SHARP WELL-POSEDNESS FOR THE CHEN-LEE EQUATION

. We study the initial value problem associated to a perturbation of the Benjamin-Ono equation or Chen-Lee equation. We prove that results about local and global well-posedness for initial data in H s ( R ), with s > − 1 / 2, are sharp in the sense that the ﬂow-map data-solution fails to be C 3 in H s ( R ) when s < − 12 . Also, we determine the limiting behavior of the solutions when the dispersive and dissipative parameters goes to zero. In addition, we will discuss the asymptotic behavior (as | x | → ∞ ) of the solutions by solving the equation in weighted Sobolev spaces.


1.
Introduction. This paper is concerned with the following initial value problem associated to a perturbation of the Benjamin-Ono equation or Chen-Lee equation CL u t + uu x + βHu xx + η(Hu x − u xx ) = 0, x ∈ R (or x ∈ T), t > 0, where β, η > 0 are constants. In the equation, H denotes the usual Hilbert transform given by y − x dy = i (sgn(ξ) f (ξ)) ∨ (x) for ξ ∈ R, f ∈ S(R).
This equation was first introduced by H. H. Chen and Y. C. Lee in [11] to describe fluid and plasma turbulence and as a model for internal waves in a two-fluid system. The fourth and the fifth terms represent the instability and dissipation, respectively. The parameter η represents the importance of instability and dissipation relative to dispersion and nonlinearity. H. H. Chen, Y. C. Lee and S. Qian in [16,15], and B. -F. Feng and T. Kawahara, in [8], investigated the initial value problem as well as stationary solitary and periodic waves, associated with Chen-Lee equation, from a numerical standpoint. R. Pastrán in [12] proved using the Fourier restriction norm method that the initial value problem CL is locally well-posed in H s (R) for any s > −1/2, globally well-posed in H s (R) when s ≥ 0 and ill-posed in H s (R), if We will employ weighted Sobolev spaces defined by r . (4) Here L 2 r (R), r ∈ R is the collection of all measurable functions f : R → C such that 1.2. Main results. First, we recall the local well-posedness result from Theorem 1.1, in [12], and we state our result about global well-posedness which improves Theorem 1.2 in [12].   . When the dispersive parameter β is zero and η > 0, we have the following Cauchy problem associated to the Chen-Lee non-dispersive equation φ ∈ H s (R), s ∈ R. Following the ideas presented by Vento in [17] for the Dissipative Benjamin-Ono equation, we can prove that for the Chen-Lee non-dispersive equation (7), the result obtained in Theorem 1.1 is sharp with the next Theorem.
Theorem 1.4. Let s < − 1 2 be given. Then there does not exist a time T > 0 such that (7) admits a unique local solution on the time interval [0, T ] and such that the flow map data-solution φ −→ u(t) of (7) is C 2 at the origin from H s (R) to C ([0, T ]; H s (R)).
We study the limiting behavior of the solutions CL when η > 0 is fixed and β tends to zero with the next theorem. Theorem 1.5. Let η > 0 be fixed. Let s > −1/2 and φ ∈ H s (R). If u β is the solution of equation (1) with initial data φ, constructed in Theorem 1.1 for all β ≥ 0 in the time interval [0, T ] (remembering that T is not dependent on β), then where u 0 is the solution of equation (7) with initial data u 0 (0) = φ.
Then we study the convergence of the solutions of CL to solutions of the Benjamin-Ono (η = 0) equation, when the dispersion parameter is fixed and the dissipation η tends to zero. Theorem 1.6. Let β > 0, φ ∈ H s (R), s > 3 2 and let u η be the solution of CL satisfying u η (0) = φ. Then the limit u 0 = lim η→0 u η exists in C ([0, T ]; H s (R)) ∩ C 1 [0, T ]; H s−2 (R) and is the unique solution of the CL equation with β = 0 that depends continuously on the initial data.
Finally, we state the result about decay properties of the solution of initial value problem CL which provide a theoretical prove of the numerical result posed in [8].
The layout of this paper is organized as follows: In Section 2, we reprove the Theorem 1.1 in [12], but using dissipative methods of Dix [5], and we improve the global result from Theorem 1.2 in [12], i.e., we show Theorem 1.1. Section 3 is devoted to give a proof of Theorems 1.2 and 1.4 which say us that CL and CLND are ill-posed in H s (R) for s < −1/2. Section 4 presents the study of the behavior of the solutions of the Cheen-Lee equation when the dispersive parameter β tends to zero, we will give a proof of the Theorem 1.5, and Section 5 is dedicated to the study of the convergence of the solutions of the Cauchy problem CL to solutions of the Cauchy problem associated to the BO equation, we will give a proof of the Theorem 1.6. Finally, Section 6 is devoted to study decay properties of the solution of initial value problem CL and a proof of the Theorem 1.7.

