Beltrami fields and knotted vortex structures in incompressible fluid flows

This paper gives a survey on recent results about the existence of knotted vortex structures in incompressible fluids. This includes the proof of Lord Kelvin's conjecture on the existence of knotted vortex tubes in steady Euler flows and a new probabilistic approach to address Arnold's speculation that typical Beltrami fields should exhibit vortex lines of arbitrary topological complexity. We review the key tools to establish these results: the global approximation properties of Beltrami fields, the inverse localization principle in spectral theory, and the theory of Gaussian random Beltrami fields, which permits the introduction of probabilistic considerations in the picture. Finally, we include some applications of these techniques to other problems, such as the existence of vortex reconnections for smooth solutions of the 3D Navier–Stokes equation and the evolution of local hot spots under the heat equation.

have found applications in seemingly unrelated topics. The basic arena for these questions is the 3D Euler equation, the notoriously hard to analyze system of Partial Differential Equations (PDEs) that models the dynamics of an inviscid incompressible fluid and which was first written down by Leonhard Euler in 1757. Mathematically, the problem of the existence of stationary Euler flows with vortex structures of complicated topologies is extremely appealing because it involves remarkable connections between partial differential equations, dynamical systems and differential geometry. From a physical point of view, these questions arise naturally in the Lagrangian approach to turbulence and in the study of hydrodynamic instability.
From a historical perspective, interest in the study of knotted structures in fluid flows dates back to Lord Kelvin [71], who developed an atomic theory in which atoms were understood as stable knotted thin vortex tubes in the ether, an ideal fluid modeled by the Euler equation. Kelvin's theory was inspired by the transport of vorticity discovered by Helmholtz, which in particular implies that the vortex tubes are frozen within the fluid flow and hence their topological structure does not change in time. Vortex tubes were called water twists by Maxwell, and were experimentally constructed by Tait by shooting smoke rings with a cannon of his own design. The stability required by Kelvin's atomic theory led him to conjecture in 1875 that thin vortex tubes of arbitrarily complicated topology can arise in stationary solutions to the Euler equation [72] (for a modern introduction to this conjecture, see the surveys [57,65]).
The mathematical elegance of Kelvin's theory, in which each knot type corresponds to a chemical element, captivated the scientific community for two decades. Nevertheless, at the end of the 19th century, with the discovery of the electron and with the Michelson-Morley experiment providing overwhelming evidence against the existence of the luminiferous ether, it was clear that Kelvin's knotted vortex theory of the atom was erroneous. Still, his atomic theory was a great boon for the development of knot theory, and Kelvin's conjecture has long been regarded as an influential open problem in mathematics. Furthermore, the appearance of knots in physical phenomena has attracted considerable attention in areas such as electromagnetism [34,63], celestial mechanics [2,9] and quantum optics [6,15,21].
In modern times, the study of knotted vortex tubes is a central topic within topological hydrodynamics [5], a young area of research that flourished after the foundational works of Arnold [3,4] and Moffatt [55]. Arnold, in his celebrated structure theorem, classified the topological structure of analytic stationary solutions to the Euler equation when the Bernoulli function is not identically constant, and he suggested that a particular class of stationary Euler flows, called Beltrami fields, should exhibit stream lines of arbitrarily complicated topology. Moffatt introduced the concept of helicity to study the knottedness of the fluid, and gave a heuristic argument supporting the existence of stationary states with stream lines of any knot type [57].
Remarkably, the interest of Kelvin's conjecture is not merely academic. In fact, spectacular recent experiments by Kleckner and Irvine at the University of Chicago [51] have physically supported the validity of Kelvin's conjecture through the experimental realization of knotted and linked vortex tubes in actual fluids using cleverly designed hydrofoils. Furthermore, the existence of topologically complicated stream and vortex lines plays an important role in the Lagrangian theory of turbulence and in magnetohydrodynamics (MHD). As a matter of fact, the connection between Lagrangian turbulence and Beltrami fields is so direct that physicists have even coined the term 'Beltramization' to describe the phenomenon that the velocity field and its curl (that is, the vorticity) tend to align in turbulent regions of low dissipation [42,58].

The 3D Euler equation
To conclude this introduction, let us now define the mathematical objects that we will be discussing in this survey. In the first place, let us start by writing down the Euler equation, which describes the motion of an inviscid incompressible fluid. The unknowns are the nonautonomous vector field ( , ), which describes the velocity field of the fluid, and the scalar function ( , ), which is the pressure. In this paper we are interested in regular solutions to the Euler equation, so it will be assumed throughout that the solutions we consider are sufficiently smooth. The Euler equation is usually studied on ℝ 3 , on a bounded domain Ω ⊂ ℝ 3 or on ℝ 3 with periodic boundary conditions, which one can identify with the flat 3-torus 3 . In the case that the fluid fills a domain with boundary, it is customary to assume that is tangent to the boundary Ω: this is the so-called no-penetration boundary condition. The motion of the fluid particles is described by the integral curves of the velocity field, that is, by the solutions to the non-autonomous Ordinary Differential Equation (ODE) for some initial condition ( 0 ) = 0 . These curves are usually called particle paths. The integral curves of the 'frozen time' velocity field, that is, of ( , 0 ) when the time 0 is fixed, are called stream lines, and thus the stream line pattern changes with time if the flow is unsteady.
Another time-dependent vector field that plays a crucial role in fluid mechanics is the vorticity, defined by ∶= curl .
The integral curves of the vorticity ( , 0 ) at fixed time 0 , that is to say, the solutions to the autonomous ODĖ( ) = ( ( ), 0 ) for some initial condition (0) = 0 , are the vortex lines of the fluid at time 0 . Additionally, the closure of a domain in ℝ 3 that is the union of vortex lines at time 0 and whose boundary is a smoothly embedded torus (an invariant torus of the vorticity) is called a vortex tube. In this article, we shall use the term vortex structures to denote both the vortex lines and vortex tubes of the fluid at a fixed time 0 . Obviously, a vortex structure is a time-dependent set in general. However, recall that, as long as there is no blow up of the solution, the vorticity is transported by the fluid flow: This equation ensures that the vorticity at time can be written in terms of the initial vorticity 0 as (⋅, ) = * 0 , (1.1)

Structure of the paper
The survey is organized as follows. In Section 2, we state Arnold's structure theorem for analytic stationary Euler flows and provide an alternative proof thereof. Beltrami fields, whose special properties are hinted at in Arnold's structure theorem, are discussed at length in Section 3. The results about the existence of Beltrami fields with vortex structures of any prescribed topology on Euclidean 3-space and on the flat 3-torus are, respectively, presented in Sections 4 and 5. Throughout, we have strived to emphasize key ideas of the proof whose applicability goes beyond these particular results. In Section 6, we consider the question, also discussed by Arnold, of to what extent complex or chaotic vortex structures are typical for Beltrami fields. One can obtain surprisingly good results in this direction by studying Gaussian random Beltrami fields. In Section 7, we consider the problem of the creation and destruction of vortex structures in the presence of viscosity, that is, in the context of the 3D Navier-Stokes equations. To conclude, in Section 8 we present some future directions of research. For the sake of completeness, we have included the Appendix, where we discuss how some of the key ideas developed to study vortex structures of complicated topologies can be effectively applied to tackle questions that arise naturally in other fields different from fluid mechanics. † In the theory of dynamical systems, a 1 function is a first integral of a vector field if the function is constant along the integral curves of , that is, ( ( ; 0 )) = ( 0 ) for all for which the integral curve ( ; 0 ) of with initial condition 0 is defined.

ARNOLD'S STRUCTURE THEOREM
In 1965, Arnold proved a landmark result on the structure of analytic stationary solutions of the Euler equation in bounded domains that marked the birth of topological hydrodynamics. Roughly speaking, Arnold's structure theorem asserts that, under mild technical assumptions, the behavior of a stationary fluid flow whose vorticity is not collinear with the velocity is analogous to the dynamics of an integrable Hamiltonian system. In this section, we shall state a slightly weaker version of Arnold's original theorem [3,4], and include a new proof that does not make use of the vorticity. Our proof provides a different insight in the sense that it relies on the classification of compact surfaces and Hopf's index theorem, instead of on ℝ 2 -actions as in the original argument. As an aside, note that ℝ -actions (which indeed arise very naturally in this problem) are the central ingredient of the proof of the Liouville-Arnold theorem for integrable Hamiltonian systems, which lays bare the connection between integrable volume-preserving dynamical systems and stationary Euler flows with a nonconstant Bernoulli function.
Before stating Arnold's theorem, we need to introduce some notation. We say that a vector field defined on a 2-dimensional torus  is orbitally conjugate to a linear flow if there exist angular coordinates = ( 1 , 2 ) ∶  → (ℝ∕2 ℤ) 2 parameterizing  and a scalar function > 0 on  such that reads in these coordinates as for some real constants with 2 + 2 ≠ 0. By relabeling the coordinates if necessary, we can assume ≠ 0. One then says that the flow is rational when the quotient ∕ is a rational number, in which case all the integral curves of are periodic. When ∕ is not a rational number, the flow is called irrational, and all the integral curves of are then quasi-periodic. If the function can be taken constant, we say that is conjugate to a linear flow.

