H.: A KdV-type Boussinesq system: From the energy level to analytic spaces. Discrete Contin

Considered here is the well-posedness of a KdV-type Boussinesq 
system modeling two-way propagation of small-amplitude long waves on 
the surface of an ideal fluid when the motion is sensibly two 
dimensional. Solutions are obtained in a range of Sobolev-type spaces, 
from the energy level to the analytic Gevrey spaces. In addition, a 
criterion for detecting the possibility of blow-up in finite time in 
terms of loss of analyticity is derived.


1.
Introduction. Consideration is given to the following coupled systems of two nonlinear dispersive wave equations in one space dimension, namely, w t + η x + ww x + η xxx = 0, η t + w x + (wη) x + w xxx = 0, (1) and its symmetric version (see [4]), w t + η x + 3 2 ww x + 1 2 ηη x + η xxx = 0, η t + w x + 1 2 (wη) x + w xxx = 0. (2) These systems have been indicated as models for long waves of small amplitude at the surface of an ideal fluid in a long rectangular channel with a flat bottom. They are known to be a valid approximation of the full, two-dimensional Euler equations for fluid motion under the influence of gravity in suitably small amplitude, long wavelength regimes (see [1,2,4]). The dependent variable η is proportional to the deflection of the free surface from its rest position, whilst w represents the horizontal velocity of the fluid measured at a height of 2 3 h, where h is the undisturbed depth of the fluid. The system (1) is a special case of a three-parameter family of longwave systems derived in [1]. In [2], the linear and nonlinear well posedness of these systems was investigated. In general, it was found that whenever a system is linearly well posed, the same result holds for the nonlinear system. The well posedeness of (2) was studied in [4]. In these works, a local well-posedness theory was developed in the L 2 -based Sobolev class H s for s > 3 4 . Detailed numerical simulations of solutions to these systems may be found in [5] and [6].
As is apparent, these systems have a structure related to the Korteweg-de Vries equation (KdV-equation) This equation has been the subject of a great number of works over the last four decades. In particular, the last two decades have seen the development of a very satisfactory theory of existence and uniqueness for the initial-value problem for (3). The initial-value problem for (3) is known to be globally well posed in all Sobolev classes with nonnegative index [14,30]. Indeed, there exist unique global solutions in the L 2 -based Sobolev classes H s down to s ≥ − 3 4 [16,17,15,10]. It can also be shown that the initial-value problem is not well posed in a strong sense if s < − 3 4 . The property that fails for very low values of s is that the flow map from initial data to solution is not of class C 2 [15].
Considering the mature state of the theory for (3), it seems natural to develop corresponding theory for the systems (1) and (2). Here, we focus on existence of solutions in Sobolev spaces of nonnegative index. In particular, this includes localin-time existence in the space of square-integrable functions on the real line, which is the energy space for the systems in view, as well as global-in-time existence for the symmetric version (2). Additionally, it will be proved that solutions are analytic in the spatial variable whenever the initial data are analytic in an appropriate sense and, for some s > 3 2 , the solutions remain bounded in the H s -norm. Conversely, if a solution corresponding to analytic initial data loses analyticity in finite time, this will preclude the solution from being global in any Sobolev space H s , s > 3 2 . In other words, loss of analyticity is a faithful indicator of a possible finite-time blow-up. The radius of analyticity is not only useful in the study of possible singularity formation, but also for proving exponential convergence of numerical approximations obtained via spectral methods. A study pertaining to the spectral projection method for approximating solutions of the KdV equation (3) has been conducted in [22,23]. As already mentioned, numerical simulations of the systems appearing in (1) and (2) have appeared recently in [5] and [6].
Related theory for similar types of systems include the recent developments of Bona, Cohen and Wang [3] that develops local and global theory set in Sobolev classes H s where s is only restricted by s > − 3 4 , as in the KdV-case. See also the papers [28] and [29] where related systems are considered. Issues of analyticity are not addressed in these works, and the range of systems considered in [3] does not include the present, physically relevant systems.
It is worth remarking that explicit examples in the KdV context may be constructed of complex-valued solutions that begin life analytic in a strip and, when restricted to the real axis, lose smoothness in finite time (see [11], [12] and [13] and the references therein).
