Non-uniform continuity of the flow map for an evolution equation modeling shallow water waves of moderate amplitude

We prove that the flow map associated to a model equation for surface waves of moderate amplitude in shallow water is not uniformly continuous in the Sobolev space $H^s$ with $s>3/2$. The main idea is to consider two suitable sequences of smooth initial data whose difference converges to zero in $H^s$, but such that neither of them is convergent. Our main theorem shows that the exact solutions corresponding to these sequences of data are uniformly bounded in $H^s$ on a uniform existence interval, but the difference of the two solution sequences is bounded away from zero in $H^s$ at any positive time in this interval. The result is obtained by approximating the solutions corresponding to these initial data by explicit formulae and by estimating the approximation error in suitable Sobolev norms.


Introduction and the main result
We consider a model equation for surface waves of moderate amplitude in shallow water u t + u x + 6uu x − 6u 2 u x + 12u 3 u x + u xxx − u xxt + 14uu xxx + 28u x u xx = 0, which arises as an approximation of the Euler equations in the context of homogenous, inviscid gravity water waves. In recent years, several nonlinear models have been proposed in order to understand some important aspects of water waves, like wave breaking or solitary waves. One of the most prominent examples is the Camassa-Holm (CH) equation [3], which is an integrable, infinite-dimensional Hamiltonian system [1,4,7]. The relevance of the CH equation as a model for the propagation of shallow water waves was discussed by Johnson [19], where it is shown that it describes the horizontal component of the velocity field at a certain depth within the fluid; see also [5]. Building upon the ideas presented in [19], Constantin and Lannes [8] have recently derived the evolution equation (1) as a model for the motion of the free surface of the wave, and they evince that (1) approximates the governing equations to the same order as the CH equation. Besides deriving (1), the authors of [8] also establish the local well-posedness results for the Cauchy problem associated to (1). Relying on a semigroup approach due to Kato [21], Duruk [10] has shown that this feature holds for a larger class of initial data, as well as for solutions which are spatially periodic [11]. The well-posedness in the context of Besov spaces together with the regularity and the persistance properties of strong solutions are studied in [26].
Similarly to the CH equation, cf. [6,25], the model equation (1) can also capture the phenomenon of wave breaking: for certain initial data the solution remains bounded, but its slope becomes unbounded in finite time cf. [8,11]. Unlike for the CH equation, which is known to posses global solutions, cf. [2,6], it is not apparent how to control the solutions of (1) globally, due to the fact that this equation involves higher order nonlinearities in u and its derivatives than the CH equation. On the other hand, if one passes to a moving frame, it can be shown that there exist solitary travelling wave solutions decaying at infinity [14]. Their orbital stability has been recently studied in [9] using an approach proposed by Grillakis, Shatah and Strauss [15], which takes advantage of the Hamiltonian structure of (1).
In the present paper, we consider the Cauchy problem associated to (1) in the setting of periodic functions. From the local well-posedness results [11,10], we know that its solutions depend continuously on their corresponding initial data in Sobolev spaces H s with s > 3/2. Our main result states that this dependence is not uniformly continuous. This property was only recently shown to hold true for the CH equation [16,17], and was subsequently confirmed also for the Euler equations [18] and for several related hyperbolic problems such as the µ − b equation [23], the hyperelastic rod equation [20], for a modified CH system [24], and for the modified CH equation [13]. The main difficulty we encounter compared to all these references is that, as mentioned before, our equation has a higher degree of nonlinearity. Nevertheless, we were able two find two sequences of smooth initial data whose difference converges to zero in H s , but such that none of them is convergent, with the corresponding solutions of (1) being uniformly bounded on a common (nonempty) interval of existence. Approximating these solutions by explicit formulae, we then successively estimate the error in suitable Sobolev norms and use well-known interpolation properties of the Sobolev spaces and commutator estimates to show that at any time of the common existence interval the difference of the two sequences of exact solutions is bounded from below in the H s -norm by a positive constant. More precisely, denoting by u(·; u 0 ), the unique solution of (1) corresponding to the initial data u 0 ∈ H s (S) with s > 3/2, cf. Theorem 2.1, our main result states: where T u > 0, and a positive constant C > 0 with the following properties: The structure of the paper is as follows: In Theorem 2.1 we recall some properties concerning the well-posedness of (1) from [11] and determine a lower bound on the existence time of the solution in H s in terms of the initial data. Then, we introduce two sequences of approximate solutions (u ω,n ) n , ω ∈ {−1, 1}, and compute the approximation error in Lemma 3.1. The corresponding solutions u ω,n of (1) determined by the initial data u ω,n (0) are then shown to be uniformly bounded on a common interval of existence, the absolute error u ω,n − u ω,n being computed in different Sobolev norms, cf. Lemmas 4.1-4.3. We end the paper with the proof of the main result.
Notation. Throughout this paper, we shall denote by C positive constants which may depend only upon s. Furthermore, H r := H r (S), with r ∈ R, is the L 2 −based Sobolev space on the circle S := R/Z. Given r ∈ R, we let Λ r := (1 − ∂ 2 x ) r/2 denote the Fourier multiplier with symbol ((1 consisting of all bounded functions which possess bounded weak derivatives of order less than or equal to n, is endowed with the usual norm. Some useful estimates. The following commutator estimates play a crucial role in our analysis: They hold for all functions f, g ∈ C ∞ (S) and for the commutator [S, T ] := ST − T S. The Calderon-Coifman-Meyer estimate (3) follows from Proposition 4.2 in Taylor [30]. The estimate (2) is due to Kato and Ponce [22,29]. Additionally, we shall use the following multiplier inequality and f ∈ H t (S), g ∈ H r (S), cf. e.g. [28].

