## Abstract

The Kelvin–Helmholtz instability is present in the motion of a vortex sheet without surface tension. This can be seen from the linearization of the equations of motion, and there have also been proofs of ill-posedness for the full nonlinear equations. In the presence of surface tension, the linearized equations no longer exhibit an instability, and it has been believed that the full equations should then be well-posed. In this paper, I sketch a proof that the vortex sheet with surface tension is well-posed in the case of both two- and three-dimensional fluids. The proof in the case of three-dimensional fluids is the joint work with Nader Masmoudi. The method is to first reformulate the problem using suitable variables and parametrizations, and then to perform energy estimates. The choice of variables and parametrizations in the two-dimensional case is the same as that of Hou *et al*. in a prior numerical work.

## 1. Introduction

The vortex sheet is the free boundary between two fluids shearing past each other. The fluid motion is described by the Euler equations and is taken to be irrotational. Rather than being identically zero, the vorticity is measure-valued; the free boundary between the fluids is the support of the vorticity. The following are the jump conditions at the free boundary:(1.1)where *τ* is the constant coefficient of surface tension and *κ* is the mean curvature of the free boundary. The jump condition for the pressure in equation (1.1) is known as the Laplace–Young boundary condition. The remaining boundary condition is that there is a jump in the tangential component(s) of the velocity; the magnitude in the two-dimensional case is given by the vortex sheet strength, and it is similar in the three-dimensional case.

In the case of two-dimensional fluids (and thus a one-dimensional free boundary), the linearized (about the flat equilibrium) equation of motion of the vortex sheet iswhere *ξ* is the variable in the Fourier transform and *τ* is the coefficient of surface tension. We can see, then, that the motion is linearly stable if *τ*>0, as there is a maximum growth rate. In the case when *τ*=0, the motion is linearly unstable. This is known as the Kelvin–Helmholtz instability. (The linear stability analysis has also been performed when linearizing about an arbitrary state; it was found that even far from equilibrium, the problem is linearly stable in the presence of surface tension (Beale *et al*. 1993).) Thus, it has been believed that surface tension regularizes the Kelvin–Helmholtz instability, even for the full nonlinear equations. This was proved in the case of two-dimensional fluids by Ambrose (2003) and in the case of three-dimensional fluids by Ambrose & Masmoudi (in press).

The method used to demonstrate the well-posedness in the presence of surface tension is to first reformulate the equations of motion using natural variables and a favourable choice of parametrizations, and then to use the energy method. Surface tension enters the equations of motion via curvature of the interface, which is very nonlinear and involves multiple derivatives when expressed using the Cartesian coordinates of the free surface. Following the approach of Hou *et al*. (1994), we choose to instead describe the free surface in two dimensions using the tangent angle the curve forms with the horizontal and arclength. Furthermore, in two dimensions, we use an arclength parametrization of the curve. This is described in more detail in §2. In three dimensions, we generalize this approach. In this case, we describe the surface by its mean curvature and first fundamental form, and choose a parametrization which enforces certain constraints on the first fundamental form. This is described in more detail in §3.

While the unregularized vortex sheet is ill-posed, work has been done on its solutions in various senses. Most famously, Delort (1991) has proved that there exist weak solutions of the Euler equations with vortex sheet initial data when the vortex sheet strength is of a single sign. This work has been extended by Lopes Filho *et al*. (2001) to cover the case of both positive and negative vortex sheet strength in the presence of certain symmetries. The reader may also be interested in Majda (1993), Evans & Müller (1994) or Schochet (1995) on the topic of the Delort solution. There are also works on the unregularized vortex sheet as a well-posed problem in analytic function spaces: Caflisch & Orellana (1986), Lebeau (2002), Hou & Hu (2003) and Wu (2006).

The classical vortex sheet is typically understood to encompass the case in which the two fluids have the same density. The well-posedness results described in this paper apply to the more general case in which the two fluids have arbitrary (constant) densities. A special case is then the water wave, in which one of the fluids has unit density and the other has zero density (i.e. there is only one fluid present). The water-wave case has been studied in more detail. It is notable that the water-wave case is the only case of the vortex sheet problem which is well-posed without any of the typical regularizations (viscosity, surface tension, etc.). The well-posedness of water waves for a wide class of initial data was first proved by Wu (1997, 1999). In those papers, the effect of surface tension was not included. As noted previously, the well-posedness of two-dimensional water waves with surface tension was proved by Ambrose (2003). More recently, Lannes (2005) has proved the well-posedness of water waves in the case of finite depth. All these papers treated the case in which the water wave is irrotational in the bulk of the fluid. One could also study the case in which, in addition to the measure-valued vorticity on the free surface, there is also a smooth distribution of vorticity in the interior of the fluid region. This problem has been shown to be well-posed without surface tension by Lindblad (2005), and both with and without surface tension by Coutand & Shkoller (in press). However, it is worth reiterating that in the water-wave case, the problem is well-posed in any case. It is only in the presence of a second fluid that there is the Kelvin–Helmholtz instability.