Theorem 2.1. Let Ω be an analytic bounded domain of ℝ 3 and a stationary solution of the Euler equation in Ω.
We assume that is tangent to the boundary and analytic up to the boundary Ω. If and its vorticity are not everywhere collinear, then Ω is partitioned by an analytic set of codimension at least 1 into finitely many subdomains in which the dynamics of is of one of the following two types.
• The subdomain is trivially fibered by invariant tori of . On each torus, the flow of is orbitally conjugate to a (rational or irrational) linear flow. • The subdomain is trivially fibered by invariant cylinders of whose boundaries sit on Ω. All the orbits of on each cylinder are periodic.
Proof. The stationary Euler equation (1.2) implies that the Bernoulli function is analytic up to the boundary Ω. Since and are not everywhere collinear, the function is not a constant. Let us take the sets 1 ⊂ ℝ of critical values of and 2 ⊂ ℝ of values whose corresponding -level sets are tangent to Ω at some point. It is then standard (see, for example, [52]) that the analyticity of and Ω and the compactness of Ω imply that the cardinality of 1 ∪ 2 is finite. We can therefore define the analytic set consisting of all the critical level sets and the level sets tangent to the boundary. Since is a nonconstant analytic function, this set has codimension at least 1.
By the Lojasiewicz's structure theorem for analytic sets [52,Theorem 6.3.3], the domain Ω is partitioned by  into finitely many subdomains, so let be one of them. By construction, ∇ does not vanish at any point of , and hence by Equation (1.2), does not vanish either. Since is a first integral of the velocity field by Proposition 1.1, we conclude that is covered by analytic surfaces and the non vanishing field is tangent to each surface. Let Σ be one of these surfaces and denote by Σ ∶ Σ → ℝ 3 its inclusion in ℝ 3 (which is an analytic embedding). Since div = 0, it follows from [1,Theorem 3.4.12] that the induced vector field * Σ (that is, the restriction of to the surface Σ) preserves an area 2-form on Σ (sometimes called the Liouville form). Following the proof of [1,Theorem 3.4.12], this area form is defined in terms of the function as ∶= * Σ˜, where˜is any 2-form in ℝ 3 that satisfies the condition where ∧ denotes the exterior product of differential forms. Although it is not relevant for the rest of the proof (so the reader who is not familiar with the calculus of differential forms can skip it), we would like to mention that can be written using the contraction operator of forms with vector fields as .
This is a straightforward computation using Equation (2.1), the identity ∇ |∇ | 2 = 1 and the fact that * Σ ( ∧ ) = 0 for any 1-form because is constant on Σ. If Σ ⊂ Ω has no boundary, since the vector field * Σ on Σ has no zeros, Hopf's index theorem readily implies that Σ is diffeomorphic to a torus (or, in other words, the only boundaryless compact surface that supports a non vanishing tangent field is the torus). Moreover, since * Σ preserves the aforementioned area form on Σ, a direct application of a theorem due to Sternberg [70,Theorem 1] ensures that it is orbitally conjugate to a linear flow.
Assume now that Σ has a nonempty boundary, which is necessarily contained in Ω because Σ is a connected component of a regular level set of . Additionally, by construction, the intersection of Σ and Ω is transverse. Since is tangent to both Ω and Σ, and has no zeros on Σ, the boundary Σ consists of finitely many periodic orbits of , say ⩾ 1. If we add a cap for each component of the boundary of Σ we obtain an orientable compact surface without boundaryΣ, which is characterized by its genus (that is, the number of holes). Note that the caps added to each component of the boundary of Σ are not embedded in ℝ 3 , soΣ is understood as an abstract surface. The velocity field admits a natural extension toΣ simply defining in each cap as a vector field with a zero of center type whose orbits are all periodic. Since each zero in the cap contributes with index 1, Hopf's index theorem allows us to relate and as = 2 − 2 , thus implying that = 2 and = 0, so that the compact surfaceΣ is a sphere which is obtained after adding two caps to the surface Σ, which is a cylinder. Since Σ is formed by two periodic orbits of , and is a non vanishing field that preserves the area form defined in Equation (2.2), a straightforward application of the Poincaré-Bendixson theorem implies that Σ is covered by periodic orbits of .
The fact that each domain is trivially fibred by the level sets of , which can be tori or cylinders, follows from a standard argument using the flow of the vector field ∶= ∇ |∇ | 2 , which is well-defined in . Indeed, noticing that we immediately conclude that the (local) flow defined by maps the level set −1 ( ) onto the level set −1 ( + ). Therefore, if the level set Σ is a torus, all the level sets of in have no boundary and are tori, and is diffeomorphic to a trivial product 2 × (0, 1). In the case that Σ has boundary, noticing that all the level sets of in intersect Ω transversely, we conclude that they are cylinders and is diffeomorphic to 1 × (0, 1) × (0, 1). The theorem then follows. □ Remark 2.2. Arnold's method of proof (see [5] for details) shows that is actually conjugate to a linear flow on each torus. Basically, this is a consequence of the fact that the velocity field and the vorticity commute, that is, [ , ] = 0. This allows one to define a (local) ℝ 2 -action using the flows of and , and argue exactly as in the proof of Arnold-Liouville's theorem in Hamiltonian mechanics to construct 'action-angle' variables that linearize the field . The action, in fact, turns out to be essentially the Bernoulli function.
The assumption in Arnold's structure theorem that and the boundary are analytic is used to control the critical and tangential level sets of the Bernoulli function. This is not crucial; analogous results can be proved, for example, under the weaker assumption that is a smooth Morse-Bott function. The hypothesis that is defined on a bounded domain is not essential either. Indeed, using that and the vorticity commute as in Arnold's original proof, it is not hard to prove the following: Theorem 2.3. Let be a smooth stationary solution of the Euler equation in ℝ 3 , and assume that and its vorticity grow at most linearly at infinity: Arnold emphasized that the key hypothesis to prove structure theorems as above is that the velocity and the vorticity should not be everywhere collinear. Indeed, if and are aligned at all points of ℝ 3 (or a domain Ω), then Equation (1.2) implies that is a constant, so a trivial first integral of . In this case, the stationary fluid flow does not need to be integrable, and may exhibit the complexity suggested by Arnold in [3].

A CRASH COURSE ON BELTRAMI FIELDS
We say that a vector field on ℝ 3 is a Beltrami field if curl = (3.1) for some nonzero constant . Note that Equation (3.1) implies that div = 0, so obviously a Beltrami field is a stationary solution of the Euler equation with constant Bernoulli function. In particular, since the vorticity of a Beltrami field is everywhere collinear with its velocity, Arnold's structure Theorem 2.1 does not apply. Actually, Beltrami fields are sometimes defined in greater generality, as a vector field on ℝ 3 that satisfies the equations for some proportionality factor g. We refer to these vector fields as generalized Beltrami fields. It is easy to check that the function g is a first integral of , that is, ⋅ ∇g = 0, and hence the orbits of the velocity field lie on the level sets of g. When g is constant, this corresponds to a Beltrami field as defined above, which is sometimes referred to as a strong Beltrami field. For a general g, there are many obstructions for the existence of (even local) solutions to Equation (3.2), see [12,30] for a detailed study of generalized Beltrami fields with nonconstant proportionality factor and their connection with the helical flow paradox [59].
Remark 3.1. Beltrami fields also appear in the context of MHD, where they are known as forcefree fields. In MHD, due to the phenomenon of magnetic relaxation, Beltrami fields in bounded domains arise as minimizers of the 2 norm (the energy) in fixed helicity classes. See, for example, [44] for a recent account of the subject.
In Subsection 3.1, we introduce the Fourier characterization of Beltrami fields, which is key to prove the inverse localization property in Subsection 5.1 and to define Gaussian random Beltrami fields in Subsection 6.1. Subsections 3.2 and 3.3 recall the existence theorems for Cauchy-Kovalevskaya and boundary value problems for the curl operator, which are instrumental tools to prove the realization theorem stated in Section 4. Finally, in Subsection 3.4 we present the global approximation theory for Beltrami fields, which is a sort of flexibility of the equation that is fundamental to study vortex structures in this context.

The Fourier characterization
Taking the curl of (3.1), it is easy to see that must also satisfy the Helmholtz equation: To put it differently, the components of this vector field are monochromatic waves, so, in particular, Beltrami fields are analytic. An immediate consequence of this is that the Fourier transformˆ of a polynomially bounded Beltrami field, which is a tempered distribution, satisfies the equation for any compactly supported function ∈ ∞ 0 (ℝ 3 ), so the support ofˆmust be contained in the sphere of radius | |, In spherical coordinates ∶= | | ∈ ℝ + and ∶= ∕| | ∈ ≡ 1 , it is standard that this is equivalent to saying thatˆis a finite sum of the form Here ( ) is the th derivative of the Dirac measure andˆis a vector-valued distribution on the sphere, soˆ∈ ( , ℂ 3 ) for some ∈ ℝ (because any compactly supported distribution is in a Sobolev space, possibly of negative order). Of course, there exist Beltrami fields that are not polynomially bounded, such as ∶= 1 ( ). Accordingly, a vector field of the form where is a vector-valued distribution on the sphere, is a polynomially bounded Beltrami field if and only if is Hermitian (that is, ( ) = (− ), which makes real-valued) and satisfies the distributional equation on the sphere The corresponding integral, which is convergent if is integrable, must be understood in the sense of distributions for less regular . We recall that, for any real , the ( ) norm of a function can be computed as where are the coefficients of the spherical harmonics expansion of . The Fourier characterization of polynomially bounded Beltrami fields permits one to study their sharp decay at infinity. Indeed, as a consequence of a classical result due to Herglotz [48,Theorem 7.1.28], if is a Beltrami field with the sharp fall off at infinity, that is, such that then it is of the form (3.3) for some Hermitian vector-valued function ∈ 2 ( , ℂ 3 ). Here (and in what follows) denotes the ball of radius centered at the origin. Furthermore, the seminorm defined by (3.5) is equivalent to ‖ ‖ 2 2 ( ) This implies that the sharp bound at infinity for a Beltrami field is In Subsection 6.1, we shall show how to introduce a Fourier representation of Beltrami fields involving just a single Hermitian function on , instead of a vector-valued distribution satisfying Equation (3.4). This refinement turns out to be crucial to construct a probability measure in the space of Beltrami fields. Remark 3.2. As shown by Nadirashvili [60], there are no Beltrami fields in ℝ 3 with finite 2 norm even if the proportionality factor g (see Equation 3.2) is allowed to be nonconstant. Actually, the construction of smooth stationary solutions of the Euler equation in ℝ 3 with finite energy (in fact, compactly supported) has only been achieved recently [14,43]; these solutions are axisymmetric with swirl and satisfy a surprising 'localizability' property, that is, the pressure is a first integral of (see also [16] for an alternative construction of compactly supported piecewise smooth weak solutions that do not satisfy the 'localizability' property).

