We prove that the water-waves equations (i.e., the inviscid Euler equations with free surface) are well-posed locally in time in Sobolev spaces for a fluid layer of finite depth, either in dimension or under a stability condition on the linearized equations. This condition appears naturally as the Lévy condition one has to impose on these nonstricly hyperbolic equations to insure well-posedness; it coincides with the generalized Taylor criterion exhibited in earlier works. Similarly to what happens in infinite depth, we show that this condition always holds for flat bottoms. For uneven bottoms, we prove that it is satisfied provided that a smallness condition on the second fundamental form of the bottom surface evaluated on the initial velocity field is satisfied.
We work here with a formulation of the water-waves equations in terms of the velocity potential at the free surface and of the elevation of the free surface, and in Eulerian variables. This formulation involves a Dirichlet-Neumann operator which we study in detail: sharp tame estimates, symbol, commutators and shape derivatives. This allows us to give a tame estimate on the linearized water-waves equations and to conclude with a Nash-Moser iterative scheme.
1.1. Presentation of the problem
The water-waves problem for an ideal liquid consists of describing the motion of the free surface and the evolution of the velocity field of a layer of perfect, incompressible, irrotational fluid under the influence of gravity. In this paper, we restrict our attention to the case when the surface is a graph parameterized by a function where , denotes the time variable and the horizontal spatial variables. The method developed here works equally well for any integer but the only physically relevant cases are of course , and The layer of fluid is also delimited from below by a not necessarily flat bottom parameterized by a time-independent function . We denote by . the fluid domain at time The incompressibility of the fluids is expressed by .
where denotes the velocity field ( being the horizontal, and the vertical components of the velocity). Irrotationality means that
The boundary conditions on the velocity at the surface and at the bottom are given by the usual assumption that they are both bounding surfaces, i.e. surfaces across which no fluid particles are transported. At the bottom, this is given by
where denotes the outward normal vector to the lower boundary of At the free surface, the boundary condition is kinematic and is given by .
where with , denoting the outward normal vector to the free surface.
Neglecting the effects of surface tension yields that the pressure is constant at the interface. Up to a renormalization, we can assume that
Finally, the set of equations is closed with Euler’s equation within the fluid,
where is the acceleration of gravity.
Early works on the well-posedness of Eqs. (Equation 1.1)-(Equation 1.6) within a Sobolev class go back to Nalimov Reference 27, Yosihara Reference 38 and Craig Reference 10, as far as 1D-surface waves are concerned. All these authors work in a Lagrangian framework, which allows one to consider surface waves which are not graphs, and rely heavily on the fact that the fluid domain is two dimensional. In this case, complex coordinates are canonically associated to the and the incompressibility and irrotationality conditions ( -coordinates,Equation 1.1) and (Equation 1.2) can be seen as the Cauchy-Riemann equations for the complex mapping There is therefore a singular integral operator on the top surface recovering boundary values of . from boundary values of The water-waves equations ( .Equation 1.1)-(Equation 1.6) can then be reduced to a set of two nonlinear evolution equations, which can be “quasi-linearized” using a subtle cancellation property noticed by Nalimov. It seems that this cancellation property was the main reason why the Lagrangian framework was used. A major restriction of these works is that they only address the case of small perturbations of still water. The reasons for this restriction are quite technical, but the most fundamental is that this smallness assumption ensures that a generalized Taylor criterion is satisfied, thus preventing formation of Taylor instabilities (see Reference 33Reference 4 and the introduction of Reference 36). Physically speaking, this criterion assumes that the surface is not accelerating into the fluid region more rapidly than the normal acceleration of gravity. From a mathematical viewpoint, this condition is crucial because the quasilinear system thus obtained is not strictly hyperbolic (zero is a multiple eigenvalue with a Jordan block) and requires a Lévy condition on the subprincipal symbol to be well-posed; one can see Taylor’s criterion precisely as such a Lévy condition (see Section 4.1 below). In Reference 3, Beale et al. proved that the linearization of the water-waves equations around a presumed solution is well-posed, provided this exact solution satisfies the generalized Taylor’s sign condition (which is a weaker assumption than the smallness conditions of Reference 27Reference 38Reference 10). Wu’s major breakthrough was to prove in Reference 36 that Taylor’s criterion always holds for solutions of the water-waves equations, as soon as the surface is nonself-intersect. Her energy estimates are also better than those of Reference 3 and allow her to solve the full (nonlinear) water-waves equations, locally in time, and without restriction (other than smoothness) on the initial data, but in the case of a layer of fluid of infinite depth. The only existing theorems dealing with the case of finite depth require smallness conditions on the initial data when the bottom is flat Reference 10, and an additional smallness condition on the variations of the bottom parameterization when the bottom is uneven Reference 38.
