1 Introduction

Wave-structure interaction for long wave models with a freely moving bottom


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Wave-structure interaction for long wave models with a freely moving bottom

Krisztian Benyo1

AbstractIn this paper we address a particular fluid-solid interaction problem in which the solid object is lying at the bottom of a layer of fluid and moves under the forces created by waves travelling on the surface of this layer. More precisely, we consider the water waves problem in a fluid of fixed depth with a flat bottom topography and with an object lying on the bottom, allowed to move horizontally under the pressure forces created by the waves. After establishing the physical setting of the problem, namely the dynamics of the fluid and the mechanics of the solid motion, as well as analyzing the nature of the coupling inbetween these two, we examine in detail two particular shallow water regimes: the case of the (nonlinear) Saint-Venant system, and the (weakly nonlinear) Boussinesq system. We prove an existence and uniqueness theorem for the coupled system in both cases. Using the particular structure of the coupling terms we are able to go beyond the standard scale for the existence time of solutions to the Boussinesq system with a moving bottom.

1 Introduction

The water waves problem, which consists of describing the motion of waves at the surface of an inviscid, incompressible, and irrotational fluid of constant density under the action of gravity, has attracted a lot of attention in the last decades. The local well-posedness theory is now well-understood following the works of Wu [wuwellposedness2D, wuwellposedness3D] establishing the relevance of the Taylor sign condition. In the case of finite depth, which is of interest here, we refer for instance to [lanneswp, iguchishallow, alazardgravitycauchy]; the case where the bottom is also allowed to depend on time has also been treated in [alazardbottommove, iguchibottommove, melinandtsunami]. In this paper, we are interested in a particular configuration where the bottom depends on time, but instead of being in forced motion as in the above references, it evolves under the action of the hydrodynamic forces created by the surface waves. Finding its evolution is therefore a free boundary problem, which is coupled to the standard water waves problem, itself being a free boundary problem. The mathematical theory for such a configuration has not been considered yet; we refer however to [lannesbonneton] for a related problem where the moving object is floating instead of lying on the bottom as in the present paper.

Here, our goal is not to address the local well-posedness theory for this double free boundary problem, but to give some qualitative insight on its behavior by deriving and analysing simpler asymptotic models. We shall focus on a regime which is particularly interesting for applications, namely, the shallow water regime, where the typical horizontal scale of the flow is much larger than the depth at rest. For a fixed bottom, several models arise in this setting such as the Korteweg–de Vries (KdV) equation (justified in [craiglangrangian, kanonishida, schneiderlongwavelimit]), the nonlinear shallow water equations (justified in [ovsjannikov, kanonishida, alvarezlanneslong, iguchishallow]), the Boussinesq systems (justified in [craiglangrangian, kanonishida, bonachen, davidboussinesq]) – seel also [peregrineoriginal, papanicolaou, weikirbyboussinesq, chazelbottominfluence, craiglannesrough] for particular focus on topography effects – the Green-Naghdi equations [liapproximatif, alvarezlannesGN, iguchiGNtsunami], etc. We refer to [lannesbible] for more exhaustive references.

For a bottom with prescribed motion in general, the problem has already been considered, local well-posedness results ([alazardbottommove]) and long time existence results ([melinandtsunami]) have been proven recently. Numerical experiments and tentatives to adapt existing and known shallow water models for a moving bottom regime have been present for a while in literature, however lacking rigorous justifications. After observing successively generated solitary waves due to a disturbance in the bottom topography advancing at critical speed ([wusolitontrain]) they formally derived a set of generalized channel type Boussinesq systems ([wutengwaterwaves]), their work was extended later on in a formal study on more general long wave regimes ([chengeneralboussinesq]). Tsunami research has also proved to be a main motivating factor with the consideration of water waves type problems with a moving bottom (see for example [mitsotakistsunami] for an extensive numerical study). The mathematical justification of these models as approximations of the full problem was carried out not too long ago ([iguchibottommove] for Saint-Venant type systems, or [iguchiGNtsunami] for the precise Green–Naghdi system).