2.
Theory in H s (R) with s > −1/2. We recall that local well-posedness result in Theorem 1.1 was already proved by Pastrán in [12] but here we present a different proof using the dissipative methods of Dix [5] (see [7,6,14] for similar results). The main idea is to construct a contraction with the integral formulation of (1) defined on an appropriated Banach space X s T , when s > − 1 2 and 0 < T ≤ 1. We introduce X s T in order to deduce the crucial linear and bilinear estimates which are an adaptation, made by Esfahani [7] and Duque [6], of the spaces originally presented by Dix in [5] for the dissipative Burgers equation. For s < 0 and 0 ≤ T ≤ 1, we define We start giving the following technical results.
Proof. For all ξ ∈ R we have that Let w t (x) = x 2λ e η(xt 1/2 −x 2 ) , for all x ≥ 0. Note that w t (x) tends to 0 as x → ∞, and Therefore, the maximum of w t is attained in x 1 and we can deduce that This inequality completes the proof.
From Lemma 2.1 and arguing as in Proposition 1 in [14], it is easy to deduce the next proposition.
Next, we establish the crucial bilinear estimates.
for all u, v ∈ X s T .

RICARDO A. PASTRÁN AND OSCAR G. RIAÑO
Proof. Since s < 0, it follows that ξ s ≤ |ξ| s , for all real number ξ different from zero. Then we deduce that The Young inequality implies that thus we obtain To estimate the integral on the right-hand side of (18), we perform the change of variables w = t 1/2 ξ to deduce |ξ| 1+s e η(|ξ|−ξ 2 )t Therefore, we get from (18) and (19) that for all 0 ≤ t ≤ T . On the other hand, arguing as above, we have for all 0 ≤ t ≤ T that Combing (20) and (21) the proof is complete.

Remark 1.
If we consider s > s > − 1 2 . Then modifying the space X s T bỹ and using that The proof of this proposition is similar to the proof of Proposition 4 in [14].
The next lemma will enable us to estimate the term ∂ x (uv) when s ≥ 0 and u, v ∈ C ([0, T ]; H s (R)).
Remark 2. Assuming that s ≥ 0 and 0 < T ≤ 1, we have a similar result as the one obtained in Proposition 3 for the space C ([0, T ]; H s (R)). In fact, we have that To deduce this result, from Lemma 2.2 with a = 1 and inequality (19) we get Remark 3. Let s ≥ 0 and 0 < T ≤ 1. We have the same result given in Proposition 4, changing X s T by C ([0, T ]; H s (R)) and taking δ ∈ [0, 1 2 ). This result is proved using Lemma 2.2 and arguing as in the proof of Proposition 4 in [14].
We consider the application for each u ∈ X s T . By Proposition 2, together with Proposition 3 when s < 0, or by Remark 2 when s ≥ 0, there exists a positive constant C = C(η, s) such that for all u, v ∈ X s T and 0 < T ≤ 1. Where g(s) = 1 4 (1 + 2s), when s ∈ (− 1 2 , 0) and . The estimates (24) and (25) imply that Ψ is a contraction on the complete metric space E T (γ). Therefore, the Fixed Point Theorem implies the existence of a unique solution u of (8) 2. Continuous dependence. We will verify that the map φ ∈ H s (R) → u ∈ X s T , where u is a solution of (1) obtained in the step of Existence is continuous. More precisely, for s > − 1 2 , if φ n → φ ∞ in H s (R), let u n ∈ X s Tn be the respective solutions of (8) (obtained in the part of Existence) with u n (0) = φ n , for all 1 ≤ n ≤ ∞. Then for each T ∈ (0, T ∞ ), u n ∈ X s T (for n large enough) and u n → u ∞ in X s T . We recall that the solutions and times of existence previously constructed satisfy for all n ∈ N ∪ {∞}. Let T ∈ (0, T ∞ ), the above inequalities and the hypothesis imply that there exists N ∈ N, such that for all n ≥ N , we have that T ≤ T n and Therefore, combining (26), (27) with the Propositions 2, 3 when the index s is negative, or with the Remark 2 when s ≥ 0, it follows that for each n ≥ N for all t ∈ [0, T ]. Arguing as in the proof of Proposition 3 or Remark 2, we deduce that there exists a positive constant C = C(η, s) depending only on η and s, such that for all r ∈ [0, T ] and all ϑ ∈ [r, T ], where In particular, inequality (28) implies that Therefore we can iterate this argument using (28)  Therefore we can iterate this argument, using uniqueness result and the fact that the time of existence of solutions depends uniquely of the H s (R)-norm of the initial data. Thus we deduce that