A Cauchy-Kovalevskaya theorem for the curl operator
The curl operator is not elliptic and, in fact, it has no noncharacteristic surfaces because its symbol is given by the 3 × 3 antisymmetric matrix It is hence remarkable that there is a version of the Cauchy-Kovalevskaya theorem that can be used to construct Beltrami fields given an analytic surface in ℝ 3 and appropriate Cauchy data on the surface. The following result, proved in [26], turns out to be an extremely useful tool in the study of Beltrami fields: where ∶= ♭ is the metric dual 1-form of , and Σ ∶ Σ → ℝ 3 is the embedding of Σ in ℝ 3 . The advantage of this formulation is that the Cauchy datum needs to be defined only on Σ, and not on a neighborhood (as required to compute the curl in Equation 3.8).
The proof of this result makes use of the following auxiliary Cauchy problem where = 0 ⊕ 1 ⊕ 2 ⊕ 3 ∈ ⨁ 3 =0 Ω (ℝ 3 ) is a differential form of mixed degree. Here and denote, respectively, the exterior derivative and codiferential of forms, and ⋆ is the Hodge star operator. The Dirac-type first-order differential operator + is elliptic, so one can directly apply the Cauchy-Kovalevskaya theorem to problem (3.9). We refer the interested reader to [26] for the details of the proof of Theorem 3.3.
A simple model to understand Theorem 3.3 and the assumption (3.8) is the following: using Euclidean coordinates ( 1 , 2 , 3 ) ∈ ℝ 3 , let us take Σ = { 3 = 0} and the analytic Cauchy datum = 1 ( 1 , 2 ) 1 + 2 ( 1 , 2 ) 2 . Using the Beltrami equation curl = and the fact that div = 0, it immediately follows that (3.12) The Cauchy-Kovalevskaya theorem then implies that this system with Cauchy datum 1 ( 1 , 2 , 0) = 1 ( 1 , 2 ), 2 ( 1 , 2 , 0) = 2 ( 1 , 2 ) and 3 ( 1 , 2 , 0) = 0, has a unique analytic solution = ( 1 , 2 , 3 ) in a neighborhood  of Σ. Taking now the derivative of (3.11) with respect to 1 minus the derivative of (3.10) with respect to 2 , and using (3.12), we obtain in  , for some analytic function ∶ ℝ 2 → ℝ. Evaluating at the plane { 3 = 0}, it follows that We then conclude that the solution ( 1 , 2 , 3 ) of the Cauchy problem (3.10)-(3.12) is a Beltrami field if and only if the Cauchy datum satisfies the constraint which is precisely the condition (3.8) in this particular case, that is, Theorem 3.3 is key in the proof of the realization theorem of knotted vortex lines that we shall state in Section 4. Here, as a simple application that illustrates its utility, let us present an easy result on the zero set of Beltrami fields in ℝ 3 :

Proposition 3.5. Let be a (nontrivial) Beltrami field in ℝ 3 . Then its zero set
has dimension at most 1.
Proof. Since is analytic, (which is the zero set of the scalar function | | 2 ) is an analytic set. The properties of analytic sets [52] then imply that has integer Hausdorff dimension which is at most 2, and if it has a 2-dimensional component, it contains an analytically embedded 2disk ⊂ . Using that | = 0, the Cauchy-Kovalevskaya Theorem 3.3 implies that = 0 in a neighborhood of , and hence = 0 on the whole ℝ 3 by analyticity. It then follows that dim ⩽ 1, as we wanted to show. □ Remark 3.6. When one considers Beltrami fields on a 3-dimensional compact boundaryless Riemannian manifold ( , g), it can be proved [27] that for a generic set of metrics, the zero set of any Beltrami field consists of isolated, nondegenerate points. Likewise, given a compact set ⊂ ℝ 3 , it is not hard to prove using the approximation theorem that we shall introduce in Subsection 3.4, that any Beltrami field in ℝ 3 can be approximated in ( ), ⩾ 1, by a Beltrami field whose zeros in are all nondegenerate.

Boundary value problems
Given a smooth bounded domain Ω ⊂ ℝ 3 , a classical result of Giga and Yoshida [74] states that curl defines a self-adjoint operator on Ω with compact resolvent whose domain  Ω is dense in the space Here is the outward-pointing normal to the boundary and  Ω denotes the vector space of harmonic fields on Ω that are tangent to the boundary, The dimension of  Ω is finite and equal to the first Betti number 1 (Ω) of Ω, which is just the genus of the boundary Ω.
The eigenfields of curl are then vector fields on Ω that satisfy curl = in Ω, (3.13) and belong to  Ω . It is well-known that there are infinitely many positive and negative eigenvalues { } ∞ =−∞ of curl, which tend to ±∞ as → ±∞ and which one can label so that We say that this set of eigenvalues is the spectrum of the curl operator in Ω.
Remark 3.7. A natural problem in this context concerns the existence of optimal domains for the first positive (or negative) curl eigenvalue. We say that Ω is optimal for the first positive curl eigenvalue if 1 (Ω) ⩽ 1 (Ω ′ ) for any domain Ω ′ of the same volume. The answer to the analogous problem for the Dirichlet Laplacian is classical: the Faber-Krahn inequality implies that the ball is the only optimal domain for the first eigenvalue. For the curl operator, even taking into account that it plays a preponderant role in fluid mechanics and electromagnetic theory, the problem is still open. We addressed this question in [36], where we proved that there are no axisymmetric optimal domains with smooth boundary that satisfies a mild technical assumption. As a particular case, this rules out the existence of smooth optimal axisymmetric domains with a convex section.
We note that the values of the eigenfields on Ω are not prescribed, but they are only assumed to be tangent to the boundary. For example, in contrast with the well-known spectral problem for the Dirichlet Laplacian on bounded domains, it is easy to prove that cannot be identically zero on Ω. More precisely: Proof. We can assume without loss of generality that the origin of coordinates of ℝ 3 is contained in Ω. Defining the smooth vector field in Ω 3 ) the position vector, it is easy to check that Vainshtein's identity holds: To pass to the second equality, we have used that is a Beltrami field. Therefore, noticing that | Ω = 0, we can integrate div over Ω to obtain This shows that = 0 on Ω, as claimed. □ In spite of the fact that we cannot prescribe the tangential value of a Beltrami field on Ω, in the constructions of Beltrami fields with knotted vortex structures it is key to consider boundary value problems for Beltrami fields where we prescribe their harmonic projection and their normal component on Ω. The following result was proved in [24] and [18]. In its statement,  Ω denotes the projection of the field into the harmonic space  Ω ; more precisely,  Ω is a vector ∈ ℝ 1 (Ω) of components defined as Here is a scalar function on Ω and 0 ∈ ℝ 1 (Ω) is a constant vector. Moreover, the Sobolev norm of is bounded as with a constant that depends on , and Ω.

A global approximation theorem with sharp decay
An important property of Beltrami fields in Euclidean space is that, under a mild topological assumption, any local solution of the Beltrami equation (3.1) can be promoted to a global solution. This is reminiscent of the classical Runge theorem in complex analysis, which allows one to approximate a holomorphic function in a compact set by a complex polynomial. When combined with the construction of 'local' Beltrami fields (that is, defined only on subsets of ℝ 3 ) that feature a wealth of complex vortex structures, the global approximation property will enable us to show that Beltrami fields are remarkably flexible. The first (non-quantitative) global approximation theorem for the curl operator was proved in [29]. A key property of the Beltrami fields we obtained is that they exhibit sharp fall off at infinity, as characterized in Equation (3.5). This is a substantial improvement with respect to the classical theory of global approximation for elliptic PDEs, where no control at infinity of the global solutions is provided. We shall next state a quantitative version of the global approximation theorem for Beltrami fields, which quantifies the 'size' of the global Beltrami field that approximates a local Beltrami field in terms of the 2 norm of the latter and the quality of the approximation. The size of the global field (which cannot be in 2 , as there are no square-integrable solutions to the Beltrami field equation on ℝ 3 ) is measured in terms of the weighted ∞ norm Theorem 3.10. Let be a vector field that solves the Beltrami equation curl = , ≠ 0, in a bounded domain ⊂ ℝ 3 , and let ⊂ be a compact set. Assume that ℝ 3 ∖ is connected. Then for any integer ⩾ 0 and any > 0, there exists a field satisfying curl = in ℝ 3 such that and satisfies the estimate where the constant > 0 only depends on , and . In particular, has the sharp decay rate at infinity.
Remark 3.11. The topological assumption on is necessary, and typical for any Runge-type global approximation theorem.
Theorem 3.10 follows from a new quantitative global approximation theorem for the Helmholtz equation proved in [35,Theorem 2.4] building upon ideas of Rüland and Salo for quantitative approximation for elliptic equations on bounded domains [67]. The non-quantitative version of this theorem can be found in [29] and [33, chapter 3.3.2].
The idea of the proof is the following. As noticed in Subsection 3.1, each component of the Beltrami field satisfies the equation (Δ + 2 ) = 0 in . Assume that there exist solutionsõ f the equation (Δ + 2 )˜= 0 in ℝ 3 that approximate as and are bounded as Then the vector field defined as ∶= 1 2 2 (curl(curl + ))˜, with˜∶= (˜1,˜2,˜3), satisfies curl = in ℝ 3 and the estimate Here we have used that curl(curl + ) = 2 2 . Additionally, its weighted ∞ norm is bounded as in Equation (3.14). Theorem 3.10 then follows upon taking =∶ 0 . We remark that in the proof of Theorem 3.10, the global Beltrami field that we construct is a finite linear combination of explicit functions: for some (large enough) integer , where ∈ ℝ 3 are constant vectors that depend in a complicated way on the local Beltrami field . Here are the spherical Bessel functions and is a real basis of spherical harmonics. In particular, it follows from this expression that (and all its derivatives) is bounded at infinity by | | −1 , and, as we saw in Subsection 3.1, its Fourier transform ( ) (with ∈ ) is in 2 (in fact, it is given by a finite linear combination of spherical harmonics with vector-valued coefficients).

A REALIZATION THEOREM
In Section 2, we saw that Arnold's structure theorem (Theorem 2.1) ensures that the stream and vortex lines of a stationary fluid flow with nonconstant Bernoulli function are nicely stacked in a rigid structure akin to those which appear in the study of integrable Hamiltonian systems. In particular, under suitable technical assumptions, one can prove that they cannot be knotted or linked in arbitrarily complicated ways [41].
In contrast, the behavior of Beltrami fields can be shown to be very flexible. To see this, in Subsection 4.1 we state a realization theorem that combines results from [26] and [29] to show that the vortex lines and vortex tubes of a Beltrami field may exhibit arbitrary topological complexity. Note that since the vorticity of a Beltrami field is proportional to the velocity, stream lines coincide with vortex lines, up to a reparameterization.
The basic philosophy of the proof of these results is to use the methods of differential topology and dynamical systems (hyperbolicity, KAM theory) to control auxiliary constructions and those of PDEs (Cauchy-Kovalevskaya theorem, boundary value problems) to realize these auxiliary constructions in the appropriate analytical framework of stationary solutions to the Euler equation. For the sake of clarity, we divide the proof of the realization theorem into two parts: the case of vortex lines (Subsection 4.2) and the case of vortex tubes (Subsection 4.3).