Here, we present a new class of problems where the bottom is still moving, but its movement is not prescribed, instead it is generated by the wave motion. A good approach to this is to place a freely moving object on the bottom of the fluid domain. The main physical motivation of this study lies in the recent development of submerged wave energy converters (submerged pressure differential devices, [LehmannTheWC] and references therein) and oscillating wave surge converters (WaveRollers and Submerged plate devices, [waveenergyconverters]), as well as reef-evolution and submarine landslide modelling problems. Bibliography in the more theoretical approach is rather lacking, existing studies are heavily oriented to physical experiments ([abadibadou] to investigate a submerged spring-block system and its numerical simulation through an adapted level set method, for further details, see for example [cottetmaitrenumerics]), as well as numerical applications ([zoazoazoo] for instance).

The structure of the article is as follows. This first introductory section starts with the presentation of the basic free surface fluid dynamics system and possible reformulations in the water waves setting. This is done first in the case of a fixed bottom, then in the more general case when the bottom has a prescribed motion. After that, the equations governing the motion of an object lying on the bottom are established, they derive from Newton’s equation and take into account the hydrodynamic force exerted by the fluid and a dynamic friction force. Then we introduce the characteristic scales of the variables of the system in order to derive the nondimensionalised equivalents of the different equations and formulae, preparing for the study of the asymptotic models.

In Section 2, we detail the first order asymptotic regime with respect to the shallowness parameter ; the resulting approximation is the well-known (nonlinear) Saint-Venant equations, in the presence of a solid moving on the bottom of the fluid domain. A key step is to derive an asymptotic approximation of the hydrodynamic force exerted on the solid. Then we establish a local in time well-posedness result for the coupled system.

In the third section, we elaborate our study on a second order asymptotic regime with respect to the shallowness parameter . This study concerns the so called long wave regime where the size of the waves and of the solid are assumed to be small compared to the mean fluid height. The resulting approximation is the so called (weakly nonlinear) Boussinesq system. A local in time well-posedness is shown for this coupled system as well. The standard existence time for a Boussinesq system with a moving bottom is with respect to the nonlinearity parameter , due to the presence of a source term involving time derivatives of the topography, which can potentially become large (as remarked in [melinandtsunami]). By a precise analysis of the wave-structure coupling we are able to extend the existence time to the time scale. This time scale is therefore intermediate between the aforementioned scale, and the scale that can be achieved for fixed bottoms ([alvarezlanneslong, cosminboussinesq]).

1.1 The case of a fixed bottom

As a basis for our model and our computations, we shall consider a fluid moving under the influence of gravity. The fluid domain (depending on the time ) is delimited from below by a fixed bottom and from above by a free surface. In our case the fluid is homogeneous with a constant density , moreover it is non-viscous, incompressible, and irrotational.

Figure 1: The water waves setting for a fixed bottom

To clarify the upcoming notations, the spatial coordinates take the form with denoting the horizontal component and the vertical one. Regarding differential operators, or as a subscript refers to the operator with respect to that particular variable, the absence of subscript for a space dependent operator means that it is to be taken for the whole space . In general we shall work in arbitrary horizontal dimension , even though the physically relevant cases are and only.

In what follows, we shall denote by the free surface elevation function and describes the bottom topography variation at a base depth of . We remark that by supposing that the bottom is flat, we may infer that . With this notation at our disposal we may establish that the fluid domain is

Let us also introduce the height function that describes the total depth of the fluid at a given horizontal coordinate and at a given time .

In order to avoid special physical cases arising from the fluid domain (such as islands or beaches), throughout our analysis we will often make use of the following (or similar) minimal water height condition


we refer to [poyferreshore] for an analysis of the water waves equation allowing vanishing depth, and to [lannesmetivier] where the evolution of the shoreline is considered for the one dimensional nonlinear Saint-Venant and Serre–Green–Naghdi equations.

