Water waves with compact vorticity

# Traveling water waves with compactly supported vorticity

Jalal Shatah Courant Institute of Mathematical Sciences
New York University
New York, N.Y. 10012
Samuel Walsh  and  Chongchun Zeng School of Mathematics
Georgia Institute of Technology
Atlanta, GA 30332
###### Abstract.

In this paper, we prove the existence of two-dimensional, traveling, capillary-gravity, water waves with compactly supported vorticity. Specifically, we consider the cases where the vorticity is a -function (a point vortex), or has small compact support (a vortex patch). Using a global bifurcation theoretic argument, we construct a continuum of finite-amplitude, finite-vorticity solutions for the periodic point vortex problem. For the non-periodic case, with either a vortex point or patch, we prove the existence of a continuum of small-amplitude, small-vorticity solutions.

The third author is funded in part by NSF DMS 1101423 and a part of the work was completed while visiting Courant Institute, New York University in 2011 and IMA, University of Minnesota in 2012.

J. Shatah]shatah@cims.nyu.edu S. Walsh]walsh@cims.nyu.edu C. Zengh]zengch@math.gatech.edu

## 1. Introduction

Consider the water wave problem of infinite depth modeled by the free boundary problem of the incompressible Euler equation

 (1.1a) ∂tv+(v⋅∇)v+∇p+ge2=0, (1.1b) ∇⋅v=0

on a moving domain

 Ωt:={(x1,x2)∈R2:x2<1+η(t,x1)}

for some profile function , with the boundary conditions on the free surface

 (1.2a) ηt=−η′v1+v2 (1.2b) p=α2κ,α>0

where is the mean curvature of the surface

 κ(x1)=−η′′(1+(η′)2)32.

Here is the constant gravitational acceleration. The kinematic boundary condition (1.2a) is the requirement that the normal velocity of the boundary agree with that of the fluid and condition (1.2b) means that we include the surface tension in our consideration.

Our goal in this paper is to seek traveling water waves in the form of

 (1.3) v=v(x1−ct,x2),η=η(x1−ct),Ω={(x1,x2)∈R2:x2<1+η(x1)}

with compactly supported vorticity. We shall consider both cases

1. and decay at infinity and

2. and are -periodic in , i.e. and without loss of generality

 (1.4) ∫πL−πLηdx1=0.

Since the arguments for both of these two cases follow a similar procedure, we will focus on the localized case (Loc) and then give outlines for the case .

The overwhelming majority of the research on water waves has been done in the irrotational setting, both for mathematical convenience and on physical grounds. The source of the convenience is clear enough: for velocity fields with gradient structure, the dynamics in the bulk of the fluid are captured simply by Laplace’s equation. This allows the problem to be pushed to the boundary, where it can be recast in a number of ways as a nonlocal equation (e.g, as an integral equation following Nekrasov [15, 17], or using Dirichlet-to-Neumann operators as in Babenko .) Moreover, because Kelvin’s circulation theorem states that an initially irrotational flow will remain irrotational absent external forcing, these considerations are physical. Today, the existence theory for steady water waves is very well-developed (cf., e.g., surveys in [12, 18, 21, 23]), though some important open problems still remain.

Yet rotational waves are completely commonplace in nature. Indeed, the presence of wind forcing, temperature gradients, or even a slight heterogeneity in the density, generates vorticity. The study of rotational steady waves essentially begins with Dubreil-Jacotin in 1934 (cf. ), but the entire topic lay dormant until the relatively recently. The main breakthrough came in 2004, when Constantin and Strauss developed a systematic existence theory for two-dimensional, periodic, finite-depth, traveling gravity waves (cf. ). Both the work of Constantin–Strauss and Dubreil-Jacotin begin with a simple observation: If there are no stagnation points in the flow, then one can use the stream function as a vertical coordinate to fix the domain. Doing so, one ultimately arrives at a quasilinear elliptic PDE on a strip (the quasilinearity coming from the change of coordinates), with a nonlinear boundary condition. Shear flows, i.e., those where the free surface and all of the streamlines are flat, can be easily described in the new coordinates, and so one can build small-amplitude solutions by a perturbative argument. Then, using a degree theoretic continuation method, Constantin and Strauss were able to obtain finite amplitude solutions. Following the publication of , many authors have been able to generalize this approach, and we now have a bounty of analogous results for other physical regimes (cf., e.g., [25, 13, 29, 30]), for a class of weak solutions (cf. ), and even for some types of waves with stagnation points (cf. [26, 10, 11]); a recent account of the field is given in .

