Well-Posedness for a Whitham–Boussinesq System with Surface Tension

We regard the Cauchy problem for a particular Whitham–Boussinesq system modelling surface waves of an inviscid incompressible fluid layer. The system can be seen as a weak nonlocal dispersive perturbation of the shallow water system. The proof of well-posedness relies on energy estimates. However, due to the symmetry lack of the nonlinear part, in order to close the a priori estimates one has to modify the traditional energy norm in use. Hamiltonian conservation provides with global well-posedness at least for small initial data in the one dimensional settings.


Introduction
Consideration is given to the following one-dimensional Whitham-type system where D = −i∂ x and tanh D are Fourier multiplier operators in the space of tempered distributions S (R). The positive parameter κ stands for the surface tension here. The space variable is x ∈ R and the time variable is t ∈ R. The unknowns η, v are real valued functions of these variables. We pick the initial values η(0), v(0) corresponding to the time moment t = 0 in Sobolev spaces as follows where s 1/2. System (1.1) has the Hamiltonian structure with the skew-adjoint matrix and the energy functional well defined on H 1 × H 1/2 . The latter conserves on solutions together with momentum I(η, v) that has the same view as in the pure gravity case In case of the trivial surface tension κ = 0, System (1.1) was proposed in [6] as an approximate model for the study of water waves to provide a two-directional alternative to the well-known Whitham equation [23]. The latter was proved to be consistent with the KdV equation [18] in the long wave regime [19]. We also refer to [10] for another version of the fully-dispersive Boussinesq type. Importance of such models is supported by experiments [4]. The unknown η denotes the deflection of the free surface from its equilibrium position, corresponding to the vertical level z = 0. The bottom is assumed to be flat and located at the level z = −1. The variable v is associated with the free surface velocity as explained in [6].
The initial value problem for Model (1.1) was studied in [5,9] in the case of vanishing surface tension κ = 0. In the same framework existence of solitary waves was proved in [8]. A natural extension of the existing results is to consider the case of non-trivial capillarity κ > 0. Note that the term 1 + κD 2 could be applied to −v x in the first equation instead, as it is done in [12], for example, to regularise the system regarded in [21]. However, the case regarded here is physically more relevant [7]. Indeed, repeating the Hamiltonian perturbation analysis from [6] to the full Hamiltonian with the surface tension, that can be found in [7], one naturally arrives to (1.1), (1.3).
It turns out that surface tension changes the nature of the equations. Indeed, the multiplication operator (η, v) → vη is not bounded in the natural Sobolevbased energy space arising from the linear equations, that is H s+1/2 (R) × H s (R). And moreover, the dispersive properties of the corresponding linear semigroup are insufficient to counterbalance the loss of 1/2 derivatives. As a result the proof of well-posedness demands a technique different from the one used in [9].
As to additional initial conditions, apart from inclusions given in (1.2), one has to impose a restriction essentially similar to the one used in [9], namely, smallness of the H 1 × H 1/2 -norm of (η 0 , v 0 ). This is important for the global-in-time existence. The meaning of this condition is that the total energy H(η 0 , v 0 ) should be positive and not too big. We point out that this condition cannot be significantly weakened even for the proof of the local result, which is also different from the non-capillarity situation. More precisely, for the local regular (s is large enough in (1.2)) well-posedness result it is enough to assume non-cavitation instead.
Analogously, one defines non-cavitation at a particular time moment.
