On the role of continuous symmetries in the solution of the three-dimensional Euler fluid equations and related models.

We review and apply the continuous symmetry approach to find the solution of the three-dimensional Euler fluid equations in several instances of interest, via the construction of constants of motion and infinitesimal symmetries, without recourse to Noether's theorem. We show that the vorticity field is a symmetry of the flow, so if the flow admits another symmetry then a Lie algebra of new symmetries can be constructed. 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 extra symmetries can be constructed. Next, we study the stagnation-point-type exact solution of the three-dimensional Euler fluid equations introduced by Gibbon et al. (Gibbon et al. 1999 Physica D 132, 497-510. (doi:10.1016/S0167-2789(99)00067-6)) along with a one-parameter generalization of it introduced by Mulungye et al. (Mulungye et al. 2015 J. Fluid Mech. 771, 468-502. (doi:10.1017/jfm.2015.194)). 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. This article is part of the theme issue 'Mathematical problems in physical fluid dynamics (part 2)'.

Introduction

Consider a smooth (i.e. of class ) three-dimensional vector field where and denote local Cartesian coordinates of a point in the spatial domain . 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 , or , or , where is the torus. In what follows, we will assume for simplicity that is divergence-free: The given vector field defines a three-dimensional flow in by the following nonlinear, non-autonomous system of ordinary differential equations for the variables : Solutions to this system subject to initial conditions are known as pathlines, or characteristics, of . As is of class for , it is then Lipschitz, so pathlines with different initial conditions do not cross, but this may be violated if at some time the velocity field becomes non-regular.

In physical terms one could interpret the field 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 , for solving system (1.1), in order to establish non-trivial results when is a fluid velocity field satisfying the three-dimensional Euler fluid equations or related models.

Section 2 is devoted to a review of the infinitesimal symmetry approach for general three-dimensional 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 three-dimensional 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 one-parameter family of models introduced in [2] that generalize the case of Gibbon et al. [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 §5.

The continuous symmetry approach

In this section, we will present a simple exposition of the main tools of interest in the solution of the ordinary differential equation (ODE) system (1.1). These tools are ultimately based on the formalism of tensor bundles, where tensorial objects of various ranks are transported by the vector field . 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.

Basic tools: constants of motion and symmetries

The most basic tool for solving the ODE system (1.1) is the constant of motion.

Definition 2.1 (Constant of motion).

A smooth scalar function of position and time is a constant of motion for system (1.1) if it satisfies the linear partial differential equation where . In words, a constant of motion takes a fixed numerical value along a given pathline of the vector field .

Remark 2.2.

Note that if one finds a constant of motion with a non-trivial dependence on the space and time variables, then in principle the number of independent variables of system (1.1) can be reduced by one unit via the formula , where is a pathline. By iterating this process, when enough functionally independent solutions to this equation exist, one can in principle solve the system (1.1) explicitly. The difficulty now lies of course in solving the linear partial differential equation (2.1).

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 is called a finite contemporaneous symmetry for system (1.1) if, under the mapping , system (1.1) maps to the system in the same form, It is crucial in this definition that the form be maintained: the exact same functions are used in (1.1) and (2.2). In words, a finite contemporaneous symmetry maps pathlines of into pathlines of .

The latter word definition is key to the utility of finite symmetries for solving the ODE system (1.1): 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 (1.1) define a group , whose product rule is induced by the composition operation of functions: 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 , 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 (1.1). For simplicity, from here on we will restrict the discussion to the subgroup of volume-preserving finite contemporaneous symmetries, namely mappings satisfying the additional constraint . In terms of infinitesimal contemporaneous symmetries defined below, this extra restriction translates into the divergence-free condition .

Definition 2.4 (Infinitesimal contemporaneous symmetry).

Consider a one-parameter family of volume-preserving finite contemporaneous symmetries depending on the continuous parameter , and such that is the identity transformation. Then, the smooth divergence-free vector field defined by is called an infinitesimal contemporaneous symmetry for system (1.1).

A geometrical interpretation follows directly from the above definition: the functions where , satisfy (2.2) up to and including terms of order whenever satisfy (1.1). In words, an infinitesimal contemporaneous symmetry maps pathlines of into nearby pathlines of .