## 2. The two-dimensional vortex sheet

We assume the free boundary, **X**=(*x*, *y*), to have normal velocity *U* and tangential velocity *V* such thatWe use *θ* and *s* to denote the tangent angle which the curve **X** forms with the horizontal and the arclength, respectively. We have *θ*=arctan(*y*_{α}/*x*_{α}) and . We can infer the following evolution equations for *θ* and *s*_{α}:(2.1)(2.2)As mentioned previously, we will take an arclength parametrization of the curve. To enforce the arclength parametrization, we use equation (2.2) to define *V*. In particular, we set *s*_{αt}=*L*_{t}/2*π* in the horizontally periodic case and *s*_{αt}=0 in the whole-line case. Integrating, we find *V*. In what follows, we will specialize to the whole-line case, so that *s*_{α}=1.

We will need the Hilbert transform, *H*. This is a singular integral operator, given by the equationThe symbol of the Hilbert transform isAn important property of the Hilbert transform is *H*^{2}*f*=−*f* when . We will also make use of the operator(The operator *D*_{α} is the differentiation with respect to *α*; we sometimes denote differentiation with subscripts, and sometimes with the operators.) Note that *Λ* is self-adjoint, and thatis equivalent to ‖*f*‖_{1/2}.

Before continuing, we note that in the two-dimensional case, it is sometimes convenient to represent the fluid domain as being the complex plane instead of ^{2}. Thus, we introduce the mapping *Φ*, which is simply the identification of ^{2} to , i.e. *Φ*(*a*, *b*)=*a*+*ib*. We will denote the conjugate of a complex number *w* as *w*^{*} and the image under *Φ* of the free surface **X** by *z*, i.e. *z*=*Φ*(**X**).

As usual, we are able to recover the fluid velocity from the vorticity by inverting the curl. In the case of a vortex sheet, the vorticity is an amplitude *γ,* known as the vortex sheet strength, times the Dirac mass supported on the free boundary. The Biot–Savart law then leads us to the Birkhoff–Rott integral(2.3)The normal velocity of the free surface must be determined by the fluid dynamics. This implies that . Thus, we see that when we wish to make sense of the evolution equation for *θ*, we need to understand the term . To understand **W**, we use a Taylor expansion(2.4)

We introduce the operator [**X**], which isNote that the integral in this definition is in fact not a singular integral, as the singularities in the terms actually cancel. We can now write **W** as(2.5)We can prove that the operator [**X**] is bounded from *H*^{s−1/2} to *H*^{s+1} when **X** is in *H*^{s+1}. (That [**X**] is smoothing by 3/2 of a derivative is important. There is another important smoothing operator in the well-posedness proof; it is the commutator of the Hilbert transform and multiplication by a smooth function.) We can now see the following formulae:(2.6)Here, the notation *O*(*H*^{j}) denotes a function whose norm in *H*^{j} is bounded by ‖**X**‖_{s+1}+‖*γ*‖_{s−1/2}. The energy function that we will consider soon is equivalent to this norm.

An evolution equation for *γ* can be inferred from the Euler equations; since we are considering potential flow, we use the Bernoulli equation in particular. We getThe relationship between the curvature, *κ*, and *θ* is *κ*=*θ*_{α}/*s*_{α}. We are able to simplify the evolution equation by making use of the formulae for *V*_{α} and stated previously.

We get the following evolution equations:(2.7)and(2.8)

The initial-value problem is specified by equations (2.7) and (2.8), together with the initial conditions(2.9)where *θ*_{0} is taken such that the associated curve **X** satisfies the non-self-intersection condition(2.10)for some positive constant .

