Bifurcation analysis for axisymmetric capillary water waves with vorticity and swirl

Abstract We study steady axisymmetric water waves with general vorticity and swirl, subject to the influence of surface tension. This can be formulated as an elliptic free boundary problem in terms of Stokes' stream function. A change of variables allows us to overcome the generic coordinate‐induced singularities and to cast the problem in the form “identity plus compact,” which is amenable to Rabinowitz's global bifurcation theorem, whereas no restrictions regarding the absence of stagnation points in the flow have to be made. Within the scope of this new formulation, local curves and global families of solutions, bifurcating from laminar flows with a flat surface, are constructed.


INTRODUCTION
In the last decades, there has been a lot of progress on the two-dimensional steady water wave problem with vorticity (see, e.g., Refs. 1-5 and references therein). The corresponding threedimensional problem is significantly more challenging, due to the lack of a general formulation that is amenable to methods from nonlinear functional analysis. This is related to the fact that in two dimensions, the vorticity is a scalar field that is constant along streamlines, while in three dimensions, it is a vector field that satisfies the vorticity equation, including the vortex stretching term. One approach to at least gain some insight is to investigate flows under certain geometrical assumptions to fill the gap between two-dimensional and three-dimensional flows. This is one of the motivations for studying the axisymmetric Euler equations, which in many ways behave like the two-dimensional equations. Indeed, for the time-dependent problem, in the swirl-free case, these possess a global existence theory for smooth solutions similar to two-dimensional flows; see Refs. 6 and 7 and references therein (note, however, the recent remarkable result 8 on singularity formation of nonsmooth solutions). The steady axisymmetric problem is also of considerable physical importance, as it can be used to model phenomena such as jets, cavitational flows, bubbles, and vortex rings (see, e.g., Refs. 9-17 and references therein).
In this paper, we study axisymmetric water waves with surface tension, modeled by assuming that the domain is bounded by a free surface on which capillary forces are acting, and that in cylindrical coordinates ( , , ), the domain and flow are independent of the azimuthal variable . In the irrotational and swirl-free setting, such waves were studied numerically by Vanden-Broeck et al. 18 and Osborne and Forbes, 19 who found similarities to two-dimensional capillary waves, including overhanging profiles and limiting configurations with trapped bubbles at their troughs. The small-amplitude theory is intimately connected to Rayleigh's instability criterion for a liquid jet 20 (see also Refs. 21 and 18), which says that a circular capillary jet is unstable to perturbations whose wavelength exceeds the circumference of the jet. Indeed, this instability criterion is satisfied precisely when the dispersion relation for small-amplitude waves has purely imaginary solutions, while steady waves are obtained when the solutions are real 18 (i.e., for smaller wavelengths). According to Hancock and Bush, 21 a stationary form of such steady waves may be observed at the base of a jet that is impacting on a reservoir of the same fluid. If the reservoir is contaminated, the wave field is moved up the jet and a so-called "fluid pipe" with a quiescent surface is formed at the base. We also note that in recent years, there has been increased interest in waves on jets in other physical contexts, such as electrohydrodynamic flows 22 and ferrofluids. [23][24][25] In this paper, we consider liquid jets with both vorticity and swirl. A motivation for this is that a viscous boundary layer in a pipe typically gives rise to vorticity, which may have a significant effect on the jet flowing out of the pipe. As an idealization, we assume that the jet extends indefinitely in the -direction and ignore viscosity and gravity. In the irrotational swirl-free case, the problem can be formulated in terms of a harmonic velocity potential. In contrast, we formulate the problem in terms of Stokes' stream function, which satisfies a second-order semilinear elliptic equation known alternatively in the literature as the Hicks equation, the Bragg-Hawthorne equation, or the Squire-Long equation (cf. Ref. 16). This equation is also known from plasma physics as the Grad-Shafranov equation (cf. Ref. 26). The first aim of the paper is to construct smallamplitude solutions using local bifurcation theory in this more general context. In contrast to Ref. 18, this means that the bifurcation conditions are much less explicit and that we require qualitative methods. The second aim is to construct large-amplitude solutions using global bifurcation theory and a reformulation of the problem inspired by the recent paper 5 on the two-dimensional gravity-capillary water wave problem with vorticity.
We now describe the plan of the paper. First, in Section 2, we start by introducing the main problem we are going to study. This means that we start with the incompressible Euler equations and recall its axisymmetric version. In Section 3, we discuss regularity issues and trivial solutions of the axisymmetric incompressible Euler equations. Regarding regularity issues and to reformulate the problem in a secure functional-analytic setting, we avoid coordinate-induced singularities by introducing a new variable in terms of the Stokes stream function and view it (partly) as a function on five-dimensional space; this trick to overcome this kind of coordinate singularities is well known and goes back to Ni. 15 Then, we study local bifurcations in Section 4 in the spirit of the theorem by Crandall-Rabinowitz, mainly by introducing the so-called good unknown; the main result of this section is Theorem 2. In addition, in Section 5, we take a closer look at the conditions for local bifurcation. First, we establish spectral properties of the corresponding Sturm-Liouville problem of limit-point type and with boundary condition dependent on the eigenvalue. After that, we investigate some specific examples in more detail. Finally, in Section 6, we close the paper by investigating global bifurcations; see Theorem 4.
Since we require the radius to be a graph of the longitudinal position along the water surface, our theoretical framework, in contrast to Ref. 5, does not allow for overhanging waves, and we leave it to further research to include this possibility. This would clearly be a desirable extension in view of the numerical results in Refs. 18 and 19.

