SUFFICIENTLY STRONG DISPERSION REMOVES ILL-POSEDNESS IN TRUNCATED SERIES MODELS OF WATER WAVES

. Truncated series models of gravity water waves are popular for use in simulation. Recent work has shown that these models need not inherit the well-posedness properties of the full equations of motion (the irrotational, incompressible Euler equations). We show that if one adds a suﬃciently strong dispersive term to a quadratic truncated series model, the system then has a well-posed initial value problem. Such dispersion can be relevant in certain physical contexts, such as in the case of a bending force present at the free surface, as in a hydroelastic sheet.


1.
Introduction. In studying the motion of free surfaces in fluid dynamics, many approximate models have been introduced in order to make problems more tractable. It is not uncommon to consider the full equations of motion to be the irrotational, incompressible, Euler equations. These equations may be written in the Craig-Sulem-Zakharov formulation, in which the Dirichlet-to-Neumann operator plays a central role [17], [38]. Craig and Sulem introduce an approximate system by expanding the Dirichlet-to-Neumann operator as a series in powers of the height of the free surface, truncating the series, and substituting the truncation for the full operator in the equations of motion. In this way, by varying the order of truncation, one may develop a quadratic model, a cubic model, a quartic model, and so on.
The resulting truncated series models have proved popular for computing, as all terms in the evolution equations then can be expressed as polynomials in the unknowns, with spatial derivatives and Hilbert transforms present. These systems are readily computed using pseudospectral methods. Some examples of applications of this approach are [15], [16], [18], [21], [30].
The problem of gravity water waves is famously known to be well-posed, with this having been proved by Wu for a wide class of initial data [36], [37], and subsequently (2) with p ≥ 0. Clearly, when p = 0, system (2) reduces to (1).
The view of ill-posedness of (2) for p = 0 taken in [3] is that the term (Λv) 2 on the right-hand side of the evolution equation for v is a parabolic term of indefinite sign, which thus allows for catastrophic growth. This can be compensated for with various smoothing mechanisms; the second author, Bona, and Nicholls considered a viscous regularization in [2] (the particular viscous regularization used was related to the work of Dias, Dyachenko, and Zakharov [19]). This viscous term provides a gain of one derivative when making energy estimates, and thus allows control of the first derivative term (Λv) 2 . As stated above, however, the water wave problem is typically considered to be an inviscid problem, and the present work instead focuses instead on dispersive regularization. Note that the dispersion relation for (2) is of order (p+1)/2. We specifically are interested in the question of how strong dispersion must be (i.e., how large p must be) to regularize the system. We demonstrate the existence and uniqueness of solutions in Sobolev spaces for the initial value problem associated to (2) for p ≥ 3.
While the analysis of this paper is for the quadratic truncated series models of water waves, we believe that the conclusions of this paper go forward for the cubic and higher models as well; recall that the results of [3] are for both the quadratic and cubic cases. While the quadratic model may not capture all the physics of ocean wave interactions, the resolution of analytical questions of well-posedness and illposedness do not rely so closely on this feature of the nonlinear terms; instead, for solutions which do exist, the nature of the nonlinear terms may affect the dynamics. For cubic and for higher-order models, the same problematic backward parabolic term identified above is still present, and additional terms, all of which have a commutator structure (see e.g. [35]) and thus are of comparable regularity to the terms already included, do not appear to affect the well-posedness properties of the system. With higher dispersion (i.e. for sufficiently large p), we again expect the methods of the present paper to compensate for the backward parabolic term, with the presence of the cubic terms having little effect on this.
We also mention that there are other ways of forming a truncated series model, as compared to the steps which led to (1), which is truncation in the evolution equations. A primary alternative is to use the Hamiltonian structure of the water wave system, make expansions in the Hamiltonian, and truncate there. Then, from the truncated Hamiltonian, evolution equations could be derived. We do not expect such differences to be fundamental from the point of view of well-posedness and ill-posedness of the resulting models. To see that Hamiltonian structure is independent of well-posedness, it suffices to observe that the unregularized vortex sheet problem is famously ill-posed [12], but also possesses a Hamiltonian structure [10]. As a further remark on the relationship between Hamiltonian structure and well-posedness theory, we note that the Hamiltonian structure for the water wave problem has not been used in any of the well-posedness theorems such as [36], [24], or [4].