We enunciate our main theorem regarding infinitesimal symmetries:

Theorem 2.5.

A smooth divergence-free vector field defined as in equations (2.3) and (2.4) is an infinitesimal contemporaneous symmetry if, and only if, it satisfies the following linear partial differential equation: where denotes the vector whose components are all equal to zero.

Proof.

The proof is a direct exercise and will be omitted. ▪

Lie derivatives

Given a vector field , the equations that determine constants of motion (2.1) and infinitesimal contemporaneous symmetries (2.7) are best written in terms of the linear operator , called the Lie derivative along the vector field , 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 [9-11] for applications in fluids) we simply define the Lie derivative in the two cases of interest:

Definition 2.6.

Let be two smooth divergence-free vector fields and let be a smooth scalar function. The Lie derivative of along is defined as the smooth scalar function Similarly, the Lie derivative of along is defined as the smooth divergence-free vector field We say that two vector fields commute if .

Remark 2.7.

In the case of vector fields, it follows from (2.8) that the Lie derivative defines a Lie algebra of divergence-free vector fields via the bilinear binary operation known as the Lie bracket, , satisfying where 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 hypotheses of definitions 2.1 and 2.4, a smooth scalar is a constant of motion and a smooth divergence-free vector field is an infinitesimal contemporaneous symmetry if and only if and

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

The gradient of a constant of motion

Taking the gradient of equation (2.1) gives an equation for the gradient of a constant of motion: where is the square matrix with components . One has the result:

Lemma 2.9.

Suppose the gradient of a scalar function satisfies equation (2.12). Then the scalar function is a constant of motion for system (1.1), up to an additive function of time that can be found.

Retracing the steps leading to (2.12) from (2.1) we find , implying for some scalar function . Thus, the scalar clearly satisfies , which means that is a constant of motion.

Unsurprisingly, equation (2.12) 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 (2.12) where 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 (1.1), in what is known as the inverse problem of the calculus of variations. See [5-7] and references therein for details, including Lie-advected higher-order tensors such as 2-forms (called symplectic forms), antisymmetric tensors (called Poisson structures) and tensors (called recursion operators). Also, note that Cauchy invariants (and generalized versions thereof) correspond to Lie-transported -forms that are exact, such as the gradient, which is an exact 1-form. See [10,14,15] and references therein for details.

Zeroes of an infinitesimal symmetry and regularity of the flow

We have an important new result on the time evolution of the zeroes of infinitesimal symmetries:

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

Let be a smooth vector field and let be a solution of the corresponding system (1.1) for , with initial position . Let be a smooth infinitesimal contemporaneous symmetry of system (1.1). Then, for any we have if and only if .

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 .

Consider the vector function defined by . Using the chain rule, its total time derivative satisfies . By virtue of (1.1) we have and thus . As is an infinitesimal symmetry, from (2.7) we get . The latter is a non-autonomous linear homogeneous first-order differential equation for the vector . If the vector field is smooth then so are the matrix components of this equation, . From uniqueness of solutions to first-order ODEs we get if and only if , which gives the desired result. ▪

This theorem is quite relevant to the scenario of finite-time singularity in fluid equations when we interpret as the velocity field. It shows that zeroes of infinitesimal symmetries can only be created when a solution becomes non-regular (see remark 4.12 for an application). A similar result can be obtained for the gradient of a constant of motion : the zeroes of follow the pathlines, and no new zeroes can be generated at any time .

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 (1.1), definition 2.3. This is the Lie algebra of infinitesimal symmetries for system (1.1).

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

Let , be two smooth divergence-free infinitesimal contemporaneous symmetries for system (1.1). Namely, and satisfy the PDE system (2.11). Then the smooth divergence-free vector field defined by is an infinitesimal contemporaneous symmetry for system (1.1).

The proof is a direct exercise and will be omitted. ▪

The above theorem is very important in practice as it allows for the construction of a succession of infinitesimal symmetries for system (1.1) 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: . For this case, we have an even better result:

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

Let , be two smooth divergence-free infinitesimal contemporaneous symmetries for system (1.1) such that they are not linearly dependent (i.e. such that they are not proportional via a simple numerical factor). Assume these symmetries commute: . Then there are two cases:

If the vector product is not zero everywhere on for all times, then where is a non-trivial constant of motion for system (1.1), defined at least locally in .

