Global Bifurcation Theory for Periodic Traveling Interfacial Gravity-Capillary Waves
We consider the global bifurcation problem for spatially periodic traveling waves for two-dimensional gravity-capillary vortex sheets. The two fluids have arbitrary constant, non-negative densities (not both zero), the gravity parameter can be positive, negative, or zero, and the surface tension parameter is positive. Thus, included in the parameter set are the cases of pure capillary water waves and gravity-capillary water waves. Our choice of coordinates allows for the possibility that the fluid interface is not a graph over the horizontal. We use a technical reformulation which converts the traveling wave equations into a system of the form “identity plus compact.” Rabinowitz’ global bifurcation theorem is applied and the final conclusion is the existence of either a closed loop of solutions, or an unbounded set of nontrivial traveling wave solutions which contains waves which may move arbitrarily fast, become arbitrarily long, form singularities in the vorticity or curvature, or whose interfaces self-intersect.
We consider the case of two two-dimensional fluids, of infinite vertical extent and periodic in the horizontal direction (of period ) and separated by an interface which is free to move. Each fluid has a constant, non-negative density: in the upper fluid and in the lower. Of course, we do not allow both densities to be zero, but if one of the densities is zero, then it is known as the water wave case. The velocity of each fluid satisfies the incompressible, irrotational Euler equations. The restoring forces in the problem include non-zero surface tension (with surface tension constant ) on the interface and a gravitational body force (with acceleration , possibly zero) which acts in the vertical direction. Since the fluids are irrotational, the interface is a vortex sheet, meaning that the vorticity in the problem is an amplitude times a Dirac mass supported on the interface. We call this problem “the two-dimensional gravity-capillary vortex sheet problem.” The average vortex strength on the interface is denoted by .
In , two of the authors and Akers established a new formulation for the traveling wave problem for parameterized curves, and applied it to the vortex sheet with surface tension (in case the two fluids have the same density). The curves in  may have multi-valued height. This is significant since it is known that there exist traveling waves in the presence of surface tension which do indeed have multi-valued height; the most famous such waves are the Crapper waves , and there are other, related waves known , , . The results of  were both analytical and computational; the analytical conclusion was a local bifurcation theorem, demonstrating that there exist traveling vortex sheets with surface tension nearby to equilibrium. In the present work, we establish a global bifurcation theorem for the problem with general densities. We now state a somewhat informal version of this theorem:
(Main Theorem) For all choices of the constants , , , (not both zero) and , there exist a countable number of connected sets of smooth111Here and below, when we say a function is “smooth” we mean that its derivatives of all orders exist. non-trivial symmetric periodic traveling wave solutions, bifurcating from a quiescent equilibrium, for the two-dimensional gravity-capillary vortex sheet problem. If or then each of these connected sets has at least one of the following properties:
it contains waves whose interfaces have lengths per period which are arbitrarily long;
it contains waves whose interfaces have arbitrarily large curvature;
it contains waves where the jump of the tangential component of the fluid velocity across the interface or its derivative is arbitrarily large;
its closure contains a wave whose interface has a point of self intersection;
it contains a sequence of waves whose interfaces converge to a flat configuration but whose speeds contain at least two convergent subsequences whose limits differ.
In the case that and then each connected set has at least one of the properties (a)-(f), where (f) is the following:
it contains waves which have speeds which are arbitrarily large.
We mention that in the case of pure gravity waves, it has sometimes been possible to rule out the possibility of an outcome like (e) above; one such paper, for example, is . The argument to eliminate such an outcome is typically a maximum principle argument, and this type of argument appears to be unavailable in the present setting because of the larger number of derivatives stemming from the presence of surface tension. In a forthcoming numerical work, computations will be presented which indicate that in some cases, outcome (e) can in fact occur for gravity-capillary waves .