The physical relevance of various values of p is naturally of interest. In the case p = 2, the system (2) has the correct dispersion relation for the presence of surface tension, and the system may be regarded as a quadratic truncated series model for pure capillary waves. Note that in this case, p = 2, the question of well-posedness or ill-posedness is at present unresolved. The case p = 4, however, is relevant in cases in which an elastic bending is present at the free surface. A variety of results have been shown recently for such waves, known as hydroelastic waves. Well-posedness of the initial value problem for 2D hydroelastic waves has been demonstrated by the authors and Siegel [6], [27]. The equations of motion in this case are based upon the Cosserat model of elastic shells, as developed by Plotnikov and Toland [31]. A number of studies have been made of traveling hydroelastic waves [9], [20], [34]. The system (2) with p = 4 can be derived as a quadratic truncated series model for hydroelastic waves; see [26] for details.
In summary, we prove existence and uniqueness of solutions for the initial value problem for (2) for p ≥ 3, which thus includes the hydroelastic case. We do this by using paradifferential calculus as developed by Bony [11] (see also [22], [28]). Our approach specifically is influenced by the use of paradifferential calculus by Alazard, Burq, and Zuily for the initial value problem of capillary-gravity water waves [1]. The main ingredient in the present proof is, as in many other works on water waves such as [4], [25], is a change of variables which allows for better regularity properties when making energy estimates. As we are using a paradifferential method, this is thus a paradifferential change of variables. The authors believe that a corresponding proof would be possible using the traditional product rather than the paraproduct, but the use of paradifferential calculus lowers the regularity requirements on the initial data in comparison. Our main result is the following: Theorem 1.1. Let p ≥ 3, s > max{5/2, (p + 1)/2}, and (u 0 , v 0 ) ∈ H s+(p−1)/2 (R) × H s (R) be given. Then there exists time T > 0 such that the Cauchy problem for (2) with initial data (u 0 , v 0 ) has a unique solution We mention that the inclusion of additional lower-order terms, such as adding back the traditional gravity term −gu to the right-hand side of the evolution equation for v in (2), is not expected to significantly alter the existence theory presented here for p ≥ 3. The paradifferential change of variables allows for control of the leading terms, and this is possible because of the presence of the Λ p u term with p ≥ 3; any lower-order terms such as a gravity term, then, are more routine to treat. In any case, for clarity of exposition, we omit such terms as we can then focus on the question of how strong the dispersion must be to balance the backward parabolic term.
The plan of the paper is as follows: we provide some preliminary results in Section 2. We prove existence of solutions in Section 3, and uniqueness in Section 4. We make some concluding remarks in Section 5.
This work was completed as part of the first author's doctoral dissertation, which was written under the supervision of the second author. The authors gratefully acknowledge support from the National Science Foundation through grant DMS-1515849.

2.
Preliminaries. In this section we present lemmas which will be useful to us throughout the sequel. First we give some commutator estimates and related results. Then, we introduce fundamental concepts and results of paradifferential calculus.
2.1. The Hilbert transform and related commutators. The Hilbert transform on R may be defined via its symbol as follows: Here, as usual, we denote the Fourier transform of a function g asĝ. In the sequel, many commutators of the form [H, φ]f = H(φf ) − φH(f ) will arise, and we thus need appropriate estimates. Towards this end, the following lemma will be helpful.
Lemma 2.1. Assume that F (ξ, η) is piecewise continuous function, and let If there exists M > 0 such that either or |F (ξ, η)| 2 dξ ≤ M 2 for all η holds, then T F : L 2 × L 2 → L 2 , with the estimate Proof. Case 1. We assume that the bound (6) holds. By the Schwartz inequality, We integrate with respect to ξ, using Tonelli's Theorem: Case 2. We assume instead that the bound (7) holds, and we set w(ξ) = T F (f, g)(ξ).
Let h ∈ L 2 be given. We compute as follows, using the triangle inequality, Tonelli's Theorem and the Schwartz inequality: Repeating the steps of the proof of Case 1, we are able to conclude Since h was an arbitrary element of L 2 , this implies that w ∈ L 2 , with the estimate This completes the proof of the lemma.
Next we define the L 2 -based Sobolev spaces, H s .
is the space of tempered distributions u with locally integrable Fourier transform, and with finite norm, with the norm defined as follows: where ξ = (1 + |ξ| 2 ).