We remark that equations (2.7) and (2.8) make clear that the motion is ill-posed in the absence of surface tension. If *τ* is taken to be zero, then we are left with a quasi-linear elliptic system, and it is no surprise then that the initial-value problem is ill-posed. In the case under consideration (i.e. *τ*>0), we instead have a semi-linear hyperbolic system for which energy estimates can be performed.

As we have stated previously, we are interested in proving estimates for the system with **X** in *H*^{s+1}. This implies that we should take *θ* in *H*^{s}. The energy functional will be equivalent to ‖*θ*‖_{s}+‖*γ*‖_{s−1/2}.

Define *E*^{k} by , withThe energy is then

We calculate the time derivatives. We get(2.11)where *R*_{1} is a collection of terms which can be bounded in terms of the energy in a routine way. Of course, *R*_{1} includes the contribution from terms referred as *O*(*H*^{s}) earlier. The next time derivative is(2.12)Again, the term *R*_{2} is made up of terms which can be bounded in terms of the energy in a routine way.

The first term on the right-hand side of equation (2.11) and the first term on the right-hand side of equation (2.12) cancel when added together; this requires only the fact that *Λ* is self-adjoint and an integration by parts. Furthermore, the second term on the right-hand side of equation (2.11) can be integrated by parts and the result is then bounded in terms of the energy. Similarly, the third term on the right-hand side of equation (2.12) can be integrated by parts and seen to be a commutator; this term is then bounded in terms of the energy. The only term, then, from which is not bounded in terms of the energy is the second term on the right-hand side of equation (2.12). This will cancel with a term from . We have(2.13)Upon integration by parts, we see that the first term on the right-hand side of equation (2.13) cancels with the second term on the right-hand side of equation (2.12). There are additional terms left over, but these are bounded in terms of the energy. We have provedfor some positive constant *c* and some *p*>1.

This energy estimate leads to the main result of Ambrose (2003).

*Given an initial condition* , *such that the associated curve* **X** *satisfies the condition* *(2.10)*, *there exists a unique solution to the initial-value problem for the vortex sheet with surface tension, given by equations* *(2.7)–(2.9)*. *There exists a positive time T*, *such that the solution* (*θ*, *γ*) *is in C*([0, *T*]; *H*^{s}×*H*^{s−1/2}).

In Ambrose (2003), a continuous-dependence theorem is also proved; let *B* and *B*′ be the space *H*^{s}×*H*^{s−1/2} and *H*^{1}×*H*^{−1/2}, respectively.

*If* (*θ*, *γ*) *and* (*θ*′, *γ*′) *are solutions of the initial-value problems* *(2.7) and (2.8)* *with the corresponding initial values* (*θ*_{0}, *γ*_{0})∈*B and* , *then there exists a positive constant c such that*

Of course, the continuous dependence can be extended to higher Sobolev spaces by interpolation, since the energy estimate provides a uniform bound in higher spaces.

## 3. The three-dimensional vortex sheet

In the case of a two-dimensional free surface in the three-dimensional fluids, there is no direct analogue of arclength. However, we can still make a favourable choice of parametrizations. We denote the free surface asWe have the unit tangent and normal vectorsWe will sometimes use the notation . We say that the surface moves with normal velocity *U* and tangential velocities *V*_{1} and *V*_{2}, such that(3.1)As in the lower-dimensional case, we must take *U* to be the normal component of the Birkhoff–Rott integral, but we are free to choose *V*_{1} and *V*_{2}. We choose them to enforce the conditions(3.2)The choice of conditions (3.2) for this problem was suggested to the author by J. Shatah. We use the name *E* for the quantity **X**_{α}.**X**_{α}. The striking feature of this choice of parametrizations is a gain of regularity in *E*. If we have **X**∈*H*^{s+1}, then one might expect *E,* defined in terms of first derivatives of **X**, only to be in *H*^{s}. This turns out not to be the case; in fact, *E* will also be in *H*^{s+1}. This can be seen from the calculation(3.3)We see that the right-hand side is in *H*^{s−1}; inverting the Laplacian, we conclude that *E* is in *H*^{s+1}, as claimed.

The conditions (3.2), taken together with the equation of motion (3.1), lead to a pair of elliptic equations which must be satisfied by the tangential velocities

As in the case of two-dimensional fluids, we will need a non-self-intersection condition. We again must have the condition(3.4)for some positive constant , satisfied uniformly by the initial data. We also remark that we must have a condition on the initial parametrization, i.e. we must havefor some positive constant , uniformly in .