The non-cavitation condition is a physical condition meaning that the elevation of the wave should not touch the bottom of the fluid for System (1.1) to be a relevant model. For convenience we have also included boundedness from above in this definition. We exploit the definition for providing with more general local existence formulation at high regularity level. However, in the low regularity case this condition cannot be controlled without imposing a stronger assumption, as we shall see below. We turn now to the formulation of the main results.
Theorem 1 Let s > 3/2. Suppose that the initial data (1.2) satisfies the noncavitation condition. Then there exist T > 0 and a unique solution The time moment T depends on s, κ and the norm η 0 , v 0 H s+1/2 ×H s . With respect to the capillarity and the initial data norm, the time of existence T is a non-increasing function. Moreover, the solution depends continuously on the initial data with respect to C H s+1/2 × H s -norm.
It is worth to emphasize here that the time of existence does not shrink as the surface tension parameter goes to zero. Making a bit stronger assumption on the initial data (1.2), one obtains a stronger result.
of System (1.1) with the initial data (η 0 , v 0 ). Moreover, the solution depends continuously on the initial data with respect to C H s+1/2 × H s -norm on any finite time interval [0, T ].
As we shall see below, the smallness of H 1 × H 1/2 -norm plays an essential role in proving the following two statements. The Cauchy problem (1.1) and (1.2) is locally well-posed for 1/2 < s 3/2. The solution can be extended to the global one for any s > 1/2. Whereas for the local result in the case s > 3/2, it is enough to impose a weaker assumption, namely, the noncavitation of η 0 .
The proof is essentially based on the energy method, that is natural to apply to quasilinear equations. The scaling H s+1/2 (R)×H s (R) is needed to rule out the linear terms, after the differentiation the corresponding energy norm. The main difficulty lies in the lack of symmetry of the nonlinearity. Indeed, a direct time differentiation of the norm η, v H s+1/2 ×H s leads to the term J s−1/2 ∂ x η ηJ s+1/2 v, where J σ stands for the Bessel potential of order −σ (see the proof of Lemma 6 below). Note that this term cannot be handled by integration by parts or commutator estimates, and so cannot be estimated via the energy norm. To overcome this difficulty we modify the energy norm adding the cubic term η J s−1/2 v 2 . The linear contribution of the derivative of this term will cancel out the mentioned inconvenient term. Meanwhile, the contribution coming from the nonlinear terms can easily be controlled. As we point out below a hint on the choice of the modifier comes from Hamiltonian (1.3). Note that after adding the cubic term the energy loses coercivity, and so one has to impose an additional condition. Either the noncavitation for big s or the smallness for small s of the initial data, both propagating through the flow of System (1.1), is enough to ensure that the modified energy is coercive. Additionally, consideration is also given to a system posed on R 2+1 of the form ⎧ ⎨ So the corresponding symbol K(ξ ) = √ tanh(|ξ |)/|ξ |. We complement (1.4) with the initial data (1.5) As above the variables η and v stand for the surface elevation and the surface fluid velocity, respectively. The system enjoys the Hamiltonian structure with the skew-adjoint matrix which in particular, guarantees conservation of the energy functional The noncavitation definition in the two dimensional problem has exactly the same view as in Definition 1 with the real line R substituted by the plane R 2 .
of System (1.4) associated with this initial data. The time of existence T is a non-increasing function of the surface tension κ and the initial data norm η 0 , v 0 H s+1/2 ×H s ×H s . Moreover, the solution depends continuously on the initial data with respect to C H s+1/2 × H s × H s -norm.
Note that the theorem has the local character, in the opposite of the one dimensional case.