We can now state and prove a useful commutator estimate.
Proposition 1. Let s ∈ R be given.
We next generalize Proposition 1 somewhat in Proposition 2. The proof is similar to the proof of Proposition 1, so we omit it; details can be found in [26], however. Proposition 2. Let s ∈ R be given, and let γ ∈ R and σ ∈ R satisfy s ≤ γ and s < γ + σ − 1/2. For any φ ∈ H γ and for any f ∈ H σ , we have We can then multiply and divide, using ξ − η 1−s η −σ ξ − η s−1 η σ . Similarly to before, then we define and we again denote φ 1 (ξ) = ξ s−1 φ(ξ) and f 1 (ξ) = ξ σ f (ξ). Since we again use Lemma 2.1, we compute the following: Define the function g by g(ω) = ω s , and let real numbers a and b such that 0 < a < b be given. Then, for some θ ∈ (0, 1), This implies the following: Here, to be able to use the fact that g is non-decreasing, we have used the assumption s ≥ 1. Continuing, we find the following: Therefore when σ−1 > 1/2, the above is uniformly bounded for all ξ. Our conclusion is Theorems of paradifferential calculus. Our approach to proving our main theorem is to use the tools of paradifferential calculus, which was introduced by J.-M. Bony [11]. In this section we review some of the basic theory of paradifferential calculus, stating theorems without proof. We refer the reader to [1], [11], [22], [28], and [33] for the general theory. More specifically, the results we now state all may be found in either the book of Metivier [28] or the paper of Alazard, Burq, and Zuily [1].

Remark 1.
We now state these results of paradifferential calculus in some generality, i.e., in d space dimensions. In the sequel, we apply all of the results with d = 1, as the free surface we consider is the one-dimensional boundary of a two-dimensional fluid.
Now we give the definition of the symbol classes in paradifferential calculus; the reader could also refer to Definition 3.1 in [1].
, which are C ∞ with respect to ξ for ξ = 0 such that, for all α ∈ N d and ξ = 0 , the function x → ∂ α ξ a(x, ξ) belongs to W ρ,∞ (R d ) and there exists a constant C α such that Remark 2. We consider symbols which are not C ∞ at the origin ξ = 0 since some symbols a(x, ξ) we consider will be not smooth at this point (such as a(x, ξ) = |ξ|). This is not a problem since the low frequencies are irrelevant in the smoothness analysis and only contribute to remainders as noted in Section 6.4 of [28]. For such symbols, it is natural to introduce a cutoff function ψ so that if a ∈ Γ m ρ , then Given a symbol a(x, ξ), we define the paradifferential operator T a by Here, χ and ψ are fixed C ∞ functions which satisfy the following: (i) there exist 1 and 2 such that 0 < 1 < 2 < 1 and where ρ is the smallest integer which is greater than ρ.
The following theorem appears as Theorem 3.6 in [1].
Now, we give the two most important theorems in paradifferential calculus. The first theorem is about the composition of operators; this can be found as Theorem 6.1.4 in [28]. Let m ∈ R and ρ > 0.

DISPERSION AND TRUNCATED SERIES MODELS OF WATER WAVES 3131
Note the following inequality: . Then, rewriting the inequality (22), we find In the case that ρ = 1, notice that by equation (23), In particular, in the sequel, there is a useful special case. Let a = a(ξ) be of order m, independent of x, and in Γ m ρ for any ρ > 0. (24) and (25), it follows that, for all µ ∈ R, The second of the most important theorems of paradifferential calculus is about adjoint operators; for this, we can refer the reader to Theorem 6.2.4 in [28]. Denote by (T a ) * the adjoint operator of T a and by a * (x, ξ)the adjoint of a(x, ξ).
Similarly to the composition theorem, the special case ρ = 1 is of interest; then, by equation (28) If a = a(x) is independent of ξ, the paradifferential operator T a is called a paraproduct. There are many useful results with respect to paraproducts. First, we quote a result concerning the case in which the symbol is only bounded (see Proposition 5.2.1 in [28]). Theorem 2.9. For all a(x) ∈ L ∞ and for all s, there is a constant C such that Of course, the relationship between L ∞ and Sobolev spaces is that a(x) ∈ L ∞ if a ∈ H r with r > d/2. One additional nice feature of paraproducts is that we may consider the case in which the symbol a(x) is not in L ∞ but merely in the Sobolev space H r with r < d/2. The following appears as Lemma 3.11 in [1].
for some constant K independent of a and u.