Though they differ in many essential details, all of these works rely on the fact that the vorticity is constant along streamlines for steady flows away from stagnation. Indeed, even the papers dealing with the stagnant case, where this does not follow from physical principles, impose it a priori. Another common feature is that they all use shear flows as their class of trivial solutions for the bifurcation argument. Taken together, these two facets make it impossible to construct waves with localized vorticity. To see why conceptually, simply note that in a shear flow, every point sits on a unique streamline extending to in the direction of wave propagation. Along that streamline, the vorticity is constant, and hence it cannot be localized unless it happens to vanish identically. Naturally, small amplitude perturbations of these flows will share this property. Indeed, if there are no stagnation points, all of the streamlines are unbounded. With stagnation, it is possible that some of the streamlines become closed upon bifurcation — a phenomenon referred to as cats’ eyes — but some unbounded streamlines will necessarily persist.

Thus a vast gulf exists between the well-studied irrotational steady waves, where of course the vorticity vanishes identically, and the current literature on rotational waves, where it cannot be allowed to vanish at infinity. The objective of the present paper is to address this gap, and, in a sense, our approach weds the two outlined above. We will be able to describe traveling waves where point vortices or eddies are suspended in an otherwise irrotational flow. In particular, the traveling waves with compactly supported vortex patch found in this paper have finite energy.

The vorticity of a 2-d velocity field is defined to be the distribution

 (1.5) ω:=∂x1v2−∂x2v1.

When is a finite measure, we may define the vortex strength to be , and in particular,

 (1.6) ϵ:=∫Ωωdx, if ω∈L1(Ω).

In the interior of the fluid, the incompressible Euler equation can be expressed in its vorticity formulation which takes the following form for traveling waves

 (1.7) −c∂x1ω+∇⋅(ωv)=(−ce1+v)⋅∇ω=0.

It means that the vorticity is transported by the flow. We consider two problems

1. , a point mass away from the fluid boundary, without loss of generality, whose concentration point is taken as the origin.

2. and is smooth on which is near the origin.

In fact, we consider both the localized (Loc) and periodic (Per) cases for the point vortex problem (PtV) and only the localized case (LoC) for the vortex patch problem (VPa). The periodic case (Per) can also be considered for the vortex patch problem (VPa), but the computation is too tedious and we simply skip it in this manuscript. Vorticity equation (1.7) can be interpreted in the distribution sense when as in the case of vortex patch (VPa).

If for a solution of the incompressible Euler equation is a point mass concentrated at for some , the general principle that the vorticity is only transported following the fluid flow suggests that might remain as such a point vortex for all . The question is, as the singular vorticity generates a singularly rotational part

 ϵ|x−¯x|2(¯x2−x2,x1−¯x1)T

of , following what vector field should the vortex point move? Since the above singular vector field is purely rotational and does not move that particle at away, it is reasonable to expect that the dynamics of the vortex point is governed only by the remaining smooth part of i.e.

 ∂t¯x=(v−ϵ|x−¯x|2(¯x2−x2,x1−¯x1)T)|x=¯x.

This well known result is rigorously established by considering a family of solutions whose initial vorticity limiting (weakly) to a -function (cf. [16, Theorem 4.1, 4.2]). For traveling waves where , this translates to the following weaker form of (1.7)

 (1.8) ce1=(v−ϵ|x|2(−x2,x1)T)|x=0.

Indeed, our own analysis of the vortex patch confirms this for we found that the traveling speed for the travel vortex patch satisfies (1.8) as the diameter of the patch converges to zero.

As another indication how traveling waves with point vortex or vortex patch are different from those close to shear flows, it is worth pointing out that (1.8) and (1.7) imply that the vortex point or some point in the vortex patch is a stagnation point. That is, the physical horizontal speed coincides with the traveling speed . Usually stagnation only occurs as a limiting case in traveling waves constructed via bifurcation from shear flows. They often coincide with the development of a singularity.