If the vector product is zero everywhere on for all times, then , where is a constant of motion for system (1.1) whose gradient is not zero everywhere and is orthogonal to and .

(i) If is not zero everywhere on for all times, we first want to find a scalar function such that . The compatibility condition for finding is on . The function may not be globally defined, but at least it can be found to be smooth on some neighbourhood , where is the ball of radius and centred at . From elementary vector calculus identities we have . But since the fields are divergence-free. Therefore, . So exists locally.

Second, we want to show that satisfies (2.12). The LHS of this equation is Looking at the first two terms of the last member, both the time derivative and the gradient satisfy the Leibniz product rule, so Now, as and are infinitesimal symmetries they satisfy (2.7), so we get By inspection, the vector is an anticommutative algebraic bilinear function of the vectors , such that . Therefore, , where is some scalar function. By taking and we get readily . As is divergence-free, we get and thus . Therefore, satisfies (2.12).

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

(ii) If the vector product is zero everywhere on for all times, then on for all times, whereby hypothesis is a scalar function that is not a simple numerical constant, thus is not zero everywhere. Let us apply the operator to the equation . Using the Leibniz product rule for the Lie derivative we get Recalling now that and are infinitesimal symmetries for system (1.1), namely that they satisfy (2.7), we readily obtain which, at the positions where is non-zero (generically this is almost everywhere), implies and thus is a constant of motion for system (1.1). Now, imply . ▪

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 [16-19]. 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 (1.1) 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.

The three-dimensional 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 . The notations and will be used interchangeably. We will set either , or , or , where is the torus.

The velocity field is thus divergence-free () and satisfies the three-dimensional Euler fluid equations: where the pressure is determined from the condition . We will be interested in the solution of this nonlinear partial differential equation for the field , under known smooth initial data . At the same time we will be interested in the formalism of the previous sections, in terms of the solutions of the ODE system (1.1), namely the pathlines of the velocity field . This velocity field may admit constants of motion and infinitesimal symmetries, namely solutions of equations (2.10) and (2.11).

The vorticity field is clearly divergence-free and taking the curl of equation (3.1) one obtains the well-known Helmholtz vorticity formulation [9,21,22]: From theorem 2.5 it is then direct to prove:

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

Let be a smooth divergence-free velocity field satisfying the three-dimensional Euler fluid equations (3.1). Then the smooth divergence-free vorticity field is an infinitesimal contemporaneous symmetry for the system (1.1). Namely, the vorticity vector field satisfies (2.7) or, equivalently, (2.11).

Remark 3.2.

Historically, equation (3.2) was first obtained in its 2-form formulation in [21]. Similar transport equations for 3-forms were found: helicity in three-dimensional Euler equations [22], potential vorticity in atmospheric models [23-25]. Of course, global helicity was shown to be conserved in three-dimensional Euler equations much earlier [26,27]. Further, the relation between Lie-advected -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.

Symmetries of steady three-dimensional Euler flows

By definition, steady three-dimensional Euler flows are time-independent solutions to the three-dimensional Euler fluid equations (3.1). Of course, the initial data must be carefully selected so that the solution remains static in time. The ODE system (1.1) becomes autonomous. The vorticity is thus time-independent, and from equation (3.2) and the definition of Lie derivative we get , for all . On the other hand, since itself is time-independent and , we deduce that is an infinitesimal symmetry for the ODE system (1.1). In conclusion, steady Euler flows possess two commuting symmetries, given by the static vector fields and . Theorem 2.12 gives:

Corollary 3.3.

Let be a steady three-dimensional Euler flow, with vorticity field . Then there exists a constant of motion satisfying .

This is a direct application of theorem 2.12 on the symmetries and . ▪

Remark 3.4.

The statement in this corollary is well known. In fact, is the Bernoulli function, equal to , where is the pressure field. Moreover, when for some this corollary does not apply, in the sense that 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 non-integrable [9].