The free surface Euler equations

To describe the fluid motion under these assumptions, the free surface Euler equations are to be considered. Denoting by the velocity field and by the fluid pressure, the homogeneous, incompressible, irrotational Euler equations take the following form:


valid in the entire fluid domain .

Here, in the equations denotes the gravitational acceleration as a downwards pointing vector. Moreover denotes the density of the fluid (constant due to the homogeneity assumption). It is useful to introduce the horizontal and vertical components of the velocity field, that is .

The boundary conditions can be resumed as follows:

  • the kinematic (or no-penetration) boundary conditions (that is, the fluid particles do not cross neither the bottom nor the free surface);

  • there is no surface tension along the free surface, so the pressure at the surface is given by the atmospheric pressure, and assumed to be constant.

A mathematical restatement of the aforementioned conditions is the following:

  • denoting by the unit normal vector of the fluid domain pointing upward, we have the following reformulation for the no-penetration condition for the bottom


    and for the free surface

  • denoting by the atmospheric pressure, we have that


The system of equations (1.2)-(1.2)’’ and (1.3) together form the free surface Euler equations for the fluid domain .

The free surface Bernoulli equations

Due to the incompressibility and irrotationality conditions (second and third equations in (1.2)), one may propose an alternative form of the equations by utilising the velocity potential , since with the knowledge of this potential one may recover the velocity field as the gradient, that is


With this velocity potential, denoting by the upwards normal derivative, the boundary conditions take the form


So from the Euler equations (1.2), with the aforementioned boundary conditions, the velocity potential is recovered as a solution of the following Laplace equation


Finally, by the momentum conservation part of the Euler system (first equation of (1.2)), we get that


in the domain . So the free surface Bernoulli equations are the system of equations (1.4)-(1.4)’’ and (1.6).

Based on equation (1.6), we can recover the pressure in terms of the velocity potential:


This relation allows to compute the hydrodynamical force exerted on the solid by the fluid (derived from Newton’s second law in Section 1.3).

The Zakharov / Craig–Sulem formulation

Following a somewhat historical introduction, we present another formulation of the equations (also referred to as the water waves problem). This formulation is attributed to Zakharov in his studies regarding gravitational waves [zakharovformula] and is based on the fact that the variables and fully determine the flow. More precisely, the water waves problem reduces to a set of two evolution equations in and ,


where solves the boundary value problem (1.5).

In more general terms, one can introduce the so-called Dirichlet-Neumann operator associated to the Laplace problem (1.5), but we will not pursue further this path, for more details we refer to the works of Craig and Sulem [craigsulem1, craigsulem2]. For a comprehensive and detailed analysis as well as the well-posedness of the water waves problem under this formulation, we refer to [lannesbible] and references therein.

1.2 The case of a moving bottom with prescribed motion

Now we consider the more general case where is time dependent, although a given function of (and of course ), that is the bottom is changing in a given fashion. This indicates the evolution of both the upper and lower boundaries of the fluid domain , although in a different way, since as opposed to the free surface, the evolution of the bottom is given. For the sake of clarity we shall precise that the fluid domain now is

which also indicates that the height function has the form .

Figure 2: The water waves setting for a moving bottom

The Euler / Bernoulli frameworks

One may notice that the fact that now depends on the time doesn’t change anything for the equations themselves in the Euler framework, thus the free surface Euler equations (1.2) and (1.3) hold just like before. However some adjustments have to be made for the boundary conditions, namely the no-penetration condition for the bottom becomes a kinematic condition similar to the one imposed on the free surface, so we replace (1.2)’ by


For the free surface Bernoulli equations, (1.4) and (1.6) are still valid. On the other hand the Neumann boundary condition for the bottom becomes nonhomogeneous in the Laplace equation used to recover the velocity potential. That is, we are left with the following Laplace equation instead of (1.5):


Similarly to the free surface Euler equations, the boundary condition on the bottom (1.4)’ is replaced by


The Zakharov / Craig–Sulem framework

As for the reformulation due to Zakharov, Craig, and Sulem, the recovery of the potential is based on the Laplace problem (1.9) now with non-homogeneous Neumann condition at the bottom. This can give rise to a natural decomposition of into a ”fixed bottom” and a ”moving bottom” component which could be used to define the Dirichlet-Neumann and Neumann-Neumann operators for this problem assuming sufficient regularity for the limiting functions. Thus one can formulate the water waves problem for moving bottom, which has already been studied; for details we refer to the article of Alazard, Burq, and Zuily [alazardbottommove] for the local well-posedness theory or to [iguchibottommove] for specific studies motivated by earthquake generated tsunami research.