To decouple the linear part of these systems, define new dependent variables , so that the resulting systems for U and V are and Since w and η can be easily recovered from U and V , our development of a well posedness theory will be focused on (4) and (5), with initial data U 0 and V 0 . The core of the argument used to show well posedness will be to obtain a contraction mapping in a conormal weighted space-time Sobolev space of Bourgain type. Banach spaces of this kind have been used extensively by many authors as a means to overcome the derivative loss inherent in dispersive equations with derivative nonlinearity. However, one problematic issue is that U and V are members of different Banach spaces, making the proof of the several required nonlinear estimates somewhat delicate. These difficulties are nevertheless overcome by carefully subdividing the domain of integration in Fourier space, as will be shown in Section 3.
As an overall guide to the rest of the paper, the main theorems to be proved are now stated. For σ ≥ 0 and s ≥ 0, define G σ,s to be the subspace of L 2 = L 2 (R) for which is finite. Note that for σ = 0, the definition yields the usual Sobolev spaces, denoted by H s . With the definition of G σ,s in hand, we are ready to state the theorems. Theorem 1.1. Let σ ≥ 0 and s ≥ 0. For initial data U 0 , V 0 ∈ G σ,s , there exists a positive time t 0 and a solution U, V ∈ C([−t 0 , t 0 ]; G σ,s ) of the initial-value problems associated to (4) and (5).
It will appear from the proof of Theorem 1.1 that the solution depends continuously on the initial data, but uniqueness holds only in a smaller class of functions. As mentioned before, the special case σ = 0 yields existence of a solution for initial data in H s , and in particular when s = 0, we obtain existence in the energy class L 2 . As the symmetric version (2) formally conserves the L 2 -norm, it transpires that local existence of a solution immediately yields global-in-time existence.
The second main theorem to be proved states that the radius of analyticity σ when regarded as a function of time cannot decrease faster than algebraically. This shows in particular that if the initial data are analytic in some strip about the real axis and if for some s > 3 2 , the Sobolev H s -norm of the solution stays finite, then the solution will remain analytic for all time.
The proofs of these two theorems are developed in the following way. In the next section, notation and some basic properties of function spaces will be recorded. In Section 3, some linear estimates are recalled, laying the groundwork for the multilinear estimates which form the heart of the analysis. The multilinear estimates will be established in Section 4. Finally, a proof of Theorem 1.1 will given in Section 5, whilst Theorem 1.2 is established in Section 6.
2. Functional setting. The Fourier transform u 0 of a function u 0 belonging to the Schwartz class is taken to be For a function u(x, t) of two variables, the spatial Fourier transform is denoted by whereas the notationû(ξ, τ ) designates the space-time Fourier transform
The following notation is used to signify the L p x -L q t norms; The usual modification applies for L ∞ -norms. Sometimes, L p -L q norms are used in Fourier variables, and this will be indicated by using subscripts, viz. L p ξ L q τ . A class of analytic functions suitable for our analysis is the analytic Gevrey class G σ,s , introduced in the context of nonlinear evolution equations by Foias and Temam [19]. G σ,s may be defined as the domain of the operator A s e σA in L 2 , and as was mentioned in the introduction, the analytic Gevrey norm of a function u 0 = u 0 (x) is defined by It is straightforward to check that a function in G σ,s is the restriction to the real axis of a function which is analytic on a strip symmetric about the real axis of width 2σ. This strip, {z = x + iy : |y| < σ}, will be denoted by S σ . If s = 0, we write simply G σ for G σ,0 . The following proposition shows that if u 0 ∈ G σ and ε is such that 0 < ε < σ, then u 0 and all of its derivatives are bounded on the smaller strip S σ−ε . Proposition 2.1. Let ε with 0 < ε < σ and n ∈ N be given, and suppose that u 0 ∈ G σ . Then there exists a constant c depending on ε and n, such that Proof. This is a direct consequence of the Sobolev embedding theorem and the inequality which holds for n ∈ N. The inequality (6) follows from the relation for a constant c n,ε .
To efficiently exploit the dispersive effects inherent in (1), we consider a space that is a blend between the analytic Gevrey space and a space of Bourgain-type.
For σ = 0, X ± σ,s,b coincides with the spaces X ± s,b introduced by Bourgain, and Kenig, Ponce and Vega, whose norms are v 2 These spaces are effective because the Fourier weight is well adapted to the linear part of the evolution equations. To see how this works, let W (t) ± be the solution groups associated with the homogeneous linear problems It follows immediately that When b > 1 2 , the spaces X ± σ,s,b are included in the space of continuous functions with values in G σ,s .
Proof. It follows directly from (7) and the Sobolev embedding theorem that the inequalities This proves the proposition.
The product spaces denoted impressionistically by Finally, for any function space Z, Z 2 will denote the product space Z × Z.