As an example, consider a velocity field defined in Cartesian coordinates by , where is a constant, and where either or . The vorticity is and we find , where is a constant of motion (non-trivial when ). Of course, when the flow is Beltrami, and a particular case of an ABC flow [28,29]. The pathlines of the velocity field satisfy the equations and are simply straight lines (i.e. geodesics) on the sub-manifold defined by the equation . For this simple flow, a general type of infinitesimal symmetry is thus given by where are arbitrary scalar functions. Note that this general symmetry has no -component. Is it possible to find a new infinitesimal symmetry with a non-zero -component? The answer depends on the topology of the spatial domain :

If then . Note that the values of for which the pair is incommensurate, cover densely the region . In these cases, the pathlines are infinitely long and cover densely the 2-torus parametrized by . Let us call the sub-manifolds ‘ergodic’ in such cases. On the other hand, embedded in the region are an infinite number of points for which the pair is commensurate, with closed pathlines on the 2-torus . We call the sub-manifolds regular in these cases. Let be an arbitrary regular sub-manifold, where . Then, the open region contains, for arbitrary small enough , ergodic sub-manifolds . 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 , and defined globally there, with a non-zero component along the -direction. To see this, suppose that such a symmetry exists, call it , and assume without loss of generality that its -component is equal to 1 at . Let be such that and such that be ergodic. Recall that is arbitrarily small. Thus, the symmetry vector field connects the point with the point . Using equation (3.3), let us denote the pathline passing through by , with since is regular. Now, let us denote the pathline passing through by , which is not closed because is ergodic. But by definition the symmetry 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 is not continuous on .

If then and the pathlines are just straight lines that extend without bound. All sub-manifolds are regular, and therefore the limitation of the previous case disappears. The following is an infinitesimal symmetry: . The commutator of the vorticity with this symmetry is . As this is zero when , we get two sub-cases:

When , i.e. when the flow is Beltrami, we can write , giving the constant of motion , which provides an equation for the pathlines on .

When , the symmetry is of the form (3.4) so there is no new information. In this case, the following time-dependent symmetry does commute with the vorticity: . Thus, we can write , giving the constant of motion , which provides an explicit time parameterization for the pathlines on .

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

Symmetries of non-steady three-dimensional Euler flows of stagnation-point type

We consider flows defined over the spatial domain and use to parametrize and to parametrize . An exact solution of the three-dimensional Euler equation (3.1) is [1] where, with reference to the plane , the ‘horizontal’ velocity is determined uniquely via the Helmholtz decomposition from the equations where the scalar functions and 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 and where denotes the average of over and is thus a function of time only. Equations (3.6) imply at all times, which is consistent with (3.7) and (3.8).

Due to the linear dependence of the -component of the velocity on the coordinate, solutions of the form (3.5) were termed ‘of stagnation-point type’ by Ohkitani & Gibbon [30]. As explained there, these solutions include well known and important subclasses such as the Burgers vortex [31] and solutions found by Stuart [32] and Childress et al. [33]. Physically, stagnation points are interesting as they play a key role in practical situations where the flow is impinging perpendicularly on a wall, so particle pathlines must change their tangent vectors from normal to parallel to the wall and the velocity field is zero at one point, called ‘stagnation point’. In our context, the wall is replaced with the symmetry plane . On that plane, our velocity field is clearly parallel to the plane as its -component goes to zero there, so in the generic case of non-uniform smooth velocity fields on , stagnation points may exist on the plane.

Because (3.5) is an exact solution of (3.1), the full three-dimensional vorticity is an infinitesimal symmetry for the non-autonomous system (1.1). Let us try to find another symmetry for this system, of the form , where is unknown. Plugging this into (2.11) and using (3.5) we get By inspection and comparing with (3.7) we see that a solution is , giving the symmetry This is the out-of-plane vorticity. Thus, by linearity, the in-plane vorticity is a symmetry: From theorem 2.11, successive Lie derivatives of along produces new symmetries: with , so and commute and are collinear (see equations (3.13) and (3.11)), which by virtue of theorem 2.12 gives the constant of motion .