Very few papers deal with the well-posedness of the water-waves equations in Sobolev spaces in the three-dimensional setting (i.e. for a 2D surface). In Reference 22, the generalization of the results of Reference 3 to the three-dimensional setting is proved. More precisely, the authors show, in the case of a fluid layer of infinite depth, that the linearization of the water-waves equations around a presumed solution is well-posed, provided this exact solution satisfies the generalized Taylor’s sign condition. As in Reference 3, the energy estimates provided are not good enough to allow the resolution of the nonlinear water-waves equations by an iterative scheme. In Reference 37, S. Wu (still in the case of a fluid layer of infinite depth) solved the nonlinear equations. Her proof relies heavily on Clifford analysis in order to extend to the 3D case (some of) the results provided by harmonic analysis in 2D. In the case of finite depth, no results exist.
1.2. Presentation of the results
In this paper, we deliberately chose to work in the Eulerian (rather than Lagrangian) setting, since it is the easiest to handle, especially when asymptotic properties of the solutions are concerned. Inspired by Reference 29Reference 13 we use an alternate formulation of the water-waves equation (Equation 1.1)-(Equation 1.6). From the incompressibility and irrotationality assumptions (Equation 1.1) and (Equation 1.2), there exists a potential flow such that and
where we used the notation and Finally, Euler’s equation ( .Equation 1.6) can be put into Bernouilli’s form
As in Reference 13, we reduce the system (Equation 1.7)-(Equation 1.10) to a system where all the functions are evaluated at the free surface only. For this purpose, we introduce the trace of the velocity potential at the surface
and the (rescaled) Dirichlet-Neumann operator (or simply when no confusion can be made on the dependence on the bottom parameterization which is a linear operator defined as ),
which is an evolution equation for the elevation of the free surface and the trace of the velocity potential on the free surface Our results in this paper are given for this system. .
The first part of this work consists in developing simple tools in order to make the proof of the well-posedness of the water-waves equations as simple as possible. It is quite obvious from the equations (Equation 1.11) that the Dirichlet-Neumann operator will play a central role in the proof; we give here a self-contained and quite elementary proof of the properties of the Dirichlet-Neumann operator that we shall need. A major difficulty lies in the dependence on of the operator It is known that such operators depend analytically on the parameterization of the surface. Coifman and Meyer .Reference 9 considered small Lipschitz perturbations of a line or plane, and Craig et al. Reference 12Reference 13 perturbations of hyperplanes in any dimension. Seen as an operator acting on Sobolev spaces, is of order one. In Reference 13, an estimate of its operator norm is given in the form:
for all integer (estimates in Sobolev spaces are also provided). In order to obtain this estimate, the authors give an expression of -based as a singular integral operator (inspired by the early works of Garabedian and Schiffer Reference 17 and Coifman and Meyer Reference 9 on Cauchy integrals) and use a multiple commutator estimate of Christ and Journé Reference 6. Estimate (Equation 1.12) has the interest of being “tame” (in the sense of Hamilton Reference 21; i.e., the control in the norms depending on the regularity index is linear), but is only proved for flat bottoms and requires too much smoothness on a control of : is needed in (Equation 1.12), and hence of with , if one works in a Sobolev framework. A rapid look at equations ( ,Equation 1.11) shows that one would like to allow only a control of in (i.e., and should have the same regularity). Using an expression of involving tools of Clifford Algebras Reference 18 and deep results of Coifman, McIntosh and Meyer Reference 8 and Coifman, David and Meyer Reference 7, S. Wu obtained in Reference 37 another estimate with a sharp dependence on the smoothness of :
