On the ill-posedness result for the BBM equation

We prove that the initial value problem (IVP) for the BBM equation is ill-posed for data in $H^s(\R)$, $s<0$ in the sense that the flow-map $u_0\mapsto u(t)$ that associates to initial data $u_0$ the solution $u$ cannot be continuous at the origin from $H^s(\R)$ to even $\mathcal{D}'(\R)$ at any fixed $t>0$ small enough. This result is sharp.


Introduction
Let us consider the initial value problem (IVP) u(x, 0) = u 0 (x) (1.1) where u = u(x, t) is a real valued function. This model describes the propagation of onedimensional, unidirectional small amplitude long waves in nonlinear dispersive media [4]. This model, widely known as the Benjamin-Bona-Mahony (BBM) equation, is the regularized counterpart of the Korteweg-de Vries (KdV) equation and is extensively studied in the recent literature, see for example [1], [2], [5], [6] and references therein.
Moreover, for R > 0, let B R denote the ball of radius R centered at the origin in H s (R) and let T = T (R) > 0 denote a uniform existence time for the IVP (1.1) with u 0 ∈ B R . Then the correspondence u 0 → u that associates to u 0 the solution u of the This Theorem improves the earlier known results by Benjamin et. al. [4] where the IVP (1.1) was shown to be globally well-posed for data in H k , k ∈ Z and k ≥ 1.
Moreover, the authors in [6] also proved that the IVP (1.1) for given data in H s (R), s < 0 is ill-posed in the sense that, the flow map u 0 → u(t) is not even C 2 . The exact ill-posedness result proved in [6] reads as follows. Cauchy problem, the flow-map acts smoothly from H s to itself (see for eg., [7], [10] and [11]). Motivated by the work of Bourgain [8], Takaoka [19] showed that the nonlinear Schrödinger equation with derivative in a nonlinear term is ill-posed in H s (R), s < 1/2.
Utilizing the same techniques Tzvetkov [20] proved that the KdV equation is locally ill-posed in H s (R) for s < −3/4 if one requires only C 2 regularity of the flow-map in the notion of well-posedness. Several dispersive models are proved to be ill-posed in certain function spaces using this notion, see for example [9], [12], [13], [14], [15], [16] and references therein. This result improves the ill-posedness result proved in Bona and Tzvetkov [6] and is sharp.
The remainder of this article is organized as follows. In the rest of this section we will introduce the notations that will be used throughout this work. In Section 2 we will sketch the proof of the earlier ill-posedness result, viz. Theorem 1.2. Finally, Section 3 is devoted to supply the proof of the main result of this work, viz. Theorem 1.3.
We use H s (R) to denote the L 2 -based Sobolev space of order s with norm Various constants whose exact values are immaterial will be denoted by C. Finally, we there exists a constant C > 0 such that A > CB and A ∼ B if A B and A B.

Earlier Ill-posedness Result
In this section we will describe, in brief, the ideas presented in [6] to prove the ill-posedness result stated in Theorem 1.2.
where ǫ > 0 is a parameter. The solution u ǫ (x, t) of (2.1) depends on the parameter ǫ.
We can write (2.1) in the equivalent integral equation form as Differentiating u ǫ (x, t) in (2.2) with respect ǫ and evaluating at ǫ = 0 we get and If the flow-map is C 2 at the origin from H s (R) to C([−T, T ]; H s (R)), we must have The main idea to complete the proof of Theorem 1.2 is to find an appropriate initial data u 0 for which the estimate (2.5) fails to hold. For this, Bona and Tzvetkov [6] considered the following initial data defined via the Fourier transform where I = [N − γ, N + γ] with N ≫ 1 and γ = N −σ , for 0 < σ ≪ 1. For this particular initial data, the estimate (2.5) fails to hold for any s < 0 thereby finishing the proof.
In this work we will renormalize the above example and exploit the analyticity of the flow-map obtained in Theorem 1.1 to prove the main result, Theorem 1.3.

Proof of the Sharp Ill-posedness Result
In this section we provide the proof of the main result of this work. As discussed earlier, we will consider the renormalized form of the counter example constructed in Bona and Tzvetkov [6] and follow the scheme introduced in [3] and [17] to accomplish the proof.
Proof of Theorem 1.3. Let N ≫ 1 and define φ N via the Fourier transform as Simple calculation shows that φ N L 2 (R) ∼ 1 and φ N H s (R) → 0, for any s < 0.
As pointed out in the previous section, the second iteration in the Picard scheme is the following, Now, using φ N in place of h and computing the Fourier transform in x, we obtain where, .
From (3.15), we have that Also, we have that Now, if we fix 0 < t < 1, take ǫ small enough and then N large enough, and take an account of (3.14); the estimate (3.19) yields that ǫ 2 I 2 (φ N , φ N , t) is a good approximation of u(ǫφ N , t) in H s (R) for any s < 0.
If we choose ǫ ≪ 1, from (3.16), (3.17) and (3.18), we get (3.20) If we fix the ǫ ≪ 1 chosen earlier and choose N large enough, then for any s < 0, the estimate (3.20) yields, (3.21) Note that, u(0, t) ≡ 0 and φ N H s (R) → 0 for any s < 0. Therefore, taking N → ∞ we conclude that the flow-map u 0 → u(t) is discontinuous at the origin from H s (R) to C([0, 1]; H s (R), for s < 0. Moreover, as φ N ⇀ 0 in L 2 (R), we also have that the flow-map is discontinuous from L 2 (R) equipped with its weak topology inducted by H s (R) with values even in D ′ (R).
Remark 3.1. In the periodic case, i.e., for x ∈ T, there is analytical well-posedness result for given data in Sobolev spaces without zero Fourier mode ( i.e., with zero xmean) H s (T), s ≥ 0, see [18]. Now, for N ≫ 1, if we define a sequence of functions a n , by a n =    1, |n| ∼ N 0, otherwise, (3.22) and φ N by φ N (n) = a n , then clearly φ N L 2 (T) ∼ 1 and φ N H s (T) → 0, for any s < 0.
If we proceed with the calculations exactly as above considering the Sobolev spaces H s (T) without zero Fourier mode, we can obtain a similar ill-posedness result for s < 0, in the periodic case too.