3. Linear estimates. Since the analysis is based on boundedness in Y σ,s,b of an integral operator given by a variation-of-constants formula, certain estimates of the solutions of the corresponding linear problem are needed. These estimates are addressed now. Let ψ be an infinitely differentiable cut-off function such that 0 ≤ ψ ≤ 1 everywhere and and, for T > 0, let ψ T (t) = ψ(t/T ). and Proof. The proof of (8) is immediate from the definition of the X σ,s,b -spaces and the linearity of the operator e σA . For the proof of (9), one follows the proof of Lemma 2.1 in [20] step by step, separating the cases T ≤ 1 and T > 1.
The second kind of linear estimates needed are Kato-type smoothing inequalities and maximal function inequalities. For a suitable function f , define F + ρ and F − ρ via their Fourier transforms according to Lemma 3.2.
[Bourgain] Let ρ > 1 4 be given. Then there is a constant c, depending on ρ, such that A For the proof of this lemma, the reader is referred to [14].
4. Bilinear estimates. The purpose of this section is the proof of several useful bilinear estimates. The first theorem contains the estimates to be used in the proof of local existence of solutions of (1) and (2).
Then there exists a constant c depending only on s, b, and b ′ such that Proof. In the proof, we focus on the case s = 0 and σ = 0 and comment on the situations wherein s > 0 or σ > 0 at the end. The first two estimates can be proved as in [27]. Of the remaining four inequalities, the fourth and the sixth are most demanding. Since the proofs are virtually the same, only the latter estimate will be proved. First note that the sixth inequality can be written more explicitly as .
If the functions f and g are defined by respectively, then the sought for bilinear estimate is equivalent to This inequality will be obtained by duality. Letting dµ = dξ 1 dτ 1 dξdτ , and supposing h(ξ, τ ) to be an arbitrary element of the unit ball of L 2 (R 2 ), we need to estimate the quantity After changing variables, the domain of integration is split into the two subdomains where |ξ| 2 + |ξ 1 | 2 ≥ 2 and |ξ| 2 + |ξ 1 | 2 < 2 (a choice whose genesis will be obvious in a moment). Thus the integral in (16) can be written as the sum I 1 + I 2 , where Focusing on I 1 , observe the relation Noticing that ξ 2 + ξξ 1 + ξ 2 1 ≥ |ξ||ξ 1 |, and recalling that |ξ| 2 + |ξ 1 | 2 ≥ 2 in the present case, it is clear that (17) holds. This implies that one of the cases always occurs. In case (a), I 1 is bounded by Changing to the original variables, it transpires that where Lemmas 3.2 and 3.3 have been used in the last step. It appears immediately that case (b) is similar to case (a). In case (c), we follow a similar reasoning as in case (a) to see that I 1 is bounded by Changing back to the original variables, it is then required to estimate the integral Split the domain of integration into two further subregions, namely the regions where |ξ 1 | > |ξ − ξ 1 | and |ξ 1 | ≤ |ξ − ξ 1 |, respectively. In the region where |ξ 1 | > |ξ − ξ 1 |, the integral just displayed is dominated by This integral can be further bounded by Since the last two factors in the integral (18) have identical structure, the analysis in the region where |ξ 1 | ≤ |ξ − ξ 1 | is the same. With the integral I 1 appropriately bounded, we turn to the estimation of I 2 . It is clear that I 2 can be dominated by Split the domain of integration into the two subregions D 1 = {|ξ 1 | > |ξ − ξ 1 |} and D 2 = {|ξ 1 | ≤ |ξ − ξ 1 |} as in case (c) above. Again using Lemmas 3.2 and 3.3 along with the trivial estimate (15), the integrand appearing in (19), when integrated over D 1 and D 2 , respectively, has the upper bounds as advertised.
In the case when s or σ are nonzero, define the functions f and g as then use the inequality e (1+|ξ|) ≤ e (1+|ξ1|) e (1+|ξ−ξ1|) and observe that positive values of s actually aid the procedure of the proof.
Next, a bilinear estimate to be used in the proof of large time estimates in Section 6 is stated and proved.