Following , we start from the formulation of the problem introduced by Hou, Lowengrub, and Shelley, which uses geometric dependent variables and a normalized arclength parameterization of the free surface , . This formulation follows from the observation that the tangential velocity can be chosen arbitrarily, while only the normal velocity needs to be chosen in accordance with the physics of the problem. The tangential velocity can then be selected in a convenient fashion which allows us to specialize the equations of motion to the periodic traveling wave case in a way that does not require the interface to be a graph over the horizontal coordinate. The resulting equations are nonlocal, nonlinear and involve the singular Birkhoff-Rott integral. Despite their complicated appearance, using several well-known properties of the Birkhoff-Rott integral we are able to recast the traveling wave equations in the form of “identity plus compact.” Consequently, we are able to use an abstract version of the Rabinowitz global-bifurcation theory  to prove our main result. An interesting feature of our formulation is that, unlike similar formulations that allow for overturning waves by using a conformal mapping, an extension of the present method to the case of 3D waves, using for instance ideas like those in , seems entirely possible.
The main theorem allows for both positive and negative gravity; equivalently, we could say we allow a heavier fluid above or below a lighter fluid. As remarked in , this is an effect that relies strongly on the presence of surface tension. In the case of pure gravity waves, there are some theorems in the literature demonstrating the nonexistence of traveling waves in the case of negative gravity , .
A similar problem was treated by Amick and Turner . As with the present paper they treat the global bifurcation of interfacial waves between two fluids. However, they require the non-stagnation condition that the horizontal velocity of the fluid is less than the wave speed (). Thus their global connected set stops once and there cannot be any overturning waves. Their paper has some other less important differences as well, namely it treats solitary waves and the top and bottom are fixed (). Their methodology is very different from ours as well, since they handle the case of a smooth density first without using the Birkhoff-Rott formulation, and only later let the density approach a step function. Another paper  by the same authors only treats small solutions.
Global bifurcation with , that is, in the water wave case, has been studied by a variety of authors. In particular, global bifurcation that permits overturning waves in the case of constant vorticity is treated in . Another recent paper is , in which a global bifurcation theorem is proved in the case for capillary-gravity waves on finite depth, also with constant vorticity. Both of these works allow for multi-valued waves by means of a conformal mapping. Walsh treats global bifurcation for capillary water waves with general non-constant vorticity in , with the requirement that the interface be a graph with respect to the horizontal coordinate. The methodologies of all of these papers are completely different from the present work.
Our reformulation of the traveling wave problem into the form “identity plus compact” uses the presence of surface tension in a fundamental way. In particular, the surface tension enters the problem through the curvature of the interface, and the curvature involves derivatives of the free surface. By inverting these derivatives, we gain the requisite compactness. The paper  uses a similar idea to gain compactness in order to prove a global bifurcation theorem for capillary-gravity water waves with constant vorticity and single-valued height.
We mention that the current work finds examples of solutions for interfacial irrotational flow which exist for all time. The relevant initial value problems are known to be well-posed at short times , but behavior at large times is in general still an open question. Some works on existence or nonexistence of singularities for these problems are , , . For small-amplitude, pure capillary water waves, global solutions are known to exist in general .
The plan of the paper is as follows: in Section 2, we describe the equations of motion for the relevant interfacial fluid flows. In Section 3, we detail our traveling wave formulation which uses the arclength formulation and which allows for waves with multi-valued height. In Section 4, we explore the consequences of the assumption of spatial periodicity for our traveling wave formulation. In Section 5, we continue to work with the traveling wave formulation, now reformulating into an equation of the form “identity plus compact.” This sets the stage for Section 6, in which we state a more detailed version of our main theorem and provide the proof.
2. The Equations of Motion
If we make the canonical identification222Throughout this paper we make this identification for any vector in . of with the complex plane , we may represent the free surface at time , denoted by , as the graph (with respect to the parameter ) of
The unit tangent and upward normal vectors to are, respectively:
(A derivative with respect to is denoted either as a subscript or as .) Thus we have uniquely defined real valued functions and such that
for all and . We call the normal velocity of the interface and the tangential velocity. The normal velocity is determined from fluid mechanical considerations and is given by:
is commonly referred to as the Birkhoff-Rott integral. (We use “” to denote complex conjugation.)
