
$\omega$ to zero, thus providing an example of theorem 2.10 on the
conservation of zeroes of the symmetry $\mbox{\boldmath{$\eta$}}_{1}$ defined
in (32). In contrast, the blowup of the supremum of the stretching rate
$\gamma$ is not always accompanied by a blowup of $\omega$, as the case
$0<\lambda$ shows.

###  4.6 Remarks on the spatial structure of the blowup

The blowup of the infimum of the stretching rate $\gamma$ is stronger than the
blowup of its supremum. In fact, for the exact 3D Euler solution case
$\lambda=0$, the supremum blowup is marginal: the denominator is going to
infinity logarithmically, making this blowup milder than the infimum blowup.
In the case $0<\lambda$ the exponent in the supremum blowup is always smaller
than in the infimum blowup. In the case $-1/2<\lambda<0$ the exponent is the
same for both infimum and supremum, but the prefactor is always larger for the
infimum blowup.

Nevertheless, when considering the spatial structure of the blowup, the value
of the stretching rate does not matter as much as the spatial profile of the
blowup region. Because
$\gamma=-\nabla_{\mathrm{h}}\cdot{\mathbf{u}}_{\mathrm{h}}$, considering the
2D flow on the $z=0$ plane an interesting pattern emerges: a blowup of the
infimum of $\gamma$ corresponds to a positive-divergence flow, where the
particles are repelled from the singular point. This makes such singularities
“thick”, and therefore amenable to numerical simulations, and when considering
the third dimension, the flow is towards the plane, and as the out-of-plane
vorticity tends to zero, the helicity is negligible near these singularities.
See [31, 40, 30] for numerical and analytical studies of the case $\lambda=0$,
dominated by a blowup of the infimum of $\gamma$. In contrast, a blowup of the
supremum of $\gamma$ corresponds to a negative-divergence flow, where the
particles are attracted towards the singular point. These singularities are
therefore “thin” and difficult to resolve numerically unless an adaptive
spatial scheme is used. See [2] for a numerical and analytical study of the
case $\lambda=-3/2$, dominated by a blowup of the supremum of $\gamma$. When
considering the third dimension, the flow is a thin jet escaping the plane,
and since both vorticity and velocity are large, the local helicity is
singular there.

##  5 Conclusion and discussion

In the first part of this paper the continuous symmetry approach is presented
in a modern and practical way, and is applied with versatility to the task of
finding the solution of 3D Euler fluid equations (and related models) along
pathlines via the construction of constants of motion that depend on the
velocity field, either locally or non-locally. In the second part of this
paper, the solutions found are pursued further and conditions on finite-time
blowup of the main fields of interest (out-of-plane vorticity and its
stretching rate) are found and fully discussed. Our model serves as a platform
to understand the complicated structure that might be encountered in nearly
singular solutions of 3D Euler equations.

In future work we will extend the applications of the continuous symmetry
approach to 3D Euler fluid models beyond the symmetry plane and to systems
involving more interacting fields, such as magnetohydrodynamics. Moreover, we
will combine the continuous symmetry approach with the current Cauchy
invariants lore to generate a hierarchy of new Cauchy invariants, as theorem
2.11 is in fact a particular case of the result that the Lie derivative of any
Lie-conserved tensor along a symmetry is a new Lie-conserved tensor, which is
exact if the original one is.

This paper is dedicated to the memory of Professor Charles R. Doering.

##  Appendix A Proof of theorem 4.11

###### Proof of theorem 4.11.

The main ideas for this proof are: first, the terms in equations (41)–(42)
that can show singular behaviour are: the factor
$1+(\lambda+1)\gamma_{0}(X_{0},Y_{0})S(t)$, which goes to zero at $t=T_{*}$
iff the pathline starts at $(X_{-},Y_{-})$; the factor $\dot{S}(t)$, which may
go to zero at $t=T_{*}$; and the factor $\ddot{S}(t)$, which may go to
infinity at $t=T_{*}$. Second, notice that equations (41)–(42) verify
$\gamma=\frac{d\ln|\omega|}{dt}$ along pathlines, so one can prove _(iv)_
first by inserting the asymptotic solutions into (42) and then prove
_(i)–(iii)_ by direct logarithmic differentiation (this is a formal proof, but
is equivalent to a full proof in our case and far more efficient). Third, the
supremum and infimum of $\gamma$ correspond to the pathlines starting at the
supremum and infimum of $\gamma_{0}$ because the level sets of $\gamma$
maintain their order (see the short explanation after equation (42)). We now
turn to prove _(iv)_. We evaluate from (42) the vorticity at the pathline
starting at $(X_{-},Y_{-})$:
$\omega(t)=\omega(0)\left(\frac{1-S/S_{*}}{{\dot{S}}^{1/2}}\right)^{\frac{1}{\lambda+1}}$,
valid for $-1<\lambda$. When $-1/2<\lambda\leq 0$, both numerator and
denominator in this formula go to zero as $t\to T_{*}$. We can integrate
equations (47) and find:

