## Abstract

Several recent results on the regularity of streamlines beneath a rotational travelling wave, along with the wave profile itself, will be discussed. The survey includes the classical water wave problem in both finite and infinite depth, capillary waves and solitary waves as well. A common assumption in all models to be discussed is the absence of stagnation points.

## 1. Introduction

The aim of this paper is to review some recent results on the regularity of solutions to the classical water wave problem, i.e. solutions to the flow of an incompressible inviscid fluid subject to gravity and possible effects owing to surface tension. Particular emphasis is placed on the study of *rotational flows*, including discontinuous and unbounded vorticity distributions.

Water waves lie at the forefront of modern applied mathematics and there is a huge and growing literature in this area. However, the majority of the available studies concentrate on the investigation of irrotational flows. This latter—physically quite particular—situation carries a variety of deep mathematical structures, which have been successfully investigated over the last century. However, in view of Helmholtz's law, irrotational flows are quite exceptional and may only serve as a mathematically idealized approximation of the water wave phenomena that Nature confronts us with.

In recent years, there has been some effort to understand different aspects of rotational flows: the existence of such waves and bifurcation patterns [1–4], the results on symmetry properties [5–9], waves of greatest height [10] and internal stagnation points and critical layers [11,12].

In this paper, we shall focus on regularity properties of two-dimensional rotational travelling waves without stagnation points. For irrotational flows, the landmark theorem of Lewy [13], which generalizes the classical Schwarz reflection principle from complex function theory, shows that such profiles must be real analytical (see also [14]). We shall see that, in the absence of stagnation points, the streamlines beneath the profile of a rotational wave are real analytical curves, even for vorticity distributions that are merely integrable (to some power). The proposed approach is quite general and allows the regularity of solutions to a wide variety of water wave models to be studied, including Stokes, capillary and solitary waves, in both finite and infinite depth.

## 2. Mathematical formulation

Our aim is to study two-dimensional water waves travelling over a flat bed. For this, we analyse a cross section of the flow, which is perpendicular to the wave crest. To fix notation, we choose Cartesian coordinates (*x*,*y*) with the horizontal *x*-axis in the direction of the wave propagation and with the *y*-axis pointing vertically upwards. The origin of the coordinate system is located at the mean water level. This means that, in the absence of waves (i.e. for an undisturbed flow), the equation for the flat surface is *y*=0, and the impermeable bed is located at *y*=−*d* for some *d*>0. If waves are present, we denote by *η*=*η*(*t*,*x*) its free surface and let (*u*,*v*)=(*u*(*t*,*x*,*y*),*v*(*t*,*x*,*y*)) be the velocity field of the flow. We neglect effects owing to viscosity, i.e. we view water as an inviscid fluid (see [15]). The equations of motion in the fluid domain
are then given by the system of Euler's equations
2.1aHere, *P* denotes the pressure and *g* is the gravitational constant. In addition, we assume that the fluid is homogeneous and incompressible, so that conservation of mass holds true, i.e.
2.1b

Let us next discuss boundary conditions on the bottom and on the water's free surface. For both components, we have a kinematic condition, saying that the surfaces
and
move with the fluid, so that they always contain the same fluid particles. Introducing particle trajectories (*x*(*t*),*y*(*t*))∈*Ω*_{η}, this implies
and
Using the relation
we find that
2.1cIn the absence of viscous forces, but including possible effects owing to capillarity, the balance of forces on the free surface implies that
2.1dwith the constant atmospheric pressure *P*_{a}, the non-negative surface tension coefficient *σ*≥0 and the curvature *κ* of *Γ*_{η}, oriented in such a way that *κ*(*z*)≥0, if *Γ*_{η} is convex in *z*. Equation (2.1d) is a dynamical boundary condition and is usually called the *Laplace–Young law*.

Our aim is to study travelling wave solutions to problem (2.1). Therefore, denoting the (constant) wave speed by *c*>0, we presuppose that
for and , and relocate our consideration to the moving frame (*x*−*ct*,*y*). In addition, we are interested in periodic waves. Therefore, all functions are assumed to be 2*π*-periodic with respect to the horizontal *x*-variable. For the sake of simplicity, we suppress any further notation referring to this periodicity—also in the function spaces we shall use hereafter.