Statement of the result
To state the result, we need to introduce some notation. By a tube (or toroidal domain) we mean a smooth bounded domain of ℝ 3 whose boundary is diffeomorphic to a torus. Note we are not making any assumptions on how this torus is embedded in ℝ 3 , so it can be knotted in arbitrarily complicated ways. Additionally, we say that a collection of vortex lines or tubes of a vector field is structurally stable if any divergence-free vector field on ℝ 3 which is sufficiently close to in the 5 norm will also have this collection of vortex lines and tubes, up to a diffeomorphism. In the case that  consists only of closed curves, the diffeomorphism Φ can be made -close to the identity (for any a priori fixed > 0). When  contains tubes, it becomes apparent from the proof that the diffeomorphism can be written as Φ = Φ 2 • Φ 1 • Φ 0 , where each diffeomorphism Φ has a controlled behavior: the effect of Φ 0 is to transform all the tubes in  into thin tubes, Φ 1 is -close to the identity, and Φ 2 ( ) ∶= is a rescaling of ratio 0 < ⩽ 1, which simply adjusts the eigenvalue of the Beltrami field.
Sketch of the proof of Theorem 4.1. As a consequence of the global approximation theorem for Beltrami fields (Theorem 3.10), the problem reduces to the construction for some 0 ≠ ′ ⩽ (we assume > 0), of a (local) Beltrami field satisfying the equation curl = ′ in a neighborhood  of a set  ′ isotopic to , such that  ′ is a set of structurally stable vortex structures of .
Indeed, assume that such a Beltrami field exists. Then Theorem 3.10 implies that for any integer and any positive constant there exists a vector field˜that satisfies curl˜= ′˜i n ℝ 3 , with sharp decay, that approximates as where ⊂  is any compact set whose interior contains  ′ . The structural stability implies that there exists a diffeomorphism Φ 1 ∶ ℝ 3 → ℝ 3 , which is close to the identity (and equal to the identity in the complement of  ), such that Φ 1 ( ′ ) is a set of vortex structures of˜.
Finally, defining the vector field it follows that satisfies curl = in ℝ 3 , it obviously has sharp decay, and Φ() is a set of vortex structures of , where Φ ∶ ℝ 3 → ℝ 3 is the (orientation preserving) diffeomorphism defined as The proof of the existence of a local Beltrami field as in the proof of Theorem 4.1 is presented in Subsections 4.2 and 4.3, the former being focused on the construction of in a neighborhood of a closed curve , and the latter concerning the construction in a neighborhood of a tube  .

Knotted vortex lines
We recall that a knot is a closed curve smoothly embedded in ℝ 3 and a link is a union of pairwise disjoint knots. Let us sketch the ideas of the construction of the local Beltrami field in the case that  consists of a knot . Details can be found in [26]. It is well-known that, perturbing the knot a little by means of a small diffeomorphism, we can assume that is analytic. Since the normal bundle of a knot is trivial, we can take an analytic ribbon Σ around . More precisely, there is an analytic embedding ℎ of the cylinder 1 × (− , ) into ℝ 3 whose image is Σ and such that ℎ( 1 × {0}) = .
In a small tubular neighborhood  of the knot we can take an analytic coordinate system adapted to the ribbon Σ. Basically, and are suitable extensions of the angular variable on the knot and of the signed distance to as measured along the ribbon Σ, while is the signed distance to Σ.
The reason for which this coordinate system is useful is that it allows us to define a vector field in the neighborhood  that is key in the proof: simply, is the field dual to the closed 1-form From this expression and the definition of the coordinates it follows that is an analytic vector field tangent to the ribbon Σ. In coordinates ( , ), the vector field on Σ = { = 0} takes the form for some positive function g with g( , 0) = 1, see [26, section 5] for details. In particular, it then follows that is an attracting hyperbolic periodic orbit of the pullback of to Σ.
Let us now consider the Cauchy problem curl = , | Σ = . (4.1) Since is tangent to Σ and ( * Σ ) = 0, the Cauchy-Kovalevskaya theorem for Beltrami fields (Theorem 3.3 and Remark 3.4) allows us to ensure that there is a unique analytic field in a neighborhood of the knot which solves the Cauchy problem (4.1).
It is obvious that the knot is a periodic orbit of the local Beltrami field . As a matter of fact, it is easy to check that it is hyperbolic (that is, its maximal Lyapunov exponent is positive). The idea is that, by construction, the ribbon Σ is an invariant manifold under the flow of that contracts into exponentially. As the flow of preserves volume because div = 0, there must exist an invariant manifold that is exponentially expanding and intersects Σ transversally on , which guarantees its hyperbolicity.
Accordingly, is a structurally stable periodic orbit. More concretely, by the hyperbolic permanence theorem [47] any field that is close enough to in the ( ) norm, ⩾ 1, has a periodic orbit diffeomorphic to , and this diffeomorphism can be chosen -close to the identity (and different from the identity only in  ).
The Beltrami field is then as desired. Note that in this case, ′ = and  ′ = , so the diffeomorphism Φ in the statement of Theorem 4.1 is -close to the identity (and equal to the identity in the complement of a neighborhood of the knot). This is also the case if  consists only of closed curves: can be constructed in a neighborhood of each knot, and the global approximation Theorem 3.10 allows us to 'glue' all the pieces together.
In Theorem 4.1, we have only considered the case of finite links, but the case of locally finite links can be tackled similarly at the expense of losing the decay condition of the velocity field. In particular, taking into account the standard fact that the knot types modulo diffeomorphism are countable, it follows that there exists a stationary solution to the Euler equation whose stream lines realize all knots at the same time, thus yielding a positive answer to a question of Williams [73]. Remark 4.3. If one is interested only in the class of torus knots, it is not hard to construct explicit axisymmetric Beltrami fields whose vortex lines exhibit a large family of knots of this type [7]. We recall that an axisymmetric Beltrami field has the form in cylindrical coordinates ( , , ) ∈ ℝ + × 1 × ℝ, where the stream function ≡ ( , ) satisfies a particular case of the Grad-Shafranov equation, and is a real constant. It is easy to find explicit solutions to this equation.
Remark 4.4. As first observed by Moffatt [55], the existence of knotted vortex lines in a fluid flow is somehow related to a remarkable conserved quantity of the evolution of the Euler equation called helicity. Remarkably, and as conjectured by Arnold and Khesin [5], the helicity is the only 'regular' functional that is invariant under volume-preserving diffeomorphisms [38,49].

Invariant tori
To describe tubes we first observe that any toroidal domain of ℝ 3 can be retracted onto a core knot . Accordingly, any tube  is isotopic to a thin tube  ( ) ≡  defined as the -thickening of a given curve in ℝ 3 , that is, the set of points that are at distance at most from .
The following result establishes the existence of the local Beltrami field introduced in the proof of Theorem 4.1 in the case that  consists of a single tube  . The isotopic set  ′ is the thin tube  and the proportionality factor ′ is of order 3 . Details can be found in [29] and [24].  on the domain  that satisfy the boundary condition ⋅ = 0 and whose harmonic projection is   = 1, in the notation of Subsection 3.3. By Hodge theory [10], as the first Betti number of  is 1 ( ) = 1, there is a unique harmonic field ℎ in  modulo an inessential multiplicative constant. By Theorem 3.9, this problem has a unique solution provided that ′ is outside some discrete set, and in particular whenever | ′ | is smaller than some -independent constant. This is because the first eigenvalue of curl in  is of order −1 . Suitable estimates (which need to have the sharp dependence on the small parameter ) show [29, Theorem 6.8] that the field becomes close to ℎ for small ′ , in the sense that for a constant that depends on but not on (here the norm is computed using suitable rescaled coordinates in  , as introduced below). Since becomes analytic after a small perturbation, we can safely assume that  is analytic as well, and therefore the Beltrami field can be analytically extended to a neighborhood  of  , cf. [25, appendix A].
By construction, the boundary  is an invariant torus of , so it remains to check that it is structurally stable. This relies on KAM arguments, which require fine information on the behavior of in the tube  . A useful simplification is suggested by the estimate (4.2): if we take small nonzero values of ′ , it should be enough to understand the behavior of the harmonic field ℎ, since the local solution is going to look basically like this field.
To analyze the harmonic field ℎ, we introduce coordinates adapted to the tube  , which essentially correspond to an arc-length parameterization of the knot and to -rescaled rectangular coordinates in a transverse section of the tube defined using a Frenet frame. Thus, we consider an angular coordinate , taking values in 1 ∶= ℝ∕ ℤ (with the length of the knot ), rectangular coordinates = ( 1 , 2 ) taking values in the unit 2-disk , and the corresponding polar coordinates 1 = cos , 2 = sin . The Euclidean metric formally reads in these coordinates as an ( 2 ) perturbation of the model flat metric 2 + 2 | | 2 . The concrete expression of the metric, which depends on the curvature and torsion of the curve , is crucially used in the proof of the result, but we will not need it at the level of this discussion.
Since the vector field with ∶= 1 − cos , can be readily shown to be irrotational and tangent to the boundary, by the Hodge decomposition it follows that the harmonic field can be written as where and 0 ∶= − div ℎ 0 = − −3 ( sin + ′ cos ) .
The explicit expressions above (which depend on the geometry of the curve) are important, but we will omit them so as not to obscure the main points of the proof. To obtain these expressions, which depend on and the curvature and torsion of , we need fine estimates for the Sobolev norms of that are optimal with respect to the parameter .
To apply KAM theory to analyze the structural stability of the invariant torus  , we consider the Poincaré map of . To define this map, we take a normal section of the tube  , say { = 0}. Given a point 0 in this section, the Poincaré map Π associates to 0 the point where the vortex line ( ) with initial condition (0) = 0 cuts the section { = 0} for the first positive time. This map is well-defined for small enough and , thus defining a diffeomorphism Π ∶ → .
Since the vector field is divergence-free, one can prove that the Poincaré map preserves some area measure on the disk. Note that the invariant torus  manifests itself as the invariant circle of the Poincaré map. After a hard, messy, lengthy calculation which makes use of the PDE estimates we previously obtained for and ℎ, and assuming that ′ = ( 3 ), we can compute Π perturbatively as an expansion in that depends on the curvature and torsion of . Combining this with the classical Poincaré-Lindstedt method to compute normal forms of diffeomorphisms, we can obtain action-angle variables ( , ) of the form where the approximate frequency function is given by Here 0 ∶= − ∫ 0 ( ) . For the proof of this result, see [24]. These expressions show that the Poincaré map is an area-preserving perturbation of an integrable twist diffeomorphism. Moser's twist condition then reads as If this holds, then the twist is a nonzero constant of order 2 . Since the Poincaré map is an order ( 3 ) perturbation of an integrable twist map, it follows (see, for example, [69]) that for small enough all the disk but a set of measure at most 1 2 is covered by quasi-periodic invariant curves of Π. Moreover, the Poincaré map has an elliptic fixed point (a perturbation of the point { = 0}) which is Lyapunov stable because the first Birkhoff constant, which is equal to ′ (0), is not zero and the eigenvalues ± ∶= exp(± (0)) avoid the resonances 1,2,3 and 4 (that is, ± , 2 ± , 3 ± , 4 ± are different from 1) for almost all . The proposition then follows noticing that the condition holds for a generic curve , that is, there is a diffeomorphism Φ 1 , which is -close to the identity, such that Φ 1 ( ) satisfies Equation (4.5). The tube  is then transformed onto the tube Φ 1 ( ) where all the previous analysis can be applied, thus implying that it is a vortex tube of a Beltrami field ; inside this tube, the quasi-periodic invariant curves and the fixed point of Π give rise to a set of ergodic invariant tori of of almost full measure and to an elliptic (and Lyapunov stable) periodic orbit of . □ Remark 4.6. By carefully picking an appropriate normal datum of order ( 3 ) in Theorem 3.9, this construction can be refined to show that can exhibit any prescribed number of hyperbolic periodic orbits, which are cables of the knot . The existence of hyperbolic periodic orbits is wellknown to be necessary for the vorticity to have positive topological entropy, and therefore for the existence of chaotic regions between the ergodic invariant tori that appear inside the vortex tube  , but it is certainly not sufficient.