For the following two theorems about paraproducts, the reader can refer to Theorem 5.2.8 and Theorem 5.2.9 in [28].

SHUNLIAN LIU AND DAVID M. AMBROSE
Theorem 2.11. Let r be a positive integer. There is a constant C such that for a ∈ W r,∞ , the mapping u → au − T a u maps from L 2 to H r and Theorem 2.12. Let r be a positive integer. There is a constant C such that for a ∈ W r,∞ and α ∈ N d of length |α| ≤ r, the mapping u → a∂ α x u − T a ∂ α x u maps from L 2 to L 2 and Another key feature of paraproducts is that one can replace nonlinear expressions by paradifferential expressions, accounting for some smoother error terms. The reader can refer to Theorem 3.12 in [1].
Recall the usual estimate for the product of two functions (see Theorem 8.3.1 in [22]): with the second inequality strict if α, β or −s is equal to d/2.

Existence. Recall the Cauchy problem
, p ≥ 3 and s > max{5/2, (p + 1)/2}. We will follow the classic strategy to prove existence by first showing the local existence and uniqueness of a corresponding mollified system by the Picard theorem. We will then establish uniform estimates and use the continuation property of ODEs on a Banach space to show that all the solutions (for different values of the mollification parameter) exist uniformly in time. One of our pivotal techniques is to construct an intermediary variable, and then reformulate our system before mollifying.
3.1. Reformulation of the system. Let u ∈ H s+(p−1)/2 , v ∈ H s . Inspecting the system, it will be difficult to do uniform estimates directly because of the presence of the nonlinear terms Λ(uΛv) and (Λv) 2 . So our first key step is to create paradifferential operators which allow us to rewrite these two terms. Given the para-products T Λv , T vx , we do some addition and subtraction to write our system as where Now, we will introduce an auxiliary variable w to reformulate the system (35). Let Plugging in the equations for u t and v t , the equation of w t will be After a cancellation, and denoting f 2 to be the remainder which is then the equation of w t is We conclude that if (u, v) solve the system (2) (or (35)), then (u, w) solve The system (41) will be helpful, especially when making uniform estimates.
3.2. The approximate system and preliminary uniform estimates. Let the mollification operator J be defined as J = (1 − ∆) −(p+1)/8 . In the present work, the definition of paradifferential operators tells us J = T J and J s = T J s as in classical paradifferential calculus. (Note that we are abusing notation slightly, but we do not expect this will cause any confusion; we should more properly write T ξ s instead of T J s , but we believe the meaning is clear. We do this occassionally in the sequel for similar operators.) However the errors J − T J and J s − T J s correspond to the low frequencies which only contribute to remainders as we mentioned before. So, in the sequel, we do not distinguish them. Note that the operators T J , T J s are uniformly bounded operators of order zero. We introduce our approximate version of system (2): taken with the initial data (u 0 , v 0 ) ∈ H s+(p−1)/2 (R) × H s (R). For this approximate Cauchy problem, we have local existence and uniqueness by the Picard theorem. Proof. Since J is a smoothing operator, it is easy to check that the right hand side of system (35) is locally Lipschtiz from H s+(p−1)/2 (R) × H s (R) to itself. By the Picard theorem, for any > 0, there is T > 0 such that the system (42) has a unique solution (u , v ) ∈ C 1 ([0, T ]; H s+(p−1)/2 (R) × H s (R)).

SHUNLIAN LIU AND DAVID M. AMBROSE
Now, fix , and let (u, v) be the solution of the system (42) with initial data (u 0 , v 0 ). As before, we let w = v − T Λv u; we see then that (u, w) satisfies the following approximate system: with the approximate f 1 (u, v), f 2 (u, v) and g(u, v) defined as follows: Now we prove that the above three approximate quantities are uniformly bounded for any ≥ 0; that is, the following estimates hold also without approximation ( = 0). Note that in the following arguments, σ will always be taken to satisfy σ > 1/2.