From a geometric point of view, it is not immediately clear what the proper analogue of tangent angle should be. We remark that instead of having evolved *θ* in the two-dimensional case, we could have just as easily evolved its derivative, *θ*_{α}. Having used an arclength parametrization, *θ*_{α} is essentially the same as the curvature. This presents us with a good choice of variable in the three-dimensional case. The primary spatial variable for the location of the vortex sheet which we will evolve is the mean curvature. (This is also good from the point of view of the equations of motion, in that surface tension again enters via the Laplace–Young equation involving the mean curvature.) To this end, we introduce the components (*L*, *M* and *N*) of the second fundamental form of the surface **X** such that(3.5)The mean curvature, *κ*, is thenWe are able to infer an evolution equation for *κ* from equation (3.1). A convenient way of writing this equation is(3.6)

As in the lower-dimensional case, we need to evolve a quantity related to the fluid velocity; we used *γ*, the vortex sheet strength, previously. Recall that it is possible to understand *γ* as *γ*=*μ*_{α}, where *μ* is the jump in the velocity potential across the free surface. Furthermore, in the previous case, we could have just as easily evolved *μ* as we did *γ*. Thus, in the present case, we choose *μ* as our variable related to the fluid velocity. We have the evolution equation for *μ,*(3.7)A version of the derivation leading to equation (3.7) can be found in Baker *et al*. (1982).

The Birkhoff–Rott integral isIn the above equation, the variables followed by a prime are evaluated at , while the variables without a prime are evaluated at . For a discussion of the Birkhoff–Rott integral in the case of three-dimensional fluids, we refer the reader to Saffman (1995) or Caflisch & Li (1992). As in the previous case, we use a Taylor expansion to understand **W** better. However, the calculation in the present case is much more delicate owing to the properties of singular integrals in two dimensions.

In making the approximation, we are led to consider various operators which include the Riesz transforms. These transforms, *H*_{1} and *H*_{2}, are given byWe introduce the operator *Λ*=*H*_{1}*D*_{α}+*H*_{2}*D*_{β}. The symbols of the Riesz transforms and *Λ* areThe following properties of the Riesz transforms are important:

By considering the operators which bear some similarity to the introduced in §2, we are able to find the following formulae for **W**:(3.8)(3.9)(3.10)(3.11)(3.12)As before, *O*(*H*^{j}) denotes any function whose norm in *H*^{j} can be bounded by ‖**X**‖_{s+1}+‖*μ*‖_{s+1/2}.

In view of these formulae for derivatives of **W**, we can rewrite the evolution equations (3.6) and (3.7) so that they are more readily estimated. We get(3.13)

For *μ*, it is easier to write an appropriate evolution equation for *Λμ* than for *μ*. That is, the term which can be seen as contributing to the Kelvin–Helmholtz instability in the *μ* evolution equation can much more clearly be seen if we take one spatial derivative. We get(3.14)

In the lower-dimensional case, the choice of an arclength parametrization leads to a semi-linear hyperbolic system of equations of motion. In the present case, we are only able to reduce to a quasi-linear system of hyperbolic equations. However, the gain of one derivative in *E* implies that the regularity of the coefficients in the terms with the most derivatives will not be an issue. We will have to symmetrize the system in order to get the energy estimates, as is customary for this kind of system. Since we will have *κ* in *H*^{s−1} and *μ* in *H*^{s+1/2}, we now introduce *A*, which has the regularity of 3/2 of a derivative of *μ*. We will use *A* to replace *μ*; note that both *A* and *κ* will now be in *H*^{s−1}. In addition, we pay particular attention to the placement of factors of . We haveFor a two-vector *v* and a scalar *u*, we define the operators and ^{*} byWe can now write the evolution equations for *κ* and *A* in their final form as(3.15)(3.16)The vector **T** is the vector of transport speeds,The term (*κ*) is the source of the Kelvin–Helmholtz instability. (As we have indicated previously, if we had *τ*=0, then the leading-order term in the *A*_{t} equation would be (*κ*) and the evolution equations would then be a quasi-linear elliptic system. The initial-value problem would then be ill-posed. This is the Kelvin–Helmholtz instability.) The definition of the operator iswhere *g* is the operator, *g*=*μ*_{α}*H*_{1}+*μ*_{β}*H*_{2}.