THE FINITE ENERGY CASE: BELTRAMI FIELDS ON THE TORUS
A Beltrami field on the flat 3-torus 3 ∶= (ℝ∕2 ℤ) 3 (or, equivalently, on the cube of ℝ 3 of side length 2 with periodic boundary conditions) is a vector field on 3 satisfying the equation

curl =
for some real number ≠ 0. To put it differently, Beltrami fields on the torus are the eigenfields of the curl operator. It is easy to see that such an eigenfield is divergence-free and has zero mean, that is, ∫ 3 = 0. Since Δ + 2 = 0, it is well-known (see, for example, [23]) that the spectrum of the curl operator on the 3-torus consists of the numbers of the form = ±| | for some vector with integer coefficients ∈ ℤ 3 . For concreteness, we will henceforth assume that > 0; the case of negative eigenvalues is completely analogous. Since has integer coefficients, one can label the positive eigenvalues of curl by a positive integer such that = 1∕2 . Let us define  ∶= { ∈ ℤ 3 ∶ | | 2 = } and note that the set  is invariant under reflections (that is, − ∈  if ∈  ). By Legendre's three-square theorem,  is nonempty (and therefore is an eigenvalue of the curl operator) if and only if is not of the form 4 (8 + 7) for nonnegative integers and .
The Beltrami fields corresponding to the eigenvalue must be of the form for some ∈ ℂ 3 . Conversely, this expression defines a Beltrami field with eigenvalue if and only if = − (which ensures that is real valued) and 1∕2 × = . (5. 2) The multiplicity of the eigenvalue is given by the cardinality ∶= # . A famous class of Beltrami fields on the torus are the ABC fields [5], which can be written in terms of three real constants as = ( sin 3 + cos 2 , sin 1 + cos 3 , sin 2 + cos 1 ) .
They correspond to the lowest positive eigenvalue = 1, which has multiplicity 6. The dynamics of this family of Beltrami fields on 3 has been extensively studied in the literature, see, for example, [11].
In what follows, we will restrict our attention to the positive integers , which we will henceforth call admissible, that are not congruent with 0, 4 or 7 modulo 8. When is congruent with 7 modulo 8, Legendre's three-square theorem immediately implies that  is empty. The reason to rule out numbers congruent with 0 or 4 modulo 8 by declaring them non-admissible is more subtle: a deep theorem of Duke [17], which addresses a question raised by Linnik, ensures that the set  ∕ 1∕2 becomes uniformly distributed on the unit sphere as → ∞ through integers that are congruent to 1, 2, 3, 5 or 6 modulo 8. This ensures that as → ∞ through admissible values, for any continuous function on the 2-sphere . A particular case is when goes to infinity through squares of odd values, that is, when = (2 + 1) 2 and → ∞ (in this case is congruent with 1 modulo 8). The global approximation Theorem 3.10 does not hold for Beltrami fields on 3 . An obvious reason is that, locally, can take any value, while Beltrami fields on 3 correspond to eigenvalues of the form as described above. If one goes to the proof of Theorem 3.10, one can check it fails because, as 3 is compact, one cannot 'sweep' certain singularities that appear in the approximation process due to the use of Green's functions. This is not just a technical issue, but a fundamental obstruction in any approximation theorem. Because of this, in the construction of Beltrami fields on the torus with complex behavior we will not employ approximation arguments but what we call the inverse localization property for eigenfunctions of the Laplacian [39,40]. This is ultimately the reason for which we restrict our attention to eigenvalues associated to admissible integers .

Inverse localization of eigenfunctions
A rescaling argument shows that Laplace eigenfunctions with a sufficiently large eigenvalue on any 3-manifold behave, locally on sets of diameter −1∕2 , essentially as solutions to the Helmholtz equation in ℝ 3 with parameter = 1 do in balls of diameter 1. However, if one picks a certain solution to the Helmholtz equation on ℝ 3 , in general one cannot decide whether there is an eigenfunction on the manifold whose behavior on a disk of the right size is essentially given by that Helmholtz solution. One can certainly construct approximate eigenfunctions with this property, given by quasimodes, but in general, it is unclear if one can do this with actual eigenfunctions because the eigenspaces are typically of multiplicity 1. One can hope, however, that if the multiplicity of the large eigenvalues grows fast enough as → ∞, then perhaps one can pick eigenfunctions which do capture the complexity of an arbitrary solution to the Helmholtz equation on suitably small balls. This turns out to be the case for the eigenfunctions of the torus, and to prove this we will need to exploit the equidistribution property of admissible integers proved by Duke (Equation 5.3).
For the statement of the inverse localization theorem [39,40], let us fix an arbitrary point 0 ∈ 3 and take a patch of normal geodesic coordinates Ψ ∶ → centered at 0 . Here (respectively, ) denotes the ball in ℝ 3 (respectively, the geodesic ball in 3 ) centered at the origin (respectively, at 0 ) and of radius > 0; we shall drop the subscript when = 1.

Theorem 5.1. Let be a solution to the Helmholtz equation Δ + = 0 in ℝ 3 . Fix a positive integer and a positive constant . For any large enough admissible integer , there is an eigenfunction of the Laplacian on 3 with eigenvalue such that
Proof. The global approximation theorem for the Helmholtz equation, cf. [33, chapter 3.3.2] (compare also with the results presented in Subsections 3.1 and 3.4 for Beltrami fields), ensures that for any ′ > 0, there exists another solution 1 to the Helmholtz equation on ℝ 3 that can be represented as where is a smooth complex-valued function on satisfying ( ) = (− ), which approximates the solution in 2 : Since the set  ∕ 1∕2 becomes uniformly distributed on the unit sphere , Equation (5.3) implies that for any large enough admissible integer , the functioñ In turn, standard elliptic estimates allow us to promote the uniform estimate Without loss of generality, we will take the origin as the base point 0 , so that we can identify the ball with through the canonical 2 -periodic coordinates on the torus. In particular, the diffeomorphism Ψ ∶ → that appears in the statement of the theorem can be understood to be the identity.
Since ∈  ⊂ ℤ 3 , it follows that the function is 2 -periodic (that is, invariant under the translation → + 2 for any vector ∈ ℤ 3 ). Therefore, it defines a well-defined function on the torus, which we will still denote by . Moreover, is an eigenfunction of the Laplacian on 3 with eigenvalue , The theorem then follows provided that ′ is chosen small enough for ′ < . □ Remark 5.2. An analogous inverse localization theorem holds for eigenfunctions of the Laplacian on the round 3-dimensional sphere, 3 [39,40]. In this case, the eigenvalues are of the form ( + 2), where is a nonnegative integer, and the theorem holds for any large enough (that is, there is no need to restrict ourselves to a subset of admissible integers).
This result implies an inverse localization theorem for the curl operator in 3 . Indeed, if is a Beltrami field in ℝ 3 that satisfies curl = , applying Theorem 5.1 to each component of , we obtain a vector field˜on 3 such that Δ˜+˜= 0 and The operations˜• Ψ −1 and Δ˜have to be understood componentwise. Finally, a straightforward computation shows that the vector field ∶= curl curl˜+ curl2 2 satisfies curl = on 3 , and its localization • Ψ −1 ( ⋅ 1∕2 ) at 0 is close to as thus proving the following: Corollary 5.3. Let be a Beltrami field in ℝ 3 , satisfying curl = , and fix any positive real and integer . Then for any large enough admissible integer there is a Beltrami field on 3 , satisfying curl = , such that Remark 5.4. In view of Remark 5.2, the inverse localization property also holds for Beltrami fields on 3 .

Knotted vortex structures
Using the inverse localization technique, we obtained in [39] a realization theorem of knotted vortex structures for high-frequency Beltrami fields on 3 (and on 3 ) which is analogous to Theorem 4.1. In the statement, we use the notation introduced in Subsection 4.1.

Theorem 5.5. Let  be a finite union of pairwise disjoint, but possibly knotted and linked, closed curves and tubes in 3 . We assume that  is contained in a contractible subset of 3 . Then, for any large enough admissible integer there exists a Beltrami field satisfying the equation curl = and a diffeomorphism Φ ∶ 3 → 3 (connected with the identity) such that Φ() is a union of vortex lines and vortex tubes of . Furthermore, these vortex structures are structurally stable.
Remark 5.6. In [23], we obtained quantitative bounds depending on for the 2 norm of the Beltrami field and for the stability of the vortex structures (that is, bounds for the size of perturbations that do not destroy the vortex structures). As we shall see in Section 7, this refinement of Theorem 5.5 is key in our study of the phenomenon of vortex reconnection in the Navier-Stokes equations.
The effect of the diffeomorphism Φ is to uniformly rescale a contractible subset of 3 that contains  to have a diameter of order −1 . The proof of Theorem 5.5 is essentially as follows. First, we shrink the set  into the ball ⊂ 3 . The Realization Theorem 4.1 implies that there is a Beltrami field in ℝ 3 with a set  ′ of vortex structures diffeomorphic to Ψ() ⊂ ; by Remark 4.2, the set  ′ is also contained in . Then, Corollary 5.3 implies the existence, for any large enough admissible integer , of a Beltrami field in 3 satisfying curl = , whose 'localization' is -close to in . Since the set of vortex structures  ′ is structurally stable, taking ⩾ 5 and small enough, Theorem 5.5 follows.