We remark that the formulation (1.8) of the water waves problem still holds, only now we have to use the velocity potential obtained from (1.9) instead of (1.5). Since our study focuses on shallow water regimes, it is convenient to bypass the aforementioned technicalities by introducing the following variable:

Definition 1.1.

The vertically averaged horizontal component of the velocity is given by


where solves (1.9).

The interest of this new variable is that a closed formulation of the water waves problem in terms of and (instead of and ) can be obtained, see for example [lannesbonneton]. For our case, it is sufficient to observe that

Proposition 1.1.

If solves (1.9) and is defined as in (1.10), then


assuming sufficient regularity on the data concerning , , and as well as the minimal water depth condition (1.1).

Remark 1.1.

Let such that they satisfy the minimal water depth condition (1.1). Moreover, let . Then the Laplace equation (1.9) can be solved with and relation (1.11) holds true, where


is a known fluid domain. For more details, we refer to Chapter 2 of [lannesbible].

With this, the water waves problem with a moving bottom takes the following form


This system seemingly depends on three variables, namely , and , but in fact the Laplace-equation provides a connection between the latter two. Exploiting this connection to express (asymptotically) one variable with the other gives rise to various well-known asymptotic equations under the shallow water assumption. In Section 1.4 detailing the nondimensionalisation of the system we shall provide the necessary tools as well as some references concerning this asymptotic expansion.

1.3 The coupled problem with a freely moving object

The aim of this paper is to understand a particular case in which the bottom of the domain contains a freely moving object, the movement of which is determined by the gravity driven fluid motion. We will work with a flat bottom with the presence of a freely moving solid object on it (see Figure 1.3).

Figure 3: The coupled water waves setting in the presence of a solid

For the solid we suppose it to be rigid and homogeneous with a given mass . The surface of the object can be characterised by two components: the part of the surface in direct contact with the fluid, denoted by and the rest, that is the part in direct contact with the flat bottom, denoted by . For convenience reasons we shall suppose that is a graph of a function with compact support for any instance of .

The solid moves horizontally in its entirety, we denote by the displacement vector, and the velocity (with ). We make the additional hypothesis that the object is neither overturning, nor rotating so its movement is completely described by its displacement vector, which will be restrained to horizontal movement only. In particular, this means that the object is not allowed to start floating, the domain has a constant (nonzero) area.

Under these assumptions a simplified characterisation of the function describing the bottom variation is possible:


where corresponds to the initial state of the solid at (so that we have ).

Taking into account all the external forces acting on the object, Newton’s second law provides us with the correct equation for the movement of the solid. The total force acting on the solid is

Here we made use of the fact that the force emerging from the contact of the solid with the bottom may be decomposed in two components: the normal force, perpendicular to the surface of the bottom, expressing the fact that the bottom is supporting the solid, and the (kinetic or dynamic) friction force, the tangential component, hindering the sliding of the solid. By making use of the three empirical laws of friction [berthelotmechanic], most notably the third law often attributed to Coulomb regarding the existence of a coefficient of kinetic friction (describing the material properties of the contact materia), we may reformulate the tangential contact force as follows


where is a purely mathematical, small constant serving as a regularising term in order to avoid a singularity in the equation when the solid stops, that is when is equal to . Normally, when the solid stops, the kinetic friction detailed just before turns into static friction, a tangential force component preventing the solid from restarting its movement. The static friction has its own coefficient, which is usually greater than , and its direction is determined by the horizontal force components rather than the velocity.

