The shoreline problem for the one-dimensional shallow water and Green-Naghdi equations
The Green-Naghdi equations are a nonlinear dispersive perturbation of the nonlinear shallow water equations, more precise by one order of approximation. These equations are commonly used for the simulation of coastal flows, and in particular in regions where the water depth vanishes (the shoreline). The local well-posedness of the Green-Naghdi equations (and their justification as an asymptotic model for the water waves equations) has been extensively studied, but always under the assumption that the water depth is bounded from below by a positive constant. The aim of this article is to remove this assumption. The problem then becomes a free-boundary problem since the position of the shoreline is unknown and driven by the solution itself. For the (hyperbolic) nonlinear shallow water equation, this problem is very related to the vacuum problem for a compressible gas. The Green-Naghdi equation include additional nonlinear, dispersive and topography terms with a complex degenerate structure at the boundary. In particular, the degeneracy of the topography terms makes the problem loose its quasilinear structure and become fully nonlinear. Dispersive smoothing also degenerates and its behavior at the boundary can be described by an ODE with regular singularity. These issues require the development of new tools, some of which of independent interest such as the study of the mixed initial boundary value problem for dispersive perturbations of characteristic hyperbolic systems, elliptic regularization with respect to conormal derivatives, or general Hardy-type inequalities.
1.1. Presentation of the problem
A commonly used model to describe the evolution of waves in shallow water is the nonlinear shallow water model, which is a system of equations coupling the water height to the vertically averaged horizontal velocity . When the horizontal dimension is equal to and denoting by the horizontal variable and by a parametrization of the bottom, these equations read
where is the acceleration of gravity. These equations are known to be valid (see [ASL08a, Igu09] for a rigorous justification) in the shallow water regime corresponding to the condition , where the shallowness parameter is defined as
where corresponds to the order of the wavelength of the waves under consideration. Introducing the dimensionless quantities
these equations can be written
The precision of the nonlinear shallow water model (1.1) is , meaning that terms have been neglected in these equations (see for instance [Lan13]). A more precise model is furnished by the Green-Naghdi (or Serre, or fully nonlinear Boussinesq) equations. They include the terms and neglect only terms of size ; in their one-dimensional dimensionless form, they can be written
where is the (dimensionless) water depth and the (dimensionless) horizontal mean velocity. The dispersive operator is given by
and the nonlinear term takes the form
Under the assumption that the water-depth never vanishes, the local well-posedness of (1.2) has been assessed in several references [Li06, ASL08b, Isr11, FI14]. However, for practical applications (for the numerical modeling of submersion issues for instance), the Green-Naghdi equations are used up to the shoreline, that is, in configurations where the water depth vanishes, see for instance [BCL11, FKR16]. Our goal here is to study mathematically such a configuration, i.e. to show that the Green-Naghdi equations (1.2) are well-posed in the presence of a moving shoreline.
This problem is a free-boundary problem, in which one must find the horizontal coordinate of the shoreline (see Figure 1) and show that the Green-Naghdi equations (1.2) are well-posed on the half-line with the boundary condition
Time-differentiating this identity and using the first equation of (1.2), one obtains that must solve the kinematic boundary condition
When , the Green-Naghdi equations reduce to the shallow water equations (1.1) which, when the bottom is flat (), coincide with the compressible isentropic Euler equations ( representing in that case the density, and the pressure law being given by ). The shoreline problem for the nonlinear shallow water equations with a flat bottom coincide therefore with the vacuum problem for a compressible gas with physical vacuum singularity in the sense of [Liu96]. This problem has been solved in [JM09, CS11] () and [CS12, JM15] (). Mathematically speaking, this problem is a nonlinear hyperbolic system with a characteristic free-boundary condition. Less related from the mathematical viewpoint, but closely related with respect to the physical framework are [dP16] and [MW17], where a priori estimates are derived for the shoreline problem for the water waves equations (respectively without and with surface tension).
Though the problem under consideration here is related to the vacuum problem for a compressible gas, it is different in nature because the equations are no longer hyperbolic due the presence of the nonlinear dispersive terms and . Because of this, several important steps of the proof, such as the resolution of linear mixed initial boundary value problems, do not fall in existing theories and require the development of new tools. The major new difficulty is that everything degenerates at the boundary : strict hyperbolicity (when ) is lost, the dispersion vanishes, the energy degenerates ; the topography increases the complexity since it makes the problem fully nonlinear as we will explain later on. An important feature of the problem is the structure of the degeneracy at the boundary. As in the vacuum problem for Euler, it allows to use Hardy’s inequalities to ultimately get the estimates which are necessary to deal with nonlinearities. The precise structure of the dispersion is crucial and used at many places in the computations. Even if they are not made explicit, except at time , the properties of are important. The dispersion appears as a degenerate elliptic operator (see e.g. [BC73] for a general theory). A similar problem was met in [BM06] in the study of the lake equation with vanishing topography at the shore: the pressure was given by a degenerate elliptic equation. To sum up in one sentence, all this paper turns around the influence of the degeneracy at the boundary.