With this ansatz, the system (2.1) takes the following form:
2.2We remark that system (2.2) is a free boundary problem for the velocity field (*u*,*v*) and that the determination of the wave profile *η* represents an essential part of the analysis. In addition, once a solution (*u*,*v*,*η*) is known, then the pressure *P* can be recovered from Bernoulli's law (see (3.14)).

We are interested in regularity properties of solutions to the above free boundary problem. It turns out that we have to distinguish between the cases *σ*>0 with surface tension effects and *σ*=0 in which effects owing to capillarity are neglected,
2.3We focus our attention first on systems (2.2) and (2.3), but we shall discuss further modelling aspects such as deep-water flows and solitary water waves subsequently.

## 3. The vorticity function

Let be given and denote by the usual Sobolev spaces. The scale of Besov spaces that is needed in our analysis is denoted by (see [16]). By an *L*_{r}-solution of (2.3), we mean a quadruple (*u*,*v*,*P*,*η*) among the class^{1}
3.1satisfying (2.3) in *L*_{r}(*Ω*_{η}), and thus almost everywhere in *Ω*_{η}.

In order to further analyse *L*_{r}-solutions (*u*,*v*,*P*,*η*) of the system (2.3), it is helpful to introduce a stream function *ψ*(*x*,*y*) for (2.3). A *stream function* *ψ* for the velocity field (*u*−*c*,*v*) is a submersion such that its level-lines are tangential to the vector field (*u*−*c*,*v*), i.e.
3.2The continuity equation (2.1b) ensures that, given , the function
3.3satisfies
3.4Therefore, if we now *assume* that
3.5we see that the first requirement of (3.2) is satisfied and that *ψ*, defined by (3.3), is in fact a stream function for (*u*−*c*,*v*). To lay down the size of the constant *m*, we proceed as follows. By the kinematic boundary conditions (2.1c), the stream function *ψ* is constant on the boundary components *Γ*_{η} and *Γ*_{0}. We now choose *m*>0 such that
Assumption (3.5) particularly ensures that there exist no stagnation points in the flow. A further consequence of (3.5) is the fact that the partial *hodograph transformation*
where , is a global diffeomorphism with
The hodograph transformation *T* also provides an easy way to parametrize the streamlines located between *Γ*_{0} and *Γ*_{η}. Indeed, given *μ*∈(−*m*,0), let *Γ*:=*ψ*^{−1}(*μ*) denote the corresponding streamline. Then we have . Hence, introducing the height function by the relation *h*(*q*,*p*)=*y*+*d*, we obtain
3.6

Observe further that (3.4) and (3.1) imply that , so that *T* and *h* are of class as well. We fix *r*>2 and note that by Sobolev's embedding theorem *T* is of class *C*^{1+β}, i.e. we have
3.7for any *β*∈(0,1−2/*r*).

The *vorticity* *ω* of the flow is defined by
If *ω*≡0 then the flow is called *irrotational*. By Helmholtz's theorem (see [17]), we know that the vorticity of each particle in a two-dimensional flow is preserved in time (see also corollary 3.3 below). Therefore, from a physical point of view, irrotational flows represent a quite specific scenario.

We remark that

Our aim is to derive a system of equations for the stream function *ψ* that is equivalent to (2.2). In order to do so, we first have to quantify the vorticity in terms of *ψ*. Note that our assumption on the regularity of the solution (3.1) ensures that *ω*∈*L*_{r}(*Ω*_{η}). Consequently, writing *ω**=*ω*°*T*^{−1} for the pulled back vorticity with respect to the *C*^{1+β}-diffeomorphism *T*, we have *ω**∈*L*_{r}(*Ω*). Denoting the coordinates in *Ω* by (*q*,*p*)=*T*(*x*,*y*), we are going to show that
where is the space of test functions and is the space of distributions in *Ω*. The duality pairing of is denoted by 〈⋅,⋅〉.

### Lemma 3.1 (Matioc [18])

*In the sense of distributions it holds that ∂*_{q}*ω**=0.