These results relate the fields and . In order to solve the evolution equations, it would be desirable to find an infinitesimal symmetry depending on the field only. Let us try to find again a symmetry , but this time look for solutions of equation (3.10) of the form where is a constant to be determined and are functions of time to be determined. Replacing we get: and using equation (3.8) we obtain , which we can solve by eliminating the coefficients of 1, and to get the system which gives the solution and , so the new symmetry is As and are collinear, theorem 2.12 gives the constant of motion . We want to find a constant of motion involving the field only. So far the only two symmetries that depend just on are and . Their Lie bracket is proportional to . Thus, theorem 2.12 gives the constant of motion . This constant determines the out-of-plane pathlines. Finally, note that as well as do not depend on . 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 . We get, preliminarily, , which is integrated to give , where is to be determined. Using (2.1) and the second equation in (3.15) we get . Thus This constant of motion allows one to solve for along the pathlines and establish conditions for blowup, in a result that is equivalent to the solution presented in [34] (see also [35]). The constant of motion is then used to solve for along the pathlines. We will not go further in the solution here because this is a particular case of a general model that we study in detail in the next section.

A new family of models of three-dimensional Euler flows of stagnation-point type: continuous symmetries and proof of finite-time blowup

The exact solution of the three-dimensional Euler fluid equation considered in the previous subsection is a limited model for a general solution that has a symmetry plane at : in a more general solution such as the one obtained from a numerical simulation on a domain [36-39], due to periodicity the velocity is not a linear function of anymore, and so the ansatz (3.5)–(3.8) 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 is a function of time only, not of the in-plane coordinates . For this reason, a minimal generalization of this model was introduced by Mulungye et al. [2], so that this pressure curvature would show some modulation on the plane, while maintaining the Lie structure of infinitesimal symmetries for the model. One can motivate this model in two seemingly different but equivalent ways: first, as done in [2], we renounce the search for an exact solution of the three-dimensional Euler equations on the whole spatial domain, dropping the ansatz (3.5), and simply focus on the three-dimensional Euler equations restricted to the symmetry plane , where the required closure for the term is constructed via a ‘sparse’ modification, as in SINDy models [40]: , with a free dimensionless parameter . The horizontal velocity field is the relevant unknown. This leads to equations (3.6) and (3.7) as before, while the scalar function satisfies now where, as usual, denotes the average of over , and . Note that the case corresponds to the exact solution of the three-dimensional Euler equations discussed in the previous subsection. In this first motivation, the main equations of the model are (3.6), (3.7) and (4.1). They can be solved along pathlines on the symmetry plane using continuous symmetry methods. However, in this motivation the role of the coordinate is unclear. Thus we introduce and adopt a second, new motivation, completely equivalent to the first one from the viewpoint of equations (3.6), (3.7) and (4.1), but where we insist on an exact solution, this time of modified three-dimensional Euler equations on the whole spatial domain : where is the unit vector in direction . An exact solution of this new model is given by ansatz (3.5)–(3.7) along with (4.1), and the case corresponds exactly to the original three-dimensional Euler equations. The new term in (4.2) is a continuous deformation of the convective derivative term, a familiar procedure in more tractable 1D fluid models [41-44]. It keeps the variational structure intact (e.g. symmetries and constants of motion), which allows for the study of the role of initial conditions on the finite-time singularity of the solutions, even for the original problem .

Symmetries and constants of motion for the new model

The new model comes at a price: the full vorticity 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 , and the method for finding these symmetries is quite similar. The relevant symmetries are From here on we assume . The case has an elementary solution and will be omitted.

As and are collinear, we obtain the constant of motion Also, the Lie derivative of along is another symmetry (from theorem 2.11) and satisfies . As this is proportional to , from theorem 2.12(ii) we get the constant of motion (also valid for ) Defining now the re-scaled symmetry , we obtain . As and are not collinear, theorem 2.12(i) gives , where and is to be determined. To determine it we simply compute the convective time derivative of using (2.1), (3.5) and (4.5), giving , which is integrated to yield the final form of the constant of motion :

Solving for the fields along the pathlines

The constant of motion (4.6) allows one to solve for along pathlines , solutions of the system (1.1). Setting initial conditions at and noting that satisfies the second-order ODE (4.5), we set for simplicity . Defining we get: and solving for we get This solution either blows up in a finite time or remains regular forever, depending on and .