THE GENERICITY OF KNOTTED STRUCTURES AND CHAOS
The Realization Theorem 4.1 ensures the existence of Beltrami fields in ℝ 3 that exhibit structurally stable vortex structures of arbitrary topological complexity. However, the method of proof provides no information whatsoever about to what extent this complex behavior is typical for Beltrami fields or if they exhibit chaotic invariant regions generically. According to Arnold [3], the typical portrait of a Beltrami field should consist of a positive volume of KAM tori coexisting with transverse homoclinic intersections.
Inspired by Arnold's speculations, we recently introduced [37] a theory of Gaussian random Beltrami fields which permits one to estimate the probability that a Beltrami field exhibits certain complex dynamics. Since each component of a Beltrami field is a monochromatic wave, as seen in Subsection 3.1, our starting point is the celebrated Nazarov-Sodin theory for Gaussian random monochromatic waves [61], which yields asymptotic laws for the number of connected nodal components of the wave. This theory was developed to provide the rigorous foundation to the works of Bogomolny and Schmit and Berry on universality laws in the contexts of quantum chaos and percolation.
However, there are several difficulties to extend Nazarov-Sodin theory to the Beltrami context. From the analytic viewpoint, the first one stems from the fact that there is no scalar characterization in Fourier space for a general Beltrami field. The reason is that the three components of the field are not independent, so the reduction to a Fourier formulation with independent variables is not trivial. Second, we want to obtain asymptotic laws for sophisticated geometric objects such as periodic orbits, invariant tori or chaotic sets, which are much subtler than nodal sets. To define suitable counting functionals we have to use tools from dynamical systems: KAM theory, hyperbolicity and horseshoes. Finally, the very clever (and non-probabilistic) 'sandwich estimate' in Nazarov-Sodin theory, has no reasonable analogue in the Beltrami context. The main reason is that both the Faber-Krahn inequality and the Kac-Rice formula do not work to estimate the geometry of the aforementioned dynamical quantities.

Random Gaussian Beltrami fields
Following the approach of Nazarov and Sodin, to define a Gaussian random Beltrami field, we make use of its Fourier representation, as introduced in Subsection 3.1: where the ℂ 3 -valued Hermitian distribution satisfies the distributional equation (3.4). For the sake of simplicity, here and in what follows we assume that the frequency of the Beltrami field is = 1.
We now observe that the ℂ 3 -valued Hermitian polynomial , satisfies the Beltrami condition (3.4) on . The reason for which we have introduced the nonvanishing normalization factor ( 1 ) is that, as we shall see later on, this will ensure that the covariance matrix of the Gaussian field is normalized.
Interestingly, one can use the complex field ( ) and a single scalar function to approximate an arbitrary Beltrami field, up to a small error [37]. This should be compared with the Approximation Theorem 3.10: where are normally distributed independent standard Gaussian random variables and is an orthonormal basis of (real-valued) spherical harmonics on . Note that is Hermitian because of the identity (− ) = (−1) ( ). We now define a Gaussian random Beltrami field as ∶= .
The following result describes the convergence of this expression for and characterizes the regularity of the function in Sobolev and Besov spaces. In the statement, the action of the differential operator ( ) on a scalar real-valued function in ℝ 3 is defined in the obvious way, which coincides with ( ) = −(curl curl + curl)( ( 1 ) , 0, 0) .
Here 1 ∶= − 1 . Note that, by the definition of , this field is real-valued.

Proposition 6.3.
With probability 1, the function is in −1− ( )∖ −1 ( ) and in −1 2,∞ ∖ −1+ 2,∞ , for any > 0. In particular, almost surely, is a ∞ vector field and can be written as denotes the Bessel function of the first kind and order . This series converges in uniformly on compact sets almost surely, for any .
It is obvious that the vector-valued Gaussian field has zero mean. A key quantity to study random fields is the covariance kernel, , which maps each pair of points ( , ) ∈ ℝ 3 × ℝ 3 to the symmetric 3 × 3 matrix

( , ) ∶= [ ( ) ⊗ ( )] .
We showed in [37] that ( , ) is translationally invariant, that is, it has the form ( , ) = ( − ) for some matrix-valued function . With the introduction of the factor ( 1 ) in the definition of the complex vector ( ), the random field is normalized so that its covariance matrix is the identity on the diagonal: ( , ) = .
By Bochner's theorem, there exists a nonnegative-definite matrix-valued measure , which is called the spectral measure of the Gaussian random field, such that is the Fourier transform of . As shown in [37], the spectral measure is supported on the unit sphere and reads as To conclude, note that the random field defines a Gaussian probability measure on the space of vector fields on ℝ 3 , where is any fixed positive integer. To apply KAM-theoretic arguments later on, we always assume that ⩾ 4. This space is endowed with its usual Borel -algebra of point evaluations. More precisely, denoting by Ω the sample space of the random variables , it is not hard to prove that the random field is a measurable map from Ω to (ℝ 3 , ℝ 3 ). In [37], we established the following properties of the probability measure .
(i) It is translationally invariant. (ii) It is ergodic with respect to translations: if Φ is an 1 random variable on the probability space ( (ℝ 3 , ℝ 3 ), , ) , then both -almost surely and in 1 ( (ℝ 3 , ℝ 3 ), ). Here denotes the translation operator, that is, for ∈ ℝ 3 its action on a vector field is ( ) = ( + ). (iii) Its support is the space of Beltrami fields. In particular, for any Beltrami field , any compact set ⊂ ℝ 3 and each > 0,

Knots and chaos appear almost surely
To state the main results on the typical behavior of Gaussian random Beltrami fields, we need to introduce some notation. The readers who are not familiar with the theory of dynamical systems may consult all the necessary background in [46]. In increasing order of complexity, we define the following. (iv) h ( ) denotes the number of horseshoes of contained in the ball . We recall that a horseshoe is a compact hyperbolic invariant set on which the time-1 flow of (a suitable reparameterization of) is topologically conjugate to a Bernoulli shift.
The first main result we proved in [37] shows that, with probability 1, a Gaussian random Beltrami field exhibits a large volume of ergodic invariant tori, and simultaneously features many horseshoes and periodic orbits of arbitrary topology. (ii) with probability 1, the volume of ergodic invariant tori of isotopic to a given embedded torus  ⊂ ℝ 3 and the number of periodic orbits of isotopic to a given closed curve ⊂ ℝ 3 satisfy the volumetric growth estimate An immediate but illuminating corollary of this result is the following, which is certainly consistent with Arnold's view of complexity for Beltrami fields: Corollary 6.5. With probability 1, a Gaussian random Beltrami field on ℝ 3 exhibits infinitely many horseshoes coexisting with an infinite volume of ergodic invariant tori of each isotopy type. Moreover, the set of periodic orbits contains all knot types.
If one considers a simpler quantity such as the number of zeros of a Gaussian random Beltrami field, one can obtain an asymptotic distribution law similar to that of the nodal components of a random monochromatic wave [61], whose corresponding asymptotic constant can even be computed explicitly: Here z = 0.00872538 … is an explicit constant and the convergence is both in 1 and almost surely.

Key ideas behind the proof
The proof of Theorem 6.4 involves a delicate interplay between deterministic tools from dynamical systems and probabilistic techniques. First, in order to have a good control on the number of periodic orbits and the volume of ergodic invariant tori, we need to introduce certain dynamical quantities from the theories of hyperbolic dynamics and KAM: o ( ; [ ], ) with  = ( 1 , 2 , Λ 1 , Λ 2 ), which is the number of hyperbolic periodic orbits of contained in , of knot type [ ], whose periods and maximal Lyapunov exponents are in the intervals ( 1 , 2 ) and (Λ 1 , Λ 2 ), respectively; t ( ; [ ],  ) with  ∶= ( 1 , 2 , 1 , 2 ), which is the (inner) measure of the set of Diophantine invariant tori of contained in , of knot type [ ], whose frequencies and twists are in the intervals ( 1 , 2 ) and ( 1 , 2 ), respectively. See [37] for a precise definition of all these dynamical quantities. These carefully chosen functionals are lower semicontinuous, which is the key to solving certain measurability issues that arise in the proof of the probabilistic estimates: The second deterministic ingredient we need to prove that typical random Beltrami fields are chaotic is one example of a Beltrami field that features a horseshoe. Such an example was not available in the literature; in fact, the non-integrable ABC flows are known to be chaotic on 3 as a consequence of the non-contractibility of the domain [11], but not on ℝ 3 . In [37], we constructed an integrable Beltrami field having a heteroclinic cycle between two hyperbolic periodic orbits. Then, using Melnikov analysis, we perturbed this field within the Beltrami class to produce a transverse heteroclinic intersection, which is known to imply the existence of a horseshoe. We then established the following: Since the probability measure is supported on Beltrami fields, which are divergence-free, an immediate consequence of the Realization Theorem 4.1, of Proposition 6.8 and of the lower semicontinuity of the functionals established in Proposition 6.7, is the positivity of the probability that a random Beltrami field exhibits a horseshoe, a periodic orbit isotopic to or a positive volume set of ergodic invariant tori isotopic to  : for some large enough 0 > 0 and a constant 0 > 0. This implies that The key tool to pass from these estimates to Theorem 6.4 is the ergodicity of the probability measure presented in Subsection 6.1 and a weak version of the lower bound appearing in the Nazarov-Sodin sandwich inequality [61]. This new estimate relates the number of dynamical objects (periodic orbits, invariant tori or horseshoes) of the Gaussian random Beltrami field that are contained in an arbitrarily large ball with ergodic averages of the same quantity involving the number of invariant sets contained in balls of fixed smaller radius.
To state the sandwich inequality, we need to introduce some notation. For any subset Γ ⊂ ℝ 3 , we denote by ( , ; Γ) the number of connected components of Γ that are contained in the ball ( ) of radius centered at . Also, if where ∈ ℝ 3 , is a countable set of points, then we define as the number of points of  contained in ( ). We will set ( ; Γ) ∶= (0, ; Γ) and similarly  ( ; ). Lemma 6.9. Let Γ be any subset of ℝ 3 whose connected components are all closed and let  be a countable set of points of ℝ 3 . Then the functions  (⋅, ; ) and (⋅, ; Γ) are measurable, and for any 0 < < one has Theorem 6.4 follows by combining the second sandwich estimate in Lemma 6.9, applied to the different dynamical quantities, with the probability bounds (6.2)-(6.3), and with the ergodic theorem. To apply the ergodic theory, we must use the properties of , as presented in Subsection 6.1.
It only remains to show how to prove Theorem 6.6 on the asymptotics of the zeros of a random Beltrami field. For this, we make use of the Kac-Rice formula. The first observation is that, almost surely, the zeros of are nondegenerate and hence isolated. Therefore, defining  ∶= { ∈ ℝ 3 ∶ ( ) = 0}, the random variable Φ ( ) ∶=  ( ;  )∕| | is finite with probability 1. The Kac-Rice formula then enables us to compute the expected value of Φ ( ), which is independent of > 0: where z is the constant in the statement of Theorem 6.6. See [37] for the details about the computation of this conditional expectation. Theorem 6.6 then follows combining the first sandwich estimate in Lemma 6.9 with the ergodic theorem for .