### Proof.

Pick . The determinant of the Jacobian *DT* of *T* is given by
3.8where we used assumption (3.5) to derive the latter inequality. Using (3.8) and the fact that *ω**∈*L*_{1,loc}(*Ω*_{η}), we conclude that
3.9Letting *φ*:=*φ**°*T*, we have
and
and therefore
Combining this with (3.9), integration by parts yields
Invoking, finally, Euler's equations, we get
and the proof is completed. ■

### Corollary 3.2 (existence of a vorticity function)

*Given a solution of* (2.3) *of class* (3.1), *there exists a* *γ*∈*L*_{r}((−*m*,0)) *such that*
3.10*The function* *γ* *is called the* *vorticity function* *for the flow* (2.3).

### Proof.

Note that (3.10) is equivalent to the assertion that *ω*=*γ*°*T* and thus to *ω**=*γ*. Pick with and set
We know that *ω**∈*L*_{r}(*Ω*) with . Hence Fubini's theorem ensures that *γ*∈*L*_{r}((−*m*,0)). Now let be given and define
3.11Then for all *p*∈(−*m*,0). Consequently,
is a test function on *Ω* with . Thus, lemma 3.1 implies that . Now (3.11) and the definition of *γ* yield 〈*ω**,*φ**〉=〈*γ*,*φ**〉. Since was chosen arbitrarily, we find that *ω**=*γ*, which completes the proof. ■

Let us briefly discuss the case *σ*>0. In this situation, the dynamical boundary condition (2.1d) requires more regularity if we wish to have solutions that satisfy system (2.2) pointwise. A possible choice of a suitable function space is, for example,
3.12where *α*∈(0,1) is fixed. The proof of corollary 3.2 then shows that, given a solution (*u*,*v*,*P*,*η*) to (2.2) among the class (3.12), there is a vorticity function *γ*∈*C*^{α}([−*m*,0]) such that

Also note that the hodograph transformation *T* and height function *h* are of class *C*^{2+α} in this case.

As a consequence of the existence of a vorticity function, we provide a simple proof of Helmholtz's theorem for system (2.3).

### Corollary 3.3 (Helmholtz's law for (2.3))

*Assume that* *γ* *is of class* *C*^{1} *and that* (*x*(*t*),*y*(*t*)) *is a* *C*^{1}-*trajectory of a particle in the flow* (2.3), *defined on some interval* *J*. *Then* *ω*(*x*(⋅),*y*(⋅)) *is constant on* *J*.

### Proof.

Let us first note that
3.13Furthermore, our regularity assumptions on *γ* and on the particle trajectory (*x*(⋅),*y*(⋅)) imply in combination with corollary 3.2 that with
Hence the proof is completed. ■

We are now prepared to complete the system for the stream function. First, we recall that Bernoulli's principle ensures that
3.14is constant throughout the fluid domain *Ω*_{η}. On the free surface *Γ*_{η}, we therefore get
for some positive constant *Q* and where *σ*≥0. Recalling the kinematic boundary conditions in terms of *ψ*, we find that solutions to (2.2) satisfy the following nonlinear elliptic free boundary value problem:
3.15It turns out that the hodograph transformation *T* is quite useful to study (3.15). Indeed, a direct calculation shows that the height function in case *σ*=0 and *h*∈*C*^{2+α}(*Ω*) in case *σ*>0 satisfies the following nonlinear boundary value problem:
3.16with constants *Q*>0 and *σ*≥0. Let *h* be a solution of (3.16). In view of (3.6), the study of the regularity of the streamlines of (3.15) is reduced to investigate the mappings [*q*↦*h*(*q*,*p*)], where *p*∈[−*m*,0] is fixed.

## 4. Regularity results

We are now going to present several regularity results on the wave profile and the streamlines beneath a periodic travelling water wave. As mentioned earlier, we have to distinguish between the cases *σ*>0 with surface tension effects and *σ*=0 in which effects owing to capillarity are neglected.

Let us first study the case *σ*=0.

### Theorem 4.1

*Consider a solution (u,v,P,η) of (2.3) of class (3.1) with a vorticity function γ∈L*_{r}*((−m,0)). Then the following assertions hold true:*

*Each streamline beneath the wave profile is a real analytical curve.**The wave profile is a smooth curve.**If γ is real analytical then the wave profile is analytical as well.*

### Proof.

(a) Let denote the height function corresponding to the solution (*u*,*v*,*P*,*η*). Recalling (3.6), we have to show that the mapping *q*↦*h*^{0}(*q*,*p*) must be real analytical for all *p*∈[−*m*,0).