DESCRIPTION OF THE PROBLEM AND THE GOVERNING EQUATIONS
We consider periodic axisymmetric capillary waves traveling at constant speed along the -axis. The fluid is assumed to be inviscid and incompressible. In a frame moving with the wave, the flow is therefore governed by the steady incompressible Euler equations where ⃗ = ⃗( ⃗) and = (⃗) denote the velocity and the pressure, respectively, and Ω is the fluid domain. In cylindrical coordinates ( , , ), that is, = cos , = sin , and = , the velocity field ⃗ is expressed as where the vectors and ⃗ = (0, 0, 1) form an orthonormal basis. Note that we allow for nonzero swirl, ≠ 0. From the incompressibility and the axisymmetry of the flow, it follows that we can introduce Stokes' stream function Ψ( , ), such that Moreover, the quantity is constant along streamlines, which we express as = (Ψ) where is an arbitrary function. The steady Euler equations are then equivalent to the Bragg-Hawthorne equation where is an arbitrary function and cf. Ref. 16 (Chapter 3.13). Note that the corresponding vorticity vector is given by We next consider the boundary conditions. Assume that the fluid domain is given by Ω = {( , ) ∈ ℝ 2 ∶ 0 < < + ( )} and its boundaries by Ω  = {( , ) ∈ ℝ 2 ∶ = + ( )} (free surface) and Ω  = {( , ) ∈ ℝ 2 ∶ = 0} (center line). Although the latter could be considered as part of the domain, it is sometimes convenient to consider it as a boundary due to the appearance of inverse powers of in the equations. On the free surface = + ( ), we have the kinematic boundary condition ⃗ ⋅ ⃗ = 0, where ⃗ = ⃗ − ′ ( )⃗ denotes a normal vector. Expressed in terms of Ψ, this takes the form Ψ + Ψ = 0 on Ω  . In addition, we have the dynamic boundary condition = − on Ω  , where is the mean curvature of Ω  and > 0 is the coefficient of surface tension. Using Bernoulli's law, we can eliminate the pressure and express this as on Ω  , where is the Bernoulli constant. At the center line Ω  , the identity Ψ = shows that Ψ = 0. Summarizing, we have following boundary value problem: in Ω, where and are arbitrary functions of Ψ.

The equations
The last two boundary conditions in (10) mean that Ψ is constant on both Ω  and Ω  . We normalize Ψ such that it vanishes on Ω  and assign the name to its value on Ω  . Thus, we deal with the equations where and are constants. The fluid velocity is given by Following a trick of Ni, 15 we first introduce the function via and will later work in five dimensions (see Section 3.4 for details). In terms of , the equations read Notice that we no longer need to impose a condition on = 0, provided that is continuous at = 0, since then (11d) is automatically satisfied for Ψ given by (13).