Remark 1
The same results hold in the periodic case as well. The proof is similar up to some small changes in the commutator estimates [15].
In the next section some important inequalities are recalled. In Section 3 we introduce the modified energy and obtain the corresponding energy estimate for System (1.1). In Section 4 we obtain the energy estimate for the difference of two solutions of System (1.1). Note that Sections 3 and 4 provide with the motivation for studying the parabolic regularisation later in Section 5, where the corresponding energy estimate is deduced for the regularised system. In Section 6 a priori estimates are obtained. Finally, in Section 7 we comment on the last steps in the proof of Theorem 2, omitting only the thorough discussion of the initial data regularisation. In Section 8 we discuss some peculiarities of the two dimensional problem. In the last section we study System (1.1) with κ 1.

Preliminary Estimates
We start this section by recalling all the necessary standard notations. For any positive numbers a and b we write a b if there exists a constant C independent of a, b such that a Cb. The Fourier transform is defined by the formula on Schwartz functions. By the Fourier multiplier operator ϕ(D) with symbol ϕ we mean the line F (ϕ(D)f ) = ϕ(ξ ) f (ξ). In particular, D = −i∂ x is the Fourier multiplier associated with the symbol ϕ(ξ ) = ξ . For any α ∈ R the Riesz potential of order −α is the Fourier operator |D| α and the Bessel potential of order −α is the Fourier operator J α = D α , where we exploit the notation ξ = 1 + ξ 2 . The L 2based Sobolev space H α (R) is defined by the norm f H α = J α f L 2 , whereas the homogeneous Sobolev spaceḢ α (R) is defined by f Ḣ α = |D| α f L 2 . We also exploit the notation H ∞ (R) = ∩ α∈R H α (R). Introduce the operator where κ is the surface tension. Note that κ > 0 is a fixed constant. We implement the notation K = K 0 = √ tanh D/D used in [9]. Its inverse K −1 and K κ both have the domain H 1/2 (R) and are equivalent to the Bessel potential J 1/2 . Below we will need to compare J , |D| and K −2 and so we prove the following simple estimates.
Proof By the Plancherel identity it is enough to check the following inequalities where the middle one is trivial. The rightmost inequality follows from The leftmost one follows from the tanh-definition via exponents and the obvious e 2ξ + e −2ξ 2 + 4ξ 2 .
Throughout the text we make an extensive use of the following bilinear estimates. Firstly, we state the Kato-Ponce commutator estimate [14].
for any f, g defined on R.

By the commutator [A, B] between operators A and B we mean the operator
Secondly, we state the fractional Leibniz rule proved in the appendix of [16].
for any f, g defined on R. Moreover, the case σ 2 = 0, p 2 = ∞ is also allowed.
We also state an estimate, firstly appeared in [17] in a weaker form, and later sharpened in [22].

Lemma 4 Suppose a, b, c ∈ R. Then for any f ∈ H a (R), g ∈ H b (R) and h ∈ H c (R) the following inequality holds
Proving a global-in-time a priori estimate we will use the following limiting case of the Sobolev embedding theorem, that in the one dimensional case d = 1 reads as follows.

Lemma 5 (Brezis-Gallouet inequality)
(2.5) Inequality (2.5) was firstly put forward and proved in H 2 (R 2 ) in the work by Brezis, Gallouet [2]. It was extended to more general Sobolev spaces and any dimension in [3], but in a slightly different form. For the sake of completeness, we provide here with the proof based on the idea introduced in [2].
where R > 0 is an arbitrary positive number. In the first integral I 1 we multiply and divide f by 1 + ξ 2 1/4 . Afterwards, we apply the Hölder inequality to get and similarly, . Now it is left to choose R depending on f, s. If f H s 1 then taking R = f H s we immediately obtain the desired inequality (2.5). In the case f H s > 1, we estimate the second integral as follows and to come to (2.5). In the last case f H s > 1 and α s = 1/(s − 1/2) > 1, we can take R = f α s H s to bound I 2 (R) by the same constant. Note that I 1 (R) in this case is bounded as and so we again obtain Inequality (2.5).

Modified Energy
As we shall see in the proof of the next lemma, a direct use of H s+1/2 × H s -norm as the energy does not allow us to close the estimates, and so we modify it as follows.
Firstly, for each κ > 0 and s 1/2 we introduce the norm which is obviously equivalent to the standard norm in H s+1/2 (R) × H s (R). Such choice will be convenient later for analysis of dependence of solution on the capillarity κ. Secondly, we define the modified energy where the pair (η, v) represents a possible solution of System (1.1). Note that in the limit case s = 1/2 this quantity coincides with the Hamiltonian given in (1.3), . This gives us a small hint for the choice of the right cubic modifier that is basically a guess.
Then there exists C s > 0 such that for any κ > 0 and Proof We have already noticed that E 1/2 (η, v) is a conserved quantity, which proves the statement for the limit case s = 1/2. Assuming s > 1/2 we calculate the derivatives and the derivative of velocity norm Summing up these derivatives and simplifying the corresponding expression via integration by parts, we obtain The second integral I 2 can be estimated with the help of Lemma 4, by setting Applying Hölder's inequality to the third integral I 3 , we get We would like to point out here that the first integral I 1 cannot be estimated via the energy norm (3.1), using only integration by parts or commutator estimates. Turning our attention to the modifier of energy E s , we calculate its time derivative as follows Let I 4 , . . . , I 8 represent these integrals, respectively. The first summand, that we notate by I 4 , is estimated easily as The third integral in (3.3), notated by I 6 , is estimated in a similar way