Note that the water wave problem is invariant with respect to reflection in . For simplicity in this paper, we consider symmetric traveling wave solutions where and are even in and is odd in .

Our main theorems are outlined in the statement below, while their more precise versions are given in Section 2.

###### Main Theorem.
1. For , there exists a unique traveling water wave solution which is even in with small amplitude and small velocity whose vorticity is given by a delta mass of strength away from the surface.

2. This solution curve of traveling water waves with a point vortex can be extended globally with one of the possibilities:

• either or becomes unbounded,

• the point vortex location becomes arbitrarily close to the water surface along the solution curve, or

• there exists a nontrivial irrotational traveling water wave, with gravity and surface tension, with an interior stagnation point.

3. For , there exist traveling water waves which is even in with small amplitude and small velocity whose vorticity of total strength is compactly supported in a small disk-like region away from the water’s surface.

The construction of small traveling water waves with a point vortex, which is given in section 3, is based on a fairly simple implicit function theory argument.

In section 5, we use a degree theoretic global bifurcation argument to extend the local bifurcation curve beyond the neighborhood of and into the finite-amplitude, speed, and vorticity regime. Part (2) is the result of this process, an alternative theorem in the spirit of Rabinowitz . The global bifurcation curve must either be unbounded, with the separation between the point vortex and the surface limiting to along some sequence, or contain a nontrivial irrotational traveling water wave with gravity and surface tension, whose velocity at some point is equal to the wave speed . Traditionally with global bifurcation arguments, one begins with a statement of this type, and then uses a priori estimates and nodal arguments to rule out one or more of the alternatives. This was the procedure used by Amick, Fraenkel, and Toland to prove the famous Stokes’ conjecture (cf. ), and it was the means by which Constantin and Strauss showed that the limiting waves of their global continuum must approach stagnation (cf. ). But, both of these results are for waves without surface tension, which is important because it renders the maximum principle dependent nodal arguments tractable. For rotational capillary waves, one of the authors has proved some theorems that go beyond simply the Rabinowitz-type (cf. ), but the large-amplitude regime is still mostly open. Intuitively, though, point vortices are nearly irrotational so that one might hope that Theorem 2.2 can be extended to match the state-of-the-art for irrotational waves. This is a very interesting question, but beyond the scope of the current paper.

The third part of the main theorem, proved in Section 4, deals with the other class of vorticity that we study, the vortex patch. These are solutions where and is supported in a compact region . One can view the point vortex as a limit of vortex patches when the size of the patch taken to zero. We will require that be continuous on , smooth in , but not necessarily across . Reformulating the vorticity equation in terms of the relative stream function, what results is a nonlinear elliptic free boundary problem, very much in the same vein as Constantin–Strauss. However, rather than perturbing from a shear flow, our point of bifurcation will be radial solutions on a ball. Moreover, we use conformal mappings to fix the support of the patch rather than streamline coordinates. By construction, then, the streamlines on which the vorticity is nonzero will be closed. On the other hand, the boundary motion of the air–water interface is dictated by (2.5) and (2.8), just as in the irrotational setting. To couple the two requires that some matching be done on .

A few remarks about the hypotheses and possible extensions. For the vortex patch problem, the interplay between the interior dynamics and the boundary motion are more intricate, which is why we need the optimal regularity furnished by fractional order Sobolev spaces. Concerning the evenness assumption, while it turns out to simplify the computation, it should be noted that this type of symmetry is often expected for traveling waves. Indeed, it is known that in a number of regimes, all traveling waves (with monotonic profiles) have an axis of even symmetry (cf., e.g., [8, 24, 4, 24, 27]).

A global continuation along the lines Theorem 2.2 seems an order of magnitude more difficult to carry out for the surface . The main obstruction is that, as can be seen in section 5, the global theory requires being able to prove compactness properties of the linearized operator not only at , but anywhere along the continuum. While this is not a problem for the free surface equations, the elliptic PDE for the vortex dynamics requires significantly more finesse. We therefore do not consider this issue in the present work.