The local well-posedness result
Using the above notation, we observe that the evolution problem associated to (1) can be rendered as the following Cauchy problem: where R := R(u) is defined as Relying upon the local well-posedness results established in [11] for the quasilinear Cauchy problem (5), we determine in the following theorem a lower bound for the maximal existence time of the solutions in terms of Sobolev norms of the initial data. Additionally, we obtain a bound on the H s -norm of the local strong solutions on this particular existence interval.
Theorem 2.1. Let s > 3/2 be given. Then, we have: is continuous. (ii) Given u 0 ∈ H s (S), the maximal existence time of the solution u(·; u 0 ) of (5) satisfies where C is a positive constant.
Before proceeding with the proof, one can show by using integration by parts shows [8,11] that the H 1 -norm of the solutions of (1) is preserved in time when s ≥ 2. Based upon this observation and relying on Theorem 2.1 (i), we then find that Proof of Theorem 2.1. The assertion (i) follows from the local well-posedness results established in [11]. For (ii), we first pick u 0 ∈ H s (S) with u 0 = 0 and denote by T the maximal existence time of the associated solution u = u(·; u 0 ). In order to determine a lower bound for T , we first show that u 2 H s satisfies a differential inequality. We proceed as in [27,29] and pick a Friedrichs mollifier 1 J ε ∈ OPS −∞ , ε ∈ (0, 1). Since J ε is itself a Fourier multiplier, the time evolution of the H s -norm of J ε u is given by where The latter equality is based on the observation that To estimate the first term, we use the following bound which was derived in Taylor [29], by means of the Kato-Ponce estimate (2): Employing the Cauchy-Schwartz inequality and the algebra property of H r (S), r > 1/2, the term I 2 can be estimated as follows: H s ). Finally, we combine these estimates and let ε tend to 0 to find that Recalling that the H 1 -norm of u is preserved in time, we get u 0 H 1 ≤ u(t) H s for all t ∈ [0, T ), and together with (9) we find that We conclude that It follows that the constant T 0 defined by the relation (7) is a lower bound for T , and that u(t) H s ≤ 2 u 0 H s for all t ≤ T 0 . This proves the claim.

Approximate solutions for the evolution equation
In the following we consider approximate solutions of the evolution equation (1) of the form where ω ∈ {−1, 1} and n ∈ N \ {0}. When n is very large, the term involving the cosine has a high spatial frequency whereas the other term is constant. Before we estimate the error of these approximate solutions, observe that for all α, σ ∈ R and n ∈ N \ {0}. Indeed, the functions φ n := e in· / √ 2π, n ∈ Z, form an orthonormal basis of L 2 (S), and therefore a direct computation (see also [17,Lemma 1]) shows that We emphasize that in contrast to [17], due to additional terms appearing in (1) the precise computation of (12) is very important when estimating the norm of u ω,n in H s (S). In view of (12) and noting that 1 H σ = √ 2π, we obtain the bound Substituting the approximate solution u ω,n into the equation (1) the following expression for the error is found: Lemma 3.1 (Estimating the error of approximate solutions). Given s > 3/2, there is a positive constant C such that for all 1/2 < σ ≤ 1, ω ∈ {−1, 1}, and n ∈ N \ {0}.