Regularity issues
Quite naturally, the fluid velocity ⃗ should be at least of class 1 (in Cartesian coordinates). Written in terms of , Equation (12) reads Due to Ref. 27, ⃗ is of class 1 , provided that ( 2 )∕ is of class 1 and is of class 2 , both viewed as functions on {( , ) ∈ [0, ∞) × ℝ ∶ ≤ + ( )}, and, moreover, ( 2 )∕ , , and ( ) vanish at = 0. In view of and it is therefore sufficient to assume and Furthermore, we need that the right-hand side of (14a) is in a Hölder class 0, if is 0, . To this end, it is sufficient that both ′ and ( ) ∶= ( )∕ (continuously extended to = 0 by (0) ∶= ′ (0)) are locally Lipschitz continuous in view of Moreover, the nonlinear operator  introduced later should be of class 2 . Hence, we need that , , and ′ are locally of class 2,1 ; notice that this condition on already implies the desired property of as above. Also, to construct trivial solutions, we will need a Lipschitz property of and ′ . Overall, we impose the following assumptions on and :

Trivial solutions
We now have a look at trivial solutions of (14), that is, solutions of (14) independent of . Therefore, we consider the (singular) Cauchy problem Here, ∈ ℝ is a parameter, which will later serve as the bifurcation parameter, and (22c) is imposed due to (18). Notice that, in view of (15), there is a one-to-one correspondence of the parameter and the velocity at the symmetry axis via ⃗ = −2 ⃗ at = 0.
To solve (22), we rewrite (22), making use of (22a), (22b), and + 3 = −3 ( 3 ), as the integral equation By Lipschitz continuity of and ′ , it is straightforward to see that the right-hand side of (23) gives rise to a contraction on ([0, ]) if > 0 is small enough. Thus, Equation ( Finally, motivated by the flattening considered below, we define where is the unique solution of (22) as obtained above.

Working in 5D and flattening
In the following, for a function = ( , ) on some Ω ⊂ ℝ 2 , we denote by  the function given by and defined on the set Ω  , which results from rotating Ω around the -axis in ℝ 5 = {( , ) ∈ ℝ 4 × ℝ}. Conversely, any axially symmetric set in ℝ 5 can be written as Ω  for a suitable Ω ⊂ ℝ 2 , and any axially symmetric functioñon Ω  equals  for a certain function on Ω, that is, =  −1̃, where  −1 is defined on the set of axially symmetric functions. Thus, it is easy to see that satisfies (18) and solves (14a) and (14c) if and only if  ∈ 2 (Ω  ) and solves with Δ 5 denoting the Laplacian in five dimensions. Observe that, unlike in (10) or (14a), there is no longer a term that is singular on the symmetry axis (due to (20))-this is the main motivation for working with instead of Ψ and in five dimensions. To transform (26) into a fixed domain, we consider the flattening ( , ) ↦ ( , ) = ( ∕( + ( )), )-from now on, we always assume that > − . Thus, introducing̃viã( , ) =  ( , ), (26) is transformed intõ where Ω 0 ∶= [0, 1) × ℝ and here and throughout this paper, repeated indices are summed over. It is straightforward to see that is a uniformly elliptic operator, provided that + is uniformly bounded from below by a positive constant.
As for Bernoulli's equation (14b), we do not have to take a detour and increase the dimension, because in (14b), no singular term appears, at least whenever the surface does not intersect the symmetry axis. Therefore, here we consider the flattening we call [ ] the inverse map. Then, with̄( , ) = ( , ), that is,