The structure of the paper is as follows. In Section 2, we start to reformulate the problems into forms more suitable for our analysis. In Section 3, we give the proof of local bifurcation for traveling waves with a point vortex. The global theory of the point vortex case is developed using a degree theoretic argument in Section 5. We address the vortex patch problem in Section 4. Additionally, some auxiliary lemmas and technical facts are collected in an appendix. Finally, though we shall always define a notation when it is introduced, we include a table outlining our conventions at the end of the appendix.

## 2. Framework and main theorems

In this paper, we seek traveling waves near the trivial state, namely and where is given in the definition of in (1.3). In the interior of , the incompressible Euler Equation of traveling waves is equivalent to the vorticity equation (1.7) (possibly in a weak sense). In addition, there are two more conditions on , the Bernoulli condition, and the kinematic condition, which come from (1.2a) and (1.2b), respectively. We will introduce the stream function and the velocity potential to reformaute the problem.

Stream functions. On the one hand, in two space dimensions, it is convenient to write a divergence free vector field as the skew gradient of a stream function

 v=∇⊥Ψ=:(−∂x2ψ,∂x1ψ)T,ΔΨ=ω.

In the periodic case (Per), even though the domain is homotopic to , and thus not simply connected, the decay of at and the periodicity of ensure that is single valued. Based on the convolution with the Newtonian potential, the vorticity naturally generates a corresponding part of the stream function where satisfies

 ΔG=ωϵ−δ(⋅−2e2)

along with -periodicity in in the periodic case (Per). More explicitly, in the localized case (Loc),

 (2.1) G=(12πϵlog|⋅|)∗ω−12πlog|⋅−2e2|.

In the periodic case (Per) we need to sum up the above stream functions generated by the vorticity in each period and thus for ,

 (2.2)

The ‘P.V.’ above means the principle value which ensures certain convergence of the summation. In the point vortex case (PtV) when , obviously the above convolutions simply yield a logarithm. While note as is assumed, the purpose of the second logarithm term in the summand above is to ensure that has better decay at , namely in all cases if is compactly supported. Physically, it corresponds to the standard trick of introducing a phantom point vortex in the air region with equal but opposite vortex strength to correct for the lack of integrability of the Newtonian potential in . With given as above, there exists a harmonic function on such that

 (2.3) v=∇⊥Ψ=∇⊥(ψH+ϵG).

We will seek traveling wave solutions with .

Velocity potentials. On the other hand, an irrotational vector field can be (maybe locally) written as a gradient field. In the localized case (Loc) where is simply connected, obviously any has a conjugate harmonic function such that . In the periodic case (Per) where , the decay assumption on as and the harmonicity of imply that the circulation vanishes along any closed curve. Therefore, also has a single valued conjugate harmonic function .

Since the vorticity of the traveling wave solutions under our consideration is supported away from the free surface , is also irrotational in a neighborhood of . Therefore, there exists a function (multi-valued in the periodic case (Per)) such that and we have the decomposition

 (2.4) v=∇Φ=:∇φH+ϵ∇Θ.

Bernoulli equation in terms of steam functions. The incompressible Euler equation of irrotational velocity fields and boundary condition (1.2b) imply the Bernoulli equation of traveling waves on

 −c∂x1Φ+12|∇Φ|2+gx2+α2κ=bon {x2=1+η(x1)}

where is a constant. Though the velocity potential might be multi-valued as pointed out above, note here only is present which is single-valued. This fact allows us to write the Bernoulli equation in terms of the stream function only

 c(∂x2ψH+ϵ∂x2G)+12|∇ψH+ϵ∇G|2+gx2+α2κ=bon {x2=1+η(x1)}.

Let be the trace of on the free surface i.e.,

 ψ(x1)=ψH(x1,1+η(x1))

and thus is the harmonic extension of to the fluid domain (cf. Lemma A.1). From the Bernoulli condition, we arrive at the following equation of only the variable :

 (2.5) 0=c(G(η)ψ+ϵ(−η′,1)T⋅∇G)+12(G(η)ψ+ϵ(−η′,1)T⋅∇G)2−12(1+(η′)2)(ψ′−η′G(η)ψ+ϵ(1+(η′)2)∂x1G)2+g(η+1)+α2κ(η)−b,

where is evaluated at and

 G(η):=√1+(η′)2N(η),

and is the Dirichlet-to-Neumann operator on ; we recapitulate some of the properties of these operators in Lemma A.1. The constant can be determined explicitly. In the localized case (Loc), by taking , we obtain

 (2.6) b=g.