Proof. We rewrite f 1, (u, v) by addition and subtraction; using the triangle inequality, we have the following: Then we will estimate the above four terms. First, applying the derivative and using the triangle inequality, We remark that σ will be chosen at the end of the proof; for now, it is sufficient to know that we will take σ > 1/2, as mentioned previously. By Proposition 3, we may estimate the first term on the right-hand side of (46) as follows: And by Proposition 1, the second and third terms on the right-hand side of (46) may be estimated as follows: This concludes our consideration of T 1, . : By the definition of T 2, and by the triangle inequality, To estimate T 2, , we estimate the three terms on the right-hand side of (49) individually. First, by the properties of the Hilbert transform, So the first term on the right-hand side of (49) is bounded by The above three terms are less than C u s+(p−1)/2 v 2+σ , by Theorem 2.14, by the fact that [T Λv , ∂ x ] is of order 0 according to (25), and by Theorem 2.12 respectively. Next, the second and the third terms on the right-hand side of (49) are less than C u s+(p−1)/2 v 2+σ by Theorem 2.14 and Theorem 2.12 respectively. We thus conclude that T 2, ≤ C u s+(p−1)/2 v 2+σ .
Finally, we estimate T 3, and T 4, . Noting that T Λv and T vx are in Γ 0 1 , by Theorem 2.7 and (25), we get Therefore, we conclude Letting σ = s − 2, this completes the proof. Proof. We first deal with g (u, v). By addition and subtraction and the triangle inequality, By Theorem 2.13 (taking α = 1 + 1/2 < 1 + σ and β = s − 1 there), the first two terms on the right-hand side of (52) can be bounded as follows: Again, by Theorem 2.7 and (25), the last two terms on the right-hand side of (52) can be estimated as follows: Now we can make the conclusion that By the triangle inequality, By Theorem 2.9 and since p ≥ 3, the first term on the right-hand side of (53) can be bounded as follows: The second term on the right-hand side of (53) is also less that C v 2 2+σ u s+(p−1)/2 because by Theorem 2.9, and by Proposition 1 By Lemma 2.10 , the third term on the right-hand side of (53) can be bounded as follows: Finally, we make the estimates − gΛ p u 3/2−(p−1)/2 ≤ C u 2+p/2 Since s > 5/2, we have σ = s−2 > 1/2, moreover 2+p/2 < s+(p−1)/2. Therefore, we have shown that there exists a nondecreasing function such that 3.3. Uniform estimates. The main result of this section is the following proposition.
Proposition 4. Let s > max{5/2, (p+1)/2}. There exists a nondecreasing function C(x) such that, for all ∈ (0, 1], for (u, v) satisfying and such that (u, v) is the unique solution of the Cauchy problem for the approximate system (42) with initial data (u 0 , v 0 ), the estimate is satisfied, with the norm Instead of dealing with the original equation of (u, v) directly, we make uniform estimates by means of the corresponding energy estimates for (u, w). Now, recall if (u, v) is the solution of the approximate system, then (u, w) satisfies system (43).
Define the energy with respect to (u, w) : where Our first step is to prove the following claim.
The proof of Proposition 5. First, we take the time derivative of u 2 0 : d dt Second, we take the time derivative of 1 2 w 2 s : d dt Notice that first term on the right-hand side of (58) cannot simply be bounded in terms of the energy; it involves a total of 2s + p derivatives on a product of u and w.
On such a product, we are able to bound at most a total of 2s+(p−1)/2 derivatives of u and w by the energy, so this is too many. This contribution to the energy will be dealt with via a cancellation. To that end, we take the time derivative of E 1 : d dt Again, the first term on the right-hand side of (59) has too many derivatives to be bounded in terms of the energy. In fact, this term cancels with the previous term (from (58)) upon adding (simply by using the self-adjointedness of the relevant operators).
Observe that to finish the proof of our Proposition 5, we turn out to need to estimate four terms which remain when adding (58) and (59) (after observing the remarked-upon cancellation). Note that we must bound w in terms of (u, v); for this purpose, we may simply use the definition w = v − T Λv u and Theorem 2.9 to find the estimate w s ≤ v s + v s u s .