Reformulation
For later reasons, it is convenient to work with functions satisfying = 0 on = 1 instead of functions̄with variable boundary condition at = 1. Thus, we introduce, for any ∈ ℝ, the function In terms of , (28) and (31), combined with (30) and = ( ) ∶= 2 (1), read and Henceforth, we search for solutions ( , , ) of (33) and (34). Our goal is to rewrite (33) and (34) in the form "identity plus compact," namely, as ( , ) = ( , , ) with  compact. Meanwhile, we also clarify what exactly is, namely, we define it as an expression in ( , , ). An advantage of rewriting the problem in this form is that it facilitates the use of Rabinowitz' global bifurcation theorem, which was originally formulated for such problems. Although it is probably possible to apply some later version adapted to problems that are not of the form "identity plus compact" [see, e.g., Ref. 28 (Theorem II.5.8)], it is not immediately clear how to verify the required hypotheses. Indeed, the fact that and appear in a mixed form in the PDE formulation (33), (34) makes it difficult to analyze the linearization. Varholm 3 studied the gravity-capillary water wave problem using a global bifurcation theorem for analytic operators, assuming that the vorticity distribution is real analytic (which would amount to requiring that and are real analytic in our problem). In this case, he was, for example, able to show that the linearization at a solution is Fredholm of index zero by using a certain transformation (the " -isomorphism"; see Section 4.2) that simplifies the linearization. If one drops the assumption that the vorticity distribution is real analytic and uses some global bifurcation theorem based on degree theory, one has to verify this Fredholm property not only at the solution set, but also in a whole open neighborhood. It is not clear how to adapt Varholm's argument to this situation. On the other hand, when the problem is reformulated in the form "identity plus compact," the Fredholm property is automatically satisfied.
To derive the new formulation, we first fix 0 < < 1 and introduce the Banach space equipped with the canonical norm Here, the indices "per," "e," and "0" denote -periodicity ( ∶= 2 ∕ in the following), evenness (in with respect to = 0), and zero average over one period. First, for here, notice that the right-hand side of (38a) is an element of 0, (Ω  0 ) (cf. (20) and the discussion there) and that (38) is invariant under rotations about the -axis, so that has to be axially symmetric.
Second, we rewrite (34) as an equation for , using  = ( , , ) instead of -notice that this change does not affect the equivalence of the whole reformulation to the original equations because clearly (33) is equivalent to = ( , , ): on = 1. To apply −2 ∶ 0, 0,per (ℝ) → 2, 0,per (ℝ), the inverse operation to twice differentiation, to this relation, the right-hand side needs to have zero average over one period. Therefore, we view as a function of ( , , ) via ( , , ) ∶= 1 Here and in the following, ⟨ ⟩ denotes the average of an -periodic function over one period, and  denotes the evaluation of a function at = 1.

Lemma 2.  and thus  is of class
Proof. The other operations in the definition of  being smooth, the property that  is of class 2 follows from the property that  is of class 2 ; this, in turn, is guaranteed by assumption (21). Now let ( , , ) ∈ ℝ ×  be arbitrary. In the following, the quantities can change from line to line, but are always shorthand for a certain expression in its arguments that remains bounded for bounded arguments. Moreover, let > 0 and suppose ‖( , , )‖ ℝ× ≤ . Since is of class 1 with respect to and is elliptic uniformly in due to + ≥ , we see that by applying a standard Schauder estimate. This shows that  2 is compact on ℝ ×  because of the compact embedding of 2, As for  1 , we immediately find, in view of the obtained estimates for ,

Computing derivatives
We now want to calculate the partial derivative  ( , ) and, in particular, its evaluation at a trivial solution. For simplicity, we always write  for  ( , , ) , that is, the partial derivative of  with respect to evaluated at ( , , ) and applied to a direction . The same applies similarly to expressions such as  , , and so forth. Linearizing the operator , which only depends on and not on , leads to Since formally linearizing an equation such as = gives + = , we see that  is the unique solution of Similarly,  is the unique solution of Evaluated at a trivial solution ( , 0, 0), we can simplify as follows: In the following, we denote Moreover, since  = 0 here, we have and Next, we turn to  1 . After a lengthy computation, we get the following results for the partial derivatives of  1 evaluated at a trivial solution ( , 0, 0), noticing that  =  = 0 at such points: where  is the projection onto the space of functions with zero average. It will be convenient to introduce the abbreviation Notice that − ( ) is the -component of the velocity at the surface of the trivial laminar flow corresponding to in view of (15). With this, we can rewrite