Evaluating at the pathline at we get and using equation (4.6) to eliminate we obtain

A crucial quantity is the Jacobian of the back-to-labels transformation in the plane parameterized by . We define where are the - and -components of the pathline with initial positions . A computation of the area of the 2-torus gives an identity to be satisfied by the Jacobian: or in terms of averages: where denotes an average over the ‘label’ variables . Noticing that is positive as long as there is no blowup, we can interpret as a probability density on the 2-torus.

From the fact that we readily obtain an evolution equation for : and using equation (4.7) we get . Finally, replacing this solution into condition (4.9) and defining the increasing function we get a remarkable first-order ODE for : Once the solution of this ODE is found, then can be obtained and so the basic fields 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 plays no role.

Singularity time and blowup asymptotics

Combining equations (4.7) and (4.8) with equation (4.10) we obtain and along the pathlines in terms of : and A singularity occurs at the first time when for some , because . The behaviour of and at may affect the blowup asymptotics. From (4.12) we get , and thus the level sets of keep the same ordering at all times. Note that system (4.11)–(4.13) combined with (3.6) to solve for the in-plane velocity, allows for general (not necessarily continuous) initial distributions : bounded and of zero mean on .

Theorem 4.1 (Singularity time).

Let be non-zero, bounded on and of zero mean. Then there is a first time at which a singularity occurs, defined by the condition Moreover, an explicit formula for this singularity time (or blowup time) is

Because is non-zero, bounded and of zero mean, it has a positive global maximum and a negative global minimum on . Noting that , it follows from (4.11) to (4.14) that when and . Thus, eventually for some , which we call the singularity time. Integrating the ODE (4.11) using separation of variables we obtain (4.15).

Remark 4.2.

We have in general when , but we do not know whether is finite or infinite: this will depend on the type of initial condition as well as on the value of . The singularity time is finite if and only if the integral (4.15) 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 (4.11) near .

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

Lemma 4.3.

If satisfies then , i.e. the singularity occurs at a finite time.

The function is nothing but the multiplicative inverse of the integrand in (4.15). By definition, if . Thus if then the integral converges. ▪

Lemma 4.4 (Formula for ).

The second time derivative of the function , solution of (4.11), is

The result follows directly after time differentiation of equation (4.11). ▪

Definition 4.5 (Positions of supremum and infimum of ).

Let the initial condition be non-zero, bounded on and of zero mean. We define this function’s supremum and infimum , and their respective positions and , as the (not necessarily unique) points on such that

As the non-negative terms , appearing in the averages over in equations (4.11), (4.15) and (4.16), are close to zero near the singularity time and near the position of supremum/infimum of , 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 near the position of its supremum or infimum. We call ‘conditional results’ everything else.

Unconditional blowup results: singularity time and asymptotics

Theorem 4.6 (Unconditional finite-time blowup for ).

Let . Let the initial condition be non-zero, bounded on and of zero mean. Then the singularity occurs at a finite time, i.e. .

If and is bounded then the integrand of the area integral in equation (4.11), at , is bounded and non-negative. As is of zero mean and bounded, this integrand is strictly positive in a subset of of non-zero measure. Therefore, the area integral is positive and bounded and thus . From lemma 4.3 this implies . ▪

Theorem 4.7 (Unconditional blowup asymptotics for ).

Let . Let the initial condition be non-zero, bounded on and of zero mean. Let be any position of the supremum of . Then the stretching rate blows up at time at the point , corresponding to the pathline with initial position . Explicitly we have: As for the out-of-plane vorticity , it blows up at time if and only if the initial vorticity satisfies . The blowup occurs at the same point at which blows up and we have, explicitly: where . At any other pathline not starting at the position of the supremum of , the stretching may blow up when , but it would do so asymptotically more slowly than its supremum, while the vorticity at any other pathline is finite at all times.