$S_{*}-S(t)\sim\begin{cases}S_{*}^{2}\,\left[4\pi^{2}\lambda^{2}\Delta\right]^{\frac{\lambda+1}{2\,\lambda+1}}\,\left(2\,\lambda+1\right)^{\frac{1}{2\,\lambda+1}}\,(T_{*}-t)^{\frac{1}{2\,\lambda+1}}\,,&\quad-1/2<\lambda<0\,,\\\
S_{*}\exp\left[2\,W_{-1}\left(-\pi\sqrt{S_{*}\Delta}\,(T_{*}-t)^{1/2}\right)\right]\,,&\quad\lambda=0\,,\end{cases}$
(52)

where $W_{-1}$ is the negative branch of the Lambert $W$ function, solution of
$W_{-1}(z)\mathrm{e}^{W_{-1}(z)}=z$ defined for small $z<0$ and such that
$\lim_{z\to 0^{-}}W_{-1}(z)=-\infty$. Using these and equations (47) it is
direct to calculate the required ratio when $-1/2<\lambda<0$, thus proving the
first line in (50). When $\lambda=0$ we have, defining
$z=-\pi\sqrt{S_{*}\Delta}\,(T_{*}-t)^{1/2}$,

$\frac{1-S/S_{*}}{{\dot{S}}^{1/2}}\sim\frac{\mathrm{e}^{2\,W_{-1}\left(z\right)}}{2\pi
S_{*}\sqrt{\Delta}\left|\log(1-S(t)/S_{*})\right|^{-1}}\sim\frac{|W_{-1}\left(z\right)|\,\mathrm{e}^{2\,W_{-1}\left(z\right)}}{\pi
S_{*}\sqrt{\Delta}}=\frac{z^{2}}{\pi
S_{*}\sqrt{\Delta}\,|W_{-1}\left(z\right)|},$

which proves the second line in (50). When $0<\lambda$ the denominator
$\dot{S}^{1/2}$ is finite at $T_{*}$, as shown in the proof of theorem 4.6.
The numerator can be approximated to lowest order and we get

$\frac{1-S/S_{*}}{{\dot{S}}^{1/2}}\sim\frac{\dot{S}(T_{*})}{S_{*}\dot{S}(T_{*})^{1/2}}(T_{*}-t)=\frac{\dot{S}(T_{*})^{1/2}}{S_{*}}(T_{*}-t),$

and from (40) we get
$\dot{S}(T_{*})={\gamma_{-}^{-2}}\left\langle\left({\gamma_{0}(\cdot)}+{|\gamma_{-}|}\right)^{-\frac{1}{\lambda+1}}\right\rangle_{0}^{-2(\lambda+1)}$.
Recalling $S_{*}^{-1}=(\lambda+1)|\gamma_{-}|$ and simplifying leads to the
third line in (50). Now, we consider the vorticity at a pathline starting at
${\mathbf{X}}_{0}\neq(X_{-},Y_{-})$: from (42) we get
$\omega(t)=\omega(0)\left(\frac{1+(\lambda+1)\gamma_{0}({\mathbf{X}}_{0})S}{{\dot{S}}^{1/2}}\right)^{\frac{1}{\lambda+1}}$,
and now the numerator does not go to zero as $t\to T_{*}$. The ratio will
blowup if $\dot{S}$ tends to zero as $t\to T_{*}$. Thus, when $0<\lambda$ the
vorticity remains finite as $\dot{S}(T_{*})$ is finite. But when
$-1/2<\lambda\leq 0$ we have blowup:

$\frac{1+(\lambda+1)\gamma_{0}({\mathbf{X}}_{0})S}{{\dot{S}}^{1/2}}\sim\frac{1+(\lambda+1)\gamma_{0}({\mathbf{X}}_{0})S_{*}}{{\dot{S}}^{1/2}}=\frac{|\gamma_{-}|+\gamma_{0}({\mathbf{X}}_{0})}{|\gamma_{-}|{\dot{S}}^{1/2}},$

which after using Eqs. (47) and (52) gives directly the first line in (51)
when $-1/2<\lambda<0$. When $\lambda=0$ we get, in terms of the $z$ variable
defined earlier,

$\frac{|\gamma_{-}|+\gamma_{0}({\mathbf{X}}_{0})}{|\gamma_{-}|{\dot{S}}^{1/2}}\sim\frac{|\gamma_{-}|+\gamma_{0}({\mathbf{X}}_{0})}{2\pi
S_{*}\sqrt{\Delta}|\gamma_{-}|\left|\log(1-S(t)/S_{*})\right|^{-1}}\sim\frac{(|\gamma_{-}|+\gamma_{0}({\mathbf{X}}_{0}))|W_{-1}\left(z\right)|}{\pi\sqrt{\Delta}},$

thus proving the second line in (51). This concludes the proof of _(iv)_.

