Global well-posedness for the derivative nonlinear Schr\"{o}dinger equation in $H^{\frac 12} (\mathbb{R})$

We prove that the derivative nonlinear Schr\"{o}dinger equation is globally well-posed in $H^{\frac 12} (\mathbb{R})$ when the mass of initial data is strictly less than $4\pi$.


Introduction
In this note, we study the Cauchy problem to the derivative nonlinear Schrödinger equation (DNLS): This equation was derived by [12,13] for studying the propagation of the circular polarised nonlinear Alfvén waves in plasma, and has been extensively studied since then. It is well-known that (1.1) is completely integrable (see [9,8,17]), and thus has infinite number of conservation laws. In particular, in this paper we will use the following three conservation laws: if u is a H 1 -solution of (1. Equation (1.1) has been extensively studied. On the well-posedness, Hayashi and Ozawa [5,6,7,14] proved local well-posedness in H 1 (R), and moreover global well-posedness for initial data in H 1 satisfying (1. 2) The condition above appears naturally in the sharp Galiardo-Nirenberg inequality to ensure an apriori estimate of H 1 -norm by mass and energy conservation. Later, Local well-posedness in H s for s ≥ 1/2 was obtained by Takaoka [15], and this result is sharp in the sense that the solution map fails to be uniformly continuous in a ball of H s if s < 1/2. Low regularity global well-posedness was also studied, for example, global well-posedness in H s (R) under (1.2) was obtained in [16,2,3] for s > 1/2, and finally in [11] for s = 1/2. On the long-time behavior and modified scattering theory, see [4] and references therein.
A natural question is whether blowup occurs for (1.1). To the authors' knowledge, this problem is still open. See [10] for a numerical blowup analysis on a class of DNLS. Recently, the second author [19] showed the global well-posedness in H 1 under a weaker condition improving his previous result [18]. This result shows a striking difference between DNLS and other mass critical equations like focusing generalized KdV and quintic focusing nonlinear Schrödinger equation. The key ingredient is the use of the momemtum conservation. The purpose of this paper is to prove the low-regularity global well-posedness under (1.3). The main result is We explain the ideas of the proof of the theorem. Inspired by [19], we derive directly an apriori estimate using the conservation laws of mass, momentum and energy as well as the sharp Gagliardo-Nirenberg inequality, and thus provide a simplified proof of the result of [19]. We do not prove by contradiction and can get a clear bound of H 1 -norm. Then we combine it with the I-method to prove the theorem.

Apriori estimate
To prove the theorem, it suffices to control the H 1/2 -norm of the solution. For convenience, we use the following gauge transformation. If u is a solution to (1.1) with Then v solves It's easy to see the map u → v is a bijection in H 1/2 . Indeed, by fractional Leibniz rule we get From now on, we only consider the equation (2.2) and we need to control the Under the gauge transformation, the conservation laws reduce to: for solution v of (2.2) then We denote · p = · L p x for 1 ≤ p ≤ ∞. By the sharp Galiardo-Nirenberg inequality then we get . Thus under the condition (1.2) we can get the apriori bound on v H 1 .
However, as observed in [19] the momentum conservation for (2.2) played a significant role. Inspired by [19] we derive directly a-priori estimate using the momentum and the following sharp GN inequality (see [1]): Then and thus

Now by the sharp GN inequality we have
Thus, . Therefore Proof. Let x = v 4 4 . Then (2.8) gives a estimate of the form Since a > 0, thus we get Then by (2.7) and mean value inequality we have Therefore by (2.11) we prove the lemma.
With this lemma, we can get that if v is a H 1 -solution of (2.2) satisfying (1.3), then v x 2 ≤ C. Therefore, global well-posedness of (2.2) in H 1 under (1.3) follows immediately.

Proof of the main theorem
In this section we prove Theorem 1.1 using the I-method as the previous works [3,11]. The main difference is that we need to use the momentum conservation.
First we recall the definition of I-operator. Let N ≫ 1 be fixed, and the Fourier multiplier operator I N be defined as Here m N (ξ) is a smooth, radially decreasing function satisfying 0 < m N (ξ) ≤ 1 and where the implicit constants are indenpendent on N.
Next we use the rescaling. For v 0 ∈ H 1/2 , let v be the solution to (2.2). For λ > 0, let Then v λ is a solution of (2.2) with the initial data v λ (0) = v 0,λ (x and Thus choosing λ ∼ N, we can make where ε 0 will be determined later.
We recall a variant local well-posedness obtained in [11]. (3.8) By the above lemma, we need to control the growth of Iv λ (t) H 1 . By mass conservation we have Iv λ (t) L 2 x ≤ v λ L 2 x ≤ C. It suffices to control ∂ x Iv λ 2 . We will use (2.9) since Iv λ Moreover, If N → ∞, I N tends to the identity operator. Thus P I (v λ ) and E I (v λ ) increases slowly in t if N is large enough. Indeed, in the previous works the growth of E I (v λ ) was already studied. Collecting the results obtained in [11] (see Section 7), we have On the modified momentum we have the following estimate. Indeed, since the momentum lies in the regularity of H 1/2 , we can estimate it in a simple way.

Lemma 3.3. We have
Proof. By the definition of momentum, we need to bound For the first term I, since By the definition of I-operator, we have Using the Hölder inequality, the Sobolev's embedding, and the fractional Leibniz inequalities, we get The similar terms in (3.12) can be handled in the same way. Thus we prove the lemma.