5.
Global well-posedness. Since Pastrán proved in [12] that CL is globally well posed for all φ ∈ H s (R) when s ≥ 0, we shall prove that CL is globally well posed in X s T when − 1 2 < s < 0. In fact, let s ∈ (−1/2, 0), φ ∈ H s (R) and u ∈ X s T be the solution of the Cauchy problem (1) obtained in above steps. Let T ∈ (0, T ) fixed, we have that Since u ∈ C ((0, T ]; H ∞ (R)), it follows that u(T ) ∈ L 2 (R). Thus, Theorem 1.2 in [12] implies thatũ, the solution of (8) with initial data u(T ), is global in time.
The global result follows from the above estimate.

RICARDO A. PASTRÁN AND OSCAR G. RIAÑO
3. Ill-posedness results. Without using the Fourier restriction norm method we have deduced for the Cauchy problem (1) local and global well-posedness in H s (R), when s > −1/2. We shall prove that this result is sharp in the sense that the flow-map data-solution fails to be C 3 in H s (R) for s < − 1 2 . We recall that Pastrán proved in [12] that the assumption of C 2 regularity in H s (R) for the flow-map of the Cheen-Lee equation fails when s < −1.
Proof of Theorem 1.2. Let s < − 1 2 , suppose that there exists T > 0 such that the Cauchy problem (1) is C 3 at the origin. When φ ∈ H s (R), we have that Φ(·)φ is a solution of the equation (1) with initial data φ. This means that Φ(·)φ is a solution of the integral equation Since the Cauchy problem (1) is supposed to be well-posed, we know using the uniqueness that Φ(t)(0) = 0. Thus, with this result and using the integral equation, we see that The assumption of C 3 regularity implies that which would lead to the following inequality We will show that (30) fails for an appropriated function φ. We define φ by his Fourier transform as where I N = [N, N + 2γ], N 1 and γ = N , with 0 < 1 fixed. We note that φ H s ∼ s 1.
Proof of Theorem 1.4. Let φ ∈ H s (R) and u 1 (t), u 2 (t) as in the proof of Theorem 1.2.

Convergence of solutions of CL to solutions of CLND.
In this section we study the convergence of the solution of the Chen-Lee equation when the dispersion β tends to zero and the dissipation η > 0 is fixed. To emphasize the dependence of the semigroup associated with the linear part of the equation (1) with the parameter β, we will use throughout this section the following notation We observe that the results given in Theorem 1.1 hold for the initial value problem (7), since the constants and arguments involved are not dependent upon the parameter β. In fact, Theorem 1.1 applies with the same proof when β ∈ R and η > 0.
Proof of the Theorem 1.5. As in the proof of Theorem 1.1 we consider X s T = X s T , if s ∈ − 1 2 , 0 and when s ≥ 0, we take X s Therefore, for 0 < T 1 ≤ T the triangle inequality implies that v β Since the solutions constructed in Theorem 1.1 satisfy u β X s ϑ ≤ γ, for all β ≥ 0, we obtain from Proposition 3 for s < 0, or by Remark 2 when s ≥ 0, that where g(s) = 1 4 (1 + 2s), if s ∈ (− 1 2 , 0), and g(s) = 1 4 , for all s ≥ 0. So, taking T 1 > 0 such that T 1 ≤ (4C s,η γ) − 1 g(s) and combining (40) with (41), we obtain We will estimate I β when β tends to zero. Using the mean value inequality and Lemma 2.1 with λ = 1, we deduce that for all s > − 1 where f 1 and f 2+|s| 2 are defined as in (12). Therefore, from (43), (44) and the definition of the norm in X s T1 , we conclude that lim β→0 + I β = 0. Hence, from (42) we deduce that Finally, we can iterate this process to conclude the result in the whole interval [0, T ].