for all real numbers large enough. If estimate (Equation 1.13) is obviously better than (Equation 1.12), it has two drawbacks. First, it is not tame, and hence not compatible for later use in a Nash-Moser convergence scheme. Second, its proof requires very deep results, which make its generalization to the present case of finite and uneven bottom highly nontrivial. In this paper, we prove in Theorem 3.6 the following estimate:
for all and where , is a fixed positive real number. This estimate has the sharp dependence on of (Equation 1.13) and is tame as (Equation 1.12). Moreover, it is sharper than the above estimates in the sense that only the gradient of is involved; this will prove very useful here. Estimate (Equation 1.14) also holds for uneven bottoms and its proof uses only elementary tools of PDE: since the fluid layer is diffeomorphic to the flat strip we first transform the Laplace equation ( ,Equation 1.7) with Dirichlet condition at the surface and homogeneous Neumann condition at the bottom into an elliptic boundary value problem (BVP) with variable coefficients defined in the flat strip The Dirichlet-Neumann operator . can be expressed in terms of the solution to this new BVP (see Prop. 3.4). We give sharp tame estimates for a wide class of such elliptic problems in Theorem 2.9. Choosing the most simple diffeomorphism between the fluid domain and as in Reference 12Reference 2 and applying Theorem 2.9 to the elliptic problem thus obtained, we can obtain, via Prop. 3.4, a tame estimate on However, this estimate is not sharp since instead of . as in (Equation 1.14), one would need a control of We must therefore gain half a derivative more to obtain ( .Equation 1.14). The trick consists in proving (see Prop. 2.13) that there exists a “regularizing” diffeomorphism between the fluid domain and the flat strip .
We also need further information on the Dirichlet-Neumann operator. In Theorem 3.10, we give the principal symbol of for all :,
where and where the constant involves the , of a finite number of derivatives of -norm Note in particular that for 1D surfaces, . while for 2D surfaces it is a pseudo-differential operator (and not a simple Fourier multiplier). We then give tame estimates of the commutator of , with spatial (in Prop. 3.15) and time (in Prop. 3.19) derivatives. Finally, we give in Theorem 3.20 an explicit expression of the shape derivative of i.e. the derivative of the mapping , and tame estimates of this and higher derivatives are provided in Prop. ,3.25.
Note that all the above results are proved for a general constant coefficient elliptic operator instead of in (Equation 1.7). This is useful if one wants to work with nondimensionalized equations. This first set of results consists therefore in preliminary tools for the study of the water-waves problem; we would like to stress the fact that they are sharp and only use the classical tools of PDE.
We then turn to investigate the water-waves equations (Equation 1.11). The first step consists of course in solving the linearization of (Equation 1.11) around some reference state and in giving energy estimates on the solution. Using the explicit expression of the shape derivative of the Dirichlet-Neumann operator given in Theorem ,3.20, we can give an explicit expression of the linearized operator Having the previous works on the water-waves equations in mind, it is not surprising to find that . is hyperbolic, but that its principal symbol has an eigenvalue of multiplicity two (i.e., it is not strictly hyperbolic). In the works quoted in the previous section, this double eigenvalue is zero. Due to the fact that we work here in Eulerian, as opposed to Lagrangian, variables, this double eigenvalue is not zero anymore, but , being the dual variable of and , being the horizontal component of the velocity at the surface of the reference state It is natural to seek a linear change of unknowns which transforms the principal part of . into its canonical expression consisting of an upper triangular matrix with double eigenvalue and a Jordan block. Prop. 4.2 gives a striking result: this a priori pseudo-differential change of unknown is not even differential, and the commutator terms involving the Dirichlet-Neumann operator that should appear in the lower-order terms all vanish! This simplifies greatly the sequel.