A few words about generic Beltrami fields on
We described in Section 5 the eigenvalues ( a positive integer) of the curl operator on 3 , and their corresponding Beltrami fields, which are of the form for some ∈ ℂ 3 satisfying that = − and Equation (5.2). In terms of the Hermitian complex field ( ) introduced in Equation (6.1), it is easy to check that the vector must be of the form = ( ∕ 1∕2 ) (6.5) unless = (± 1∕2 , 0, 0). Here ∈ ℂ is an arbitrary complex number. Analogously to the Euclidean case, we can define a Gaussian random Beltrami field on the torus with eigenvalue as where the real and imaginary parts of the complex-valued random variable are standard Gaussian variables. We assume that these random variables are independent except for the constraint = − (the inessential normalization factor (2 ∕ ) 1∕2 has been introduced for convenience). As before, this induces a Gaussian probability measure on the space of -smooth vector fields on the torus, ⩾ 4.
Exploiting the fact that the covariance kernel of the localization of the field at a point 0 ∈ 3 tends to the covariance kernel of the Gaussian random Beltrami field on ℝ 3 as → ∞ through admissible values, we obtained in [37] asymptotic laws for typical high-frequency Beltrami fields on 3 . We recall that the aforementioned localization of the field was introduced in Subsection 5.1. In fact, the convergence of the covariance kernels is the probabilistic counterpart of the (deterministic) inverse localization property presented in Subsection 5.1.
In parallel with Subsection 6.2, for any closed curve and any embedded torus  , we, respectively, denote by z , h , o ([ ]) and t ([ ]) the number of zeros, horseshoes, periodic orbits isotopic to and ergodic invariant tori isotopic to  of the field , as well as the volume (that is, inner measure) of these tori, which we denote by t ([ ]). Although the results proven in [37] are considerably more precise, to avoid technicalities we shall just state a qualitative result that is nonetheless illustrative of the generic behavior of high-frequency Beltrami fields on the torus:

VORTEX RECONNECTION IN VISCOUS FLUIDS
The dynamics of a viscous incompressible fluid is described by the Navier-Stokes equations, where the viscosity is a fixed positive constant. Throughout this section, we assume that the spatial variable takes values in the torus 3 . When = 0, it is well-known that the vorticity is transported by the velocity field, as described in Equation (1.1), so the vortex structures do not change their topology as long as the solution remains smooth. However, in the presence of viscosity, the vorticity is no longer transported along the flow, and the diffusive term gives rise to a different phenomenon known as vortex reconnection. In short, one says that a vortex reconnection has occurred at time if the vortex structures at time and at time 0 are not homeomorphic, so there has been a change of topology [13,50]. For example, certain vortex tubes can break and new vortex tubes or vortex lines, possibly knotted or linked in a different way, can be created.
There is overwhelming numerical and physical evidence of the fact that vortex reconnection occurs [13,50,75]. We highlight the recent experimental results presented in [51,68], where the authors study how vortex lines and tubes of different knotted topologies reconnect in actual fluids using cleverly designed hydrofoils. In contrast with the wealth of heuristic, numerical and experimental results on this subject, the first mathematically rigorous scenario of smooth solutions to the 3D Navier-Stokes equations whose vortex structures exhibit topological changes was constructed very recently in [23]. We remark that this vortex reconnection scenario operates in the linear regime of the equation, where the dissipative term is dominant, in contrast with the highly nonlinear vortex reconnection process that is observed in turbulent flows at high Reynolds number.

Existence of vortex reconnections
To state the result, let us prescribe at will the following data. The following theorem [23] establishes the existence of vortex reconnections for smooth global solutions of the Navier-Stokes equations. In the statement, we say that two sets  1 and  2 of 3 are isotopic if there exists a diffeomorphism Φ ∶ 3 → 3 , connected with the identity, such that Φ( 1 ) =  2 . Remark 7.2. By structurally stable we mean that the vortex reconnection phenomenon occurs for any initial datum that is close enough in the 5 ( 3 ) norm to the initial velocity 0 of the theorem, and the existence of non-homeomorphic vortex structures occurs not only between the times and ±1 with odd, but also between any nonnegative times and ±1 for which | − | + | ±1 − ±1 | is small enough. This ensures that the vortex reconnection is experimentally observable.
The proof that this scenario of vortex reconnection occurs [23] is indirect, meaning that we prove that there has been a change in the topology of the vortex structures of the fluid but we cannot describe the way in which these structures break. Informally, this theorem does not provide a movie of how the reconnection process happens but only some significant snapshots of it. In particular, we do not know if the vortex reconnection happens essentially as in the well-known model [75] of two rotating columns of fluid that meet at an angle that evolve into a configuration similar to cutting both columns in half at their point of contact and then reconnecting each end of the first column with one of the second.
As an aside, note that in the context of quantum fluids, which are described by the Gross-Pitaevskii equation, with a complex-valued function in ℝ 3 , we have also established [35] the existence of vortex reconnections. The quantum vortices at time are defined as the connected components of the zero set of (⋅, ). In stark contrast with Theorem 7.1 for the Navier-Stokes equations, here we are able to track the evolution of the quantum vortices during the whole process, which permits one to describe the reconnection events in detail and to verify that they exhibit the properties observed in the physics literature. The way we describe the process is through certain 2-dimensional surfaces in space-time that we employ to effectively encode any reconnection cascade. These surfaces provide a universal way of describing generic reconnections that crucially uses the scalar nature of the problem. Alas, for vectorial equations one cannot hope to find an analogous way of encoding reconnections.

Sketch of the proof
Let us give some heuristic ideas about the proof of Theorem 7.1. It hinges on choosing an initial datum that is the superposition of a finite number of Beltrami fields  that oscillate at different large frequencies: Here are small constants satisfying ≪ −1 ≪ ⋯ ≪ 1 ≪ 1. The Beltrami fields  satisfy the equation curl  =  in 3 with frequencies 0 ≫ 1 ≫ ⋯ ≫ ≫ 1. The Beltrami fields  for odd are constructed using the Realization Theorem 5.5, which ensures the existence of a Beltrami field  with high eigenvalue exhibiting a set of vortex structures isotopic to  ; the eigenvalues are taken to be odd integers to ensure 2 is admissible. To implement this argument, we crucially need the quantitive stability and norm bounds for knotted high-frequency Beltrami fields on the torus discussed in Remark 5.6. In contrast, the Beltrami fields  for even are explicit and read in terms of the Cartesian coordinates ( 1 , 2 , 3 ) as  ∶= (sin 3 , cos 3 , 0) .
Since the integral curves of these fields lie on the tori 3 = const, where  is a linear field, it follows that they are all non-contractible.
An essential property of these families is that they are 'robustly non-equivalent'. Indeed, any uniformly small perturbation of a member of the first family (that is, odd) exhibits a set of vortex structures isotopic to  , whereas in the second family (that is, even) all the vortex structures are non-contractible. This is proved using suitable estimates for Beltrami fields with sharp dependence on the eigenvalue and the KAM theory.
The global existence of the solution with initial datum 0 follows from a suitable stability theorem for the Navier-Stokes equations and the fact that 0 is a small perturbation of a Beltrami field. More precisely, the solution to the Navier-Stokes equations with initial datum 0 ∶=  0 is global and decays exponentially fast as tends to infinity: We showed in [23] that any initial condition 0 that is close enough to 0 also gives rise to a global exponentially decaying solution to the Navier-Stokes equations, that is, for all ⩾ 0. For our application we take ⩾ 5 (to apply KAM theory). Finally, the analysis of the solution to the Navier-Stokes equations with initial datum 0 , which allows us to establish the existence of vortex reconnection, involves a delicate interplay between the large frequencies of the fields and their relative sizes. This ensures that, at time , the vortex structures of the fluid are related to those of the field  in the sense that (⋅, ) = ( + small) for some nonzero constant . More precisely, a careful choice of the constants and , combined with the aforementioned stability theorem, allows us to prove that during the time interval [0, ] the evolution of the Navier-Stokes equations with initial condition 0 is governed by the heat equation modulo a small error: Therefore, at = 0 the field 0 is a small perturbation of  0 , and at time we have the rescaled field All the constants are then carefully chosen so that the second and third summands in this equation are suitably small. The theorem then follows from the properties of the carefully constructed vector fields  . Remark 7.3. A scaling argument shows that the frequencies we need to take in the proof of Theorem 7.1 are much larger than −1∕2 . Therefore, there is no hope of making this scenario of vortex reconnection work in the vanishing viscosity limit. There are several interesting problems related to the geometry of 3D magnetohydrostatic (MHS) equilibria. We first observe [56] that, formally, the MHS equations are the same as the stationary Euler equation, where the velocity field of the fluid plays the role of the plasma magnetic field, and the Bernoulli function stands for the plasma pressure. Motivated by problems related to the confinement of plasmas in stellarators, the physicist H. Grad posed the following outstanding question [45] on the structure of MHS equilibria. For consistency with the previous sections, we formulate it in the language of fluid mechanics.

SOME FUTURE DIRECTIONS OF RESEARCH
Problem 2. Let Ω ⊂ ℝ 3 be a smooth bounded domain diffeomorphic to a solid torus. Assume that there exists a smooth steady Euler flow in Ω, tangent to the boundary, whose Bernoulli function is constant on Ω and its level sets are nested toroidal surfaces that foliate Ω. Are Ω and the stationary solution necessarily axisymmetric?
From a dynamical viewpoint, the following problems concerning periodic orbits of steady Euler flows on the 3-sphere remain open. The first one was asked by Williams in [73], while the second was answered affirmatively by Rechtman [64] in the analytic category.  The last problem that we shall state concerns the phenomenon of vortex reconnection for smooth solutions of the Navier-Stokes equation. In Theorem 7.1, we showed the existence of solutions of Navier-Stokes on 3 that exhibit a change of topology of their vortex lines and tubes during the time evolution. However, in this scenario the solutions operate in the linear regime of the equation, and the method of proof does not allow us to track the evolution of the vortex structures to describe how they break when they meet and then how they reconnect. A rigorous understanding of this process should shed some light on the role of the nonlinear term and the changes experienced by the vortex structures.
Problem 5. To introduce a model of bifurcation for dynamical systems compatible with the 3D Navier-Stokes evolution where the bifurcation parameter is the time, which captures the typical reconnection scenario of vortex lines observed by physicists near the reconnection time.