We first note that, from the proof of theorem 4.6 we have and . Thus, looking at the formulae for and along the pathlines, (4.12) and (4.13), we see that the blowup depends on the factor , which has a simple zero as a function of at , because . Thus, this factor will produce a blowup only if the initial position of the pathline is at a supremum of . Regarding the blowup of , the only question remaining is whether the term is finite, or at least asymptotically smaller than the first term in (4.12). From equation (4.16) we see that if then the exponent is non-negative and thus is finite. For , this exponent is negative but we can bound: leading to , a bound that blows up. But the first term in (4.12) blows up as , which is asymptotically faster than because . Thus, the blowup of in (4.12) is dictated by the first term, with the desired result. As for the blowup of in (4.13) at the pathline starting at , note that . The desired result follows after direct algebraic manipulations. At any other pathline, remains finite. ▪

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

For the singularity time and blowup asymptotics depend on the initial condition for the stretching rate, specifically on the local profile of near the positions and of definition 4.5. For simplicity we consider only one type of initial condition.

Definition 4.8 (Generic type of function).

Let the function be non-zero, bounded and of zero mean. This function is said to be of generic type if it attains its supremum at a single point and its infimum at a single point , such that the matrix of second derivatives is negative definite at and positive definite at . In words, is of generic type if it has a unique global maximum and a unique global minimum, and its level contours near these points are asymptotically ellipses.

Theorem 4.6 already establishes a finite-time blowup if . For other values of we have:

Theorem 4.9 (Conditions on singularity time for generic initial conditions and ).

Let the initial condition be non-zero, bounded on , of zero mean and of generic type. These statements follow:

If then the flow is regular at all times: , and .

If then a singularity occurs at a finite time: , and .

If then and thus a singularity occurs at a finite time: .

By looking at the RHS of (4.11), the exponent inside the area integral is negative, so the contribution to the singularity when , if any, must come from the region in that is near the global minimum of definition 4.5. After a rotation and rescaling we get: where is small but fixed and is the matrix of second derivatives of evaluated at the global minimum. We first consider the case . There, the integral over converges as so we deduce , which by lemma 4.3 implies , thus proving statement (iii). Second, for the integral is elementary, and keeping the leading term (at ) we get these asymptotic expressions as : In all these cases at . Recalling that , the above expressions can be used to determine if converges or not. Clearly, for , so statement (i) is proven. By contrast, for and also for , thus statement (ii) is proven.

Remark 4.10.

The statements of this theorem depend sensitively on the profile of near the position of its infimum. For example, if one considers an initial condition such that 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 : in (i) the new range is ; in (ii) the new range is ; in (iii) the new range is . Thus, the case of exact solution of three-dimensional Euler equations, , suddenly becomes regular () for this type of initial condition.

Theorem 4.7 shows unconditional blowup asymptotics for . Theorem 4.9 shows there is no blowup with generic initial conditions for . As for other values of we have:

Theorem 4.11 (Conditions on blowup asymptotics for generic initial conditions and ).

Let . Let the initial condition be non-zero, bounded on , of zero mean and of generic type. Let and be the respective positions of the infimum and supremum of . Let be the determinant of the matrix of second derivatives of at the position of its infimum. Then the stretching rate has the following asymptotic behaviour as :

At the pathline with initial position , the field blows up as

At the pathline with initial position , the field blows up more mildly as where is the negative branch of the Lambert function.

At any pathline starting at neither nor , the stretching blows up as in (4.20).

The out-of-plane vorticity vanishes at the pathline with initial position : while, at any other pathline starting at , it remains bounded if and blows up if :

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 is always accompanied by a collapse of the out-of-plane vorticity to zero, thus providing an example of theorem 2.10 on the conservation of zeroes of the symmetry defined in (4.3). By contrast, the blowup of the supremum of the stretching rate is not always accompanied by a blowup of , as the case shows.

Remarks on the spatial structure of the blowup

The blowup of the infimum of the stretching rate is stronger than the blowup of its supremum. In fact, for the exact three-dimensional Euler solution case , the supremum blowup is marginal: the denominator is going to infinity logarithmically, making this blowup milder than the infimum blowup. In the case , the exponent in the supremum blowup is always smaller than in the infimum blowup. In the case , 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 , considering the two-dimensional flow on the plane an interesting pattern emerges: a blowup of the infimum of 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 [30,34,35] for numerical and analytical studies of the case , dominated by a blowup of the infimum of . By contrast, a blowup of the supremum of 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 , dominated by a blowup of the supremum of . 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.

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 three-dimensional 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 three-dimensional Euler fluid equations.

In future work, we will extend the applications of the continuous symmetry approach to three-dimensional 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.