In order to relieve our notation, let us consider the Banach spaces
Recall that and that all functions are 2*π*-periodic in the *q*-variable. Recall also that *r*>2.

Let us now introduce the open set
Observe that by construction *h*_{0} belongs to . Further, given , we define *F*(*h*)=(*F*_{1}(*h*),*F*_{2}(*h*))∈*Y* ×*Z* by
4.1

Denoting by a real analytical dependence of functions, we see that
4.2Also observe that *F*(*h*^{0})=0.

Furthermore, if we denote by *τ*_{a} the translation by the amount *ap* in the *q*-variable, that is,
then
Using these relations, a direct calculation shows that
4.3where the operator *K*=(*K*_{1},*K*_{2}) is given by
and
Note that (4.3) holds true for any and any with |*a*| sufficiently small to ensure that . Further remark that
4.4Beside the operators *F* and *K*, we need a further operator , defined by
4.5with *λ*>0 chosen such that
4.6This last condition guarantees that the Fréchet derivative *D*_{1}*Φ*(*h*^{0},0), defined by
is an isomorphism from *X* onto *Y* ×*Z*. Indeed, a direct calculation gives
with
and leading-order coefficients
and
(see [18,19]). To verify that *D*_{1}*Φ*(*h*^{0},0) is injective, assume that there is a *h*∈*X*\{0} such that (*L*,*T*+*λ*∂_{p})*h*=0. We may assume that *h* attains its positive maximum at a point , since if *h* is non-positive we may consider −*h*. In view of *Lh*=0, the strong maximum principle (see [20,21]) implies that *h* must be constant if (*q*_{0},*p*_{0}) lies in *Ω*. This contradicts the boundary condition *h*=0 on *p*=−*m*. Hence, we conclude that *p*_{0}=0. But then we have *h*_{p}(*q*_{0},0)≥0 and *h*_{q}(*q*_{0},0)=0. These relations, in combination with our choice of *λ*>0, lead to a contradiction to the hypothesis (*T*+*λ*∂_{p})*h*=0. The surjectivity of (*L*,*T*+*λ*∂_{p}) follows from the fact that this operator is Fredholm of index 0 (see [18,19]).

By (4.2) and (4.4), we see that . Moreover, since *F*(*h*^{0})=0 and we deduce in view of (4.3) that
4.7

Thus, it follows from the implicit function theorem for real analytical mappings (see [22]) that there is *ε*>0 and a unique such that *Φ*(*a*,*φ*(*a*))=0 for all *a*∈(−*ε*,*ε*). Taking into account (4.7), by uniqueness, we deduce that *τ*_{a}*h*^{0}=*φ*(*a*) for *a*∈(−*ε*,*ε*). In particular, given , we have
Real analyticity being a local property, we conclude that for any *p*∈[−*m*,0). This completes the first part of the proof.

(b) If we take the *q*-derivative of a solution *h*^{0} to the boundary value problem (3.16), one verifies that is the solution of a uniformly elliptic boundary value problem with leading coefficients being Hölder continuous and subject to a uniformly oblique boundary condition which also possesses Hölder continuous coeffients (see [23]). Thus, by elliptic *L*_{r}-theory (see [24]) we conclude that . This procedure can be repeated to obtain for any . Thus, and in particular .

(c) This has been shown in Constantin & Escher [19], using regularity results on elliptic free boundary problems as presented in Kinderlehrer *et al*. [25] directly for the system (3.15) in the case *σ*=0. An argument for solutions to the nonlinear system (3.16) has been provided in Henry [23,26].

This completes the proof. ■

### Remark 4.2

It is worthwhile to add some concluding remarks and comments.

Theorem 4.1 was first established in Constantin & Escher [19] for solutions (

*u*,*v*,*P*,*η*) in the class In this case, the corresponding vorticity function is of class . Subsequently, it was shown in Matioc [18] that it is possible to extend the class of admissible vorticity distributions to . The above proof of theorem 4.1 is a slight extension of the results in Matioc [18].Given

*σ*>0, the assertions of theorem 4.1 hold true for solutions of the system (2.2) belonging to the regularity class mentioned above in remark (a) and corresponding vorticity functions in the class*C*^{1+α}. This has been shown in Henry [23].The idea to introduce auxiliary parameters into a parabolic evolution equation in order to study regularity properties of its solutions goes back to Masuda [27] and independently to Angenent [28]. Still for nonlinear parabolic problems, this approach was further developed [29–31].