APPENDIX: SOME APPLICATIONS OUTSIDE FLUID MECHANICS
In this appendix, we discuss how some of the key ideas developed to study topologically complicated vortex structures have found application in the study of problems that arise in other areas different from fluid mechanics. Specifically, the global approximation theorem with sharp decay for Beltrami fields and for the Helmholtz equation presented in Subsection 3.4 can be extended to other linear equations. This allows one to follow the same strategy as in the proof of the Realization Theorem 4.1 to show the existence of geometrically complex phenomena in solutions to various PDEs, including nonlinear equations as long as one stays in the linear regime. Moreover, the inverse localization theorem for eigenfunctions on the torus that we introduced in Subsection 5.1 can also be proved for some spectral problems in Euclidean space. In what follows, we give a short survey of these topics; a more comprehensive introduction can be found in [32].

A.1 Approximation theorems in nonlinear elliptic PDEs
The global approximation theory for linear elliptic PDEs, which extends the classical theorems of Runge and Mergelyan on holomorphic approximation in complex analysis, was developed by Lax [53], Malgrange [54] and Browder [8] during the 1950s. Roughly speaking, this theory establishes a sort of flexibility for solutions that satisfy a linear elliptic equation on a bounded domain ⊂ ℝ : the local solution can be uniformly approximated in compact subsets of by a global solution to the same equation provided that ℝ ∖ does not have any bounded connected components. This result was generalized in [28] to include unbounded domains and better-than-uniform approximation, which allowed us to prove realization theorems for noncompact level sets of harmonic functions in ℝ . A major drawback of the global approximation theory is that it works only for linear equations. The proof obviously does not carry over to the nonlinear case because we need to write the solutions to certain inhomogeneous equations using a fundamental solution. Nevertheless, the global approximation theorems turn out to be very useful to study nonlinear equations when they operate in the linear regime. We already saw this application in Section 7 when we studied the vortex reconnections for solutions of the Navier-Stokes equations.
In the context of elliptic PDEs, we also explored this idea in [19,31]. Let us focus on the paradigmatic case of the Allen-Cahn equation in ℝ , Obviously, the linearization of Equation (A.1) at the trivial solution ∶= 0 is the Helmholtz equation. To pass from Helmholtz to Allen-Cahn, one resorts to an iterative procedure. Using a fundamental solution ( ) of the Helmholtz equation, which satisfies the distributional equation Δ + = , one can set up an iterative scheme as where is a small positive constant and is a solution to the Helmholtz equation with sharp decay at infinity. This decay is crucially used to show the convergence of the scheme.
One can show [31] that converges as → ∞ to some function in the weighted space Moreover, the functions and are close in the following sense: This argument allows us to construct solutions to the Allen-Cahn equation with nodal sets of prescribed topology using solutions to the Helmholtz equation whose nodal sets are structurally stable (and hence their isotopy types are not changed by 1 -small perturbations). To construct the latter, one of the key tools is the global approximation theorem with sharp decay for the Helmholtz operator mentioned in Subsection 3.4. The following theorem illustrates the kind of results that one can prove using this strategy: Theorem A.1. Let Σ be any compact hypersurface of ℝ , ⩾ 3. Then there is a bounded entire solution of the Allen-Cahn equation in ℝ such that its nodal set −1 (0) has a structurally stable connected component isotopic to Σ.

A.2 Approximation theorems in parabolic PDEs
Recently, we extended the approximation theory of Lax, Malgrange and Browder to the context of linear parabolic equations [20]. In this setting, the flexibility granted by the approximation theorems has found important applications to study the movement of local hot spots and isothermic hypersurfaces. To keep things simple, let us focus on the heat equation. Given a compactly supported function 0 ∈ ∞ (ℝ ), we denote by ∶= Δ 0 the only solution to the Cauchy problem for the heat equation The following approximation theorem [20] ensures that a solution of the heat equation in a bounded domain of spacetime can be promoted to a global solution, up to a small error, given by a compactly supported initial datum. In the parabolic setting, the necessary topological hypothesis on the domain is written in terms of the set ( ), which denotes the intersection of with the time-slice, that is, ( ) ∶= { ∈ ℝ ∶ ( , ) ∈ } .
Theorem A.2. Let be a bounded domain whose closure is contained in ℝ +1 + , ⩾ 3, such that ℝ ∖ ( ) is connected for all ∈ ℝ. Given a function satisfying the heat equation − Δ = 0 on , an integer , a small constant > 0 and any compact subset ⊂ , there exists a function 0 ∈ ∞ (ℝ ) such that ∶= Δ 0 approximates as An analogous approximation theorem can be proved for the Schrödinger equation [35], although the proof is substantially different and presents new key technical subtleties. Without getting into details, let us mention that, unsurprisingly, overcoming these difficulties involves dealing with oscillatory integrals. In the particular case that the spacetime domain is of the form × ℝ, where ⊂ ℝ , and the local solution satisfies certain mild decay condition for large times, the approximation theorem for the Schrödinger equation can be stated in a quantitative way [35] similar to the quantitative approximation theorem for Beltrami fields (Theorem 3.10). The approximation theorem for the Schrödinger equation is crucially used in the proof of the existence of quantum vortex reconnections for the Gross-Pitaevskii equation, which we discussed in Subsection 7.1.
Finally, we would like to mention that recent works [62,66] have established, in the context of the Navier-Stokes equation, that global solutions with localized initial data can be approximated by solutions on large enough domains with Dirichlet or periodic boundary conditions. Although the methods of proof are totally different from the ones used in [20,35] (in particular, they apply to a nonlinear PDE), they can be understood as a sort of 'inverse approximation' theorem (from a global solution to a local one with boundary conditions).

A.3 Inverse localization in spectral theory
As stated in Subsection 5.1, it is well-known that the Helmholtz equation arises, as a rescaled limit, from any well-behaved eigenvalue problem. More precisely, let us consider the spectral problem −Δ g + = in a Riemannian manifold ( , g), where Δ g is the associated Laplace-Beltrami operator, and is a smooth (real-valued) potential. If the spectrum of this operator is discrete and unbounded, it is not hard to check that the localization at a fixed point 0 ∈ Here is a second-order differential operator whose coefficients depend on the metric g and the potential , and Ψ denotes a patch of normal geodesic coordinates (see Subsection 5.1). Accordingly, as → ∞, the localization tends to a solution of the Helmholtz equation in Euclidean space, that is, Δ + = 0.
Theorem 5.1 shows that a converse statement also holds for eigenfunctions of the Laplacian on the torus: given any solution to the Helmholtz equation in ℝ 3 , we can 'transplant' it into the localization of an eigenfunction of the Laplacian on 3 with large enough eigenvalue. More generally, this result holds for any dimension ⩾ 2 [40].
Motivated by a conjecture of Sir Michael Berry [6] concerning the nodal set of eigenfunctions of Schrödinger operators in ℝ 3 , we proved in [21] that the inverse localization property also holds for eigenfunctions of the harmonic oscillator, that is, a Schrödinger operator with potential ( ) = | | 2 . Let us discuss the main ideas in some detail.
First, recall that the eigenfunctions of the harmonic oscillator are the 2 (ℝ 3 , ℂ) functions satisfying the equation −Δ + | | 2 = in Euclidean space. It is well-known that the eigenvalues are of the form = 2 + 3 with a nonnegative integer, and that the multiplicity of the corresponding eigenspace is 1 2 ( + 1)( + 2). In this setting, the inverse localization principle has this concrete realization: Theorem A.3. Let be an even or odd solution of Δ + = 0 in ℝ 3 . Fix > 0 and an integer . Then for any large enough integer , there is an eigenfunction of the harmonic oscillator with eigenvalue = 2 + 3, such that The reason to assume that is even or odd (that is, (− ) = ± ( )) is that all eigenfunctions of the harmonic oscillator have a definite parity, which is then a property inherited by their localizations at the origin. Also note that the eigenfunction is localized in balls of radius −1∕2 centered at 0 = 0. This is because the operator −Δ + | | 2 is not translationally invariant, but spherically symmetric, so the origin is a special point. Compare this with the inverse localization for eigenfunctions on the torus or the sphere (cf. Section 5), which holds for any point on the manifold (because the Laplacian commutes with all the isometries of the space).
The proof of Theorem A.3 makes use not only of the fact that the multiplicity of the eigenvalues grows as → ∞, but of the specific analytic form of the eigenfunctions, which allows us to connect their asymptotic expressions with spherical Bessel functions, and hence with solutions to the Helmholtz equation.
This inverse localization result paves the way to establishing the existence of eigenfunctions of the harmonic oscillator with nodal sets of arbitrarily complicated topologies, which eventually enables a proof of Berry's conjecture. But this is another story, so we refer the interested readers to [21].
We conclude this section by mentioning that a variation of the inverse localization principle also holds for the hydrogen atom [22], which is described by the eigenfunctions of the Schrödinger in ℝ 3 . Interestingly, the statement is rather different from Theorem A.3 because the eigenvalues are of the form = − 1 2 with a positive integer, so they accumulate at 0 instead of at ∞ as in the case of the harmonic oscillator. Therefore, one does not need to rescale the space variable, and the limit model is not the Helmholtz equation, but the equation in ℝ 3 . In this setting, roughly speaking, the phenomenon that occurs is that any solution to Equation (A.3) can be approximated by an eigenfunction of the hydrogen atom with eigenvalue close enough to 0. In contrast with the harmonic oscillator, this approximation takes place in balls of diameter of order 1 contained in ℝ 3 ∖{0}. This can be regarded as evidence of the fact that the key ingredients to have a sort of inverse localization principle are the degeneracy of the eigenvalues and the specific asymptotic properties of their eigenfunctions, rather than a high-frequency analysis. Incidentally, we find it striking that the only quantum operators for which we have been able to establish a sort of inverse localization property (the harmonic oscillator and the hydrogen atom) are superintegrable as classical systems. We do not see an obvious explanation for it.

A C K N O W L E D G E M E N T S
We thank the reviewers for their careful reading of the manuscript and useful suggestions. This work has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation program through the Grant Agreement 862342 (Alberto Enciso). It is partially supported by the Grants CEX2019-000904-S, RED2018-102650-T and PID2019-106715GB GB-C21 (Daniel Peralta-Salas) funded by MCIN/AEI/10.13039/501100011033.

J O U R N A L I N F O R M AT I O N
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