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.
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. (Equation1.1)-(Equation1.6) within a Sobolev class go back to Nalimov Reference27, Yosihara Reference38 and Craig Reference10, 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,Equation1.1) and (Equation1.2) 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 ( .Equation1.1)-(Equation1.6) 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 Reference33Reference4 and the introduction of Reference36). 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 Reference3, 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 Reference27Reference38Reference10). Wu’s major breakthrough was to prove in Reference36 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 Reference3 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 Reference10, and an additional smallness condition on the variations of the bottom parameterization when the bottom is uneven Reference38.
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 Reference22, the generalization of the results of Reference3 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 Reference3, the energy estimates provided are not good enough to allow the resolution of the nonlinear water-waves equations by an iterative scheme. In Reference37, 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.
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 Reference29Reference13 we use an alternate formulation of the water-waves equation (Equation1.1)-(Equation1.6). From the incompressibility and irrotationality assumptions (Equation1.1) and (Equation1.2), there exists a potential flow such that and
where we used the notation and Finally, Euler’s equation ( .Equation1.6) can be put into Bernouilli’s form
As in Reference13, we reduce the system (Equation1.7)-(Equation1.10) 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 (Equation1.11) 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 .Reference9 considered small Lipschitz perturbations of a line or plane, and Craig et al. Reference12Reference13 perturbations of hyperplanes in any dimension. Seen as an operator acting on Sobolev spaces, is of order one. In Reference13, 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 Reference17 and Coifman and Meyer Reference9 on Cauchy integrals) and use a multiple commutator estimate of Christ and Journé Reference6. Estimate (Equation1.12) has the interest of being “tame” (in the sense of Hamilton Reference21; 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 (Equation1.12), and hence of with , if one works in a Sobolev framework. A rapid look at equations ( ,Equation1.11) 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 Reference18 and deep results of Coifman, McIntosh and Meyer Reference8 and Coifman, David and Meyer Reference7, S. Wu obtained in Reference37 another estimate with a sharp dependence on the smoothness of :
for all real numbers large enough. If estimate (Equation1.13) is obviously better than (Equation1.12), 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 (Equation1.13) and is tame as (Equation1.12). 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 (Equation1.14) 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 ( ,Equation1.7) 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 Reference12Reference2 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 (Equation1.14), one would need a control of We must therefore gain half a derivative more to obtain ( .Equation1.14). 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 (Equation1.7). 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 (Equation1.11). The first step consists of course in solving the linearization of (Equation1.11) 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 Reference3Reference22Reference36Reference37. 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 ( .Equation1.11) 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: ),
Let and be such that with , ( depending only on Assume moreover that ).
Then there exists and a unique solution to the water-waves equations Equation1.11 with initial conditions and such that .
Organization of the paper. Section 2 is devoted to the study of the Laplace equation (Equation1.7) 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 (Equation1.14) 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 ).
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.
For all open sets we denote by , , and the canonical norms of , and respectively. When no confusion is possible on the domain we write simply , , and .
Throughout this section, we denote by any diffeomorphism between and the flat strip which we assume to be of the form ,
and we denote its inverse by ,
We always assume the following on :
One has with and Moreover, there exists . such that on .
Finally, we need the following definition:
The most simple diffeomorphism between and is given by
and hence If . and it is clear that , is with -regular, , and ,.
The following lemma shows that the constant coefficients elliptic equation on can equivalently be formulated as a variable coefficients elliptic equation on .
Suppose that the mapping given by ,Equation2.6 satisfies Assumption 2.2. Let with satisfying Equation2.2. Then the equation holds in if and only if the equation holds in where , and are deduced from and via formula Equation2.7, and with ,
Moreover, one has, for all ,
By definition, in if and only if
By definition of one also has ,
Integrating by parts yields therefore that is equal to
and thus to
By definition of and one has , for all Differentiating this identity with respect to . and respectively yields
Using these expressions in the above expressions gives the equality
where is as given in the statement of the lemma. Since one clearly has
We now prove the coercivity of One has, for all .,
and owing to (Equation2.2) we have therefore
The matrix is invertible, and its inverse is given by
so that can be bounded as
Together with (Equation2.11), this estimate yields the result of the lemma.
The next lemma shows how the boundary conditions are transformed by the diffeomorphism .
The first assertion of the lemma is straightforward. We now prove the second. By definition,
Replacing by its expression given in Lemma 2.5, one obtains easily that
which ends the proof of the lemma.
Suppose that the mapping given by ,Equation2.6, satisfies Assumption 2.2. Then is a (variational, classical) solution of Equation2.3 if and only if given by Equation2.7 is a (variational, classical) solution of
where is as given in Lemma 2.5.
The next section is therefore devoted to the study of the well-posedness of such variable coefficients elliptic boundary value problems on a flat strip. Before this, let us state a lemma dealing with the smoothness of the coefficients of Its proof is given in Appendix .A.