We now turn to the proof of _(i)_ , which concerns the pathline starting at
$(X_{-},Y_{-})$. From logarithmic differentiation of (50), we get

$\gamma({\mathbf{X}}(t),t)=\frac{d}{dt}[\ln\omega({\mathbf{X}}(t),t)]\sim\begin{cases}\frac{d}{dt}\ln\left(T_{*}-t\right)^{\frac{1}{2\,\lambda+1}}\,,&-\frac{1}{2}<\lambda<0\,,\\\
\scriptstyle-\frac{d}{dt}\ln{\left|W_{-1}\left(-\pi\sqrt{S_{*}\Delta}\,(T_{*}-t)^{1/2}\right)\right|}+\frac{d}{dt}\ln{\left(T_{*}-t\right)}\,,&\lambda=0\,,\\\
\frac{d}{dt}\ln\left(T_{*}-t\right)^{\frac{1}{\lambda+1}}\,,&0<\lambda\,,\end{cases}$

which leads to (48) as the last term in the $\lambda=0$ case above dominates
over the first term.

We now turn to the proof of _(ii)–(iii)_ , corresponding to pathlines starting
at ${\mathbf{X}}_{0}\neq(X_{-},Y_{-})$. As explained earlier, the asymptotic
behaviour of $\gamma$ along these pathlines is similar because, from equation
(41) the first term is bounded while the second term, being proportional to
$\ddot{S}/\dot{S}$ and thus independent of the initial point (thus ensuring
_(iii)_ follows from _(ii)_), blows up because $\ddot{S}$ blows up (and
moreover, when $-1/2<\lambda\leq 0$, because $\dot{S}$ tends to zero). As the
level sets of $\gamma$ keep their ordering, the supremum of $\gamma$ is always
attained at the pathline starting at $(X_{+},Y_{+})$. When $-1/2<\lambda\leq
0$, logarithmic differentiation of (51) gives

$\gamma({\mathbf{X}}(t),t)=\frac{d}{dt}[\ln\omega({\mathbf{X}}(t),t)]\sim\begin{cases}\frac{d}{dt}\ln\left(T_{*}-t\right)^{-\frac{|\lambda|}{(\lambda+1)(2\,\lambda+1)}}\,,&-\frac{1}{2}<\lambda<0\,,\\\
\scriptstyle\frac{d}{dt}\ln{\left|W_{-1}\left(-\pi\sqrt{S_{*}\Delta}\,(T_{*}-t)^{1/2}\right)\right|}\,,&\lambda=0\,,\end{cases}$

proving directly the first line in (49), namely when $-1/2<\lambda<0$. The
$\lambda=0$ case gives, using the identity
$\mathrm{d}\ln|W_{-1}(z)|=(1+W_{-1}(z))^{-1}\mathrm{d}\ln|z|$,

$\lambda=0:\qquad\gamma({\mathbf{X}}(t),t)\sim\frac{1}{1+W_{-1}(z)}\frac{d}{dt}\ln|z|\sim\frac{-\frac{1}{2}(T_{*}-t)^{-1}}{W_{-1}(z)},$

giving the second line in (49). Finally, the case $0<\lambda$ requires the
calculation of $\ddot{S}$ asymptotically, which blows up, while
$\dot{S}(T_{*})$ is finite, so
$\gamma({\mathbf{X}}(t),t)\sim-\frac{1}{2(\lambda+1)}\frac{\ddot{S}(t)}{\dot{S}(T_{*})}$.
Now, from (45) for $0<\lambda$ evaluated near the singularity we get,
following a similar calculation as the one leading to (46),

$\ddot{S}\sim\frac{(\lambda+1)\gamma_{-}}{\pi\sqrt{\Delta}}(\dot{S}(T_{*}))^{2+\frac{1}{2(\lambda+1)}}\int_{0}^{\epsilon}(1-S(t)/S_{*}+(\lambda+1)S_{*}r^{2}/2)^{-\frac{\lambda+2}{\lambda+1}}r\mathrm{d}r\,,$

where $\epsilon$ is small but fixed. The integral is elementary and the term
at $r=0$ is divergent as $t\to T_{*}$ so it dominates, and we get
$\ddot{S}\sim\frac{(\lambda+1)\gamma_{-}}{\pi
S_{*}\sqrt{\Delta}}(\dot{S}(T_{*}))^{2+\frac{1}{2(\lambda+1)}}\left(1-S(t)/S_{*}\right)^{-\frac{1}{\lambda+1}}.$
Writing to lowest order
$1-S(t)/S_{*}\sim\frac{\dot{S}(T_{*})}{S_{*}}(T_{*}-t)$ and recalling that
$\dot{S}(T_{*})={\gamma_{-}^{-2}}\left\langle\left({\gamma_{0}(\cdot)}+{|\gamma_{-}|}\right)^{-\frac{1}{\lambda+1}}\right\rangle_{0}^{-2(\lambda+1)}$
and $S_{*}^{-1}=(\lambda+1)|\gamma_{-}|$ we get the third line in (49), thus
ending the proof. ∎

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