Well-posedness for one-dimensional derivative nonlinear Schr\"odinger equations

In this paper, we investigate the one-dimensional derivative nonlinear Schr\"odinger equations of the form $iu_t-u_{xx}+i\lambda\abs{u}^k u_x=0$ with non-zero $\lambda\in \Real$ and any real number $k\gs 5$. We establish the local well-posedness of the Cauchy problem with any initial data in $H^{1/2}$ by using the gauge transformation and the Littlewood-Paley decomposition.


1.
Introduction. In the present paper, we consider the following Cauchy problem for the derivative nonlinear Schrödinger equation u(0, x) = u 0 (x), (1.2) where u = u(t, x) : R 2 → C is a complex-valued wave function, both λ = 0 and k 5 are real numbers. A great deal of interesting research has been devoted to the mathematical analysis for the derivative nonlinear Schrödinger equations [3,4,6,7,8,9,10,11,13,18,21]. In [13], C. E. Kenig, G. Ponce and L. Vega studied the local existence theory for the Cauchy problem of the derivative nonlinear Schrödinger equations iu t + u xx + f (u,ū, u x ,ū x ) = 0, (t, x) ∈ R 2 , with small data u(0, x) = u 0 (x) in H 7/2 where f is a polynomial having no constant or linear terms with the lowest order term of degree being greater than or equal to 3. Subsequently, it was improved to H 3 by N. Hayashi and T. Ozawa [11].
If the nonlinearity consists mostly of the conjugate waveū, then it can be done much better. In the case f = (ū x ) k , A. Grüenrock, in [8], obtained local wellposedness when s > s c = 3/2 − 1/(k − 1), s 1, and k 2 was an integer. In particular, the global well-posedness in H 1 is obtained when f = i(ū x ) 2 with the help of the Bourgain spaces (cf. [2,23]). In [21], H. Takaoka discussed the derivative nonlinear Schrödinger equation of the form u t − iu xx + |u| 2 u x = 0, (t, x) ∈ R 2 , and obtained the local well-posedness in H s for s 1/2 by performing a fixed point argument in an adapted Bourgain space X s,b which yields a C ∞ -solution map.
A very similar equation to (1.1) is the generalized Benjamin-Ono equation where u is a real-valued function, H is the Hilbert transformation defined by and k 2 is an integer, the symbol· (or F ) denotes the spatial Fourier transform. For this equation, L. Molinet and F. Ribaud [16,17] obtained the local well-posedness in the Sobolev space H s for s > 1/2 if k = 2, 4, s 3/4 if k = 3 and s 1/2 if k 5 by using Tao's gauge transformation. In [14], C. E. Kenig and H. Takaoka have shown the global well-posedness for the case k = 2 in H s for s 1/2 by combining the gauge transformation with a Littlewood-Paley decomposition and following the compactness argument with a priori estimates with the help of the preservation of the Hamiltonian and the L 2 -mass.
In the present paper, we shall generalize the above results to the derivative nonlinear Schrödinger equation with k 5 by using some ideas in [14]. However, we have to reconstruct new and complicated estimates for the case k 5 which is quite different from the case k = 2.
We first state the main result of this paper as follows, though we shall define later the function space X T at the end of this section. For convenience, we now introduce some notations. For nonnegative real numbers A, B, we use A B to denote A CB for some C > 0 which is independent of A and B. A ∼ B means A B A, and A ≪ B denotes A CB for some small C > 0 which is also independent of A and B.
Throughout this paper, we often use the Littlewood-Paley theorem (cf. [20,23]) for 1 < p < ∞. We also use more general operators P ≪N and P N which are defined by and P ≫N , P N and P ∼N which can be defined in a similar way. The Littlewood-Paley operators commute with derivative operators (including |∇| s and i∂ t − ∂ xx ), the propagator S(t) = e −it∂ 2 x , and conjugation operations, are self-adjoint, and are bounded on every Lebesgue space L p and homogeneous Sobolev spaceḢ s if 1 p ∞. Furthermore, they obey the following Sobolev and Bernstein estimates for R with s 0 and 1 p ∞ (which is similar to those of three dimensions [5]): which can be verified by combining the Bernstein multiplier theorem [1] and the interpolation theorem of Sobolev spaces. We define the Lebesgue spaces L q T L p x and L p x L q T by the norms .
In particular, we abbreviate L q T L p x or L p x L q T as L p x,T in the case p = q. We also use the elementary inequality [5] , for all 2 q, p ∞ and arbitrary functions f N , and the dual version where p ′ is the conjugate number of p given by 1/p + 1/p ′ = 1. It is easy to verify that they also hold if we replace the norm L q T L p x by the norm L p x L q T in both side of the above inequalities.
Let · = (1 + |·| 2 ) 1/2 . We use the fractional differential operators D s x and D x s defined by Thus, we can introduce the resolution space. For T > 0, we define the function space X T in a similar way as in [14] by 2. Gauge transformation. We transform the equation (1.1) by introducing the following complex-valued function v N : By computation, we have For the second term, we integrate by parts and have Thus, v N obeys the following differential-integral equation 3. Preliminaries. In order to prove the a priori estimate for the equation of v N , we need the linear estimates associated with the one-dimensional Schrödinger equation. We first recall the Strichartz estimates, smoothing effects and maximal function estimates. For the proofs, one can see [13,14].
and T ∈ (0, 1), We also need the L q T L p x and L p x L q T estimates for the linear operator f → t 0 S(t − τ )f (τ )dτ . For the proofs, one can see [14].
where p ′ is the conjugate number of p ∈ [1, ∞], i.e. 1/p + 1/p ′ = 1, and 1 p(θ) Next, we recall the Leibniz' rule for a product of the form e iF g where F is the spatial primitive of some function f . For the proof, we refer to [14,17]. . Let α ∈ (0, 1), p, p 1 , p 2 , q, q 1 ∈ (1, ∞), q 2 ∈ (0, ∞] 4. Bilinear estimates. In this section, we prove the following space-time estimate which is crucial to the proof of the nonlinear estimates.
Proof. By the Littlewood-Paley decomposition, we can write Now, we derive the estimates for I 1 , I 2 and I 3 , respectively. From the Hölder inequality, the Bernstein type inequalities and the real interpolation theorem, we have Applying the Sobolev embedding theorem and the Hölder inequality to the first term, and Bernstein estimates to the second term, we can see that it is bounded by For I 2 or I 3 , it is suffice to consider one of them, e.g. I 2 , in view of symmetry.
For the case N 1 1 ≪ N 2 , from the Hölder inequality and the Littlewood-Paley theorem, we can get For the case N 1 ≪ N 2 1, we have, by the Hölder inequality and the Littlewood-Paley theorem, that For N 2 1, we have, by the Sobolev embedding theorem, that where ε = (p − 1)/2p. Thus, (4.5) can be bounded by Noticing that and for N 1 ≫ 1, ε = 1/p and p 4 we can bound (4.6) by in view of the Hölder inequality. Thus, we have obtained Therefore, we have the desired result (4.1) for any real number p 4.