The real-valued quantity is called in  “the unnormalized vortex sheet-strength,” though in this document we will primarily refer to it as simply the “vortex sheet-strength.” It can be used to recover the Eulerian fluid velocity (denoted by ) in the bulk at time and position via
The quantity is also related to the jump in the tangential velocity of the fluid. Specifically, using the Plemelj formulas, one finds that:
In the above, the “” and “” modifying mean that the limit is taken from “above” or “below” , respectively. If we let be the component of which is tangent to at , then the preceding formula shows:
which is to say that is a scaled version of the jump in the tangential velocity of the fluid across the interface.
As shown in , evolves according to the equation
Here is the Atwood number,
Note that can be taken as any value in the interval Lastly, is the tangent angle to at the point . Specifically it is defined by the relation
Observe that we have the following nice representations of the tangent and normal vectors in terms of :
As observed above, the tangential velocity has no impact on the geometry of . As such, we are free to make anything we wish. In this way, one sees that equations (2) and (7) form a closed dynamical system. In , the authors make use of the flexibility in the choice of to design an efficient and non-stiff numerical method for the solution of the dynamical system. In the article , is selected in a way which is helpful in making a priori energy estimates, and in completing a proof of local-in-time well-posedness of the initial value problem. We leave arbitrary for now.
3. Traveling waves
We are interested in finding traveling wave solutions, which is to say solutions where both the interface and Eulerian fluid velocity propagate horizontally with no change in form and at constant speed. To be precise:
holds for all and . Then is a traveling wave solution with speed .
First we prove (i). Since , we have which immediately gives (9). Then, since we have and thus
And so we have (10).
Now we prove (ii). Suppose gives a traveling wave solution. The reparameterization which yields (11) can be written explicitly. Specifically, condition (9) implies that is a parameterization of . Clearly , and we have the first equation in (11).
Now let be the corresponding vortex sheet-strength for the parameterization of given by . Since we have a traveling wave, we have (10). Define
Then for we have
However, for a point , the Plemelj formulas state that
where the “” and “” signs modifying in the limit indicate that the limit is taken from “above” or “below” , respectively. But, of course, is identically zero so that
which in turn implies . Since this is true for any and any , we see that , the second equation in (11).
We additionally assume that is parameterized to be proportional to arclength, i.e.
for all . One may worry that the enforcement of the parameterization such that in (11) is at odds with this sort of arclength parameterization. However, notice that implies that which in turn implies that (and thus ) does not depend on time. Then the reparamaterization of given by where has . Thus it is merely a convenience to assume (14). We will select a convenient choice for later. Arguments parallel to the above show that implies that and thus we will view as being a function of only.
Now we insert the ansatz (11) and the arclength parameterization (14) into the equations of motion (2) and (7). First, as observed in , we see that elementary trigonometry shows that and (2) are equivalent to
Notice this last equation selects in terms of the tangent angle . That is to say (16) should be viewed as the definition of . On the other hand (15) should be viewed as one of the equations we wish to solve. Using (3), we rewrite it as
The above considerations transform (7) to:
The last part of this expression may be rewritten as follows. Observe that
So (19) simplifies to
which we rewrite as
Note that we have not specified as one of the dependencies of . This may seem unusual, given the prominent role of in computing the Birkhoff-Rott integral . However, given in (14) one can determine solely from the tangent angle , at least up to a rigid translation. Specifically, and without loss of generality, we have
In this way, we view as being a function of , and .
In short, we have shown the following:
It happens that under the assumption that the traveling waves are spatially periodic, (23) can be reformulated as “identity plus compact” which, in turn will allow us to employ powerful abstract global bifurcation results. The next section deals with how to deal with spatial periodicity.