The good unknown
Before we proceed with the investigation of local bifurcation, we first introduce an isomorphism, which facilitates the computations later and is sometimes called  -isomorphism in the literature (e.g., in Refs. 2 and 3). The discovery of the importance of such a new variable (here ) goes back to Alinhac, 29 who called it the "good unknown" in a very general context, and Lannes, 30 who introduced it in the context of water wave equations. As explained in Ref. 31(p. 8), the general idea is that if and are unknown variables, related by an unknown change of variables Φ, that is, = •Φ, then formally = •Φ + ( ′ •Φ) Φ. Hence, the "good unknown" for the linearized problem in the new variables is actually − ( ′ •Φ) Φ. In particular, ( − ( ′ •Φ) Φ)•Φ −1 satisfies the linearized problem in the original variables. In our setting, the latter linearization is formal due to the free boundary, but it has a simpler structure than the linearization in the flattened variables.
and assume that ( ) ≠ 0. Then is an isomorphism. Its inverse is given by Proof. Both  ( ) and [ ( )] −1 are well defined, and a simple computation shows that they are inverse to each other. ■ Let us now consider a trivial solution ( , 0, 0). In view of the  -isomorphism, we introduce For given , we denote by = [ ] the unique solution of We notice that Indeed, from we infer that the function ∶= − and  = 0 at | | = 1. Thus, recalling (54), (55), (58), (59), we can rewrite and because of Notice that, under the assumption ∈ 2, per (Ω 0 ),  2 ( ) is the unique solution of and  1 ( ) is (in the set of -periodic functions with zero average) uniquely determined by

Range
Before we proceed with the investigation of the transversality condition, we first prove that the range of  can be written as an orthogonal complement with respect to a suitable inner product. This will be helpful later. To this end, we introduce the inner product for 1 , 2 ∈ 1 0,per (ℝ), 1 , 2 ∶Ω 0 → ℝ with  1 ,  2 ∈ 1 per (Ω  0 ), whereΩ 0 ∶= [0, 1) × (0, ) is one periodic instance of Ω 0 and ∇ ∶= ( 1 ∕ , … , 4 ∕ , ) ; to avoid misunderstanding, we point out that the index "0" in 1 0,per (ℝ) means "zero average" as before and not "zero boundary values." This inner product is positive definite on the space Notice that if 2 ∈ 2 per (ℝ) and that if  2 ∈ 2 per (Ω 0 ), using that 2 2 is the surface area of the 3-sphere. Using (69), (72), and (73), we now compute for smooth , ∈ ⟨( 2 2 2 ( )  , making use of ⟨ ⟩ = 0. Noticing that the terms at the beginning and at the end of this computation only involve at most first derivatives of and , an easy approximation argument shows that this relation also holds for general , ∈ . Moreover, since the last expression is symmetric in and , we can also go in the opposite direction with reversed roles and arrive at the symmetry property ⟨( 2 2 2 ( )  , Thus, the range of  is the orthogonal complement of with respect to ⟨⋅, ⋅⟩. Indeed, one inclusion is an immediate consequence of the symmetry property and the other inclusion follows from the facts that we already know that , being a compact perturbation of the identity, is Fredholm with index zero and that  gains no additional kernel when extended to functions of class 1 .

Transversality condition
Assuming that the kernel is spanned by the function ( , ) = −( ) 2 , ( ) cos( ), we have to investigate whether  is not in the range of , which is equivalent to by the preceding considerations. Differentiating (69) and (70) with respect to , for general , it holds where  is the unique solution of whenever  1 = 0. Now let ( , ) = −( ) 2 , ( ) cos( ) and notice that = −( ) 2 , solves after integrating by parts. Thus, we have proved the following.
Applied to our problem, we obtain the following result. Theorem 2. Assume (80) and that there exists 0 ∈ ℝ with ( 0 ) ≠ 0 such that the dispersion relation with  given by (82), has exactly one solution 0 ∈ ℕ and assume that the transversality condition ) cos( 0 ).

Spectral properties
In view of the defining Equation (79) for̃, and the dispersion relation ( , ) = 0 and writing =̃, , we study the eigenvalue problem which is a singular Sturm-Liouville problem on (0,1). Here and in the following, we denote Proof. It is easy to see that is of limit point type at 0, because ( ) = −2 ∉ 2 3 (0, 1) solves = 0.
Since ∈ ∞ (0, 1), is also of limit point type at 0 according to Ref. 32(Corollary 7.4.1). Thus, (i) is proved. As for (ii), the first statement is an immediate consequence of being of limit point type at 0; see Ref. 32(Lemmas 10.2.3, 10.4.1(b)). Plugging in ( ) = 1 and then ( ) = (which both belong to ( )) yields the second statement. ■ As a consequence, the following result holds; in particular, this explains why we could leave out ′ (0) = 0 in (104).