In the periodic case (Per), (1.4) implies

 b=b(ϵ;η,ψ,c)=g+12πL∫πL−πL[c(G(η)ψ+ϵ(−η′,1)T⋅∇G)+12(G(η)ψ+ϵ(−η′,1)T⋅∇G)2−12(1+(η′)2)(ψ′−η′G(η)ψ+ϵ(1+(η′)2)∂x1G)2+α2κ]dx1.

In the above integral, the term integrates to due to the periodicity of . Moreover,

 ∫πL−πL[G(η)ψ+ϵ(−η′,1)T⋅∇G]dx1=∫∂ΩN⋅∇(ψH+ϵG)ds=∫Ωωdx=1.

Therefore we obtain in the periodic case (Per)

 (2.7) b=b(ϵ;η,ψ,c)=g+cϵ+12πL∫πL−πL[12(G(η)ψ+ϵ(−η′,1)T⋅∇G)2−12(1+(η′)2)(ψ′−η′G(η)ψ+ϵ(1+(η′)2)∂x1G)2]dx1.

Kinematic equation. In terms of and , the kinematic condition (1.2a) for traveling waves is written as

 (2.8)

Equations (2.5) and (2.8) in terms of the velocity are derived, e.g., in  for the unsteady problem, here we have simply adapted them to the steady regime and transformed it using the stream function which is more convenient when the interior vorticity is treated.

Point vortex problem. In the localized case (Loc), and thus

 (2.9) G(x)=12πlog|x|−12πlog|x−2e2|.

As discussed in Section 1, (1.7) is replaced by (1.8). In the localized case (Loc) where , according to (2.1) and the symmetry of in , we obtain

 (2.10) c=−(∂x2ψH)(0)−ϵ4π.

In the periodic case (Per) where , (2.2) implies instead

 (2.11) G(x)=12πP.V.∑k∈Z(log|x−2kπLe1|−log|x−(2kπL,2)T|)

and

 (2.12) c=−(∂x2ψH)(0)−ϵ4π∞∑k=−∞1k2π2L2+1.

In the point vortex case (PtV), we will seek solutions with even in and odd in , so that (2.5), (2.8), (2.6), and (2.10) in the localized case (Loc), or (2.5), (2.8), (2.7), and (2.12) in the periodic case (Per), are satisfied.

To state the main theorems in the point vortex case (PtV), we introduce the following spaces of even functions. For the localized case (Loc), let

 Hke(R):={f∈Hk(R):f is even in x1},

be the corresponding homogeneous space, and

 (2.13) X:=Hke(R)×(˙Hke(R)∩˙H12e(R))×R.

For the periodic problem (Per), we simply replace with ,

 (2.14) Xper:=Hkm(LS1)×Hkm(LS1)×R

where

 Hkm(LS1):={f∈Hk(LS1):f% has mean 0, even in x1}.

Our first theorem establishes the existence and uniqueness of a curve bifurcating from the trivial solutions.

###### Theorem 2.1 (Point vortex local bifurcation).

Consider the traveling water wave problem with a point vortex at the origin (2.5), (2.8), (2.6), and (2.10) in the localized case (Loc), or (2.5), (2.8), (2.7), and (2.12) in the periodic case (Per). The following statements hold.

• There exists and a -curve of solutions

 Cloc={(ϵ,η(ϵ),ψ(ϵ),c(ϵ)):|ϵ|<ϵ0}⊂R×X (or R×Xper)

for any with

 (0,η(0),ψ(0),c(0))=(0,0,0,0).

Moreover, in a sufficiently small neighborhood of in (or ), comprises all solutions.

• In the localized case (Loc), the solutions have the asymptotic form

 c(ϵ)=−ϵ4π+o(ϵ2),|ψ(ϵ)|Hk=O(ϵ3)

and

 |η(ϵ)−ϵ24π2(g−α2∂2x1)−1(x21−1(1+x21)2)|Hk=O(ϵ3).
• In the periodic case (Per), the solutions have the asymptotic form

 c(ϵ)=ϵ~c0+o(ϵ2),|ψ(ϵ)|Hk=O(ϵ3),|η−ϵ2η∗|Hk=O(ϵ3)

where

 ~c0=−14π∞∑k=−∞1k2π2L2+1,η∗=−(g−α2∂2x1)−1(~c0∂x2G+12(∂x2G)2−12πL∫πL−πL~c0∂x2G+12(∂x2G)2dx1),

is defined as in (2.11), and is evaluated at .