The fourth integral in (3.3) equals
where the first integral can be treated with interpolation in Sobolev spaces and the second integral by the fractional Leibniz rule as follows The last integral in (3.3), that we notate by I 8 , is bounded by It is left to regard the second integral in (3.3), denoted by I 5 , and the integral I 1 appeared after the differentiation of the energy norm (3.1). Firstly, let us note that and so summing I 1 , I 5 together one can easily obtain Applying the Kato-Ponce estimate to the first commutator one obtains Taking p 1 (s) = 1 1−s , p 2 (s) = 2 2s−1 for s ∈ ( 1 2 , 1) and p 1 = p 2 = 4 in case s 1 one deduces after implementing the Sobolev embedding. Similarly, with the implicit constant independent of κ > 0.
As we shall see below, the non-cavitation condition is convenient to work with only in the case of high regularity s > 3/2. Then the time interval on which the condition holds true can be easily estimated through the first equation in (1.1). Our goal is to study well-posedness in spaces of low regularity as well. So in case of s 3/2 we will have to impose a stronger condition, instead of non-cavitation, namely smallness of the initial data norm, that we can control in time with the help of the Hamiltonian conservation, as the following lemma demonstrates.

Lemma 8
There exists a constant H > 0 independent of the surface tension κ > 0 such that for any Proof We use a continuity argument. We simply write Then there exists C > 0 independent of κ > 0 such that where u = u(t) is a solution of (1.1) defined on some interval. Take H = (2C) −1 , any 0 < H and a solution with u 0 = u(0) having u 0 /2. By continuity u on some [0, T ] and so Hence the function u satisfies that u(t) does not reach the level at any time t.
As a consequence of the lemma we can control η L ∞ for any s 1/2 in time, admitting only small initial data, by the inequality which guarantees non-cavitation, in particular.

Uniqueness Type Estimate
Suppose that we have two solution pairs η 1 , v 1 and η 2 , v 2 of System (1.1) on some time interval. Define functions θ = η 1 − η 2 , w = v 1 − v 2 . Then θ and w satisfy the following system We need an a priori estimate similar to one obtained in the previous section for the difference of solutions. For this purpose we introduce the difference energy (4.2) where the implicit constant depends on κ, r, s.
Proof We follow the same arguments as in the proof of Lemma 6. The derivative of squared norm In the case r 1/2 we have the commutator estimate and so For r ∈ (0, 1/2) we apply the Leibniz rule where p 2 > 2 is such that σ 2 = r − 1/2 + 1/p 2 > 0. The last estimate is due to Sobolev's embedding. Operator J r+ 1 2 − |D| r+ 1 2 is bounded in L 2 . Thus where the first integral can be estimated by interpolation in Sobolev spaces. In the second integral the fractional derivative |D| r+ 1 2 can be approximated by J r+ 1 2 to come again to (4.3) now for 0 < r < 1/2.
Differentiation of the energy modifier gives Proof Non-cavitation implies coercivity for E r and the rest is obvious.

Remark 2
The restriction s > 1/2 appeared in the lemma and its corollary is inconvenient. It comes from the loss of Hamiltonian structure of System (4.1). This results in the fact that we can obtain only a weak solution in case s = 1/2 and probably not unique.