Proof. The first two cases have essentially been proved in [8]. The proofs of the third and fifths case are symmetric, as are the proofs of the fourth and sixth cases. We prove only the fifth case which is somewhat more straightforward than the fourth, because no change of variables is necessary. Only the case s = 0 is treated; the case s > 0 is straightforwardly reduced to the case s = 0. The elementary inequality e σ(1+|ξ|) ≤ e + σ will play a role at a certain point in the proof of the theorem. As in the proof of Theorem 1, the estimate may be rewritten in the form it is required to bound appropriately the quantity uniformly for h belonging to the unit ball B in L 2 (R 2 ). As before, the notation dµ = dξ 1 dτ 1 dξdτ has been employed. Using the inequality (20), the integral (21) is bounded by the sum of the two terms |h(ξ, τ )||ξ| To analyze I, proceed as in the proof of Theorem 4.1 to split the domain of integration into the region where |ξ| 2 +|ξ 1 | 2 < 2 and the region where |ξ| 2 +|ξ 1 | 2 ≥ 2. In the first region, the proof is the same as the corresponding part of the proof of Theorem 1. In the second region, use the inequality which is derived in a similar way as (17), to conclude that one of the cases (a) |τ − ξ + ξ 3 | ≥ 1 3 |ξ + ξ 1 ||ξ||ξ 1 |, always occurs. After noticing that the factors e −σ(1+|ξ1|) and e −σ(1+|ξ−ξ1|) are always less than 1, the estimation of I then proceeds along the lines of the proof of Theorem 4.1.
Attention is next turned to the task of estimating J. Because of the relation e (1+|ξ|) ≤ e (1+|ξ1|) e (1+|ξ−ξ1|) , it is plain that J can be dominated by Again splitting the domain of integration into the region where |ξ| 2 + |ξ 1 | 2 ≥ 2 and the region where |ξ| 2 + |ξ 1 | 2 < 2 leads to the need to estimate the integrals As before, it is shown that for J 1 , one of the cases (a), (b) or (c) in (22) must occur.
In case (a), J 1 is bounded by At this point, the domain of integration is again split into two further subregions, |ξ 1 | > |ξ − ξ 1 | and |ξ 1 | ≤ |ξ − ξ 1 |. In the region where |ξ 1 | > |ξ − ξ 1 |, the quantity in the just displayed integral is dominated by The latter integral can be further bounded above by Since the last two factors in the integral have identical structure, the analysis in the region where |ξ 1 | ≤ |ξ − ξ 1 | is the same. In case (b), J 1 is dominated by We split the domain of integration into the same two subregions as before. In the region where |ξ 1 | > |ξ − ξ 1 |, J 1 is dominated by and the latter can be bounded by In the region where |ξ 1 | ≤ |ξ − ξ 1 |, the integral is dominated by which in turn is bounded above, viz.
The proof in case (c) in (22) is similar to the proof in case (b). Finally, the estimation of J 2 is carried out as in the corresponding case in the proof of Theorem 4.1.

5.
Local-in-time existence and global-in-time existence for the symmetric version in L 2 . With the inequalities provided in the previous section, local-in-time existence and uniqueness of solutions in Y σ,s,b to the initial-value problems can be proved for any s ≥ 0 and σ ≥ 0 with the help of the contraction-mapping theorem. Consider the integral operator Let r = U 0 Gσ,s + V 0 Gσ,s . It will be proved that t 0 can be chosen so that Γ is a contraction in the ball B(2cr) ⊂ Y σ,s,b of radius 2cr centered at 0. Proof. We consider the system (4) -the proof in the case of the symmetric version (5) is the same.
First, it is proved that Γ is a mapping on B(2cr). Using (8), (9) and the nonlinear estimates, it is seen that With the same choice of t 0 , the same set of inequalities yield the contraction inequality It follows that Γ has a unique fixed point The functions U and V provide a solution of the initial-value problem associated to (4), and it is necessarily unique in the class Y σ,s,b . Continuous dependence follows immediately since the mapping Γ depends continuously on U 0 and V 0 (cf. [27,21]).
Remark 5.2. Multiplying the equations in (5) by U and V , respectively, integrating over the real line, integrating by parts and summing the result reveals that the total energy, U 2 + V 2 dx, of the symmetric system is conserved. This combined with the local-in-time well-posedness yields global-in-time existence for the symmetric version in L 2 .

Remark 5.3
Using the estimates of Section 4 combined with techniques used in [26], see [3], it is also possible to obtain existence of solutions in Sobolev classes of negative index.
Since the main focus of this paper is on analytic solutions, we do not stop here to provide an existence result in negative Sobolev classes. However, it would be interesting to find the critical Sobolev index below which well posedness becomes impossible, and to compare it to the situation for the initial-value problem associated to the KdV equation (3). For the related, but simpler systems studied in [3], well-posedness in H s × H s is known for s > − 3 4 . Presumably this result is sharp in the same sense it is for (3).