It should be noted that a solution

*h*of (3.16) does not belong to if . This follows from the first equation of (3.16). The question of whether the wave profile is analytical for any vorticity distribution belonging to seems to be open.The auxiliary term

*λ*∂_{p}in*Φ*is essential in order to guarantee that the kernel of*D*_{1}*Φ*(*h*^{0},0) is trivial, at least if*h*^{0}is not constant in the*q*-variable: indeed, assume that ∂_{q}*h*^{0}≠0 and denote by the standard translation in the*q*-variable, i.e. . Then . Hence, writing*DF*(*h*^{0}) for the Fréchet derivative of*F*at*h*_{0}, this equivariance implies that*DF*(*h*^{0})[∂_{q}*h*^{0}]=0.The existence of Stokes waves of greatest height with a sharp crest of interior angle of 120

^{°}at a stagnation point where the gradient of the stream function vanishes clearly reveals that the assumption (3.5) cannot be dropped in our approach. We refer to [14,32,33] and the references given there for an overview of the theory of Stokes waves with stagnation points. It is worthwhile pointing out that, at the crest*P*_{c}of a Stokes wave, a somewhat counterintuitive effect occurs. Although, at the crest*P*_{c}, the velocity field satisfies*u*=*c*and*v*=0, so that*P*_{c}is an equilibrium point of system (3.13), the results in Constantin and co-workers [34,35] show that a particle cannot persist at*P*_{c}, being at this location for only an instant. The reason for this ambiguity is the break-down of uniqueness: the physical relevant solution of the system is not the stationary one of (3.13).Unbounded fluid bodies: let us briefly discuss deep-water waves and solitary waves.

In the case of deep-water waves, one drops the boundary condition at

*y*=−*d*and replaces it by the following far-field condition: expressing the modelling assumption that there is no flow at great depths.For deep solitary waves, we impose the condition

Using elliptic regularity theory, the following result has been shown in Matioc [36]:

### Theorem 4.3

*Let (u,v,P,η) be a solution of**belonging to the class**where**is either**in the case of periodic deep-water flows or**in the case of deep-water solitary waves, and**Assume further that the far-field conditions formulated in (g) are satisfied, and that the flow has no stagnation points. Then all streamlines beneath the wave profile, along with the wave profile, are of class**. If the vorticity function γ is real analytical, then the wave profile is analytical as well.*Also in case of unbounded geometries, it is possible to use the translation invariance of the water wave problem to prove analyticity of the streamlines. However, in this situation, one is confronted with some technical difficulties caused by the lack of compact embeddings of the usual function spaces and the lack of the Fredholm property of elliptic boundary value problems. A possible way to overcome these obstacles is the use of weighted function spaces, e.g. as elaborated in Lockhart [37]. In the case of periodic deep-water waves, this has been studied in Matioc [36]. Roughly speaking, if [

*p*↦|*p*|^{3}*γ*(*p*)] belongs to and if the height function*h*^{0}to a solution, along with its first and second derivatives, decays appropriately as , then the streamlines are analytical.It is worth mentioning that so far the only known explicit solution for travelling water waves is available in water of infinite depth: this is Gerstner's original discovery [38]. This wave motion was re-discovered in Rankine [39]. More recently, a three-dimensional version describing explicitly edge waves propagating along a sloping beach was provided in Constantin [40]. Finally, for a modern exposition of Gerstner's wave, the papers by Constantin [41] and Henry [42] are recommended.

Compared with deep-water waves, the case of solitary waves of finite depth seems to be easier. Here, the approach presented in step (a) of the proof of theorem 4.1 works fine: if

*Q*is large enough, a suitably corrected abstract operator, comparable to the operator*Φ*defined in (4.5), can locally be resolved by the implicit function theorem and leads to analytical streamlines beneath the wave profile. For details, we refer to Hur [43].

## Acknowledgements

The author is grateful to the anonymous referees for their valuable remarks.

## Footnotes

One contribution of 13 to a Theme Issue ‘Nonlinear water waves’.

↵1 Here and in the following, denotes the circle of length 2

*π*.

- This journal is © 2012 The Royal Society