Parabolic Regularisation
For application of the energy method we need to do a parabolic regularisation of the view where μ ∈ (0, 1). We want to prove solution existence for (5.1) for any given μ, by the contraction mapping principal and so p should be big enough. However, we also do not want to spoil our energy estimates, and so p should be small enough. As we shall see below, this bounds us to p ∈ (1/2, 1]. Here the left number comes from the following lemma.

f (t)g(t)) H r dt C(T ) f C T H r g C T H s
for any functions f, g defined on [0, T ]. Here either r = s + 1/2 or r = s.
with K κ defined by (2.1). For any fixed u 0 = (η 0 , v 0 ) T ∈ X s = H s+1/2 (R)×H s (R) the function S(t)u 0 solves the linear initial-value problem associated with (5.1). Let X s T = C([0, T ]; X s ) and regard a mapping A : X s T → X s T defined by Then the Cauchy problem for System (5.1) with the initial data u 0 may be rewritten equivalently as an equation in X s T of the form Proof We need to show that the restriction of A on some closed ball B M with the center at point S(t)u 0 is a contraction mapping. Note that S(t)u X s exp(−κμt|D| p )u X s . Hence by Lemma 10 for any T , M > 0 and u, u 1 , and so taking M = u 0 X s one can find a T > 0 such that A will be a contraction in the closed ball B M . The first statement of the lemma follows from the contraction mapping principle. Smoothness of the flow map can be proved in the same spirit applying the implicit function theorem instead, and so we omit it. Some details can be found in [9].
By a standard argumentation, see for example [11], one can show that if u = (η, v) T ∈ X s T is the solution of Problem (5.4) then u ∈ C 1 (0, T ); H s−1 (R) × H s−3/2 (R) and it solves the regularised system (5.1) as well with the initial data u 0 ∈ X s . Clearly, in order to be able to use the following energy and a priori estimates, one has to pick up a smooth initial data. The justification is discussed briefly in Section 7.
Proof Following the proof of Lemma 6 one arrives at d dt E s (η, v) = I 1 + I 2 + I 1 + . . . where for p 1 and the rest integrals I 1 , . . . , I 8 are the same as in Lemma 6.
As was noticed at the end of Section 3, one has to make sure that the modified energy is coercive. An effective way to do it at the low level of regularity is to control η L ∞ via the energy conservation. One can get the same controllability for the regularised problem via the energy dissipation due to the following result.
Proof Hamiltonian (1.3) has the derivative 1 κμ where the rest integrals are of no definite sign. One has to check that I 1 , I 2 are absorbed by the first and third norms. Firstly, we rewrite I 1 in the form Applying the Hölder inequality and the fractional Leibniz rule (2.3) for |D| p/2 with L 2 -norm to the first integral, Lemma 4 to the second integral and the Hölder inequality to the third integral, one obtains Using the Sobolev embeddingḢ 1/4 → L 4 , one finally obtains H 1/2 . The second integral I 2 can be treated by the Hölder inequality as follows Here the first norm is estimated with the help of the Leibniz rule in the way where we have used Lemma 4 and the embeddingḢ 1/4 → L 4 . Thus Eventually we obtain that concludes the proof. Note that the implicit constant here does not depend on κ.
As a simple corollary with the proof similar to that of Lemma 8 one obtains the following. The dependence of δ on the parabolic regularisation power p is unimportant since below we stick only to the case p = 1.