Moreover, there is a constant
Proof. Let ω ∈ {−1, 1} and let n ≥ n 0 be arbitrary. For simplicity we set u := u ω,n . Because u(0) is smooth, we know in view of Theorem 2.1 that the Cauchy problem (18) has a unique maximal solution with existence time T := T (ω, n). In order to derive a bound on the absolute error in H k we have to prove first that this additional regularity holds up to and including the time T u . That is, we have to show that T > T u for all ω ∈ {−1, 1} and n ≥ n 0 . To this end we proceed as in the proof of Theorem 2.1 and compute that We study the last term more carefully and obtain in view of the commutator estimate (2) that for 2 ≤ p ≤ 4. In view of the embedding H s (S) ֒→ C 1 (S) and (19) we find that Next we employ the well-known interpolation inequality for r = k − 1, r 1 = s and r 2 = k and obtain that u 2 which we may integrate with respect to time to obtain This inequality shows that T > T u for ω ∈ {−1, 1} and for all n ≥ n 0 . Indeed, assuming to the contrary that T < T u , then u(t) H k → ∞ as t approaches the maximal existence time T of u ∈ H k . This is a contradiction to the fact that u is bounded in H k in view of (22). Finally, the error estimate (20) is a simple consequence of (22) and of the estimate for all n ≥ n 0 , cf. (13).
It turns out that estimate (20) can be improved when we choose k = 1 and s ≥ 2. The argument relies on the regularity properties derived in the previous Lemma 4.1.
Proof. Denoting the difference between the approximate solution and the exact solution by v := u ω,n − u ω,n , we see that v is a solution of the initial value problem whereby E is the error term defined by (14) and In view of the regularity property derived for u ω,n in Lemma 4.1, we may apply Λ 2 on both sides of (24) and find that and noting that This leads us to the following inequality Observing that the relation (13) implies sup [0,Tu] u ω,n (t) H 2 ≤ C, we find together with (19) that Taking now into account the estimates u ω,n x L∞ ≤ Cn 1−s and u ω,n x W 1 ∞ ≤ Cn 2−s for n ≥ n 0 and ω ∈ {−1, 1}, we obtain in view of the error estimate (15) the desired estimate (23) following in view of Gronwall's inequality and since v(0) = 0.
Before proving the main result, we show the analog of Lemma 4.2 in the situation when 3/2 < s < 2. The regularity properties derived in Lemma 4.1 are once again essential.
In view of (26), we have 1 2 The first term in the previous equation vanishes while applying the Cauchy-Schwarz inequality for the second and fourth term we obtain the estimates To derive a bound for the third term, we use the Calderon-Coifman-Meyer type estimate (3). We first commute the operator Λ σ ∂ x with the function u ω,n + u ω,n and obtain After integrating by parts, we estimate the first integral as follows To estimate the second integral, we apply the Cauchy-Schwarz inequality and then use the estimate (3) to find ≤C u ω,n + u ω,n H s v 2 H σ . In view of the boundedness of the family {max [0,Tu] u ω,n + u ω,n H s : n ≥ n 0 , ω = ±1} we may combine the preceding estimates and obtain that Λ σ vΛ σ ((u ω,n + u ω,n )v) x L 1 ≤ C v 2 H σ . The latter argument and the multiplier inequality (4) show that and together with the error bound (15) obtained in Lemma 3.1 we conclude that Whence, and the conclusion follows, as in Lemma 4.2, by taking into account that −2s + 1 + σ ≤ −s for all σ ∈ (1/2, s − 1].

Proof of the main result
In the remaining part we prove that the functions u n := u 1,n+n 0 and u n := u −1,n+n 0 , n ∈ N, satisfy all the properties required in Theorem 1.1. Recalling the estimate (19), which ensures that the strong solutions u ±1,n , n ≥ n 0 , are bounded in H s , proves the first claim sup  To show that the third claim of Theorem 1.1 holds, we have to derive a decay estimate for the difference between the two unknown exact solutions. The trick is to work with inequalities involving the estimates for the absolute errors deduced in the preceding lemmas. We assume first that s ≥ 2, and observe that