###### Remark 2.1.

(a) Note the above existence and uniqueness results hold for all , so the solution are actually functions.
(b) The above local uniqueness is stated in the framework of (or ). However, if a velocity field satisfies , , and outside any neighborhood of , it can be written in the form given in (2.3). Therefore, the local uniqueness holds in the space of such vector fields.

The proof, which is given in section 3, is based on a fairly simple implicit function theory argument. In section 5, for the periodic case (Per), we use a degree theoretic global bifurcation argument to extend the curve beyond the neighborhood of and into the finite-amplitude, speed, and vorticity regime. The result is the following alternative theorem in the spirit of Rabinowitz .

###### Theorem 2.2 (Point vortex global bifurcation).

For in the definition of , there exists a connected set of solutions to (2.5), (2.8), (2.7), and (2.12) with . One of the following alternatives must hold:

• There is a sequence that is unbounded in ;

• there exists a nontrivial irrotational (i.e. ) traveling wave solution with stagnation point at the original, or

• along some sequence in , we have .

Localized waves with vortex patch case. In this case, we will look for solutions of (2.5), (2.8), (2.6), and (1.7). In fact, (1.7) is essentially only required on while it should be satisfied on in the distributional sense. Since is an unknown, through a more elaborated procedure using conformal mappings in Section 4, (1.7) will be further transformed to a form more suitable for our analysis. Using a (local) bifurcation theory argument, we are able to construct a curve of small-amplitude, small vorticity, slow speed, and small patch solutions in the neighborhood of the trivial solution.

###### Theorem 2.3 (Vortex patch local bifurcation).

For any and integer or , there exists a three-dimensional surface of solutions to the vortex patch problem

 Sloc={(η(δ,ϵ,τ),ψ(δ,ϵ,τ),ω(δ,ϵ,τ),c(δ,ϵ,τ)):(ϵ,δ,τ)∈U},

where for some ,

The parameterization is such that

 (η(0,0,0),φ(0,0,0),ω(0,0,0),c(0,0,0))=(0,0,0,0),c=ϵ(−14π+O(ϵ+δ)).

Moreover, where and is an closed curve with asymptotic form:

 ∂D(ϵ,δ,τ)={δ(cosθ+τsin(2θ),sinθ−τcos(2θ))+O(δ2(δ+ϵ)):θ∈[0,2π]}.
###### Remark 2.2.

1.) Here we see that, for fixed , as , the wave speed converges to the wave speed of that of the point vortex problem.
2.) In fact, such a smooth family of solution is found for any fixed vorticity strength function from a large class of functions to be introduced in Section 4. By choosing different , potentially a large family of such traveling waves can be found which differ in the term in the above.

## 3. Small amplitude waves with a point vortex

Let us first turn our attention to the point vortex problem. We will only focus on the localized case (Loc) and the proof for the periodic case (Per) follows from exactly the same procedure. Since the traveling wave solutions we are seeking are bifurcated from the trivial solution, and are of order . In principle, the wave speed can be anything for the trivial solution, equation (2.6) implies that is also of this order, which is one of the major differences between the problems with localized vorticity and the one near the shear flows. Therefore it is more convenient to rescale

 η=ϵ~η,ψ=ϵ~ψ,c=ϵ~c.

Using abstract formalism, we can express the governing equations (2.5), (2.8), (2.6), and (2.10) as

 F(ϵ;~η,~ψ,~c)=0,

where and

 (3.1) F1:=~cϵ(G(ϵ~η)~ψ+(−ϵ~η′,1)T⋅∇G)+ϵ2(G(ϵ~η)~ψ+(−ϵ~η′,1)T⋅∇G)2−ϵ2(1+(ϵ~η′)2)(~ψ′−ϵ~η′G(ϵ~η)~ψ+(1+(ϵ~η′)2)∂x1G)2+g~η+α2ϵκ(ϵ~η),F2:=~cϵ~η′+~ψ′+(1,ϵ~η′)T⋅∇GF3:=~c+(∂x2~ψH)(0)+14π

where is defined in (2.9), is evaluated at , is replaced by as in (2.6), the space is defined as in (2.13); and is taken to be

 Y:=Hk−2e(R)×(˙Hk−1e(R)∩˙H−12e(R))×R.