A Priori Estimate
We have an a priori global bound for solutions of both systems (1.1) and (5.1) in H 1 (R) × H 1/2 (R) due to Lemma 8 and Corollary 3, respectively. Our aim is it to obtain estimates in H s+1/2 (R) × H s (R) with s > 1/2. Suppose that its initial data (1.2) either satisfies the non-cavitation condition for s > 3/2 or has small enough H 1 κ × H 1/2 -norm for s 3/2. Then there exists for some C > 0 independent of κ, μ. The time of existence T 0 is a non-increasing function of the surface tension κ and of the initial data norm η 0 , v 0 H s+1/2 κ ×H s .
Proof We closely follow the arguments in [12] since we have essentially the same energy estimates. The main difference lies in the control of coercivity of the modified energy by coercivity of the energy. These constants depend only on h 0 , H 0 . They are used to define the time set that is non-empty and closed in (0, T ) by the solution continuity. Moreover, for T = sup T we have either T < T and so T ∈ T or T = T = +∞ by the blow-up alternative (6.1). Introduce T 0 = min{T 1 , T 2 } with where C 1 , C 2 are two big positive constants to be fixed below in the proof. The idea is to show that these constants can be chosen, independently on the initial data, in such a way that T 0 ∈ T or equivalently T 0 T . Assume the opposite T < T 0 . Firstly, we will check that the non-cavitation condition holds on [0, T ]. Indeed, in the low regularity case s ∈ (1/2, 3/2] it is assumed smallness of the initial data and so H 1 κ × H 1/2 -norm of the solution stays small with time evolution by Lemma 8 and Corollary 3. In particular, the wave satisfies the non-cavitation condition. For s > 3/2 one can estimate η using the first equation in System (1.1) (or in System (5.1)) as follows with the implicit constant independent on μ ∈ (0, 1), obviously. Hence for big enough C 2 since T < T 2 . As a result the non-cavitation Without loss of generality one can assume that for s 3/2 the non-cavitation of η is governed by the same constants h, H . Let E(t) = E s (η, v)(t) be the energy defined by (3.2) and E 0 = E(0). For System (1.1) (or for System (5.1)) we have the a priori energy estimate given in its differential form by Corollary 1. It can be rewritten in the form A straightforward use of Grönwall's inequality gives for any t ∈ [0, T ] with c depending only on h, s. Note that e c(1+κ)t 1 + 1 1 + C 1 (1 + κ)E 0 for any C 1 c and 0 t T < T 1 . In particular, Thus As a result setting C 1 = max{2, c} we have Taking into account T < T and continuity of the solution one can find T < T < T , T 0 such that on [0, T ] holds which contradicts the definition of T . Therefore, we showed that T 0 T concluding the main part of the proof. It is left to specify the dependence of T 0 on the initial data and the surface tension. From its definition one can see that T 0 is non-increasing as a function of the initial data norm for each κ > 0 fixed. One can also see straightaway that T 0 is nonincreasing as a function of κ for each (η 0 , v 0 ) and s > 3/2 fixed. To the same conclusion one can easily come in the case s 3/2, taking into account that the smallness assumption imposed on the initial data norm is κ-independent according to Corollary 3.
Studying the low surface tension regime in the last section, we will appeal to the following remark.
Taking p 1 (s) = 1 2−s , p 2 (s) = 2 2s−3 for s ∈ 3 2 , 2 and p 1 = p 2 = 4 in case s 2 one deduces Secondly, in the case s = 3/2 the commutator is estimated straightforwardly as and then appealing to the Leibniz rule (2.3) we obtain Hence for s ∈ (1/2, 3/2) the sum of I 2 and I 4 is estimated as Thus gathering all the parts one obtains Knowing coercivity of the energy E s , controlled either by the smallness or by the non-cavitation of the initial data, one can deduce from the lemma that the time of existence depends only on η 0 , v 0 H s +1/2 κ ×H s , where 1/2 < s < s. Taking into account the boundedness of η, v H 1 κ ×H 1/2 , holding true at least for small initial data, one can get a stronger result thanks to the Brezis-Gallouet limiting embedding (2.5). In order to exploit it we need the following Grönwall inequality.

The Low Capillarity Regime
This section is devoted to analysis of the solution dependence on the surface tension κ ∈ (0, 1]. It allows, for instance, to validate that solutions of Systems (1.1), (1.4) with κ = 0, that are known to exist [9], do indeed approximate solutions of the same systems when κ 1. We restrict ourselves to the one dimensional case. The extension to the two dimensional situation is straightforward. Proof By the Bona-Smith argument it is enough to prove the statement for the smooth initial data u 0 = (η 0 , v 0 ) with η 0 , v 0 ∈ H ∞ (R). Moreover, it is enough to prove convergence in C [0, T ]; L 2 (R) × H 1/2 (R) . Without loss of generality we can assume that T coincides with T 0 defined in Lemma 14. Note that it can be regarded as independent of κ ∈ (0, 1] according to Remark 3. Moreover, we can assume that on the same time interval [0, T ] the solution u, corresponding to the zero surface tension, also satisfies (6.2) with the same constant C and κ = 0.