Our main result is to prove the local in time (uniformly in ) well-posedness of the shoreline problem for the one-dimensional Green-Naghdi equations. The precise statement if given in Theorem 2 below. Stability conditions are required. They are introduced in (4.5) and (4.6) and discussed there. The spirit of the main theorem is given in the following qualitative statement. Note that the case corresponds to the shoreline problem for the nonlinear shallow water equations (1.1).
For smooth enough initial conditions, and under certain conditions on the behavior of the initial data at the shoreline, there exists a non trivial time interval independent of on which there exists a unique triplet such that solves (1.2) with on , and .
1.2. Outline of the paper
In Section 2, we transform the equations (1.2) with free boundary condition (1.3) into a formulation which is more appropriate for the mathematical analysis, and where the free-boundary has been fixed. This is done using a Lagrangian mapping, together with an additional change of variables.
The equations derived in Section 2 turn out to be fully nonlinear because of the topography terms. Therefore, we propose in Section 3 to quasilinearize them by writing the extended system formed by the original equations and by the equations satisfied by the time and conormal derivatives of the solution. The linearized equations thus obtained are studied in §3.3 where it is shown that the energy estimate involve degenerate weighted spaces. The extended quasilinear system formed by the solution and its derivatives is written in §3.4; this is the system for which a solution will be constructed in the following sections.
Section 4 is devoted to the statement (in §4.2) and sketch of the proof of the main result. The strategy consists in constructing a solution to the quasilinear system derived in §3.4 using an iterative scheme. For this, we need a higher order version of the linear estimates of §3.3. These estimates, given in §4.3, involve Sobolev spaces with degenerate weights for which standard Sobolev embeddings fail. To recover a control on non-weighted norms and norms, we therefore need to use the structure of the equations and various Hardy-type inequalities (of independent interest and therefore derived in a specific section). Unfortunately, when applied to the iterative scheme, these energy estimates yield a loss of one derivative; to overcome this difficulty, we introduce an additional elliptic equation (which of course disappears at the limit) regaining one time and one conormal derivative; this is done in §4.4.
The energy estimates for the full augmented system involve the initial value of high order time derivatives; for the nonlinear shallow water equations (), the time derivatives can easily be expressed in terms of space derivatives but the presence of the dispersive terms make things much more complicated when ; the required results are stated in §4.5 but their proof is postponed to Section 9.
We then explain in §4.6 how to solve the mixed initial boundary value problems involved at each step of the iterative scheme. There are essentially two steps for which there is no existing theory: the analysis of elliptic (with respect to time and conormal derivative) equations on the half line, and the theory of mixed initial boundary value problem for dispersive perturbations of hyperbolic systems. These two problems being of independent interest, their analysis is postponed to specific sections. We finally sketch (in §4.7 and §4.8) the proof that the iterative scheme provides a bounded sequence that converges to the solution of the equations.
Section 5 is devoted to the proof of the Hardy-type inequalities that have been used to derive the higher-order energy estimates of §4.3. We actually prove more general results for a general family of operators that contain the two operators and that we shall need here. These estimates, of independent interest, provide Hardy-type inequalities for -spaces with various degenerate weights.
In Section 6, several technical results used in the proof of Theorem 2 are provided. More precisely, the higher order estimates of Proposition 6 are proved with full details in §6.1 and the bounds on the sequence constructed through the iterative scheme of §4.4 are rigorously established in §6.2.
The elliptic equation that has been introduced in §4.4 to regain one time and one conormal derivative in the estimates for the iterative scheme is studied in Section 7. Since there is no general theory for such equations, the proof is provided with full details. We first study a general family of elliptic equations (with respect to time and standard space derivatives) on the full line, for which classical elliptic estimates are derived. In §7.2, the equations and the estimates are then transported to the half-line using a diffeomorphism that transforms standard space derivatives on the full line into conormal derivatives on the half line. Note that the degenerate weighted estimates needed on the half line require elliptic estimates with exponential weight in the full line.
In Section 8 we develop a theory to handle mixed initial boundary value problems for dispersive perturbations of characteristic linear hyperbolic systems. To our knowledge, no result of this kind can be found in the literature. The first step is to assess the lowest regularity at which the linear energy estimates of §3.3 can be performed. This requires duality formulas in degenerate weighted spaces that are derived in §8.2. As shown in §8.3, the energy space is not regular enough to derive the energy estimates; therefore, the weak solutions in the energy space constructed in §8.4 are not necessarily unique. We show however in §8.5 that weak solutions are actually strong solutions, that is, limit in the energy space of solutions that have the required regularity for energy estimates. It follows that weak solutions satisfy the energy estimate and are therefore unique. This weak=strong result is obtained by a convolution in time of the equations. Provided that the coefficients of the linearized equations are regular, we then show in §8.6 that if these strong solutions are smooth if the source term is regular enough. The last step, performed in §8.7, is to remove the smoothness assumption on the coefficients.
Finally, Section 9 is devoted to the invertibility of the the dispersive operator at in various weighted space. These considerations are crucial to control the norm of the time derivative of the solution at in terms of space derivative, as raised in §4.5. We reduce the problem to the analysis of an ODE with regular singularity that is analyzed in full details.