Nonlinear estimates.
To state the estimates for the nonlinearities I N,j , we define the function space Y T equipped with the following norm: We have the following proposition for the nonlinearities.
denotes the maximal integer that is less than or equal to k).
We consider each nonlinearity separately.

Nonlinear estimates of
For the second term in (5.1), we have the following estimate.
To shift a derivative from the high-frequency function P N u x to the lowfrequency function |P ≪N u| k , we require the following Leibniz rule for P N from [14]: Thus, we have For the first term in (5.5), we have By the Littlewood-Paley theorem, we can obtain In the similar way, we have Thus, For the last two term in (5.5), in a similar way as in the proof of Proposition 4.1, we can obtain the following bound: From the Sobolev embedding theorem and (5.7)-(5.9), we obtain that (5.4) can be bounded by Thus, we can bound (5.2) by XT , which yields the desired result.
For the first term in (5.1), we have the following estimate: Proof. We split (5.1) into several terms for N ≫ 1 and k 4 Notice that where ε k > 0 is defined by ε k = 1/k. Thus, for the first term (5.12), from the fact φ N L 1 1 and Proposition 4.1, we have for k 4 XT . Therefore, we obtain, for any k 4, that XT . For (5.14), in the same way as the case (5.12), we have XT . Now, we derive the estimate for (5.13) by using the induction argument in k. For k = 4, we have From the Young inequality, the Hölder inequality, (5.6) and Proposition 4.1, we can get for k = 4 From the triangle inequality for complex number, i.e. ||z 1 | − |z 2 || |z 1 − z 2 | for z 1 , z 2 ∈ C, we can get |z 1 | θ − |z 2 | θ |z 1 − z 2 | θ for any θ ∈ (0, 1]. For k ∈ (4, 5], we have Then where ε k = (k − 4)/2k for k ∈ (4, 5]. By the same procedure, we can obtain for any k 4 holds for any ε ∈ [0, 1) in view of Proposition 4.1.
We turn to the proof of Proposition 5.1 for the nonlinearity I N,1 . We also consider the decomposition in (5.1). For convenience, we denote B N = P N (|P ≪N u| kP N u x )− |P ≪N u| k P NPN u x . From (3.5), (3.11), (3.6) and (3.7), we have By Lemma 5.1, the first term can be bounded by XT . For the second term, we split the sum M into three parts M∼N + M≪N + M≫N as in [14]. For the part of M ∼ N , it is the same as Lemma 5.1 by summing in M such that M ∼ N . For the part M ≪ N , we can add the projection operator P ∼N to e − iλ 2 R x −∞ |P≪N u| k dy since B N has Fourier support in |ξ| ∼ N . Thus, by the Hölder inequality, we have where ε ∈ (0, 1/k). By the Bernstein inequality, we have and from (5.3) and the Hölder inequality, we can get, as a similar way as in (5.4), that 14 C. C. HAO Thus, (5.18) can be bounded by For the part M ≫ N , we can add the projection operator P M to e − iλ 2 R x −∞ |P≪N u| k dy . In a similar way with the part M ≪ N , we have For the first term in (5.1), we denote it by A N , i.e. A N = P N ((|u| k − |P ≪N u| k )u x ). Similarly, from (3.5), (3.11), (3.6) and (3.7), we can get From Lemma 5.2, the first term is bounded by Noticing that (5.16), and in the same way as in dealing with the second term of (5.17), we can bound the second term of (5.19) by Therefore, we have obtained For the first term (5.20), from Lemma 3.3 and the Hölder inequality, it can be bounded by N ≫1 By the Hölder inequality, we have for k 5 and from Proposition 4.1 and the proof of Lemma 5.1, Thus, we can bound (5.20) by XT .
For the first term (5.24), noticing that M N M 1/2 N 1/2 and (5.23), we can bound it by where B N,3 = x −∞ |P ≪N u| k−2 P ≪N |u| k (u x +ū x )dy. By Hölder inequality, we get XT . From the Hölder inequality and Proposition 4.1, we have In addition, for N ≫ 1, we have P N u L ∞ Thus, in the same way as in the case I N,2 , we can bound (5.26) by 5.4. Nonlinear estimates of I N,4 . From (3.5), (3.11), (3.6) and (3.7), we have where B N,4 = |P ≪N u| k−2 P N uP ≪N uP ≪N u x . By the Hölder inequality, we have Thus, the first term in (5.29) can be bounded by XT . By the Hölder inequality, we get Noticing that B N,4 has Fourier support in |ξ| ∼ N , we can repeat the procedure which we use to deal with the second term in (5.17), and obtain that the second term in (5.29) can be bounded by 5.5. Nonlinear estimates of I N,5 . From (3.5), (3.11), (3.6) and (3.7), we have where B N,5 = |P ≪N u| 2k P N u. By the Hölder inequality, we have Thus, in a similar way as dealing with I N,1 and I N,4 , and noticing that B N,5 has Fourier support in |ξ| ∼ N , we can bound (5.30) by