We now proceed with the energy estimate. As before, the energy will be the sum of three kinds of terms, and the third term will include a higher number of factors of the dependent variables, but fewer derivatives. To simplify matters a bit, we will use the notation Then, taking *k* to be a positive integer, we defineAs before, the energy is now

Note that the energy is equivalent to the *H*^{3n+3/2} norm of *A* and *κ*. (As we have said before that we are taking *κ* in *H*^{s−1}, we have *s*=3*n*+5/2. If we instead wanted to take *s* to be an integer, we would just take a slightly different energy functional.) The term in the energy is like a *H*^{3n+1} norm of *A*, so it is of lower order; it is important to include this in the energy to get a particular cancellation. This is directly analogous to the term in the energy estimate in §2.

We take the first time derivative(3.17)Here, _{1} is the term which comes from the transport term in the evolution equation for *A*; it can be handled in the usual way. In addition, _{1} is the collection of terms which can be bounded in terms of the energy immediately. That is, _{1} is the contribution from terms such as those which were previously denoted as *O*(*H*^{s−1/2}), and terms including *a*_{t}.

The time derivative of is(3.18)We use the fact that the adjoint of ^{*} is ,(3.19)We can rearrange the parentheses to rewrite this as(3.20)Now, we see that when we add equations (3.17) and (3.20), the first terms on the right-hand side cancel exactly.

The time derivative of is(3.21)We rewrite this, simply by using adjoints, as(3.22)

There will now be a cancellation between the first term on the right-hand side of equation (3.22) and the second term on the right-hand side of equation (3.17). Counting derivatives, we see that each of these terms has 1/2 too many derivatives. (For example, the first term on the right-hand side includes 3*k* derivatives on *A* and 3*k*+7/2 derivatives on *κ*, where *k* can be as large as *n*. As each of *A* and *κ* may take 3*n*+3/2 derivatives, this is a total of 1/2 too many derivatives in the worst case.) If we rearrange the factors in one of these terms, we will see that we get a cancellation, with a remainder bounded in terms of the energy.

We make the brief calculation(3.23)Now, we rewrite (just using adjoints) the second term from the right-hand side of equation (3.17) as(3.24)We use the above formula for *κ* and get(3.25)Simply by rearranging the order of the factors, we see that the last integral can be written as(3.26)Those terms comprising _{5} include terms which can be estimated using at most 3*n*+1 derivatives on *κ*, 3*n*+3/2 derivatives on *A*, 3*n*+3 derivatives on *μ* and 3*n*+2 derivatives on *a*. All these can then be bounded in terms of the energy. To conclude the energy estimate, we note that equation (3.26) makes clear that upon adding , and , we have two cancellations, leaving alone the terms which are bounded by the energy.

We have provedwhere *C* is a continuous function. Standard techniques now allow for a proof of existence, uniqueness and continuous dependence of solutions to the initial-value problem. We have the following main theorem of Ambrose & Masmoudi (in press).

*Given an initial condition* (*κ*_{0}, *μ*_{0})∈*H*^{s−1}×*H*^{s+1/2}, *such that the associated surface* **X** *satisfies the condition* *(3.4)*, *there exists a unique solution to the initial-value problem for the vortex sheet with surface tension, given by equation* *(3.6), (3.7)* *and the associated initial conditions. There exists a positive time T, such that the solution* (*κ*, *μ*) *is in C*^{1}([0, *T*];*H*^{s−1}×*H*^{s+1/2}).

As we remarked in the two-dimensional case, a continuous-dependence theorem can also be proved.

## 4. Conclusion

The method described in this paper, in both two and three dimensions, has allowed for the proof of well-posedness of various free surface problems in fluid dynamics: the vortex sheet with surface tension, the water wave and the Hele–Shaw flow. Furthermore, the method is elementary as it relies primarily on energy estimates. The analytical method was developed only after the numerical work by Hou *et al*. (1994, 1997) in two dimensions. It appears that now the analysis will be able to further influence the computational studies of these problems. Furthermore, as there is very little knowledge about the long-time behaviour of free surfaces in fluids, it is hoped that the method of the present paper will be able to provide insight into this behaviour. Indeed, this has already happened to an extent, as the framework described in §2 for two-dimensional fluids has been used by Guo *et al*. (in press) to address the stability issues for the vortex sheet with surface tension.

## Acknowledgments

The author was partially supported by a grant from the New York University Research Challenge Fund and by National Science Foundation grants DMS-9983190, DMS-0406130 and DMS-0610898.

## Footnotes

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

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