We address the existence of solutions for the free-surface Euler equation with surface tension in a bounded domain. Considering the problem in Lagrangian variables we provide a priori estimates leading to existence of local solutions with the initial velocity in for which the trace on the free boundary belongs to .
In this paper, we address the local existence of solutions to the 3D free-surface incompressible Euler equations
Problems related to local or global existence of solutions of free surface evolution under the Euler flow, with or without surface tension, have attracted considerable attention in the last decades. For both cases different approaches have been developed; however, the search is still in progress for the lowest regularity spaces where the existence or uniqueness of solutions hold. For the history of both problems, cf. [2,17,32] and references therein.
While in the zero surface tension case the problem is known to be unstable, and thus the Rayleigh-Taylor stability condition has to be imposed, this is not necessary when the surface tension is nonzero since the surface tension provides a stabilizing effect close to the boundary.
We may divide the existing results of the rotational case, i.e., when the vorticity is nontrivial, into the Eulerian approach and the Lagrangian one. In the Eulerian approach, Schweitzer has obtained in  a local existence result with the initial velocity in and with a smallness assumption on the height of the interface. The primary tools in  are tangential and time differentiation, up to order three. In , Coutand and Shkoller used the Lagrangian formulation to obtain the local existence with initial data in . The method used by Coutand and Shkoller is, as in , differentiation in space and time up to three times; however, the Lagrangian approach allowed to bypass the smallness assumption on the initial surface. We would like to stress that simple integrations by parts are not by themselves sufficient to close the estimates; additional care, including a careful treatment of the vorticity and the pressure equations, is necessary to close the estimates. In addition, in , a harmonic change of variables was used to overcome a lack of 1/2 derivative in the estimates resulting from tangential and time differentiation.
In  the authors employ ideas inspired by the geometrical description of Euler flows as geodesics on the infinite dimensional group of volume preserving diffeomorphisms to obtain conditional a priori energy estimates for the solutions when the initial velocity belongs to . They also provide estimates which are uniform in surface tension, if additionally a Raleigh-Taylor condition is satisfied. Recently, in [20,21], using a different method, the authors established the local existence when the initial velocity belongs to for every .
Our goal is to revisit the Lagrangian approach to the free-surface rotational Euler equations and provide a priori estimates leading to local existence for the velocity in such that the trace on the free boundary belongs to , lowering the regularity requirements from  and . While the basic framework still involves time and tangential differentiation used in [17,42], we introduce two improvements which allow us to lower the required regularity. The first improvement is the use of Cauchy invariance [11,14,15,23,35,44] recently used in the zero surface tension case in [32,33]. The second improvement is a simple and direct treatment of the pressure, employing the Laplace problem with Neumann boundary conditions.
Before discussing the organization of the paper, we briefly recall the history of free surface Euler equation problems. Early works on the free surface Euler equations involve results on small analytic data [19,38,51]. The important work  considered the viscous case, employing a Lagrangian set-up, subsequently used in many works on the inviscid problem. In [47,48] Wu obtained existence of solutions of the free surface Euler equations in 2D and 3D cases respectively, both addressing irrotational, no surface-tension cases. Positive surface tension was considered by Ambrose and Masmoudi in [4,5], who also studied the zero surface tension limit. The works [17,42,43] then constructed local solutions for the nonzero-surface tension Euler equations; cf. also works [29,40,41,46,50] for the positive surface-tension Navier-Stokes system. For other works on the zero-surface tension case, see [2,3,8–10,13,18,22,27,30,31,34,36–38,45,52,53], for other works on non-zero surface tension, cf. [1,39] while for global existence of solutions, see [24–26,28,49].
The paper is organized as follows. In Section 2, we introduce the Lagrangian setting of the problem and state the main result, Theorem 2.1. Section 3 contains a preliminary lemma containing a priori estimates on the Lagrangian map η and the cofactor matrix a. Section 4 contains the proof of the main statement. It is subdivided into four subsections containing the , , estimates, and the div-curl estimate. In the final section, we collect all the available inequalities and apply the Gronwall lemma.
2.The main result
We consider the 3D Euler equation in the Lagrangian framework over a fixed domain Ω. Let be the flow map under which the initial domain configuration Ω evolves with time, such that . For simplicity, we assume that the initial domain Ω is flat, i.e.,
We now state the main result of this paper.
In this section, we give formal a priori estimates on time derivatives of the unknown functions needed in the proof of Theorem 2.1. We begin with an auxiliary result providing bounds on the flow map η and the matrix a.
Assume that . Let and . With , where C is a sufficiently large constant, the following statements hold:
(i) for ;
(ii) for ;
(iii) for ;
(iv) for and where are such that ;
(v) , for and ;
(vi) for and and ;
(vii) for ;
(viii) for every and all , we have
Above and in the sequel, if the domain of the norm is not specified, it is understood to be Ω.
Proof of Lemma 3.1.
In order to estimate the second derivative of the pressure, we need the following regularity lemma for an elliptic equation with Neumann boundary condition in a smooth (bounded) domain Ω. Assume that satisfies and that for all and , where .
Let q be an solution of the
Proof of Lemma 3.2.
The bounds on the pressure and its derivatives are obtained by solving a linear elliptic equation with Neumann boundary conditions.
Applying the Lagrangian divergence to the evolution equation (2.4) leads to
4.Local in time solutions
4.1. estimate on
Applying to (2.4), multiplying the resulting equation by , and integrating in space and time gives
Now, we bound the second integral
Next, for we proceed as in by first integrating by parts in space
Lastly, we consider . We use that , where the lower order terms are of the form , , (and the resulting integrals are clearly easy to bound). Thus, we estimate only the leading term in . We have
4.2.Tangential estimate on
Applying to the equation (2.4), multiplying by , summing for , and integrating in space and time, we get
4.3.Tangential estimate on
Applying to (2.4), multiplying by , summing for , and integrating in space and time, we get
We recall that , so in particular . By (4.47) with , we have
Recall the Cauchy invariance (cf.  for instance)
5.Closing the estimates
I.K. was supported in part by the NSF grant DMS-1311943 and DMS-1615239. We would like to thank Peter Constantin for useful discussions, and especially for Lemma 3.2 and the inequality (3.30). We also thank Marcelo Disconzi and the referee for useful suggestions.
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