6.
A priori estimates for solutions. By the scaling argument we have Thus, we may rescale where we choose γ = γ( u 0 H 1/2 ) ≫ 1, and take the time interval T depending on γ later. We now drop the writing of the scaling parameter γ and assume We now apply this to the norms X T and H 1/2 , and define new version of the norms of X T and H 1/2 , given by with the decomposition I = P 1 + P ≫1 , which implies that u 0 H1/2 2. For the low frequency part, we have the following estimates.
Lemma 6.1. Let u be a solution of (1.1)-(1.2). Then Proof. Using the integral equation of (1.1) XT , which is the desired result.
For the high frequency part, we have Lemma 6.2. Let u and v N be given in (2.1). Then Proof. By (2.1), we have For x -norm, by the interpolation theorem, we obtain for N ≫ 1, which yields the desired estimate by summing on l 2 N .

C. C. HAO
For the L ∞ x L 2 T -norm, noticing that To estimate the second term (6.1), we split the sum N2 = N2∼N + N2≁N . For N 2 ∼ N , from the Bernstein inequality, we bound (6.1) by For the part N 2 ≁ N , we split it as N2≁N = N2≪N + N2≫N . Noticing that for and for N 2 ≫ N , we can bound (6.1), in view of the Bernstein inequality and the Hölder inequality, by Therefore, summing on l 2 N , we complete the proof for the L ∞ x L 2 T -norm. For the L 2 x L ∞ T -norm, it is easy to obtain the desired result since |P N u| = |v N |. We turn to estimate the L 4 x L ∞ T -norm. It is similar with the proof for the L ∞ We also split N2 = N2∼N + N2≁N . For N 2 ∼ N , we bound (6.3) by For the part N 2 ≁ N , we split it as N2≁N = N2≪N + N2≫N . Noticing that for N 2 ≪ N ,P N N1 and for N 2 ≫ N , we can bound (6.3), in view of the Bernstein inequality and the Hölder inequality, by which yields the desired estimate by applying l 2 N -sum. Thus, we complete the proof of this Lemma.
Of course, we need the following estimate of the data.   (1 + u 0 k H 1/2 ) P ≫1 u 0 H 1/2 , which yields the desired result.
With the help of the above lemmas, we can prove the following proposition which yields the a priori estimate. Proposition 6.1. Let u be a smooth solution to (1.1)-(1.2) and 0 < T C 4 high . Then we have u X T C(C low ) + C(C low + u X T ) 3k (T 1/4 + C high ) u X T .
The high frequency part C high P ≫1 H1/2 can be absorbed into theX T -norm. Then substituting Lemma 6.1 again in estimating the low frequency part of the norm P 1 u L ∞ T H 1/2 x , we complete the proof of Proposition 6.1.

C. C. HAO
From Proposition 6.1, we have the following a priori estimate for the solution of (1.1)-(1.2) if we take T and C high small enough. Corollary 6.1. Let u be a smooth solution to (1.1)-(1.2). we have for T and C high small enough.
For the proof of Theorem 1.1, we can follow the compactness argument with the a priori estimate. Since the proof is standard, we omit the details and refer to the papers [14,15,16,17,19,22].