Obviously small traveling waves with a point vortex at the origin corresponds to a zero point of . We are now ready to prove the local bifurcation theorem for point vortices.

###### Proof of Theorem 2.1.

Recall the term has the explicit form

 ∇G=12π(x|x|2−x−2e2|x−2e2|2)|{x2=1+ϵ~η(x1)}.

Thus the term depends analytically on for . Similarly, the operators , , and are also smooth (cf. Lemma A.1), implying that is of class (in fact they are analytic, but this is beyond our needs.)

For , is clear that the following point is in the zero-set of :

 ~η0=0,~ψ0=−G|x2=1=0,~c0=−14π.

Now, denoting

 (DF)0:=(D~η,D~ψ,D~c)F(0;0,0,−14π),

a simple computation reveals that

 (DF)0=⎛⎜ ⎜⎝g−α2∂2x1000∂x100(∂x2⟨H(0),⋅⟩)|(0,0)1⎞⎟ ⎟⎠∈L(X;Y).

Each diagonal entry here is invertible (see Lemma A.1), and thus is an isomorphism. The implicit function theorem immediately implies the local existence and uniqueness of a curve of zero points of parametrized by . Direct expansion shows , and

 ~η=ϵ(g−α2∂2x1)−1(−~c0∂x2G−12(∂x2G)2+12(∂x1G)2)+O(ϵ2)=ϵ4π2(g−α2∂2x1)−1(x21−1(1+x21)2)+O(ϵ2)

where was evaluated at . This in turn yields . The proof for the localized case (Loc) is complete. ∎

## 4. Small amplitude waves with a vortex patch

We now consider traveling waves in the localized case (Loc) where and the vorticity is supported in a compact region . As outlined in section 2, this problem is equivalent to the system given by (2.5), (2.8), (2.6), and (1.7) with unknowns with as a small parameter. Since is supported on an unknown domain , we first further rewrite the problem to address this difficulty.

### 4.1. Reformulation

As one of the unknowns, we consider the vortical region as being a perturbation of , the ball of radius centered at the origin, where is fixed and small. As we assumed and , there is a positive separation between the patch and the air–water interface.

As given in (2.3), recall is the stream function of where is given in (2.1). Define the relative stream function

 f:=Ψ+cx2,

then the vorticity equation (1.7) takes the equivalent form

 (4.1) ∇f⋅∇Δf=∇⊥f⋅∇ω=0.

Before proceeding further, it will be convenient to scale the physical variables, the relative stream function, and the vorticity. With that in mind, define and by111Here and elsewhere in this section, we adopt the convention that quantities scaled in and are identified with tildes.

 (4.2) f(x)=:ϵ˜f(xδ),ω(x)=ϵδ2˜ω(xδ).

Note is thus a small perturbation of and that, as ,

 Δ˜f=˜ω in D0,∫D0Δ~fdx=∫D0˜ωdx=1.

One way to ensure that the vorticity equation (4.1) is satisfied is to require that the vortex lines and streamline coincide, i.e.

 Δ˜f=˜ω=γ(˜f) in % D0,

for some function called the vorticity strength function. As , is a vortex line () and thus a streamline where is a constant. Without loss we may assume that so that, in total, the scaled relative stream function is the solution to the elliptic PDE

 (4.3) {Δ˜f=γ(˜f)in% D0˜f=0on ∂D0.

Since the above elliptic boundary value problem can be approximated by the one with , we impose some conditions on so that it has a non-degenerate solution when . Namely, we assume

 (4.4a) γ∈CN(R),γ(0)=0,γ′(0)<0,γ>0 on R−, (4.4b) there exists a negative radial solution ˜f∗ to (4.3) with D0=B1, and (4.4c) Δ−γ′(˜f∗) is non-degenerate,

where the integer with and being given in Theorem 2.3. We will look for traveling wave solutions with close to for each such fixed .