Having transformed the linearized operator into an operator whose principal part exhibits the Jordan block structure inherent to the water-waves equations, we turn to study this operator The Lévy condition needed on the subprincipal symbol of . in order for the associated Cauchy problem to be well-posed is quite natural, due to the peculiar structure of a certain function : depending only on the reference state must satisfy for some positive constant (this is almost a necessary condition, since the linearized water-waves equations would be ill-posed if one had It appears in Prop. ).4.4 that this sign condition is exactly the generalized Taylor’s sign condition of Reference 3Reference 22Reference 36Reference 37. Assuming for the moment that this condition holds, we use the tools developed in the first sections to show, in Prop. 4.5, that the Cauchy problem associated to is well-posed in Sobolev spaces, and to give energy estimates on the solution. There is a classical loss of information of half a derivative on this solution due to the Jordan block structure, but also a more dramatic loss of information with respect to the reference state which makes a Picard iterative scheme inefficient for solving the nonlinear equation. Fortunately, the energy estimates given in Prop. ,4.5 are tame, and Nash-Moser theory will provide a good iterative scheme. Inverting the change of unknown of Prop. 4.2, tame estimates are deduced in Prop. 4.14 for the solution of the Cauchy problem associated to the linearized operator The last step of the proof consists in solving the nonlinear equations ( .Equation 1.11) via a Nash-Moser iterative scheme. This requires proving that Taylor’s sign condition holds at each step of the scheme (and of course that the surface elevation remains positive!). It is quite easy to see that it is sufficient for this condition to be satisfied that the first iterate satisfies it. Wu proved that this is always the case in infinite depth. We prove in Prop. 4.15 that this result remains true in the case of flat bottoms. For uneven bottoms, however, we must assume that the generalized Taylor’s sign condition holds for the initial data. This can be ensured by smallness conditions on the initial data, but we also give a sufficient condition stating that Taylor’s sign condition can be satisfied for initial data of arbitrary size provided that the bottom is “slowly variable” in the sense that
where is the bottom parameterization, the second fundamental form associated to the surface and , the tangential component of the initial velocity field evaluated at the bottom.
Our final result is then given in Theorem 5.3. For flat bottoms (i.e. it can be stated as: ),
Organization of the paper. Section 2 is devoted to the study of the Laplace equation (Equation 1.7) in the fluid domain, or more precisely to the equation where , is a constant coefficient, symmetric and coercive matrix. In Section 2.1, we show that this equation can be reduced to an elliptic boundary problem with variable coefficients on a flat strip, and sharp tame elliptic estimates for such problems are given in Section 2.2. We then show in Section 2.3 that among the various diffeomorphisms between the fluid domain and the flat strip, there are some that are particularly interesting, which we call “regularizing diffeomorphisms” and which allow the gain of half a derivative with respect to the regularity of the surface parameterization.
Section 3 is entirely devoted to the properties of the Dirichlet-Neumann operator. Basic properties (including the sharp estimate (Equation 1.14) mentioned above) are gathered in Section 3.1. In Section 3.2, we are concerned with the derivation of the principal part of the Dirichlet-Neumann operator, and in Section 3.3 with its commutator properties with space or time derivatives. Finally its shape derivatives are studied in Section 3.4.
The linearized water-waves equations are the object of Section 4. We first show in Section 4.1 that the linearized equations can be made trigonal and prove in Section 4.2 that the Cauchy problem associated to the trigonal operator is well-posed in Sobolev spaces, assuming that a Lévy condition on the subprincipal symbol holds. We also provide in this section tame estimates on the solution. The link with the solution of the original linearized water-waves equations is made in Section 4.3, and the Lévy condition is discussed in Section 4.4.
The fully nonlinear water-waves equations are solved in Section 5. A simple Nash-Moser implicit function theorem is first recalled in Section 5.1 and then used in Section 5.2 to obtain our final well-posedness result.
Here is a set of notation we shall use throughout this paper:
always denotes a numerical constant which may change from one line to another. If the constant depends on some parameters we denote it by ,.
For any we write ,.
For all we write , similarly, we write ; and, for all , ,.
We denote by the set of functions continuous and bounded on together with their derivatives of order less than or equal to endowed with its canonical norm , We denote also ..
We denote by the usual scalar product on .
We denote by or , the Fourier multiplier with symbol ,.
For all we denote by , the space of distributions such that where , denotes the Fourier transform of We also denote ..
If we write ,.
If is a Banach space and if then we write , If . then ,.
For all , denotes the first integer strictly larger than (so that ).
2. Elliptic boundary value problems on a strip
Throughout this section, we work on a domain defined as
where and satisfy the following condition:
(this assumption means that we exclude beaches or islands for the fluid domain, either perturbed or at rest).
We also consider a constant coefficients elliptic operator where , is a symmetric matrix satisfying the following condition:
Finally, we consider boundary value problems of the form
where is a function defined on and are functions defined on Moreover, . denotes the conormal derivative associated to of at the boundary ,
where denotes the outwards normal derivative at the bottom.
2.1. Reduction to an elliptic equation on a flat strip
Throughout this section, we denote by any diffeomorphism between and the flat strip