In order to prevent the numerous complications that would arise by implementing the physically more relevant threshold for and the associated jump in friction force, we simplify the system by regularizing the friction force, thus neglecting static effects. A more specific modelling and analysis of the transition between static and dynamic friction will be addressed in future works.

Treating the horizontal and vertical component of separately and using the fact that the solid is constrained to horizontal motion, we have that the vertical components are in equilibrium, thus


and we obtain that the horizontal movement of the solid is given by


Finally, by making use of the fact that

due to the fact that the outwards normal vector for the surface of the solid can be easily expressed by the bottom variation , since

Therefore we obtain from (1.16) that


thus, by (1.15), (1.17) writes as


So we have that Newton’s equation characterising the motion of the solid takes the following form


A key step in our study is to handle the force term exerted by the fluid, which requires the computation of the integral of the pressure on the bottom over the solid domain. For this we will establish an appropriate formula for the pressure to be used in the integral.

In both the case of the freely moving bottom (due to the moving object) and the free surface, the kinematic no-penetration condition still applies, most notably we still have that

or equivalently, on the part of the surface of the solid in contact with the fluid (), the normal component of the fluid velocity field coincides with the normal component of the velocity of the solid, that is


To sum up, the water waves problem in the presence of a solid on the bottom is given by equations (1.13) and (1.9), where in the Neumann boundary condition, the bottom function and its time derivative are given by (1.14), with arising from (1.20) and the pressure derived from (1.7).

1.4 Dimensionless form of the equations

The main part of the analysis consists of establishing and analysing the wave-structure interaction system for shallow water regimes, and for that we need first of all the correct parameters involving the characteristic orders of magnitude of our variables as well as the dimensionless equations obtained with the help of these quantities.

The different scales of the problem

First of all we present the proper dimensionless parameters relevant to the system, bearing in mind that our aim is to derive simpler asymptotic models. For that we need to introduce the various characteristic scales of the problem:

  • , the base water depth,

  • , the characteristic horizontal scale of the wave motion (both for longitudinal and transversal directions),

  • , the order of the free surface amplitude,

  • , the characteristic height of the solid (order of the bottom topography variation in general).

Figure 4: The characteristic scales of the coupled water waves problem

Using these quantities, we can introduce several dimensionless numbers:

  • shallowness parameter ,

  • nonlinearity (or amplitude) parameter ,

  • bottom topography parameter .

Our goal in this paper is to examine asymptotic models when is small (shallow water regime), and under various assumptions on the characteristic size of and .

With these parameters in our hand, we may remark that the natural scaling for the horizontal space variable is , and for its vertical counterpart it is . Moreover the natural order of magnitude for the function characterising the free surface is , equivalently, for the bottom it is . Thus the nondimensionalised form for the water depth is

Furthermore, one can establish the correct scale of the velocity potential/field through linear wave analysis, which gives rise to

As for the pressure, we choose the typical order of the hydrostatic pressure, that is . For the time parameter, from linear wave theory one can deduce the scaling as

Finally, for the parameters concerning the solid, we impose that the characteristic horizontal dimension of the solid is comparable to . Following this, by taking into account the volume integral of the density, the natural scaling for the mass of the solid is given by

Our main interest will be to express the equations principally with the different orders of magnitude of (the shallowness parameter) to pass on to the different asymptotic regimes.

Thus the proper nondimensionalised parameters are obtained by

For the sake of clarity we shall omit the primes on the variables from here on.

Given the particular structure of the asymptotic regimes we are going to examine we shall make an a priori hypothesis concerning certain parameters.

Remark 1.2.

Since all the regimes handled in this article involve the hypothesis that and are of the same order of magnitude we shall, for the sake of simplicity, assume that .

An additional precision shall be made concerning the quantities involving the bottom. The explicit form of the nondimensionalised form for the water depth is




Nondimensionalised equations

Using the previous section and in particular taking as in Remark 1.2, one easily derives the dimensionless version of (1.13), namely


where is now defined as


with , furthermore solves


the nondimensionalised equivalent of the Laplace problem (1.9).