(60) By the Hölder inequality, it is obvious that The requisite bound in terms of M (T ) follows from these inequalities, from (60), Lemma 3.2, and Lemma 3.3. Now it is similar to deal with the remaining terms, We only give details of one of these as an example. First, we have the following expansion: [J s , T vx ] H s →H 1 ≤ C v x W 1,∞ ; these estimates follow from Theorem 2.7 and the discussion which follows that theorem. Furthermore, we know the estimate T vx − T * vx L 2 →H 1 ≤ C v x W 1,∞ by Theorem 2.8. Therefore, we have the following: We introduce one more commutator of the form [T vx , J s ], so that the final integral on the right-hand side matches the integral on the left-hand side; we then make the following conclusion: Similar considerations yield the following as well: Thus, we have proved the claim: This immediately implies that By Lemma 2.10, and using the definition of w, we have the following: To finish the proof, we show that v 2−p/2 ≤ C(M 0 ) + C(M (T ))T : Lemma 3.4. There exists T 0 > 0 such that for all ∈ (0, 1], we may take T ≥ Proof. For any ∈ (0, 1] and T < T , (u , v ) ∈ C 1 ([0, T ]; H s+(p−1)/2 × H s ) is the solution of system (2). Let We get the desired contradiction. (65) We omit the proof of Proposition 7 because the proof is entirely similar to the proof of uniqueness, which we provide in the subsequent section. Both results require estimating the norm of a difference; for Proposition 7, we estimate the norm of two solutions with different values of the regularization parameter. For uniqueness, by contrast, we estimate the norm of two solutions with possibly different initial data. The only difference is that the proof of Proposition 7 requires dealing with terms involving J 1 − J 2 . The following fact allows these to be estimated in a straightforward way.
Lemma 3.5. For any 0 < 1 < 2 and m > 0, we have While we omit the full details of the proof of Proposition 7, we will provide some details after the proof of Proposition 8.

Continuity in time.
In the previous subsection, we have proved the existence of a solution for the Cauchy problem in C 0 ([0, T ]; H (p−1)/2 ×H 0 ) L ∞ ([0, T ]; H s+(p−1)/2 × H s ). By interpolation, for any s < s, it is true that (u, v) ∈ C 0 ([0, T ]; H s +(p−1)/2 × H s ). We now prove that the solution (u, v) is continuous in time in H s+(p−1)/2 × H s . To prove this, we will introduce the molli-

For any
> 0, because of the presence of J , we know that We also shall introduce the auxiliary variable w = J v − T Λv (J u). For any , w ∈ C 0 ([0, T ]; H s ) since (J u, J v) is continuous in time and v ∈ C 0 ([0, T ]; H s ), and by Lemma 2.10. We then turn to prove that in the space C 0 ([0, T ]; H s+(p−1)/2 × H s ), (J u(t), w (t)) is a Cauchy sequence. This will immediately imply the desired result for J v(t), since v ∈ C 0 ([0, T ]; H s ) and We rewrite the system for (J u, J v): where f 1, is Similarly to the system (41), the equation of (J u, J w) follows: where Now we do an energy estimate for (J u, w ) as before. We define the energy We omit the details of proof the energy estimate (the evolutionary system has the same structure as before), but the conclusion is the following bound: Now we define the difference energy as We can make the corresponding estimate for the difference energy (again, the evolutionary system has the same structure), finding the following: That is, It is clear that E 1, 2 (0) can be made small by taking 1 and 2 small. We then turn to prove that (f Proposition 8. Let s > max{5/2, (p + 1)/2} and T 0 > 0. Let both (u 1 , v 1 ) and (u 2 , v 2 ) be in C 0 ([0, T 0 ]; H s+(p−1)/2 (R) × H s (R)) and satisfy system (2), with initial data (u 0 , v 0 ) and (ũ 0 ,ṽ 0 ), respectively. Then the following estimate is satisfied: Proof. Define the differences δu = u 1 − u 2 and δv = v 1 − v 2 . Let 0 < T ≤ T 0 , and define The system satisfied by (δu, δv) is We use the paradifferential operators T Λv1 and T ∂xv1 as before. Also we define an auxiliary variable δw = δv − T Λv1 δu as the intermediate variable. We rewrite the first equation of system (73) by doing addition and subtraction: where f 1 is defined as Using the system (73) and substituting, the equation of δw is where and Proposition 9. Let f 1 , f 2 be as above. Then they satisfy the following estimate: The proof of Proposition 9. First, we rewrite f 1 (p−1)/2 by addition and subtraction: where The estimates for T 1 , T 2 , and T 3 correspond exactly to the estimates for T 1, , T 2, , and T 3, in the proof of Lemma 3.2, with J (p−1)/2 being present instead of J s+(p−1)/2 . So, Now, we estimate T 4 . By Proposition 2, since (p − 1)/2 + 1 < s + (p − 1)/2 − 1 − 1/2 for s > 5/2, We have proved (77) For f 2 , we estimate each of the Q i for 1 ≤ i ≤ 5. Firstly, the estimates for Q 1 and Q 2 are similar, and we only give the details of Q 1 . By addition and subtraction and the triangle inequality, we have the following: By Theorem 2.12, the first term of (78) is less than Λ(v 1 + v 2 ) W 1,∞ Hδv H 0 , which is in turn bounded by C(M 1 , M 2 )N . And by Lemma 2.10, letting µ = m = 3/2, the second term of (78) is less than Λδv 3/2 Λδv −1 , which is bounded by Secondly, by Theorem 2.9 and since (p − 1)/2 ≥ 1, Thirdly, by Theorem 2.9, Proposition 1 (using (p − 1)/2 ≥ 1), and Proposition 2 (using 1 < s + (p − 1)/2 − 1 − 1/2), Finally, by Lemma 2.10, This completes the proof of Proposition 9.