Lemma 8. is self-adjoint.
Proof. We first prove that is symmetric. To this end, for 1 , 2 ∈ , ∈ (0, 1), let Now if 1 , 2 ∈ ( ), we have, after integrating by parts, Clearly, is symmetric if and only if the first expression converges to 0 as → 0 (for any 1 , 2 ∈ ( )). However, the second expression converges to 0 due to Lemma 6(ii). To see that is even self-adjoint, we first note that obviously, admits the fundamental decomposition = ( 2 3  Proof. Following the proof of Ref. 33 (Theorem 2) using the -norm ‖ ‖ = √ ⟨ , ⟩ , we see that the essential spectra of and coincide. Notice that the criterion 35(Theorem XIII.7.1) applied there is purely topological and does not make use of an additional structure from an (definite or indefinite) inner product. To see that the essential spectrum of is empty, we can apply a criterion of Ref. 36; see also Ref. 33. Indeed, is obviously bounded from below on (0,1) and, moreover, Finally, it is a priori clear that each eigenvalue of cannot have (geometric) multiplicity larger than two; the case of multiplicity two is excluded by the fact that is of limit point type at 0. ■ In fact, we can say more about the location of the eigenvalues of . To this end, the following lemma turns out to be useful.

Lemma 9.
For any ∈ ( ), we have Proof. The only critical point is to ensure that no boundary terms at 0 appear after an integration by parts, which again follows from Lemma 6(ii). ■

Proposition 2.
has no or exactly two nonreal eigenvalues, and in the latter case, they are the complex conjugate of each other. Moreover, the (real part of the) spectrum of is bounded from below.
Proof. The first assertion is clear because is a 1 -space and is self-adjoint; cf. Ref. 37. To prove the second statement, we use a perturbation argument. First, notice that does not affect the domain of the associated operator. Now let 0 be the operator in the case = = 0, which yields for the resolvent holds. If and are arbitrary, we define the perturbation via ( ) = { ∈ ∶ ∈ ( ), = (1)} and Clearly, is densely defined and bounded, and we have = 0 + . Now consider a real < 0 − ‖ ‖ . Because of and the resolvent operator − is invertible in view of the Neumann series. This completes the proof. ■ Under a certain condition, we can infer even more properties of the spectrum of , as we see in what follows.

Proposition 3. Assume that
where − denotes the negative part of . Then the operator has only real eigenvalues, has exactly one eigenvalue < −‖ − ‖ ∞ , and all its other eigenvalues satisfy > −‖ − ‖ ∞ . Moreover, all eigenvalues are algebraically simple.
Proof. Let be an eigenvalue of and = ( , (1)) an associated eigenvector. Due to Lemma 9, we can calculate By assumption and since (1) ≠ 0 (otherwise, also ′ (1) = 0 and thus ≡ 0), it follows that ( Hence, cannot be neutral and has to be real. Since, additionally, by Ref. 37-noting that is a 1 -space-there exists exactly one nonpositive eigenvector of , the first assertion follows immediately. The second statement is a direct consequence of the fact that all eigenvalues are real and no eigenvectors are neutral. (where the condition is regarded to be vacuous if ± are not real). In particular, if and ′ are bounded, this condition is satisfied if " ( ) is sufficiently large" or, provided that additionally, is bounded, if simply "| ( )| is sufficiently large."

Examples
We now turn to a more detailed investigation of the conditions for local bifurcation for specific examples of and .