4. Spatial periodicity
To be precise, by spatial periodicity we mean the following:
It is clear if one has a spatially periodic curve then it can be parameterized in such a way that the parameterization is -periodic in its dependence on the parameter. That is to say, the curve can be parameterized such that
It is here that we encounter a sticky issue. As described in Lemma 2, our goal is to find and such that (23) holds and additionally (24) holds. The issue is that, given a function which is periodic with respect to , it may not be the case that the curve reconstructed from it via (22) satisfies (24). In fact, due to (14), the periodicity (24) is valid if and only if
Indeed, we first fix the spatial period . Suppose we are given a real -periodic function for which
so that the period of the curve will not vanish. Then we define the “renormalized curve” as
Of course, this function is one derivative smoother than . A direct calculation shows that
Thus is the parameterization of a curve which satisfies
Now , and the tangent and normal vectors for are given by:
These expressions are not equal to and , as was the case for and in (8).
For a given real function and parametrized curve , define the Birkhoff-Rott integral
Thus . If satisfies (29), we can rewrite this integral as
by means of Mittag-Leffler’s famous series expansion for the cotangent (see, e.g., Chapter 3 of ). Finally, for any real -periodic functions and and any constant , define
In terms of these definitions the basic equations are rewritten as follows:
(Traveling wave equations, spatially periodic version) If the -periodic functions , and the constant satisfy (26) and
then is a spatially periodic traveling wave solution with speed and period for and .
Putting , from the definitions above we have . Thus by Lemma 3 below,
Together with the first equation in (33) and the fact , this gives
If satisfies (29) and is a -periodic function, then
This lemma says that the mean value of the normal component of is equal to zero. This follows from the fact that extends to a divergence-free field in the interior of the fluid region, and from the Divergence Theorem. ∎
5. Reformulation as “identity plus compact”
5.1. Mapping properties
Let be the usual Sobolev space of periodic functions from to whose first weak derivatives are square integrable. Likewise for intervals , let be the usual Sobolev space of functions from to whose first weak derivatives are square integrable. Finally, is the set of all functions from to which are in for all bounded intervals .
By (28), is periodic. Let
Clearly is a complete metric space with the metric of We have the following lemma concerning the renormalized curve :
As already mentioned, is one derivative smoother than . A series of naive estimates leads to the bound on in (35). Next, since belongs to ,
The Cauchy-Schwarz inequality on the cosine term leads to
For , this implies that cannot vanish. Hence and the remaining bounds in (35) follow by routine estimates. The smooth dependence of , and on is a consequence of standard results on compositions. ∎
The most singular part of the Birkhoff-Rott operator is essentially the periodic Hilbert transform , which is defined as
It is well-known that for any , is a bounded linear map from to (the subscript here indicates that the average over a period vanishes). Moreover, annihilates the constant functions and , where In order to conclude that the leading singularity of the function is given in terms of , we require a “chord-arc” condition, as stated in the following lemma.
For and , let the “chord-arc space” be
and the remainder operator be
Then is a smooth map from If then there exists a constant such that for all and for all
The set is the open subset of of functions whose graphs satisfy the “chord-arc” condition. This condition precludes self-intersection of the graph. Note that this is true even in the case where since we have selected the strict inequality in the definition. Of course if , membership of in this set implies that for all .
Note that is real because is real-valued. Also note that the definition of implies that is also real. Thus
Let , and
Then the mappings and are smooth from into . Furthermore, is a smooth map from into .
is a smooth map from into .
The fact that is a smooth map from into follows from the previous corollary and the definition of . Examination of the terms in shows that all but one is a perfect derivative, and thus will have mean value zero on . The remaining term is a constant times , which also has mean zero. Thus ∎
We introduce the “inverse” operator
which is bounded from to . Indeed, it is obvious that , so we only need to demonstrate the periodicity of for any . To this end, we compute
5.2. Final reformulation
Using (37) in the first equation of yields the equation
It will be helpful to break up into the sum of its average value and a mean zero piece, so we let
Applying to both sides and using , we obtain
It will turn out that we are free to specify in advance, and so henceforth we will view as a constant in the equations, akin to , , or .
Now one of the equations we wish to solve is . We use to “solve” this equation for . Keeping in mind that , we define
Summarizing, our equations now have the form