Well-posedness and ill-posedness results for the regularized Benjamin-Ono equation in weighted Sobolev spaces

We consider the initial value problem associated to the regularized Benjamin-Ono equation, rBO. Our aim is to establish local and global well-posedness results in weighted Sobolev spaces via contraction principle. We also prove a unique continuation property that implies that arbitrary polinomial type decay is not preserved yielding sharp results regarding well-posedness of the initial value problem in most weighted Sobolev spaces.


Introduction.
In this work we shall study the initial value problem (IVP) for the regularized Benjamin-Ono (rBO) equation where H denotes the Hilbert transform, f (x − y) y dy = (−i sgn(ξ) f (ξ)) ∨ (x).
The regularized BenjaminOno equation, rBO, models propagation of long-crested waves at the interface between two immiscible fluids. In particular this equation is appropiate to describe the pycnocline in the deep ocean, and the two-layer system created by the inflow of fresh water from a river into the sea ( see [17] and references therein). A related equation associated with the same type of phenomena is the Benjamin-Ono equation which was introduced by Benjamin [3] and Ono [27] as a model for long internal gravity waves in deep stratified fluids. Both equations admit solitary wave solutions but it is well known that BO equation defines a completely integrable system, however, for the rBO equation, numerical simulations suggest that this property does not hold, see [17] and [4].
On the other hand it is interesting to notice that for both equations the dispersive part is given by a non-local operator involving the Hilbert transform , however, it is possible to find solutions with Picard iterations in the Sobolev spaces, H s (R) = 1 − ∂ 2 x −s/2 L 2 (R), for the rBO equation but it was recently shown that this scheme can not be applied for the BO equation, see [24] and [22]. Numerical studies take advantage of this fact and give evidence of the better suitability of the rBO equation over the BO equation for modelling purposes, see [17].
In [2] and [4] the initial value problem 1.1 was shown to be globally well posed in H s (R), s ≥ 3 2 . Recently, in [1], rBO equation was considered in the periodic and in the continuos setting. There it was shown that 1.1 was locally and globally well-posed in H s , s > 1 2 . It is worth to mention that in [1] it is proved that when considering negative indexes, i,e s < 0, the map data-solution for the rBO flow is not C 2 and therefore Picard's iteration fails for those rough Sobolev spaces.
Our goal in this paper is to obtain sharp well-posedness results in weighted Sobolev spaces of Regarding the BO equation 1.2, it was recently shown in [10] that arbitrary decay is not preserved, and that indeed there is an upper limit bound for the decay of solutions in order to guarantee well-posedness in those spaces which expressed in terms of the index r tells us that r should be less than 7/2 which contrasts with another wave propagation famous model, the Korteweg-de Vries equation [21], for which the Schwartz class is preserved, [18]. These type of results are of special interest from the point of view of the qualitative properties of the solitary waves for the respective flow. It is well known that for the BO the profile of its solitary waves has mild decay, r < 3 2 , whereas for the KdV the solitary waves decay exponentially. The discussion for the rBO equation in these weighted Sobolev spaces is contained in our main theorems Theorem 1.1. Let ϕ ∈ ℑ s,r (R), s > 1/2, r = 1, 2, then there exists a unique u ∈ C([0, ∞); ℑ s,r ) solution of the initial value problem (1.1) such that, ∂ t u ∈ C([0, ∞); L 2 (R)). Theorem 1.2. Let ϕ ∈ ℑ s,r (R) with s > 1/2 and 0 < r < 5/2 then there exists a unique u ∈ C([0, ∞); ℑ s,r ) solution of the initial value problem (1.1) such that, ∂ t u ∈ C([0, ∞); L 2 (R)). Theorem 1.3. Let u ∈ C([0, T ] : ℑ s,2 ) be the solution of the initial value problem 1.1 with initial data, ϕ, having mean value ϕ(x) dx ≥ 0. Then if at times t 1 = 0 < t 2 < T it holds that u(t j ) ∈ ℑ s, 5 2 , j = 1, 2 then u must be identically zero Remarks.
a) Theorem 1.1 refers to integer value powers of the weights whilst Theorem 1.2 deals with general real value powers of them. We choose to present this way our results since in the latter case the proof is a bit more involved and relies in some continuity properties of the Hilbert transform. We remark that for the BO equation similar results to Theorem 1.1 and Theorem 1.3 were first established by Iorio in [14] and [15], and those in Theorem 1.2 and Theorem 1.3 were recently obtained by Fonseca, Ponce and Linares in [10] and [11] and more generally for the dispersion generalized Benjamin-Ono equation by the latter authors in [12]. b) Theorem 1.3 establishes that the initial value problem is ill-posed whenever the decay rate r of the initial data is or goes beyond 5 2 . In particular, the result in Theorem 1.2 is sharp. Also, we notice that this unique continuation result requires information at only two different times in a similar way to what happens in the case of the generalized KdV and non-linear Scrödinger equations, see [8] and [9]. For the BO and dispersion generalized BO equations, information at three different times is indeed required, see [11] and [12]. c) The proof of Theorem 1.3 implies that if ϕ(0) = ∞ ∞ ϕ(x) dx < 0, then the L 2 -norm of the solution is conserved and u(t) 2 0 = ϕ 2 0 = −2 ϕ(0), and therefore under more general hypothesis on the initial data ϕ, ill-posedness for the I.V.P 1.1 holds.