N.B. A glossary gathers the main notations at the end of this article.
Acknowledgement. The authors want to express their warmest thanks to Didier Bresch (U. Savoie Mont Blanc and ASM Clermont Auvergne) for many discussions about this work.
2. Reformulation of the problem
This section is devoted to a reformulation of the shoreline problem for the Green-Naghdi equations (1.2). The first step is to fix the free-boundary. This is done in §2.1 and §2.2 using a Lagrangian mapping. We then propose in §2.3 a change of variables that transform the equations into a formulation where the coefficients of the space derivatives in the leading order terms are time independent.
2.1. The Lagrangian mapping
As usual with free boundary problems, we first use a diffeomorphism mapping the moving domain into a fixed domain for some time independent . A convenient way to do so is to work in Lagrangian coordinates. More precisely, and with , we define for all times a diffeomorphism by the relations
the fact that for all times stems from (1.4). Without loss of generality, we can assume that .
We also introduce the notation
and shall use upper and lowercases letters for Eulerian and Lagrangian quantities respectively, namely,
2.2. The Green-Naghdi equations in Lagrangian coordinates
when combined with the relation
that stems from (2.2), this easily yields
We thus recover the classical fact that in Lagrangian variables, the water depth is given in terms of and of the water depth at ,
In Lagrangian variables, the Green-Naghdi equations therefore reduce to the above equation on complemented by the equation on obtained by composing the second equation of (1.2) with ,
with and defined as
while the nonlinear term is given by
2.3. The equations in variables
In the second equation of (2.4), the term in the second equation is nonlinear in ; it is possible and quite convenient to replace it by a linear term by introducing
The resulting model is
(recall that and therefore ). The operators and are given by
and, denoting and ,
and the nonlinear term , with , is
Since by (2.1) we have , we treat the dependence on in the topography term as a dependence on , hence the notation and not for instance.
3. Quasilinearization of the equations
When the water depth does not vanish, the problem (2.6) is quasilinear in nature [Isr11, FI14], but at the shoreline, the energy degenerates and as we shall see, some topography terms make (2.6) a fully nonlinear problem. In order to quasilinearize it, we want to consider the system of equations formed by (3.2) together with the evolution equations formally satisfied by and , where and are chosen because they are tangent to the boundary.
After giving some notation in §3.1, we derive in §3.2 the linear system satisfied by and and provide in §3.3 -based energy estimates for this linear system. We then state in §3.4 the quasilinear system satisfied by (the fact that it is indeed of quasilinear nature will be proved in Section 4).
Throughout this section and the rest of this article, we shall make the following assumption.
i. The functions and are smooth on . Moreover, satisfies the following properties,
ii. We are interested in the shallow water regime corresponding to small values of and therefore assume that does not take large values, say, .
N.B. For the sake of simplicity, the dependance on and shall always be omitted in all the estimates derived.
3.1. A compact formulation
For all , let us introduce the linear operator defined by
so that an equivalent formulation of the equation (2.6) is given by the following lemma.
If is a smooth solution to (2.6), then it also solves
with, writing ,
As explained above, we want to quasilinearize (3.2), by writing the evolution equations satisfied by and (). We therefore apply the vector fields and to the two equations of (3.2). For the first equation, we have the following lemma, whose proof is straightforward and omitted.
If is a smooth enough solution to (2.6), then one has, for ,
For the second equation, the following lemma holds. The important thing here is that the term cannot be absorbed in the right-hand-side. As explained in Remark 7 below, this terms makes the problem fully nonlinear.
If is a smooth enough solution to (2.6), then one has, for ,
and where, writing and , one has
From the definition (2.9) of , we have
so that, applying the vector field (throughout this proof, we omit the subscript ), we get
where we used the fact that .
Before computing , we first replace by its equivalent expression (3.4). Applying we find therefore
Since moreover , one gets
and the result follows easily. ∎
3.3. Linear estimates
As seen above, an essential step in our problem is to derive a priori estimates for the linear problem
where we recall that
As we shall see, (3.8) is symmetrized by multiplying the first equation by and the second one by ; since is bounded away from zero and since controls (see the proof of Proposition 6 below), it is natural to introduce the weighted spaces
where is the water height at the initial time. We shall also need to work with the following weighted versions of the space
endowed with the norm
(the in the definition of the norm is important to get energy estimates uniform with respect to ).
The dual space of is then given by
(this duality property is proved in Lemma 2 below), with
This leads us to define the natural energy space for and its dual space by
We can now state the based energy estimates for (3.8). Note that these estimates are uniform with respect to .
where , the density of energy is
with , and . We also set
We shall repeatedly use the following uniform (with respect to ) equivalence relations
One multiplies the first equation of (3.8) by and the second by . Usual integrations by parts show that
Remarking further that
we easily deduce that
with as in the statement of the proposition. Integrating in time, using (3.17), and using a Gronwall type argument therefore gives the result. ∎
The assumption (3.15) contains two types of conditions: bounds and positivity conditions and which are essential to have a definite positive energy, thus for stability.