It is also necessary to nondimensionalize the formula describing the pressure (1.7), thus


Here we had to separate the horizontal and the vertical part of the gradient due to the different scaling parameters for the different directions.

We remark that the outwards normal derivative is given by

Thus we may reformulate Newton’s equation (1.20) in the following way


taking into consideration the characteristic scales of the variables.

2 The asymptotic regime: The nonlinear Saint-Venant equations

We shall now start our analysis for shallow water regimes, that is an asymptotic analysis with respect to the shallowness parameter for the nondimensionalised water waves problem (1.24) coupled with Newton’s equation (1.28) for the solid. With our notations, this means that we would like to consider systems that are valid for .

In this section we treat the general first order approximate system, more specifically a model with approximation that allows large wave amplitudes and large bottom variations (). So, the asymptotic regime writes as follows


2.1 Asymptotics for the fluid model

As mentioned before, the important step in deducing asymptotic models relies on how we establish the connection between the variables and . More precisely, it is possible to construct an asymptotic expansion of (depending on , and ). For details, we refer to chapter of [lannesbible]. One can equally obtain an asymptotic expansion of with respect to , depending on the aforementioned variables. Quite obviously, the equation in (1.26) reduces to at leading order in ; since the Neumann boundary condition in (1.26) is , it follows that does not depend on at leading order, and therefore

see Proposition 3.37. in [lannesbible] for a rigorous proof.

So the system (1.24) for the variables simplifies as follows


where we considered the gradient of the second equation in (1.24), and then neglected terms of order . This system is known as the (nonlinear) Saint-Venant or nonlinear shallow water system.

2.2 Formal derivation of a first order asymptotic equation for the solid motion

Our strategy is as follows: we establish an asymptotic formula of order for the pressure based on (1.27). With this at our disposal, we shall establish Newton’s equation (1.28) at order describing the displacement of the solid.

For an approximation, we shall start with the corresponding development for the velocity potential, that is


where as before, the restriction of the velocity potential on the free surface. Knowing this we recover the following for the time derivative of (based on the second equation of the water waves problem (1.24))

So by substituting the first order asymptotic expansion of the velocity potential described in (2.2) into the general nondimensionalised formula of the pressure (1.27) the corresponding approximation for the pressure takes the form

using the fact that does not depend on the variable .

So in particular, at the bottom, we find that the pressure is given by the hydrostatic formula


Thus for Newton’s equation (1.28),

Using the fact that is of compact support, the integral of its (and ’s) gradient on the whole horizontal space is , and the equation simplifies into

Notice the presence of the friction term (the first term on the right hand side). Even though it is of order , it will not pose a problem when controling the solid velocity, as we shall see in Lemma 2.5. later on, since it acts as a damping force.

So recalling that is given by (1.23) the corresponding approximative equation characterising the motion of the body is


where we made use of the following abbreviation:


2.3 The wave-structure interaction problem at first order

With (2.4) in our hand, we have all three equations for our coupled system. Indeed, notice that for the first equation in the nonlinear Saint–Venant system (2.1), the right hand side depends on , since depends on it. Thus by the chain rule the right hand side is

Our remark concerning the friction term present in the acceleration equation (2.4) becomes even more pertinent now, since we can observe a direct influence of the solid velocity (and thus an order term) in the first equation of the fluid system (2.1). This implies that a careful attention has to be paid on the velocity estimate for the solid.

To sum it up, the free surface equations with a solid moving at the bottom in the case of the nonlinear Saint-Venant approximation take the following form


In what follows, we proceed to the mathematical analysis of this system. We shall establish a local in time existence result for the coupled equations.

2.4 Local in time existence of the solution

The main result on the local well-posedness of the wave-structure interaction problem (2.6) is the following:

Theorem 2.1.

Suppose that , and that is sufficiently small. Let us suppose that for the initial value and the lower bound condition (1.1) is satisfied. If the initial values and are in with , , and , is an arbitrary initial condition for the solid motion, then there exists a solution

to (2.6) for a sufficiently small time .