5.
Convergence of solutions of CL to solutions of BO. In this section we examine the convergence of solutions of the Chen-Lee equation to solutions of the initial value problem for the integral version of the Benjamin-Ono equation when the dissipation η tends to zero and the dispersion β > 0 is fixed. In order to deduce this result we will adapt the ideas employed in the parabolic regularization method (see [2] and [6]).
We will start showing that the time of existence of solutions for CL can be chosen independent of η ∈ (0, 1).
2 is fixed, and let u η ∈ C ([0, T ]; H s (R)) be a solution of CL with η > 0. Then there exists a T s > 0 depending on φ H s , but not on 0 < η < 1, such that u η can be extended to the interval [0, T s ], and there is a function ρ(t) ∈ C ([0, T s ]; R) such that Proof. We observe that Kato's Inequality 1 and the assumption that η < 1 imply that there exists a constant depending only on s such that 1 2 H s . Then, integrating the above expression we get Hence Therefore defining and taking 0 < T s ≤ 2 Cs ln the lemma follows.
Next we will show that u 0 = lim η→0 + = u η exists and satisfies the Benjamin-Ono equation in a weak sense.
Since the application t ∈ [0, T ] → u 0 (t)u 0 x (t) + βHu 0 xx (t) is weakly continuous, Bochner-Pettis' Theorem implies that it is strongly measurable. Thus it follows that which implies that u 0 ∈ AC [0, T ]; H s−2 (R) . Finally, to prove the uniqueness, Since v is strongly differentiable with respect to t in L 2 (R), we get 1 2 Thus, integrating the above expression and applying Gronwall's inequality we obtain Uniqueness follows on taking φ = ψ.
Proof Theorem 1.6. From Lemma 5.2 we have that u 0 = lim η→0 + u η exists in the class described in this lemma, and is the weak solution of the Benjamin-Ono equation. We claim that u 0 ∈ C ([0, T ]; H s (R)). Let ψ ∈ H s (R) be such that ψ H s = 1. Therefore Thus, taking the sup over ψ H s = 1, we deduce that lim t→0 + u(t) H s = φ H s . Combining this result and the fact that u 0 (t) φ in H s (R), we conclude the continuity at the origin. Let t * ∈ (0, T ) be fixed, we obtain right continuity at t * from continuity at the origin and uniqueness. Left continuity is deduced using that u(t * − t, −x) is a solution of the Benjamin-Ono equation for any fixed t * ∈ (0, T ]. On the other hand, since uu x + βHu xx ∈ H s−2 (R), then by (49) it follows that u ∈ C 1 [0, T ]; H s−2 (R) . Uniqueness is proved as in Lemma 5.2. Finally, the proof that the solution depends continuously on the initial data is similar to the proof of Theorem 3.1 in [6]. 6. Decay properties of the solution. In this section we reduce asymptotic questions to existence theorems by solving the equation in appropriate function spaces. As we shall see below the combined effects of the nonlinearity and the nonsmoothness of the symbol of the Hilbert transform determine an upper bound for the rate of decay of the solutions of the equation (1) as |x| → ∞. First of all, we will examine the function u(t) = S(t)φ, where φ ∈ F r,r is defined in (4), in order to obtain the semigroup estimates. Next lemma provides some formulas for derivatives of the semigroup associated to the CL equation. They easily follow from a direct computation.
Proof. This result follows easily from the chain rule.
Proposition 5. Let η > 0 and β > 0 be fixed. Then, (a.) S : [0, +∞) −→ B(F r,r ), r = 0, 1, is a C 0 -semigroup and satisfies the estimate, for all φ ∈ F r,r , where Θ r (t) is a polynomial of degree r with positive coefficients that depend only on η, β and r. (b.) If r ≥ 2 and φ ∈ F r,r , the function S(t)φ belongs to C([0, ∞); F r,r ) if, and only if, In this case an estimate of the form (56) also holds Proof. The proof is similar to the proof of Theorem 2.4 in [9] or Lemma 5.2 in [1].
6.1. Theory in F 2,1 (R). Some decay properties of the solution of equation (1) for η > 0 and β > 0 are obtained similarly to those known for the Benjamin-Ono equation (see [9]). Theorem 1.7 is a unique continuation theorem for equation in (1). It implies loss of persistence for CL equation in F 3,3 , while for the Benjamin-Ono equation this occurs in F 4,4 . We begin proving a lemma which gives us some a priori estimates.
Proof. We divide the proof in two steps: 1. Local Existence. Since ξ j e ηt(|ξ|−ξ 2 ) ≤ ξ j e −ηtξ 2 /2 , for j ∈ N and ξ ≥ 2, then Using the Proposition 1 and (50), it follows that (S(t)) t≥0 is a C 0 semigroup on F 2,1 (R) and for all φ ∈ F 2,1 , where Θ η,β,1 (t) is a polynomial of degree one with positive coefficients depending only on η. The local theory for the integral equation (8) in F 2,1 (R) is similar to the proof of Theorem 1.1, but now we use (50) and (64) to get So, let u ∈ C([0, T ]; F 2,1 ) be the local solution of equation (8) with u(0) = φ, and T = T ( φ F2,1 , η, β). 2. Global Existence. To prove global existence for the initial value problem (1) in F 2,1 (R), it is enough to combine the global well-posedness result in the proof of Theorem 1.1 with the next computations.
Integrating now (73) between 0 and t, we find that The last expression implies that and hence u(t, 0) = 0, for all t ∈ [0, T ].
Integrating (83) between 0 and t, we have that