Wellposedness in
Therefore, it easily follows that This implies the result.
Next theorem in [1] shows that the I.V.P 1.1 is globally well-posed in H s (R), s > 1/2. Since the proof in [1] was only carried out in detail for the periodic case, we find interesting to include its proof in the continous setting. The key point is to obtain a priori estimates of the Sobolev norm in H s (R), s > 1/2 with the help of the a priori bound of the H 1 2 norm in Lemma 2.2 and the help of the Kato-Ponce commutator, [19], and the Brezis-Gallouet's inequality, [5], which in dimension one reads , for s > 1/2. This last inequality was also used in the same spirit by Ponce to show globalwell-posedness of the Benjamin-Ono equation in H s (R), s ≥ 3 2 , see [28].
where C 0 depends only ϕ 1 2 . The lemma above and this inequality imply that Then, there exists C 1 > 0 such that, and hence, there are constants C 2 > 0 and C 3 > 0 such that for every t ∈ [−T, T ], u(t) s ≤ e C2e C 3 t .

2.2.
Group estimates in weighted spaces. In order to obtain the group estimates, next lemma provides some formulae for derivatives of the unitary group associated to the rBO equation in Fourier space. They easily follow from a direct computation.
Moreover, for j ≥ 5 the j-derivative for F (t, ξ), F j (t, ξ), has the form: where δ is the Dirac distribution, p k (t), is a polynomial, a k y b k are constants that depend on k.
Proof. We use Leibniz rule and lemma 2.4 to conclude that For r = 1, we obtain For r = 2, we have For r = 3, we use that tδ ϕ = tδ ϕ(0) = 0, and therefore Arguing with the induction principle, we obtain the desired result.
In order to consider non-integer weights we have to introduce new harmonic analysis tools. Let us first recall the definition of the A p condition. We shall restrict here to p ∈ (1, ∞) and the 1-dimensional case R (see [25]).
For further references and comments we refer to [7] and [30]. However, even though we will be mainly concerned with the case p = 2, the characterization (2.17) will be the one used in our proof. In particular, one has that We notice that for 0 ≤ θ ≤ 1 we have that . and that Proposition 2.9. ( [12]) Given φ ∈ L ∞ (R), with ∂ α x φ ∈ L 2 (R) for α = 1, 2, then for any θ ∈ (0, 1) 3. Proof of Theorem 1.1 Proof. We restrict to the case r = 2. We use Theorem 2.5, Lemma 2.6 and Theorem A.2 in [16] (if ϕ ∈ ℑ r , then x α ∂ β x ϕ ∈ L 2 (R), for all integers α and β such that 0 ≤ α + β ≤ r. Moreover, . For existence of solutions, we consider the integral equation: In fact, (3.21) implies that, And if u ∈ X s,2 (T, M ) we have that, Next, we show that Φ is a contraction in X s,2 .
Uniqueness in ℑ s,2 is a consequence of the theory in H s (R). In order to extend this local solution, let us obtain apriori estimates for u in ℑ s,2 .
Now we estimate each integral on the right hand side of (3.22). First we have

Proof of Theorem 1.2
Proof. We will restrict ourselves to the most interesting case 2 < r < 5 2 . In this setting r = 2 + θ, with 0 < θ < 1 2 . We also consider first the local existence problem via the contraction principle as we did before. Let us fix T > 0.
We estimate the group in the weighted norm: For F 2,4 we easily obtain from (2.20), with φ being the group F , that Last, for F 2,1 we use the continuity property of the Hilbert transform with respect to the A 2 weights |x| 2θ , with 0 < θ < 1 2 , and then proceed as we did above: where ψ(ξ) = ϕ(ξ) (1+|ξ|) 3 . Therefore, collecting the information in (4.26)-(4.29), we have that (4.30) |x| With the help of the former estimates and Plancherel, we now estimate the integral equation 3.21 in the weighted Sobolev norm We estimate each term in the integrand similarly as we did in the linear part (4.32) (4.35) where ψ = v (1+|ξ|) 3 and v = u 2 . The rest of the proof follows exactly the same arguments as in the proof of Theorem 1.1, so we omit the details.
With the help of the usual estimates it is easy to verify that D 1 2 ξ G 2 ∈ L 2 (R) at any t ∈ [0, T ].
Since F (t,ξ) (1+|ξ|) 3 − 1 vanishes at ξ = 0 and it is bounded by 2, it follows that D 1 2 ξ G 1,2 ∈ L 2 (R) for any t ∈ [0, T ] as well. The same strategy can be applied by subtracting ϕ(0) and therefore we can write up G 1 as The continuity of the solution u allows us to conclude from (5.45) that u ≡ 0, and this completes the proof.