Having completed the proof of Proposition 9, we resume the proof of Proposition 8. Since the evolutionary system for (δu, δw) has the same structure as before, we may conclude d dt Then, by Gronwall's Inequality, Furthermore, we have The above inequality tells us that there exists T * > 0 small enough such that Thus on the interval [0, T * ], we conclude the uniqueness of solutions. The value of T * may be smaller than T 0 . However, we can repeat the argument to get uniqueness until time T 0 , since the value of T * only depends on M 1 and M 2 .
Now, as we have indicated previously, we remark on the proof of Proposition 7.
Proof. Since the proof is similar to the the proof of Proposition 8, we only point out the differences here. Let 0 < 1 < 2 be given. Suppose (u i , v i ) are the solutions for the approximate Cauchy problem corresponding to J i for i = 1, 2. Then they satisfy Set δu = u 1 − u 2 and δv = v 1 − v 2 . Then, these satisfy the following system: where We let δw = δv − T Λv1 δu, and we find that the system for δu and δw is The definitions of f 1 and f 2 may be inferred from (83) and (84), and are entirely analagous to the corresponding definitions in the proof of Proposition 8. To complete the proof of the proposition, we need to estimate f 1 (p−1)/2 , R 1 (p−1)/2 , f 2 0 , and R 2 0 . The estimates of f 1 (p−1)/2 and f 2 0 are again entirely analogous to the corresponding estimates in Proposition 8, so we omit them. For the estimates of R 1 and R 2 , by Lemma 3.5, it follows that Notice that we state a requirement s > p+1 2 in Theorem 1.1; here, we need this condition so that the right-hand sides of (85) and (86) go to zero with 2 .

5.
Conclusion. The second author, Bona, and Nicholls have previously presented strong evidence that quadratic and cubic truncated series models of gravity water waves are ill-posed [3]; for the quadratic case, the second author and Siegel are working on a full proof of ill-posedness [7]. That the initial value problems for these systems are ill-posed is in one sense surprising, since the full equations of motion for gravity water waves are well-posed. In another way, however, the ill-posedness is not so surprising, since the energy estimates needed to establish well-posedness, such as in [36], are quite subtle, and the design of the truncation scheme is unrelated to the energy balance. Furthermore, evidence of this ill-posedness has been observed previously in numerical and applied works, such as by Milder [29].
The ill-posedness is caused by one particular parabolic term of indefinite sign, which leads to catastrophic backwards parabolic growth. We have shown here that a sufficiently strong leading-order dispersive term can control this growth. This strong dispersion can be relevant in some physical contexts, such as hydroelastic waves [31]. We mention that while in a sense we made use of the smoothing properties afforded by this dispersion, we did not present here an explicit result on gain of regularity for solutions of our system. It is possible to do so, similarly to the proofs of gain of regularity for capillary water waves [1], [13].
Finally, we mention that the present work, along with the corresponding works [3] and [7], leave open an important question. While the evidence of [3] and [7] is that system (2) is ill-posed when p = 0, and the main theorem of the present work is that the same system has a well-posed initial value problem for p ≥ 3, the status of the initial value problem for values p ∈ (0, 3) has not yet been established. This is especially relevant in the case p = 2, which corresponds to the presence of surface tension at the free surface. This will surely be the subject of future studies.