Proof: The demonstration is based on the fixed point theorem applied to an iterative scheme presented in the following subsections. The brief outline of our proof is as follows:

  1. Reformulation of the system,

  2. Construction of the iterative scheme,

  3. Existence and a priori estimates for the iterative scheme,

  4. Convergence of the iterative scheme solutions.

Reformulation of the coupled system

Let us remark the following: the nonlinear Saint-Venant equations (2.1) admit a quasilinear hyperbolic structure. More precisely, we have the following classical reformulation:

Lemma 2.1.

The system (2.6a) may be reformulated using the variable into a single equation, that is


Proof: Let us take the following real valued matrices


where for every we have the th coordinate vector with respect to the standard Euclidean basis of .

We recall that thus implying that the matrices indeed depend on , however only through the bottom variation (1.23).

Following the notation in (2.7), after an easy computation, the additional term is the vector

From here on, we shall use the following uniform notation for the coordinate functions of :

As for the initial values, we have and , . There is no restriction necessary on the initial values concerning the solid motion.

There exists a symmetrizer defined by


such that the matrices are symmetric. Moreover, based on our imposed lower boundary condition on ,we shall establish that , which guarantees that the matrix is positive definite.

Owing to the existence of such a symmetrizer , the local well-posedness for a bottom with a prescribed motion follows from classical results [taylorpde]. In our case and additional step is needed due to the presence of the coupling with the equation describing the solid motion.

Let us make one further remark, concerning the second order (nonlinear) ordinary differential equation characterising the displacement of the solid in (2.6b). More precisely, let us define the functional as

The coupled system (2.6) has the following equivalent form


The iterative scheme

To solve the coupled system (2.10) we construct a sequence of approximate solutions through the scheme


Here the matrices and are the matrices defined in (2.8) and (2.9). In what follows we will make use of the following abbreviations

The main goal is to prove the existence and convergence of this sequence. We will follow the footsteps of a classical method, presented by Alinhac and Gérard in [alinhacgerard] for instance, detailing only the parts where additional estimates are necessary due to our coupled system.

The iterative scheme works as follows: we choose the initial elements to be . From then on, at each step () we have to solve

  • a linear symmetric hyperbolic PDE system (2.11a) to recover ,

  • and then a second order nonlinear ODE (2.11b) to obtain .

Existence and a priori estimates

Now, the aim is to establish the existence of solutions () for the iterative scheme to justify their definition in (2.11). Furthermore we shall also obtain a control of the velocity fields for our coupled system. In particular an upper bound on in a ”large norm”, partially in order to guarantee the boundedness conditions required for the existence result presented, as well as to introduce certain inequalities which will be useful for the convergence of the series.

This so called large norm shall be the following:

Definition 2.1.

For an function let us define


With this definition at our disposal, we can state the induction hypothesis (($H_{k}$)) for boundedness of solutions of (2.11):


for some constant , where a small independent constant to be defined.

Proposition 2.1.

For , with the knowledge of (($H_{k}$)), there exists a solution , of (2.11), by an adequate choice of and (independent of ), moreover

Proof: The proof goes by induction. For , is clearly verified. For the induction step, we shall treat separately the case of the PDE (part A) and the case of the ODE (part B), for the sake of clarity.

Part A: existence and energy estimate for : The initial values are bounded since they are equal to the original initial values . Since we are operating by induction with respect to , for the respective term we already have existence, moreover we also have the large norm estimates (($H_{k}$)) at hand, which in particular guarantees the uniform boundedness (independently of the index ) for small time and . Also, given the simple structure of and , they are bounded as well in Lipschitz norm.

Lemma 2.2.

For , with the initial condition and the hypothesis (($H_{k}$)) there exists a solution for the linear symmetric hyperbolic PDE system defined in (2.11a).

Proof: Notice that (2.11a) has a particular symmetric structure which may be exploited based on the following proposition:

Proposition 2.2.

Let us consider the symmetric hyperbolic differential operator