No vorticity, no swirl
As a first example, we consider the case without vorticity and swirl, that is, = = 0. By (23) and (24), the trivial solutions are given by Thus, The general solution to the ordinary differential equation is given bỹ Therefore, This dispersion relation was also obtained in Ref. 18. Clearly, to find solutions of (138), we can first choose arbitrary > 0, ∈ ℕ with > 1∕ and then such that (138) holds. This gives exactly two possible choices ± 0 for , which correspond to "mirrored" uniform laminar flows. It is important to notice that, given ( ) ≠ 0, Equation (138) is solved by at most one ∈ ℕ; consequently, the kernel of ( ) is one-dimensional if this relation is satisfied for some ∈ ℕ and is trivial if it fails to hold for all . Indeed, Equation (138) obviously cannot hold for ≤ 1; moreover, the function is strictly monotone on (1, ∞) since as 0 ∕ = 1 and 0 ≥ 1 > 0 on (0, ∞); see Ref. 38. Furthermore, it is therefore clear that the transversality condition  (−( ) 2 , ) ≠ 0 always holds in view of ( ) ≠ 0.
We have 0 = ( ± ( )) 2 (−( ) 2 , ) = ± ( ) 2 ( ) + (1 − 2 ) + 2 ± ( ), where Differentiating (155) with respect to yields Thus, if ± = 0 at some > 0, then Here, we notice that > 0 for > 0 because of Instead of presenting a lengthy, not very instructive proof of this inequality, we provide a plot of the left-hand side (multiplied by a suitable positive function) in Figure 1 to convince the reader of the validity of (162). Thus, putting everything together, ± ≷ 0 provided ± = 0. In particular, ± can have at most one critical point on (0, ∞), which, if it exists, has to be a local minimum (maximum). Since moreover ± tends to ±∞ as → ∞ by (147), we conclude that the monotonicity properties of ± can be characterized by its behavior near 0 if ≤ 1∕2 or near 1 if > 1∕2.
the dispersion relation has at most one root.
the dispersion relation has no root.
the dispersion relation has at most two roots.
If 1∕( ) ∈ ℕ and there is the additional root = 1∕( ). If, however, < 0, these statements remain true after reversing all inequalities in the conditions for ( ) and changing max − to min − .

= 0, linear
The easiest case to include nonzero swirl is to take = 0 and ( ) = for some ∈ ℝ (recall that we need (0) = 0). Having a look at, for example, (14), we find that both parameters and F I G U R E 3 The black area consists of points ( , ) with ( , ) ≤ 0 and of the horizontal lines corresponding to zeros of 0 ( ). Therefore, exactly for ( , ) in the white region, (181) has a solution ( ) ≠ 0 − lead to the same equations, so it suffices to consider > 0. A lengthy computation leads to ( ) = 2 0 ( ), where 0 and 1 are Bessel functions of the first kind and where we interpret √ 2 + 0 ( in case 2 + < 0. It is clear that necessarily 0 ( ) ≠ 0 has to hold to allow for ( ) ≠ 0. Moreover, the dispersion relation can be written as where ∶= ∕ 2 , ∶= > 0. Due to this relation and under the assumption 0 ( ) ≠ 0, which yields ( ) ≠ 0, it is clear that the transversality condition is always satisfied, provided that the corresponding kernel is one-dimensional. Since ( , ) depends in a rather complicated way on , we do not further study the dispersion relation and the possibility of multidimensional kernels rigorously; it is, however, interesting to notice that, due to (181), such a study would be independent of . Instead, we only plot in Figure 3 the set of points ( , ) for which ( , ) > 0 and 0 ( ) ≠ 0, that is, exactly the points that allow for solutions of the dispersion relation (181) with ( ) ≠ 0.

GLOBAL BIFURCATION
The theory for local bifurcation having setup, we now turn to global bifurcation, which is, of course, the main motivation of our formulation "identity plus compact." To this end, we first state the global bifurcation theorem by Rabinowitz.
Theorem 3 with chosen to be the interior of ℝ ×  . Thus, on each ℝ ×  ,  coincides with its counterpart obtained from Theorem 3. Since > 0 is arbitrary and ℝ ×  = ⋃ >0 (ℝ ×  ), it is evident that necessarily after using Sobolev's embedding theorem, the Calderón-Zygmund inequality (see Ref. 39(Chapter 9); notice that on the right-hand side, the term ‖ ‖ (Ω  0 ) can be left out because of unique solvability of the Dirichlet problem associated with ), and changes of variables via [ ] and via cylindrical coordinates in ℝ 5 and ℝ 3 . In the above,Ω denotes a periodic instance of Ω = [ ](Ω 0 ) and Ψ, are analogously defined as in the statement of (c). Moreover, the constant > 0 can change in each step.
Finally, we turn to alternative (iii). If Equation (185) holds, but not (i)(b), then clearly, we find a sequence as described in (iii) due to the compact embedding of 2, 0,per,e (ℝ) in Remark 4. In a two-dimensional situation without surface tension (and with gravity), sometimes, an alternative such as (ii) above can be eliminated. The strategy to this end typically relies on maximum principle arguments, which, however, appear to be unavailable when capillary effects are taken into account. Therefore, it is unclear whether and, if so, how (ii) can be eliminated in the present situation.
We also have the following.