Let and assume that the mapping given by ,Equation2.6, is Then one can write -regular. with , and
We have seen in the previous section that the theory of elliptic equations on a general strip of type (Equation2.3) can be deduced from the study of elliptic equations on a flat strip, but with variable coefficients. In this section, we study the following generic problem:
where we recall that denotes the conormal derivative associated to ,
We also assume that satisfies the following coercivity assumption:
The main result of this section is the following theorem.
Let , Let . , and .
The proof below shows that the quantity can be estimated more precisely than Namely, one can replace the quantities . and in both estimates of the theorem by and respectively. This remark is very useful when giving estimates on the Dirichlet-Neumann operator.
The second estimate of the theorem remains of course valid when but in that case, the first estimate of the theorem is more precise. ,
Even though is unbounded, the proof follows the same lines as the usual proofs of existence and regularity estimates of solutions to elliptic equations on regular bounded domains (Reference26Reference19), but special care must be paid to use the specific Sobolev regularity of the coefficients of We only prove the second point of the theorem since the first one can be obtained by skipping the fourth step of the proof below. .
Step 1. Construction of a variational solution to (Equation2.13). We first introduce where , is a smooth compactly supported function such that Classically, one has . and
It follows that is a variational solution of (Equation2.13) if and only if is a variational solution to
Define the space as where the closure is taken relative to the , It is a classical consequence of Lax-Milgram’s theorem that there exists a unique -norm. such that
Step 2. Regularity of the variational solution. We show that using the classical method of Nirenberg’s tangential differential quotients. For all and one has , where , is defined as
we also recall that the adjoint operator of is and that one has the product rule Using ( .Equation2.18) with instead of one gets therefore ,
By the trace theorem and Poincaré’s inequality, we get so that the r.h.s of the above inequality can be bounded from above by ,
for all such that The missing term . is obtained as usual using the equation,
where from which it follows easily that ,
Replacing and by their expressions in the above inequality, and using the estimates (Equation2.16) yields
Step 3. Further regularity. We show by finite induction on that for all , one has ,
By Step 2, this assertion is true when Let . and assume it is also true for For all . we apply ( ,Equation2.24) with to the function :
We now estimate the four terms which appear in the r.h.s. of (Equation2.25). Since one has ,
The second and third terms of (Equation2.25) are very easily controlled. For the fourth one, we use the explicit expression of use the trace theorem, and proceed as for the derivation of ( ,Equation2.26) to obtain
From (Equation2.25), (Equation2.26) and (Equation2.27) (and letting it follows that ), is bounded from above by the right-hand side of (Equation2.24). In order to complete the proof, we still need an estimate of in As in Step 2, such an estimate is obtained using ( .Equation2.21).
Step 4. Further regularity. We show by induction on that (Equation2.24) can be generalized for as
Let and Then one has .
An estimate on in is then provided as before using (Equation2.21), which concludes the induction.
Step 5. Endgame. From the variational formulation of the problem, one easily gets the following lemma, whose proof we omit.
Let , and If . solves the boundary value problem Equation2.13, then
If solves the boundary value problem (Equation2.3), then one can give precise estimates on owing to Prop. ,2.7 and 2.8, and using Theorem 2.9. However, these estimates depend strongly on the diffeomorphism chosen to straighten the fluid domain. The trivial diffeomorphism given in Remark 2.4 is not the best choice possible: in order to control the of its Sobolev component, one needs to control the -norm of the surface parameterization -norm The next proposition shows that there exist “regularizing” diffeomorphisms for which a linear control of the . suffices. -norm
Let , and let , , If there exists . such that on then there exists a diffeomorphism , of the form Equation2.6 such that
is (with -regular);
one has ;
one has and
The diffeomorphism provided by this lemma is a perturbation of the trivial diffeomorphism given in Remark 2.4. The -component remains unchanged, and the behavior at the surface is exactly the same. However, the Sobolev component is half a derivative smoother here than for the trivial diffeomorphism (where it has the smoothness of This is why we say that the diffeomorphism is “regularizing”. ).
Note that if for some and if the condition , is satisfied uniformly in then one has , and This will be used in the proof of Prop. .3.19.
If with is such that on then one can find a neighbourhood , of in such that for all , To each of these . one can associate a regularizing diffeomorphism , by Prop. 2.13. The proof shows that if is small enough, then the mapping is affine. This mapping is therefore smooth and (using the notation of the proof) one can check that for all one has , for some Hence, .
Note that the Jacobian of the mapping is equal to Therefore, if . satisfies the properties stated in the lemma, is indeed a diffeomorphism between and .
Let be given by
we look for such that satisfies
We construct such a mapping using a Poisson kernel extension of Let . be a smooth, compactly supported, function defined on and such that and For any . and , we define , as From this definition it follows also that for all . one has ,