Solving (4.3) will allow us to determine the vorticity, but before we can do that, we must represent the domain in an analytic way, which is accomplished by using near identity conformal mappings between and . Let be a conformal mapping with domain the unit ball , and satisfying

 (4.5a) ∂¯zΓ=0 in B1,Γ(0)=0,Γ′(0)=1. By identifying z=x1+ix2∈C with the point (x1,x2)∈R2, we may view the dilated domain D0 as the image of B1 under Γ, D0:=Γ(B1), and the unscaled domain D:=δD0. As we are interested in vortex patches that are perturbations of Bδ, we think of |Γ(z)−z| as being small throughout D0. Let us now briefly motivate (4.5a). The first statement is just the conformality, while the second fixes the origin. The third is made in order to eliminate a certain redundancy. Observe that for each σ>0, (δ,Γ) and (δ/σ,σΓ) each result in the same patch D. By fixing Γ′(0)=1, we exclude all but σ=1. In addition, since we look for traveling waves with the fluid domain and the stream functions even in x1, we require that D is symmetric over the x1-axis, which, stated in terms of Γ, is equivalent to (4.5b) ReΓ is odd in x1,andImΓ is even in x1.

It is clear that there is a one-to-one correspondence between a symmetric domain close to and such a near identity conformal mapping satisfying (4.5a) and (4.5b).

The connection between (4.5b) and the symmetry of the domain can be seen as follows. As is analytic, we may express it as a power series

 (4.6) Γ(z)=∞∑n=1anzn=z+∑n≥2anzn.

That and follows from (4.5a). Consider the term , for some . Denote and , which implies

 arg(anzn)=nθ+θn.

Symmetry of over the -axis translates to the requirement that

 cos(n(π−θ)+θn) =−cos(nθ+θn), sin(n(π−θ)+θn) =sin(nθ+θn).

Expanding the first of these identities yields

 (−1)n+1cos(nθ−θn)=cos(nθ+θn).

From this it is apparent that for even, , for odd. That is

 (4.7) Rea2n−1=a2n−1,Rea2n=0,for n≥1.

As a consequence, we have (4.5b). Moreover, the above property of implies that takes the form

 (4.8)

Note that the correspondence between a conformal mapping defined on and the real part of its trace on is one-to-one and onto. Let

 (4.9) β=β(θ)=Re[Γ(eiθ)−eiθ]=∞∑n=2βncos(12(n−1)π+nθ),θ∈S1.

the trivial solution corresponds to . For each , define the space

 (4.10) Xs:={β∈Hs(S1):β=∞∑n=2βncos(12(n−1)π+nθ), {βn}∞n=2⊂R}.

Obviously if and only if .

We consider to be the unknown describing the shape of the vortex patch, and write to emphasize that is defined in terms of . It is not hard to show that is smooth and bounded in the appropriate spaces, cf. Lemma B.1. Where there is no risk of confusion, we shall abuse notation and suppress this dependence entirely.

Letting , the semi-linear problem for becomes

 {Δ˜F=|∂zΓ|2γ(˜F)on B1˜F=0on S1.

We have thus managed to fix the domain. Note, however, that the above equation will not suffice as a definition of , because we need in addition that . With that in mind, we instead consider the problem of

 (4.11) ⎧⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪⎩Δ˜F=a|∂zΓ|2γ(1a˜F)on B1˜F=0on S1∫B1Δ˜Fdx=1, a>0.

Now we proceed to transform the system given by (2.5), (2.8), (2.6), and (1.7) with unknowns to a system of equations with unknowns . The key here is (1.7) which is already expressed by the above semilinear elliptic problem.

Given , by the nondegenercy assumptions on in (4.4), for in a neighborhood of , there exists a locally unique solution to (4.11). Indeed, we show in Lemma B.1 that the solution will depend smoothly on . Once and are known, the scaled vorticity support and define through

 (4.12) ˜ω∘Γ=|∂zΓ|−2Δ˜F in D0 and ˜ω=0 in (D0)c

which is expected to be the scaled vorticity. Accordingly, we define the mapping

 (4.13) ˜H:β∈Xs↦12πa∫B1log|Γ(⋅)−Γ(z)|γ(1a˜F(z))|Γ′(z)|2dz

where